Patent Publication Number: US-2005127789-A1

Title: Piezoelectric motors and methods for the production and operation thereof

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS  
      This application claims priority under 35 U.S.C. § 120 from application no. PCT/EP03/04072 filed Apr. 17, 2003, titled “Piezo Motor” listing inventors Kai Wolf, Peter C. Varadi, Bjoern B. Magnussen, Benjamin Hagemann, Dieter A. Schuler and Erick M. Davidson, and these applications claim priority under 35 U.S.C. §119 to German Priority Document no. DE 102 17 963.8 filed April 22, 2002 titled “Piezomotor” listing applicant Elliptec Resonant Actuator AG, Dortmund/DE; and to German Priority Document no. DE 102 27 509.2 filed Jun. 19, 2002 titled “Piezomotor” listing applicant Elliptec Resonant Actuator AG, Dortmund/DE. The entire disclosures of these preceding applications are incorporated herein by reference as though set forth fully herein. This application claims priority under 35 U.S.C. § 120 from application Ser. No. 09/801,194 filed Mar. 8, 2001, and application Ser. No. 10/691,362 filed Mar. 22, 2003, each titled “Vibratory Motors and Methods of Making and Using Same” and listing inventors Bjoern B. Magnussen, Peter C. Varadi, Benjamin Hagemann, Steven Schofield and Erick M. Davidson, and these applications further claim priority under 35 U.S.C. § 119(e) from U.S. patent application No. 60/191,876 filed Mar. 23, 2000 entitled “Electric Motor Using Vibration to Convert Electrical Energy Into Mechanical Motion” and listing inventors Bjoern B. Magnussen, Steven Schofield and Benjamin Hagemann; and from application No. 60/215,438 filed Jun. 30, 2000, application No. 60/215,686 filed Jun. 30, 2000, application No. 60/231,001 filed Sep. 8, 2000, and application No. 60/236,005 filed Sep. 27, 2000, each entitled “Electrical Motor Using Vibration to Convert Electrical Energy Into Mechanical Motion” and listing inventors Bjoern B. Magnussen, Benjamin Hagemann and Peter C. Varadi, with the entire disclosures of these preceding applications also incorporated herein by reference as though set forth fully herein. 
    
    
     BACKGROUND OF THE INVENTION  
      To move small components, electromagnetic motors are often used because they are relatively inexpensive. The electromagnetic motors rotate very quickly and can only apply a low force, so they are always used with a gearbox that provides the slower motion and increased power necessary for practical applications. It should be noted that the movement of driven elements referred to in this disclosure refers to a translation or rotary motion in a common direction, and does not included motion that merely moves a part alternatively back and forth to shake the part without any net movement. While the conventional electromagnetic motors are relatively inexpensive, there are a large number of moving parts which complicates assembly and reliability, and the low power and need for a gearbox not only limits their application but also makes the cost excessive for many applications. Moreover, these motors are too big, not very precise in their motion, and are noisy. There is thus a need for a simpler, quieter and less expensive motor.  
      An alternative type of small motor is a piezoelectric motor, which uses a material that can change dimension when a voltage is applied to the material. Piezoelectric ceramics are used in electromechanical micromotors to provide linear or circular motion by making frictional contact between the vibratory motor and a driven object. These piezoelectric motors are composed of at least one mechanical resonator and at least one piezoelectric actuator. When electrically excited by oscillating electrical signals, the actuator generates mechanical vibrations that are amplified by the resonator. When the resonator is brought into contact with a body, these vibrations generate frictional forces in the contact area with the body and cause the body to move. The speed, direction and mechanical power of the resulting mechanical output depend on the form and frequency of the vibrations in the contact area. These piezoelectric motors work with small changes in dimension for a given voltage, and they can vibrate at many tens of thousands of cycles per second. Various cumbersome and expensive designs have been used to obtain useful forces and motions from these small vibratory motions.  
      One type of piezoelectric motor is a traveling wave motor, which uses a wave that travels through the piezoelectric material. These motors typically are based on a disc shaped design and are expensive to produce. The shape and the cost of these motors limit their application.  
      Other types of piezoelectric motors require a specially shaped waveform in the input signal in order to cause the piezoelectric material to move in a desired direction. One such type of motor is referred to as a stick-slip drive. These motors have a piezoelectric element that moves an object in a desired direction on a support at a relatively slow rate sufficient to allow friction to move the object. The waveform applied to the piezoelectric element causes the piezoelectric to then quickly retract and effectively pull the support out from under the object causing the object to slip relative to the support. The process is repeated, resulting in motion. Since these motors require a sawtooth or similar shaped waveform to operate, they require complex electronics that increase the cost of such motors.  
      A yet further type of piezoelectric motor is the impact drive, which repeatedly hits an object in order to make it move.  
      In piezoelectric micromotors, the piezoelectric element can be used to excite two independent modes of vibration in the resonator. Each mode causes the contact area on the resonator to oscillate along a certain direction. The modes are often selected so that the respective directions of oscillation are perpendicular to each other. The superposition of the two perpendicular vibrations cause the contact area to move along curves known as Lissajous figures. For example, if both vibrations have the same frequency and no relative phase shift between the vibrations, the motion resulting from the superposition is linear. If the frequencies are the same and the relative phase shift is 90 degrees, then the resulting motion is circular if the amplitudes of each vibration are identical; otherwise the resulting motion is elliptical. If the frequencies are different, then other motions such as figure-eights can be achieved.  
      The Lissajous figures have been used to produce figure-eight motion drives. These drives require an electrical signal that has to contain two frequencies to cause a tip of the vibration element to move in a figure-eight shaped motion. The resulting electronics are complex and expensive, and it is difficult to use the figure-eight motion to create useful motion of an object.  
      In order to move another body and to create a mechanical output, circular or large-angle elliptical motions (semi-axes nearly equal) have been preferred over linear motions. Piezoelectric micromotors in the prior art thus commonly employ two perpendicular modes of vibration that have a relative phase shift of approximately ninety degrees. The modes are excited close to their respective resonance frequencies so that the resulting mechanical output is maximized. If the relative phase shift between the two modes is changed to −90 degrees, the direction in which the ellipse is traversed is reversed. The motion of the body in contact with the resonator is thus reversed as well. But these conventional motors require two piezoelectric drivers located and selected to excite the two separate resonant modes. This requires two sets of drivers, two sets of electronic driving systems, an electronic system that will reverse the phase of each driver, and the basic design places limitations on the locations of components.  
      The prior art thus includes electromechanical micromotors where a rod-like resonator has a small piezoelectric plate that is attached to the resonator. The resonator contacts the moving body at the tip of the rod. The actuator excites a longitudinal mode and a bending mode of the rod. The excitation frequency is chosen in-between the two resonance frequencies of the respective modes so that the relative phase shift is 90 degrees. The phase shift is generated by the mechanical properties of the resonator, in particular its mechanical damping properties. The resulting elliptical motion of the resonator&#39;s tip is such that one of the semi-axes of the ellipse is aligned with the rod-axis and the other semi-axis of the ellipse is perpendicular thereto. A second piezoelectric actuator is used to reverse the direction in which the ellipse is traversed, and is placed at a different location on the resonator. The second piezoelectric actuator is located in such a way that it excites the same two modes but with a relative phase shift of −90 degrees.  
      Unfortunately, this actuator requires two sets of electronics to drive the motor in opposing directions, and has two sets of driving piezoelectric plates, resulting not only in a large number of parts but also greatly increasing the complexity of the system and resulting in significant costs for these type of motors. The motor also has limited power because the driving frequency is selected to be between two resonant frequencies. There is thus a need for a vibratory motor with simpler electronics, fewer parts, and greater efficiency.  
      In other vibratory motors, a piezoelectric element has a number of electrodes placed on different portions of the element in order to distort the element in various ways. Thus, for example, two modes of vibration can be excited by at least two separate, independently excited electrodes in each of four quadrants of a rectangular piezoelectric ceramic element. A second set of electrodes is used to reverse the direction in which the ellipse is traversed. The resulting elliptical motion is such that one of the semi-axes of the ellipse is aligned with the longitudinal axis of the motor and the other semi-axes of the ellipse is perpendicular thereto. As mentioned elsewhere, the ratio of the semi-axes can be advantageously used to increase motion or reduce travel time, by making advantageous use of ratios of 5:1, 10:1, or from 20-50:1. Again though, there are a number of electronic connections and many parts to achieve this motion, resulting in a high cost for this type of motor. It is an object of some aspects of the present invention to provide a micromotor, which is cheaper and easier to manufacture than previous art.  
      This invention further relates to a piezoelectric motor comprising a piezoelectric element connected to a resonator and a driven element interacting with the resonator. Furthermore, the invention relates to a method for avoiding failure of such a piezoelectric motor. Contrary to known piezoelectric motors, it is not attempted to avoid wear in the contact portion between the resonator and the driven element, but instead it is attempted to specifically use that wear for optimizing the motor properties and for increasing the motor performance.  
      Several piezoelectric motors comprising a piezoelectric component connected to a resonator and a movable element interacting with the resonator are known in the prior art. In these motors, piezoelectric elements, which vibrate mechanically when an appropriate electric voltage is applied thereto, are interactively connected to a resonator, which in turn contacts a driven element. The resonator transforms the vibrations of the piezoelectric element into preferably elliptical oscillations of the contacting portion between the resonator and the driven element. The driven element preferably moves in a first direction when a first voltage with a first frequency is applied to the piezoelectric element, and further moves in a second preferably opposite direction when a second voltage with a second frequency is applied to the piezoelectric element.  
      By way of these piezoelectric motors, electrically operated actuators can be moved over very exactly adjustable distances. This motion is silent to human hearing and consumes especially little energy. The piezoelectric motors are particularly cheap to manufacture. These piezoelectric motors can be used where a compact design requires a motor with little space requirements or where no or only small magnetic fields may be generated during operation.  
      Piezoelectric motors are mass products that, for example, in children&#39;s toys cause certain elements to move, or that in vehicles electrically deploy or actuate a variety of elements such as the ash tray or the side view mirrors. However, in all applications, one needs to ensure that the motors do not suffer a major loss of performance during operation and thus fail after a brief period of operation or do not perform their task in a satisfactory fashion anymore.  
      One goal of the invention is, therefore, to provide a piezoelectric motor that does not have the disadvantages of the prior art, and to provide a method for reducing the risk of failure of a piezoelectric motor during its life time.  
     SUMMARY OF THE INVENTION  
      This invention preferably uses a single piezoelectric element and a mechanical resonator to achieve its desired motion. The piezoelectric element has one pair of electrical contacts. The piezoelectric element is excited using sinusoidal electrical signals with the element, resonator, and sometimes the mounting system being configured so that at least two modes of vibration are excited by the single signal to generate an elliptic motion in the area where the resonator comes into contact with the body to be moved.  
      Unlike the prior art, the semi-axes of the ellipse advantageously are neither aligned with the longitudinal axis of the resonator nor in a direction perpendicular to it. Also, the relative phase shift between the two modes need not be close to 90 degrees so as to produce a circular or nearly circular-elliptical path. The amplitudes of the respective vibrations can be different in magnitude. At a given frequency, the motor  26  (see  FIG. 1 ) moves the body  42  in one direction. When operated at a different frequency, the motor  26  moves the body  42  in a different direction or different rotation. Preferably, it moves the body  42  in the opposite direction, but this will depend on the needs of the user and the design of the motor  26 , its support, and the driven body  42 . It is possible to operate the motor  26  at even more frequencies to generate additional motions of the body such as rotation and/or translation of an axle. The movement of driven body  42  in this disclosure refers to a translation or rotary motion of the body  42  in a common direction, rather than motion that merely moves the body  42  alternatively back and forth in a cyclic path to shake the body without any net translation or net rotation.  
      Advantageously, a piezoelectric element is mounted inside a mechanical resonator in part to preload the element in compression. The combined piezoelectric element and mechanical resonator are referred to as a motor or as a vibration element. The combined piezoelectric element and resonator are configured so that a single driving frequency excites at least two vibration modes sufficiently to cause an elliptical motion in a first direction at a predetermined point on the motor that is going to be used to drive a driven object. In particular, a vibration mode is typically along the longitudinal axis of the motor, and a second vibration mode is transverse thereto so as to result in bending or torsion. The motion can be achieved by appropriately configuring the resonator and piezoelectric element, or in some cases by locating the driving piezoelectric element offset from a longitudinal axis of the resonator to cause combined axial and bending motion.  
      The motion at a distal edge  44  at a distal end  36  of the resonator is typically greatest and is preferably used, although other locations on the motor can be used in some specific embodiments. The opposing end of the motor is the proximal end  35 . The result is the distal edge moves in an elliptical path resulting from a combination of at least two vibration modes when the motor is excited by a single signal at a first frequency. The motor is further configured such that a second driving frequency excites two resonant vibration modes in the motor so that the predetermined point on the motor rotates in an elliptical path in an opposite direction as the first elliptical path. A single piezoelectric element and resonator are thus driven by a single frequency to generate a first elliptical motion at a predetermined location on the vibratory motor. The piezoelectric element is driven at a second frequency to excite two resonant vibration modes of the vibratory motor that cause the predetermined location to move in a second elliptical motion in a different, and preferably opposite direction to the first elliptical motion, sufficient to move the driven element a desired distance. The two elliptical motions are typically not overlapping. The motion can be achieved at various locations on the motor, in varying amplitudes and directions, and that allows a variety of arrangements in which the motor can drive other elements.  
      Advantageously, the motor thus requires a single piezoelectric driver, a single resonator, and two separate frequencies to move objects in two opposing directions. The selection and configuration of the piezoelectric driver and the resonator achieve resonance or near resonant vibrations of sufficient magnitude to move objects with predetermined force. The effort expended in the design results in a motor of simple design, few parts, low cost and high efficiency.  
      In a further embodiment, the motor is resiliently urged toward the driven object. Depending on the mounting arrangement, the mounting may become part of the vibrating mass and affect the resonant vibration modes of the motor in order to achieve the desired motion at the desired location on the motor that is to be in contact with the driven object.  
      A simplified vibratory system is provided that has a source of vibration in driving communication with a resonator that has a selected contacting portion located to engage the driven element during use of the system. The source of vibration is preferably a piezoelectric element, but could comprise other elements that convert electrical energy into physical motion, such as magnetostrictive or electrostrictive devices in some specific embodiments. For convenience, a piezoelectric vibration source will usually be used in this description.  
      The vibrating element and resonator are configured to move the selected contacting portion in a first elliptical motion when the resonator is excited to simultaneously resonate in at least two vibration modes by a first signal at a first frequency provided to the vibrating element, according to a specific embodiment. The resulting elliptical motion is of sufficient amplitude to move the driven element when the driven element and selected contacting portion are maintained in sufficient contact to achieve movement of the driven element. The at least two vibration modes are selected so that at least one does not include a pure longitudinal or bending mode of the resonator in order to produce the first elliptical motion. The movement of driven elements referred to in this disclosure refers to a translation or rotary motion in a common direction, rather than motion that merely moves a part alternatively back and forth to shake the part without any net translation or net rotation.  
      The piezoelectric element and resonator are preferably configured to cause the selected contacting portion to move in a second elliptical motion a desired amount when excited to simultaneously resonate in at least two vibration modes by a second signal at a second frequency provided to the piezoelectric element, according to the specific embodiment. This allows multi-degree motion of the driven element by a single vibrating element. Additional vibration modes excited by different discrete frequencies can be used to provide different motions to the same selected contacting portion, or to different selected contacting portions engaging different driven elements. In one version of a preferred embodiment, the resonator comprises an elongated member with the selected contacting portion being located on an edge of a distal end of the member.  
      A number of variations on this basic combination are described, after which some further features and advantages are discussed. One variation includes having a resilient element interposed between a base and the vibratory element and located to resiliently urge the vibratory element against the driven element during operation of the system. There are advantages to having the vibration mode produce a node on the resonator element at the first frequency, with a resilient mounting connected to the vibratory element at the node and located to resiliently urge the vibratory element against the driven element during operation of the system. The resilient mounting could also be connected to the vibratory element at a location other than the node yet still located to resiliently urge the vibratory element against the driven element during operation of the system. The resilient mounting can help determine the various vibration modes.  
      Advantageously, the piezoelectric element is held in compression in the resonator during operation of the system. Preferably, the piezoelectric element is press-fit into an opening in the resonator to place the piezoelectric element in compression during operation of the system. Further advantages of this press-fit can be achieved if the piezoelectric element is held in compression by walls of the resonator that are stressed past their yield point, during operation of the system. Further advantages are derived by having the walls curved. Advantages are also provided if the piezoelectric element has an inclined surface adjacent an edge of the piezoelectric element to make it easier to press-fit the piezoelectric element into an opening in the resonator.  
      The first and second elliptical motions each have a major and minor axis, and there are advantages to having the ratio of the major to minor axes of each elliptical motion being in the range of about 3:1 to 150:1, and preferably from about 4:1 to 30:1, and ideally from about 5:1 to 15:1. Among other advantages, faster motion can be achieved, and the system design is easier to achieve. Advantageously, one of the major or minor axes is aligned with an axis of motion of the driven element in order to maximize the motion, and preferably the major axis is aligned.  
      There are advantages to having the major axes of these ellipses inclined at an angle with respect to a predominant axis of the vibratory element, and to maintain that inclination angle over a range of driving frequencies. There are thus advantages to having the system configuration and angle of inclination selected so that an angle β between the major axis and a tangent to the driven element at the selected contacting portion and along the direction of motion, varies by about 25 degrees or less over a frequency range of about 200 Hz or greater, on either side of the first frequency. Advantageously the angle β varies by about 10 degrees or less.  
      There are also advantages to having the angle vary in order to allow greater ease in system design and to improve performance, among other factors. Thus, there are advantages to having a major axis of the elliptical motion inclined at an angle β, with the angle β being between about 5-85 degrees when the selected contacting portion is drivingly engaging the driven element. Most of these ranges omit the range when the angle β is between about 0-5 degrees, and that occurs when the same selected contacting portion is used for multiple motions. But when the selected contacting portion achieves only one direction of motion of the driven element, it is possible to more closely align the axes and achieve alignments within about 0-5 degrees of the driven motion.  
      Another feature is the ability to achieve the desired motion over a range of driving frequencies in a manner that allows the use of components with lower tolerances and thus lower costs. Thus there is provided a vibratory element having a source of vibration vibrating a resonator to amplify the vibration. The resonator has a selected contacting portion located to engage a driven element to move the driven element along a driven path during use of the vibratory element. The selected contacting portion moves in a first elliptical path when the source of vibration is excited by a first electrical signal at a first frequency. The elliptical path has a major and minor axis which are not aligned with a predominant axis of the vibrating element by a defined angle that varies by less than about 10 degrees when the first frequency varies by about 200 Hz or more on either side of the first frequency. Preferably the defined angles varies by less than 5 degrees when the first frequency varies by 200 Hz, and desirably when the first frequency varies by 2.5 kHz, or more.  
      The other features described herein can also be used with this range of driving frequencies. Thus, as before, the source of vibration is preferably a piezoelectric element, but other elements could be used. The motion can be caused by pure vibration modes or by at least two vibration modes that are superimposed, but preferably at least one of the vibration modes is not a pure longitudinal mode or pure bending mode. Advantageously the vibratory element is connected to a resilient support located to resiliently urge the selected contacting portion against a driven element during use of the vibratory element. As desired, the resilient support can be used to help define the vibration modes generating the elliptical motion.  
      Another aspect comprises a vibratory component for moving a driven element using off-resonance vibration modes. The vibratory component includes a vibratory element, such as a piezoelectric vibration source, mounted to a resonator to form a vibrating element. The vibrating element has a selected contacting portion located to engage the driven element during use. A variety of piezoelectric vibration sources can be used, including plural piezoelectric elements to achieve the desired elliptical motion of the selected contacting portion. But preferably the selected contacting portion moving in a first elliptical path has a major axis and minor axis when the vibration source is excited by a first electrical signal that causes at least two vibration modes superimposed to create the first elliptical path. Advantageously at least one of the vibration modes is other than a pure longitudinal mode and other than a pure bending mode. Further, for this particular aspect, at least one of the at least two vibration modes is off-resonance, with the first electrical signal being amplified sufficiently to cause the at least one off-resonance vibration mode to produce a motion of the selected contacting portion having sufficient amplitude that the resulting elliptical path can move the driven element during use. This off-resonance feature can be used with other features described herein, including the resilient support, press-fit piezoelectric elements, and other features to name a few.  
      One feature not mentioned earlier but also applicable to the various embodiments and features disclosed herein is the use of a large aspect ratio on the elliptical motion of the selected contacting portion. The ratio of the major axis to the minor axis is preferably about 5:1 or greater, with ratios of 15:1 and 30:1 believed to provide usable but progressively less desirable motion. As the aspect ratio increases, the driving motion become more akin to an impact drive. Nevertheless, it is believed possible to have aspect ratios of 3:1-150:1, or even more, provide usable motion using the various features and embodiments of this disclosure.  
      One further aspect is the use of vibration modes other than pure longitudinal or pure bending. Thus, there is advantageously provided a vibration source mounted to a resonator to form a vibrating element. The vibrating element has a selected contacting portion located to engage the driven element during use. The selected contacting portion moves in a first elliptical path having a major axis and minor axis when the vibration source is excited by a first electrical signal that causes at least two vibration modes that are superimposed to create the first elliptical path. In this particular aspect, at least one of the vibration modes is other than a pure longitudinal mode and other than a pure bending mode. The elliptical motion has a major axis and minor axis, one of which is aligned with the first direction an amount sufficient to cause motion of the driven element. Stated differently, the vibratory element moves the selected contacting portion in first and second elliptical paths each having a major and minor axis. At least one of the major and minor axes does not coincide with the direction of motion resulting from the elliptical path with which the axis is associated. This use of vibration modes other than pure bending or pure longitudinal can be used with other features described herein, including the resilient support, press-fit piezoelectric elements, and other features to name a few.  
      Another aspect is the use of elliptical motion that does not align with the vibration element, but rather uses an inclined driving element and driven element. There is thus provided a vibratory system for moving a driven element that includes a driven element movable in at least a first direction. The vibration source is mounted to a resonator to form a vibrating element; the vibrating element having a selected contacting portion located to engage and move the driven element. For this particular aspect, the selected contacting portion moves in a first elliptical path having a major axis and minor axis at least one of which is not aligned with a longitudinal axis of the vibrating element. Advantageously, the longitudinal axis is inclined at an angle α to a tangent to the driven element in the first direction at the selected contacting portion. The angle α is between about 10 and 80 degrees when the selected contacting portion is drivingly engaging the driven element. That angle is further refined as discussed later. This use of the inclined axis can also be used with other features described herein, including the resilient support, press-fit piezoelectric elements, and other features to name a few.  
      There are also advantageously provided methods for implementing the above apparatus and advantages. In particular, it includes a method of configuring a vibratory system having a vibrating element with a selected contacting portion drivingly engaging a driven element to move the driven element by moving the selected contacting portion in a first elliptical motion. The method comprises analyzing that elliptical motion in a localized coordinate system in which at least one of the major and minor axes of the elliptical motion are not aligned with a predominant axis of motion of the vibrating element. The method then varies the system design to incline at least one of the elliptical axes relative to a tangent to the driven element in the direction of motion at the selected contacting portion to more closely align at least one axis with the tangent by an amount sufficient to achieve acceptable motion of the driven element. The inclination is achieved by altering the elliptical motion or altering the relative orientation of the vibrating element and the driven element, or both. That inclination is maintained during operation of the vibrating system.  
      There are advantages to orienting the localized coordinate system relative to the tangent. There are further advantages in setting the angle of inclination of the major axis of the first elliptical motion, designated by an angle β 1 , to an angle that is greater than 5 degrees, and with the vibrating element and the driven element being inclined relative to each other by an angle α that is greater than about 5 degrees.  
      The method also can include the provision of a vibrating element having the selected contacting portion moving in a second elliptical motion to move the driven element in a second direction a desired amount. A further variation of this method is to analyze that second elliptical motion in a similar method to the first elliptical motion. Thus, the second elliptical motion is analyzed in a localized coordinate system in which at least one of the major and minor axes of the second elliptical motion are not aligned with a predominant axis of motion of the vibrating element. The system design is altered to incline at least one of the second elliptical axes relative to a tangent to the driven element in the second direction at the selected contacting portion to more closely align the at least one axis of the second elliptical motion with the tangent in the second direction by an amount sufficient to achieve acceptable motion of the driven element in the second direction. It is advantageous to maintain that inclination of the second elliptical axis during use of the system. The orientation of at least one of the first and second elliptical axes is typically a compromise that is selected to achieve less than optimum motion of the driven element in one direction in order to improve the motion of the driven element in the other direction.  
      The method of analysis can also orient the localized coordinate system relative to the tangent, with the angle of inclination of the major axis of the first elliptical motion being designated by an angle β 1 , and with the vibrating element and the driven element being inclined relative to each other by an angle α that is greater than about 5 degrees. The angle of inclination of the major axis of the second elliptical motion can be designated by an angle β 2 , with at least one of β 1  and β 2  being greater than 5 degrees. Preferably, at least one of the angles β 1  and β 2  is between about 5-85 degrees. Moreover, in this method the vibratory element can be resiliently mounted to a base. The other features discussed herein could be used as well.  
      The present disclosure allows the use of simplified driving systems. One driving system uses an inductive coil mounted on the piezoelectric element and acting in cooperation with the inherent capacitance of the piezoelectric element to form an L-C driving circuit. The wire coil can be integrated into the vibratory element with the coil wire being also used as an electrical connection to the vibratory element, either in series or parallel.  
      The present disclosure also allows the use of a simple driver apparatus to control the operation of the vibrating element and its mechanical resonator when the vibrating element has an inherent capacitance. As mentioned, the piezoelectric element has an inherent capacitance. The control apparatus has at least one switching element allowing the application of a predetermined signal, such as the sinusoidal signal discussed herein. Further, there is at least one electrical resonator driver circuit driving the vibrating element, where the driver circuit is electrically coupled to and activated by the switching element. Finally, there is at least one inductive coil electrically coupled to the vibrating element to form an electric resonator together with the capacitance of the vibrating element so the signal excites the driver circuit at a predetermined frequency. The circuit resonances are selected to produce with the first and second signals at the first and second frequencies used to generate the first and second (and other) elliptical motions.  
      There are advantages if the coil is either mounted to the vibratory element or mounted to a common support with the vibratory element. Preferably the coil encircles a portion of the piezoelectric element or the mechanical resonator. Further, it is useful to locate the driver circuit and switching element more than four times further away from the piezoelectric element than the coil. To make the construction even simpler, the same electrical conductor that is used to form the coil can also connect the piezoelectric element to the driver circuit—either in parallel or series.  
      Moreover, in a further embodiment there is provided a piezoelectric resonator driver circuit having a plurality of unidirectional electrical gates to drive the piezoelectric element. The driver circuit is electrically coupled to and controlled by the control element; the piezoelectric element being electrically coupled to and paired with one of the unidirectional gates. At least one electromagnetic storage element, such as an inductive coil, is electrically coupled to the piezoelectric element so that the electromagnetic storage element forms an electric resonator together with the capacitance of the vibrating element. The unidirectional electrical gates can take the form of one or more diodes arranged to prevent a negative electrical voltage to the piezoelectric element. The driver circuit preferably resonates at a modulated predetermined first resonant frequency selected to cause the vibrating element to cause the selected contacting portion to move in the first elliptical motion with sufficient amplitude to move the driven element in the first direction when the selected contacting portion engages the driven element. The driver circuit also preferably resonates at a modulated predetermined second resonant frequency selected to cause the vibrating element to cause the selected contacting portion to move in a second elliptical motion with sufficient amplitude to move a driven element in the second direction when the selected contacting portion engages the driven element. Moreover, a resistor can be electrically coupled with the inductor and piezoelectric element and/or the gate element to maintain an input voltage to the piezoelectric element within predetermined operating parameters. Advantageously the diode(s) are coupled to the resistor in an orientation to prevent a negative voltage in the piezoelectric element.  
      The control methods achieved by the control circuits broadly include placing a control element in electrical communication with the piezoelectric element and an inductor to alternate the electric signal between the inductor and piezoelectric element, with the piezoelectric element providing a capacitance to function as a switched resonance L-C circuit so the electrical signal can resonantly drive the vibrating element at a first frequency. Advantageously a portion of the inductor is formed on the resonator.  
      Further, the method for controlling the operation of the vibrating element includes placing the control element in electrical communication with the piezoelectric element and the inductor to alternate the electric signal between the inductor and piezoelectric element, with the piezoelectric element providing a capacitance to function as a switched resonance L-C circuit so the electrical signal can resonantly drive the vibrating element at a first frequency. Preferably, the method further includes selecting the first frequency and configuring the vibrating element to cause a selected contacting portion of the vibrating element to move in a first elliptical path with sufficient amplitude to move a driven element in a first direction when the selected contacting portion engages the driven element.  
      Advantageously, the voltage to drive the piezoelectric element at the first frequency is greater than the supply voltage to the circuit. Moreover, the method includes placing a resistor in electrical communication with the piezoelectric element to shape the electrical signal provided to the piezoelectric element. Further, the method preferably forms, at least a portion of the inductor around a portion of the vibratory element. Finally, the inductor and piezoelectric element preferably provide a capacitance to function as a switched resonance L-C circuit so that a second electrical signal can resonantly drive the vibrating element at a second frequency, with the second frequency being selected in conjunction with the configuration of the vibratory element and its mounting to cause the selected contacting portion of the vibrating element to move in a second elliptical path with sufficient amplitude to move the driven element in a second direction when the selected contacting portion engages the driven element.  
      This invention also includes a method of configuring a vibratory system for moving a driven element that is supported to allow the driven element to move in a predetermined manner at a predetermined rate of travel with a predetermined force. The system has a selected contacting portion of a vibratory element periodically engaging the driven element to move the driven element, with one of the selected contacting portion and the driven element being resiliently urged against the other of the placed in resilient contact with the selected contacting portion and the driven element. The resilient contact is provided by a resilient support, with the vibratory element being caused to vibrate by a vibration source that converts electrical energy directly into physical motion. The vibratory element includes the vibration source mounted in a resonator with the selected contacting portion being on the resonator.  
      The method of configuring this system comprises defining a desired elliptical motion of the selected contacting portion to produce a desired movement of the driven element. At least one of the vibratory element and the resilient support is configured to cause the resonator to vibrate in two modes of sufficient amplitude and phase that the selected contacting portion moves in an elliptical path when the vibratory source is excited by a first signal at a first frequency provided to the vibration source. The elliptical path is sufficiently close to the desired elliptical motion to achieve an acceptable motion of the driven element.  
      The method can further comprise defining a second desired elliptical motion of the selected contacting portion to produce a second desired movement of the driven element. At least one of the vibratory element and the resilient support is configured to cause the resonator to vibrate in two modes of sufficient amplitude and phase that the selected contacting portion moves in a second elliptical path when the vibratory source is excited by a second signal at a second frequency provided to the vibration source. The second elliptical path is selected to be sufficiently close to the second desired elliptical motion to achieve an acceptable second movement of the driven element. The vibration source is preferably selected to comprise a piezoelectric element. Further, the resonator can be configured to cause the desired motion of the selected contacting portion, or the resonator in combination with a resilient support can be configured to cause the desired motion.  
      In addition to the selected contacting portion moving the driven element in a first direction when the source of vibration is driven by the first signal and moving the driven element in a second direction when the source of vibration is driven by the second signal, advantages arise if the selected contacting portion further moves in the first direction when a single sinusoidal signal of a first frequency is applied, and can also move in the first direction when the first frequency is dominant and superimposed with plural sinusoidal signals of different frequencies. In these latter instances, the second signal does not occur simultaneously with the first signal or else the first and second signals are of substantially different amplitude if they do occur simultaneously.  
      The method further includes placing the piezoelectric element in compression in the resonator during operation of the system by press-fitting the piezoelectric element into an opening in the resonator. This is preferably achieved by stressing walls of the resonator past their yield point but not past their ultimate strength point. The method further includes interposing a resilient element between the base and the vibratory element to resiliently urge the vibratory element against the driven element during excitation at the first frequency. Further methods to implement the above features and advantages are disclosed in more detail below.  
      A further method includes a method for moving objects using vibratory motors having a vibration source placed in a resonator. The method comprises moving a selected contacting portion of a resonator in a first elliptical motion in a first direction by configuring the resonator to simultaneously vibrate in two modes of sufficient amplitude and phase to cause the first elliptical motion of the selected contacting portion when a single electrical signal is applied to the vibration source. The method can further comprise placing the selected contacting portion in resilient contact with a driven element to move the driven element. Additionally, the method can further comprise connecting a resilient element to the resonator to resiliently urge the resonator against a driven element.  
      Other aspects of this method include selecting a piezoelectric element for the vibration source and placing that piezoelectric element in compression by press fitting it into an opening in the resonator. The opening is preferably defined by at least two opposing walls that are stressed beyond their elastic limit when the piezoelectric element is press-fit into the opening. There are advantages if the walls are selected to be curved.  
      When a piezoelectric element is used for the vibration source, the inherent capacitance of the piezoelectric lends itself to the use of simplified control systems while still maintaining system performance. A control switch can activate a resonator driver circuit driving the vibrating element, with at least one electromagnetic storage element, such as an inductive coil, electrically coupled to the vibrating element to drive the vibrating element when the driver circuit is activated. The vibrating element increases charge when the electromagnetic storage element discharges and the coil increases its charge when the vibrating element discharges and the driver circuit is not activating the vibrating element. This construction basically places a control element in electrical communication with the piezoelectric element and an inductor to alternate the electric signal between the inductor and piezoelectric element, with the piezoelectric element providing a capacitance to function as a switched resonance L-C circuit so the electrical signal can resonantly drive the vibrating element at a first frequency selected to achieve the desired elliptical motion at the selected contacting portion. This allows the voltage to drive the piezoelectric element at the first frequency to be greater than the voltage of the electrical signal provided to the control element. The same circuit can be used to provide the electrical signal for other vibration modes of the piezoelectric element.  
      Further, the coil can be mounted to the vibratory element or mounted to the same support as the vibratory element. Advantageously, the coil can encircle a portion of the vibratory element. Moreover, the coil can be connected to the piezoelectric element in series, or in parallel. Additionally, the piezoelectric driver circuit can have a plurality of unidirectional electrical gates, such as a diode, can be paired with the piezoelectric element to prevent or at least limit any negative voltage to the piezoelectric element. In these driver circuits, the frequency is selected to achieve the desired motion of the selected contacting portion.  
      The improvements further include improved manufacturing and assembly aspects for vibratory apparatus used to move a driven element. In these aspects a vibration source is used that converts electrical energy directly into physical motion. A resonator is provided having an opening defined by at least two opposing sidewalls that are stressed beyond their elastic limit to hold the vibration element in compression. The vibration source is within that opening so that the vibration element is held in compression by the resonator under a defined preload during operation. Advantageously, the vibration source is press-fit into the opening, and comprises a piezoelectric element. Further advantages are achieved if the sidewalls are curved.  
      Moreover, it is useful to provide the piezoelectric element with at least two opposing edges that are inclined and located to engage edges of the opening to make it easier to press-fit the piezoelectric element into the opening while reducing damage to the piezoelectric element. The reduction of damage is especially desirable in view of the damage that can occur to the piezoelectric element and to the resonator if the inclined edges are absent. Preferably, there are at least two opposing edges that have surfaces substantially parallel to the abutting walls defining the opening, and an inclined surface extending therefrom to a contacting surface abutting one of the walls, with the contacting surface exerting the preload.  
      In one embodiment, a resonator has a longitudinal axis with an opening partially defined by two sidewalls on opposing sides of the longitudinal axis and two opposing end walls on the longitudinal axis. A piezoelectric element is held in compression by the opposing end walls, with each of the sidewalls being stressed beyond its elastic limit to hold the piezoelectric element in compression. The resonator has a selected contacting portion, which moves in a first elliptical motion when the piezoelectric element is excited by the various electrical signals described herein. There are advantages if the sidewalls are curved, and if at least one of the end walls or two opposing sides of the piezoelectric element that engage the end walls have edges that are inclined to facilitate press-fitting the piezoelectric element into the opening and wherein the piezoelectric element is press-fit between the end walls. The sidewalls can be curved to bow away from the piezoelectric element, or toward the piezoelectric element. Further, a portion of an elastic element for supporting the resonator can be interposed between one of the end walls and the piezoelectric element.  
      The invention also includes a method of placing a piezoelectric element in compression in a resonator, where the resonator has end walls and sidewalls defining an opening sized to receive and place the piezoelectric element in compression. The method includes increasing the distance between opposing end walls enough to allow the piezoelectric element to be forced between the end walls with a force that by itself could not force the piezoelectric element between the end walls in the original state of the opening, and thereby placing the piezoelectric element in compression while also stressing the sidewalls beyond their elastic limit. The method can further include providing an inclined surface on at least one of either the end walls or the corresponding edges of the piezoelectric element, and forcing the piezoelectric element into the opening by engaging said at least one inclined surface.  
      Moreover, the method can include pulling the opposing end walls apart while forcing the piezoelectric element into the opening. In one further embodiment, the method includes curving the sidewalls away from each other, and urging the opposing, curved sidewalls toward each other in order to move the end walls away from each other and then placing the piezoelectric element between the end walls. In another embodiment, the method includes curving the sidewalls toward each other, and urging the opposing, curved sidewalls away from each other in order to move the end walls away from each other and then forcing the piezoelectric element between the end walls. The various methods can also include interposing a resilient mount for the piezoelectric element between the piezoelectric element and one of the end walls.  
      There is also advantageously provided a piezoelectric element configured to be press-fit into an opening in a resonator. The opening is defined by sidewalls located on opposing sides of a longitudinal axis through the opening and separated by a first dimension, with opposing end walls located on the longitudinal axis and separated by a second dimension. The piezoelectric element has a first dimension that is smaller than the first dimension of the opening and has a second dimension larger than the second dimension of the opening and selected to stress the sidewalls beyond their elastic limit when the piezoelectric element is inserted into the opening. The piezoelectric element has inclined edges corresponding in location to edges of the end walls when the piezoelectric element is aligned to be inserted into the opening. The above variations can also be used with this embodiment, including curved sidewalls, a resilient support for the resonator interposed between one end wall and the piezoelectric element during use, and at least one inclined edge corresponding in location to an edge of the end wall when the piezoelectric element is aligned to be inserted into the opening.  
      There is also advantageously provided a resonator  24  for use with a piezoelectric actuator. The resonator has a continuous walled, externally accessible opening sized to receive a piezoelectric element or other source of vibration, and to hold that element in compression. The opening is optionally, but preferably defined in part by opposing sidewalls that are curved. The walls can be curved toward, or away from the opening and the piezoelectric element therein. Preferably the sidewalls are curved, and have a uniform cross section for a substantial portion of the length of the sidewall. A substantial length includes over half the length, preferably more, and ideally the entire length until the junction with the end walls is reached. Rectangular cross sections are preferred.  
      The problems of the prior art are addressed with motors preferably, but optionally, having one or more of the following basic features. There is provided a piezoelectric motor having at least one piezoelectric element that is connected to a resonator and having a driven element that interacts with the resonator, characterized in that at least during portions of the service life at least one piezoelectric motor operating state variable changes without an external influence and thus improves or lessens motor efficiency. The change of the operating state variable occurs slower than the duration of one vibration period of the excitation frequency, preferably 5 times slower than the duration of one vibration period of the excitation frequency. There is also provided a piezoelectric motor having at least one piezoelectric element that is connected to a resonator and having a driven element that interacts with the resonator, characterized in that the resonator interacts with the driven element through a contact portion and that the contact portion changes without an external influence at least during portions of the service life of the piezoelectric motor. The change of the operating state variable occurs slower than the duration of one vibration period of the excitation frequency, and preferably 5 times slower than the duration of one vibration period of the excitation frequency.  
      Further provided is a piezoelectric motor having at least one resonator, at least one piezoelectric element that has two electric terminals and is connected to the resonator, and a driven element that interacts with the resonator. The motor is characterized in that at least during portions of the service life at least one piezoelectric motor operating state variable changes without an external influence. The change of the operating state variable occurs slower than the duration of one vibration period of the excitation frequency, preferably 5 times slower than the duration of one vibration period of the excitation frequency. Preferred embodiments of the above piezoelectric motors can be found in the dependent claims.  
      There is also provided a method for avoiding failure of a piezoelectric motor having any features of combinations of features described herein or defined in the claims, wherein the change of at least one operating state variable causes an increase of the efficiency of the piezoelectric motor. There is further provided a method for avoiding failure of a piezoelectric motor having any features of combinations of features described herein or defined in the claims, wherein the change of at least one operating state variable counteracts a change in another operating state variable, the change in another operating state variable having occurred without an external influence. Preferred embodiments of the above summarized method can be found in the dependent claims.  
      There is thus provided a piezoelectric motor wherein a change of at least one operating state variable improves or reduces the motor performance. The piezoelectric motor described herein is preferably especially easily manufactured and offers a constant performance or a performance that preferably increases with the duration of operation. An initial test of the piezoelectric motor can determine the motor&#39;s minimal degree of efficiency even before regular motor operation actually begins.  
      Any variable that influences the state of operation of a motor is an operating state variable. The operating state variable is preferably the frequency of operation, the electrical current consumption, the vibration modes of the resonator, the pressure and/or the friction properties between the resonator and the driven element, the resonator&#39;s orientation and/or impact angle at the contact portion to the driven element, or the shape, position and/or the orientation of the contact portion. The change of the operating state variable preferably occurs slowly compared to the service life of the motor. It is preferable that more than one oscillation period of the excitation signal to the piezoelectric element is needed in order to change the respective operating state variable. Preferably more than 5 oscillation periods are needed. The operating state variables are preferably understood as quantities that are averaged over at least 10 periods of the excitation signal. The change occurs preferably in an irreversible manner. Preferably, the change of the operating state variable occurs without external influence. This means preferably that the change of the respective operating state variable is achieved without a change of the control signal to the piezoelectric element.  
      An operating state variable is thus a quantity that is linked to motor performance and whose rate of change is slow compared to the period of the piezoelectric motor control signal during motor operation. Operating state variables preferably cause motor performance to vary at a rate that is slower than five oscillation periods of the piezoelectric motor control signal. Operation state variables change due to regular motor operation and due to the interaction between the piezoelectric motor and a driven element during regular motor operation. Operation state variables are intrinsic to the piezoelectric drive system and are not caused by external means other than those needed for regular motor operation.  
      For example, the optimal frequency of operation, i.e., the frequency at which a sinusoidal electric signal to the piezoelectric element would cause optimal motion of the driven element, typically changes with piezoelectric motor temperature. During regular motor operation, the piezoelectric motor temperature increases due to dielectric losses in the piezoelectric element at a rate that is slow compared to the excitation frequency. The rise in temperature typically causes a reduction of the optimal frequency of operation at a similarly slow rate. In applications, the frequency of the control signal to the piezoelectric element is often adjusted in order to compensate for the varying optimal frequency of operation in order to avoid a change of motor performance. As this example shows, temperature and the optimal frequency of operation are operation state variables. The actual control signal on the other hand is not.  
      Other operation state variables are, for example, the state of wear of the resonator contact portion to the driven element and the characteristics that depend thereon, such as the shape of the contact portion or the location and orientation of the resonator with respect to the driven element.  
      A calibration adjustment on the other hand is considered an external influence as it does not occur during regular motor operation. While such an external means may also cause an operation state variable to change, such means will not be considered here. A noted exception is externally forcing a relative motion between the resonator and the driven element during a run-in phase, which occurs before regular motor operation takes place.  
      The elements of the piezoelectric motor are preferably configured so that motor operation, and particularly the run-in phase of the motor, modifies the motor performance in a predetermined manner. One the one hand, this is achieved by various changes of operating state variables mutually and almost completely compensating each other. On the other hand, this is achieved by using elements and/or materials that during the run-in phase maintain the piezoelectric motor performance at a constant level or preferably increase the performance. All in all, the motor performance can also worsen due to the change in the operating state variable.  
      In particular if one knows the run-in behavior of the elements and if one knows the dependence of the operation properties on the duration of operation of the piezoelectric motor, then the change of motor performance that is caused by the run-in procedure or by the duration of operation can be compensated, or it can also be supported if so desired, preferably by a predetermined change of one or more other operating state variables.  
      The piezoelectric element and the resonator are described in an exemplary fashion in DE 100 62 672 A1 and in the unpublished German patent application with the application number 101 41 820.5. The resonator is interactively connected to the piezoelectric element and transfers its vibrations onto the driven element. In doing so, the resonator preferably vibrates close to its resonance. The resonator preferably comprises a flat and essentially two-dimensional shape wherein the resonator shape remains nearly unchanged along the third dimension, which is preferably taken along the thickness of the resonator. An exemplary embodiment of the piezoelectric motor has already been described in WO 01/41228A1 (the disclosure of which is now U.S. Pat. No. 6,768,245) or in the parallel application with the application number PCT/EP01/03245. These patent applications are hereby incorporated by reference and thus are considered to be parts of the disclosure.  
      In a preferred embodiment, the piezoelectric element has a monolithic construction. In a particularly preferred embodiment this monolithic piezoelectric ceramic element comprises internal copper electrodes that are arranged in an interlocking comb-like fashion. Such piezoelectric elements are described in the German patent application 101 46 704.4. This patent application is hereby incorporated by reference and thus is considered to be a part of the disclosure. The piezoelectric ceramics is preferably implemented as a single piece (even though it may be a multilayered element).  
      In a preferred embodiment the piezoelectric motor is excited by a single piezoelectric element having two terminals for applying an excitation signal. In a particularly preferred embodiment the signal is a sinusoidal or a rectangular wave signal. This particularly preferred embodiment permits using signals that can more easily be generated than saw-tooth shaped signals.  
      The vibration modes of the resonator are the result of the position and the clamping of the piezoelectric element as well as of the resonator shape and its mass. The run-in phase and the duration of operation, respectively, directly influence the vibration modes. Furthermore, different regions of the resonator perform different vibrations. The contact portion between the resonator and the driven element changes during the run-in phase as well as during operation of the motor. In this manner, by way of an adapting contact portion, a region of the resonator can transfer vibrations onto the driven element that are decidedly different from the vibration modes of the contact portion at the start of operation of the piezoelectric motor.  
      There is thus provided a piezoelectric motor comprising a resonator interactively connected to a piezoelectric element and configured to exhibit different vibrations in different regions of the resonator during operation of the piezoelectric motor when an electric input signal excites the piezoelectric element to vibrate, wherein the resonator has a contact portion to the driven element and wherein the contact portion is allowed to be worn either due to piezoelectric motor run-in or due to regular piezoelectric motor operation, and wherein the wear of the contact portion consecutively exposes different regions of the resonator to interaction with the driven element, so that the consecutively different vibrations of these different, consecutively exposed regions cause the motor performance to change in a predetermined manner. In a preferred embodiment the wear of the resonator contact portion is further configured to simultaneously cause the resonator location and/or orientation to change with respect to the driven element, this change being simultaneous with the consecutive exposing of different resonator regions so that the different vibrations are always oriented in a predetermined way with respect to the driven element and the resulting motor performance is thus maintained in a predetermined manner.  
      In a preferred embodiment the piezoelectric motor also comprises a resilient element, which may also be called an elastic element, for resiliently urging the resonator contact portion against the driven element and for maintaining a predetermined resonator location and/or orientation with respect to the driven element. In a preferred embodiment the resilience of the resilient element causes that resonator location and/or orientation to change when the resonator contact portion is worn. In a further preferred embodiment the resilient element also mounts the resonator to a base.  
      In a preferred embodiment the resonator behavior during run-in and operation, respectively, is preferably measured by way of a measurement of the changing vibration modes, of the frequencies, of the degree of efficiency, etc. In this way, the run-in behavior of the resonator can be determined with simple means. Subsequently, by way of a predetermined change of an operating state variable, preferably by way of adapting the operating frequency to the changed resonance frequency, the mechanical power of the piezoelectric motor can be maintained at least at a constant level.  
      The contact portion between the driven element and the resonator is a surface, and preferably a two-dimensional surface or even a curved surface when the geometries of the two elements mutually and preferably exactly conform to each other. During operation of the piezoelectric motor, the run-in causes changes of the contact portion, which then represents a modified area whose position, shape and/or orientation is different compared to its original state. Therefore, in a preferred embodiment the change of the contact portion occurs in such a manner that the mechanical power of the piezoelectric motor is increased. It is preferable if the contact surface area of the selected contact portion increases in size compared to the initial, virgin state that existed prior to wear and/or if the contacting angle between the resonator and the driven element changes during the duration of operation of the motor.  
      The normal contact force between the resonator and the driven element occurs in their mutual contact portion and, together with the friction force, directly influences the mechanical power that is transmitted from the piezoelectric motor to the driven element. The normal contact force depends on the stiffness of the resilient element and on what distance the resilient element moves the resonator in order to compensate for wear at the resonator tip. In prior art systems, one often encounters the problem that over time this normal contact force deteriorates to the point where the motor ceases to generate a sufficient amount of mechanical power. A known countermeasure is the use of very hard, wear-resistant materials.  
      In a preferred embodiment, an element, preferably a spring, having small stiffness is used as resilient element, the element being preloaded by a large amount as can bee seen by comparing the unloaded spring in  FIG. 82   b  with the loaded and mounted spring in  FIG. 84 . Wear of the contact portion then results in a comparably small change in contact force.  
      This change occurs during the useful service life of the motor and can advantageously be influenced in a predetermined manner by configuring the surface of the contact portion to be worn such that properties can be increased, decreased, or maintained at constant levels in a predetermined manner.  
      It is advantageous to configure the contact portion to be initially small, which offers the advantage of a quick run-in of the motor, but then to be enlarged by wear, which results in a considerable wear reduction. The contact portion can advantageously grow to the extent that with a concurrent slight decrease of preload due to contact portion growth, the pressure is reduced to a point where wear practically ceases to take place. It is advantageous if the contact portion is enlarged by a factor of at least 1.5. It is even more advantageous if the contact portion grows by a factor of at least 10. It is advantageous if the contact portion reaches a size of at least ¼*LG*L 5  (cf.  FIGS. 82   b  and  83   c ) where LG corresponds to the resonator thickness and L 5  corresponds to the width of the resonator leg that abuts the contact portion.  
      The process of displacing the contact portion advantageously brings other points on the resonator into contact with the driven element. These points can exhibit different motion vectors as.  FIG. 88  illustrates. By this property, the predetermined manipulation of the motor performance over the motor&#39;s life time is also possible. By selecting an appropriate resilient element  50 , e.g. a coil spring  507 , the angle of the motion vectors can advantageously be configured as a function of the duration of operation. It is advantageous to configure the angle change that is due to contact portion wear in such a manner that, corresponding to the state of wear, the motion vectors form a predetermined angle to the driven element. This angle is advantageously such that the motor either generates an optimal driving force or exhibits a predetermined driving force history.  
      The wear of the contact portion can advantageously be larger than 0.05*LG (where LG corresponds to the thickness of the resonator per  FIG. 82   b ) without changing mechanical power by more than 25%. The wear of the contact portion can even more advantageously be larger than 0.3*LG without changing mechanical power by more than 15%. The motor can advantageously be configured to exhibit increasing motor performance over a specified duration of operation.  
      There is thus provided a piezoelectric motor comprising a resonator having a contact portion in interactive contact with a driven element, wherein the contact portion is allowed to be worn during run-in and/or during regular motor operation, wherein the contact portion is configured to be initially small in a virgin state and than to increase in size due to wear. The initial, virgin contact portion is preferably an edge of the resonator or a preformed surface shape that conforms to the driven element. The increase of contact area of the resonator contact portion causes the contact pressure, which is approximately calculated as contact force divided by contact area, to decrease, which results in reduced subsequent wear. Wear also causes the contact portion to recede into the resonator, which causes the resilient element to resiliently give way in order to maintain the resonator in contact to the driven element. As a result, the resilient force of the resilient element also decreases simultaneously with the increasing contact area, which reduces subsequent wear even more. By themselves or in combination, these two effects can bring wear and the motor performance variations that depend thereupon to a virtual standstill. This state of standstill in some sense represents an equilibrium and is preferably reached during the run-in phase of the piezoelectric motor but may also be reached later during regular operation.  
      Using a resilient element with a small stiffness also offers advantages regarding the tolerance requirements during assembly of the motor. Advantageously, resilient elements with nonlinear characteristics can also be used.  
      In a particularly preferred embodiment this resilient element comprises a coil spring that is connected to the resonator, wherein the coil spring has preferably round coil portions. Moreover, in an exceptionally preferred embodiment the coil spring is damped, preferably by a foamed material that has been pressed into the coils. In this particularly simple manner, a suspension for the piezoelectric motor is created that on one hand is capable to compensate for the run-in induced changes in the operating state variables of the piezoelectric motor and to keep the contact force in the contact portion constant. On the other hand, spring vibrations that are superposed onto the resonator vibrations are reduced in this way.  
      In a further preferred embodiment the coil spring comprises coil windings that are configured to not touch each other. In a particularly preferred embodiment this is achieved by a spring that in the region of its windings comprises a defined inclination of the planes that are spanned by the individual windings. In this manner one securely avoids that those resonator vibrations that are absorbed by the spring are afflicted by friction and cannot be transformed into piezoelectric motor driving force.  
      There is thus provided a resilient element for resiliently urging the resonator of a piezoelectric motor against a driven element, wherein the resilient element comprises a coil spring having a helical coil portion made up of windings, the helical coil portion having a sufficient pitch, and the windings having a corresponding sufficient mutual distance, that prevents the windings to touch each other at rest and when vibrating during use of the piezoelectric motor. Preventing mutual, intermittent vibration-induced contact during piezoelectric motor operation reduces the amount of energy that is absorbed by the spring and is thus not available for moving the driven element.  
      In addition, in a further particularly preferred embodiment the coil spring is pressed into recesses in the resonator. The springs are hereby pressed into the recesses by urging a flat urging tool against the spring wire. In a particularly preferred embodiment the diameter of the spring wire is somewhat larger than the width of the recess so that a form-fit is created between the resonator and the spring in such a way that there is no air gap. In a particularly preferred embodiment the resonator further comprises a bulge that extends above the resonator side and that is located directly next to the recess. In this particularly advantageous embodiment, the urging tool molds the bulge material around the wire and thus causes a partial enclosure of the wire. Therefore, a force- and form-fit between the coil spring and the resonator is achieved with particularly simple means, the means securely preventing a loss in performance due to vibration components being absorbed and reabsorbed by the joint.  
      In an exceptionally preferred embodiment the wire comprises a cross-section that at least in the region of the resonator joint is non-circular. It can also be extremely advantageous to introduce an additional joining means at the joint such as, for example, an adhesive. In this particularly simple manner, a joint to the resonator is created that also inhibits twisting.  
      The impact angle of the resonator onto the driven element, i.e., the orientation of an elongated resonator with respect to the driven element, is measured in the contact portion between the two elements. This contact portion is spanned on the one hand by the surface of the driven element at the driving point, and on the other hand by the direction of vibratory motion of the resonator at the corresponding contact point.  
      When in a particularly preferred embodiment coil springs are used that are joined to the resonator, then the resonator orientation, and therefore the impact angle, i.e., the orientation of a principal axis of the resonator with respect to the contact portion, can change during the duration of operation in order to attain a desired motor performance.  
      In a preferred embodiment the resonator and/or the driven element comprise several materials in at least the contact portion. For example, the resonator and/or the driven element can comprise a hardened core by the contact portion. This embodiment causes a relatively quick run-in phase of the piezoelectric motor because, after the relatively soft layer has quickly worn off, constant conditions prevail over an extended period of time due to the harder core.  
      In a further preferred embodiment the surfaces of the resonator and/or of the driven element comprise a surface coating with selected friction properties in their respective contact portion. After the operation of the piezoelectric motor has worn off this surface coating, the friction properties of the base material determine the mechanical power of the piezoelectric motor. In using particularly simple means, one thus can transition from one set of friction properties to another set of friction properties in order to influence the piezoelectric motor performance in a preferably positive manner.  
      If it is desired that the contact portion adapts during operation of the piezoelectric motor, it can be advantageous if the surface of the resonator and/or of the driven element comprises a coating having abrasive particles such as, for example, glass fibers or glass spheres, carbon fibers, or mineral fillers for polymers. The run-in phase of the piezoelectric motor and the phase of mutual adaptation between the respective contact portions occur particularly quickly in this case. The equilibrium whereat nearly no change of the contact portions is observed during the duration of operation is also particularly quickly reached.  
      In a further preferred embodiment it is arranged that the service life of the piezoelectric motor is terminated when at least one operating state variable or the generated driving force leaves a predefined domain so that the replacement of the piezoelectric motor is needed. In a preferred embodiment the resilient element urging the resonator to the driven element comprises a limiting element for this purpose so that no more forces are transmitted when a predetermined deflection is reached. In a further preferred embodiment the piezoelectric motor comprises a counter of operating hours that switches the piezoelectric motor off after a predetermined duration of operation is reached. In this manner, simple means guarantee that piezoelectric motors provide persistently adequate mechanical power for a particular application without uncontrolled deterioration of performance.  
      The operation hours of the piezoelectric motor can be determined by measuring the changing operating state variables. The measurement can be done, for example, using an electrical contact that opens or closes when a predetermined deviation from certain initial states is reached. By the same token, one can determine the resonance frequencies of the piezoelectric motor, which change during the duration of operation.  
      There is thus provided a piezoelectric motor that is configured to cease operation, or to fail, when a predetermined operating state is reached. The motor is thus prevented from slowly and gradually deteriorating into a domain of uncertain operational characteristics. For example, a piezoelectric motor comprising an elastic element can be equipped with a limiting contact element for closing an electrical circuit when it contacts an electrically conducting portion of the resonator. The contact element is configured such that the contact occurs when the resonator contact portion has been worn by a predetermined amount and the elastic element thus has relocated and/or reoriented the resonator by a predetermined amount. If the piezoelectric motor is configured to wear at a predetermined rate depending on the total amount of relative motion between the resonator and the driven element, predetermined motor failure will occur after a predetermined total distance of motion of the driven element.  
      The method for avoiding piezoelectric motor failure arranges for a change, and preferably an increase, of piezoelectric motor efficiency by way of a predetermined change of at least one operating state variable. Consequently the gradual loss of piezoelectric motor performance is prevented in a safe and effective manner.  
      In a preferred embodiment of the method the change of at least one operating state variable counteracts the change of another operating state variable, which has no external cause. In this manner there is thus provided a method that when executed ensures a constant efficiency of the piezoelectric motor during its entire service life.  
      A further object is a method for the run-in of a piezoelectric motor, characterized in that the resonator and the driven element are moved relative to each other with external means before the motor enters into its actual intended service so that the contact portion between the resonator and the driven element is ground in.  
      In a preferred embodiment the surface of the driven element is allowed to wear in addition to, or instead of, allowing the resonator contact portion to wear. Similar properties as for the case of resonator wear alone can than be derived. Instead of the resonator contact portion receding into the resonator due to wear at the contact portion, the surface of the driven element withdraws instead.  
      The following description and figures are for illustration only, and are not intended to limit the inventive ideas in any way.  
      Given the present disclosure, further methods will be apparent to one skilled in the art to implement the above features and advantages, and the features and advantages discussed below. Further, other objects and features of the invention will become apparent from consideration of the following description taken in connection with the accompanying drawings, in which like numbers refer to like parts throughout. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       FIGS. 1   a - 1   d  show a plan side view, side perspective view, end view, and bottom view, respectively, of a first embodiment of this invention;  
       FIG. 2  shows a top view of the vibratory element of  FIG. 1 ;  
       FIG. 3  shows an end view of  FIG. 2 ;  
       FIG. 4  shows a perspective view of a second embodiment;  
       FIG. 5  shows a side view of a third embodiment using a C-clamp configuration;  
       FIG. 6  shows a perspective view of a fourth embodiment driving multiple elements;  
       FIG. 7   a  shows a perspective view of a vibratory element containing a press-fit piezoelectric element;  
       FIG. 7   b  shows an enlarged portion of the vibratory element of  FIG. 7   a  during assembly;  
       FIG. 8  shows a fifth embodiment having a press-fit piezoelectric element;  
      FIGS.  9  shows a top view of a press-fit embodiment before deformation;  
       FIG. 10  shows a top view of the embodiment of  FIG. 9  after deformation by a cylindrical wedge;  
       FIG. 11  shows a sectional view along line  11 - 11  of  FIG. 10 ;  
       FIG. 12  shows a top view of an alternative embodiment of  FIG. 9  using a rectangular wedge;  
       FIG. 13  shows an embodiment with the piezoelectric element offset from the axis of a resonator;  
       FIG. 14  shows an embodiment with an insert offsetting the force from the piezoelectric element from the centerline of a resonator;  
       FIG. 15  shows an embodiment with the piezoelectric element skewed relative to the axis of the resonator;  
       FIG. 16  shows an embodiment with the piezoelectric element positioned between selectably positioned inert elements and compressed by a threaded fastener;  
       FIGS. 17-19  show suspension configurations for a vibratory element having a pivoted support for the vibratory element;  
       FIGS. 20-21  show suspension configurations for a vibratory element having a resilient support;  
       FIG. 22  shows a suspension configuration for a vibratory element having a pivoted support;  
       FIGS. 23-24  show configurations of a vibratory element and driven element with the longitudinal axes of the parts in parallel but offset planes;  
       FIG. 25  shows a configuration of a vibratory element and driven element with the axes of the parts inclined at an angle;  
       FIG. 26  is an end view of the configuration of  FIG. 25 ;  
       FIGS. 27-29  show configurations of two vibratory elements located in parallel but offset planes relative to the plane of the driven element;  
       FIG. 30  shows a configuration of two vibratory elements located in the same plane but offset from the plane containing the driven element;  
       FIG. 31  shows a configuration of two vibratory elements and one driven element with the driven elements located above and below the driven element and at inclined angles relative to the driven element and facing each other;  
       FIG. 32  shows a configuration of two vibratory elements and one driven element with the driven elements located above and below the driven element and at inclined angles relative to the driven element and facing the same direction;  
       FIG. 33  shows a configuration of two vibratory elements and one driven element with the driven elements located on one common side of driven element and at inclined angles relative to the driven element and facing the same direction;  
       FIG. 34  shows a configuration of two vibratory elements and one driven element with the driven elements located on one common side of the driven element and at inclined angles relative to the driven element and facing each other;  
       FIG. 35  shows a configuration of two vibratory elements and one driven element with the driven elements located on opposing sides of the driven element and at inclined angles relative to the driven element and facing the same direction;  
       FIG. 36  is an end view of the configuration of  FIG. 35 ;  
       FIGS. 37-40  show configurations of three vibratory elements and one driven element;  
       FIG. 41  shows a front view of a configuration of six vibratory elements and one driven element;  
       FIG. 42  shows a left side view of the configuration of  FIG. 41 ;  
       FIG. 43  shows a diagram of the elliptical motion of the selected contact portion;  
       FIGS. 44-51  show graphical presentations of various aspects affecting the elliptical motion of the contacting portion depicted in  FIG. 43 ;  
       FIG. 52  shows a perspective view of a vibratory element having a slot in the resonator in the same face of the resonator in which the opening is formed to receive the piezoelectric element;  
       FIG. 53  shows a perspective view of a vibratory element having a slot in the resonator and an opening with curved ends to receive the piezoelectric element;  
       FIG. 54  shows a perspective view of a vibratory element having a wider slot in the resonator;  
       FIG. 55  shows a perspective view of a vibratory element having a slot in the resonator in a face of the resonator that is different from the opening formed to receive the piezoelectric element;  
       FIG. 56  shows a perspective view of a vibratory element having an “H” shaped opening to receive the piezoelectric element;  
       FIG. 57  shows a perspective view of a vibratory element having a slot defining two beams in the resonator with the piezoelectric element being located in one beam;  
       FIG. 58  shows a perspective view of a vibratory element having a hole in the resonator to alter the performance of the vibratory element;  
       FIG. 59  shows a perspective view of a vibratory element having an enlarged mass at a proximal end of the resonator;  
       FIG. 60  shows a perspective view of a vibratory element having four sidewalls defining the opening in which the piezoelectric element is placed;  
       FIG. 61  is a cross sectional view of a vibratory element enclosing the piezoelectric element in a cavity within the resonator;  
       FIG. 62  is a side view of a vibratory element having several selected contacting portions to engage a driven element;  
       FIGS. 63-66  are electrical schematics for systems to provide electronic signals to the vibratory elements;  
       FIG. 67  is a plan side view of a piezoelectric element having specially configured ends;  
       FIG. 68  is a perspective view of the piezoelectric element of  FIG. 67 ;  
       FIG. 69  is a side sectional view of a die used to form the piezoelectric elements of  FIGS. 67-68 ;  
       FIG. 70  is a schematic view of a vibrating driving element and a vibrating driven element of a further embodiment;  
       FIGS. 71-72  are schematic views of several positioning sensing configurations;  
       FIG. 73  shows cross-sections for resonator elements;  
       FIG. 74  shows a schematic view of a vibrating element with a curved spring suspension system;  
       FIG. 75  shows a sequence for press-fitting a piezoelectric into an opening in a resonator;  
       FIG. 76  shows a pull-fit process for a piezoelectric motor assembly;  
       FIG. 77  shows a further embodiment of a piezoelectric motor assembly;  
       FIGS. 78-80  show further embodiments in which a coil is integrated with or associated with the motor or motor components;  
       FIG. 81  shows the motion of a selected contacting portion;  
       FIG. 82   a  shows a top view of a piezoelectric motor, wherein an elastic spring is joined to the resonator;  
       FIG. 82   b  shows a side view, cut along the line BB shown in  FIG. 82   d,  of the piezoelectric motor, the motor comprising a press-fit piezoelectric element;  
       FIG. 82   c  shows a side view, cut along the line BB shown in  FIG. 82   d,  of the piezoelectric motor without a piezoelectric element;  
       FIG. 82   d  shows a bottom view of the piezoelectric motor that is shown in  FIG. 82   a;    
       FIG. 82   e  shows a frontal view of a piezoelectric motor;  
       FIG. 83   a  shows a top view of a resonator having recesses for receiving a coil spring and further having a space for clamping a piezoelectric element;  
       FIG. 83   b  shows a side view, cut along the line AA shown in  FIG. 83   a,  of a resonator;  
       FIG. 83   c  shows a further top view of a resonator;  
       FIG. 83   g  shows a magnified view of the recess sketched in  FIG. 83   a;    
       FIG. 84  shows a side view of a piezoelectric motor whose resonator is urged against a driven element by way of a coil spring;  
       FIG. 85   a  shows a side view of a first coil spring;  
       FIG. 85   b  shows a top view of a first coil spring;  
       FIG. 85   c  shows a perspective view of a first coil spring;  
       FIG. 86  shows a perspective view of a second coil spring;  
       FIG. 87   a  shows a cut view along the line AA of the coil spring according to  FIG. 8   d;    
       FIG. 87   b  shows a side view of a third coil spring;  
       FIG. 87   c  shows a frontal view of a third coil spring;  
       FIG. 87   d  shows a top view of a third coil spring;  
       FIG. 88   a  shows a diagram-type overview of the vibrations performed by individual regions of a resonator, the deflections having x- and y-components during operation of the piezoelectric motor;  
       FIG. 88   b  shows a diagram-type overview of  FIG. 88   a  with a worn resonator contact portion;  
       FIG. 89   a  shows a piezoelectric motor in a linear drive configuration;  
       FIG. 89   b  shows a piezoelectric motor in a rotary drive configuration;  
       FIGS. 90   a  and  90   b  show a resonator comprising an opening with protrusions on the end walls of the opening;  
       FIGS. 91   a -  91   c  show various embodiments of the end walls of the resonator opening;  
       FIG. 92  shows a resilient element comprising a foamed material; and  
       FIG. 93  shows a schematics of spring-damper interaction. 
    
    
     DETAILED DESCRIPTION OF SPECIFIC EMBODIMENTS  
      Several embodiments of the motor will be described, following which a number of theoretical and practical operational and design aspects of the motors are described. Referring to  FIGS. 1-3 , and as described in detail at various locations, the piezoelectric motor assembly  20  has an element that converts electrical energy into macroscopic mechanical motion. This is achieved by using a single electrical signal to generate at least two vibration motions at a predetermined location of a vibration element. The at least two vibration motions result in an elliptical motion at the predetermined location. The elliptical motion is selected to cause the vibrating element to engage a driven element during a time corresponding to at least a portion of travel in direction of a long axis of the ellipse, and to disengage or slide over the driven element during a time corresponding to travel in the opposite direction. A second, single frequency results in a second elliptical motion in an opposing direction to move the driven element in an opposing direction. The desired motion is used to determine the elliptical motion needed, and the various components of the system are designed to achieve that motion. The use of a single frequency to generate elliptical motion and the simplicity of the resulting design allow a low cost, high reliability motor.  
      The motor assembly  20  has a vibration source  22  that converts electrical energy directly into physical motion. The vibration source  22  is preferably a piezoelectric element and comprises a block of piezoelectric material, or a multi-layer piezoelectric so that the motion of the various elements combine to increase the movement in desired directions. The shape of the piezoelectric  22  can vary, but it advantageously has a longitudinal axis  25  along its direction of greatest motion. The piezoelectric  22  is mounted to, and preferably inside, a resonator  24 . The piezoelectric  22  and resonator  24  comprise a vibration element  26  or motor  26 .  
      The piezoelectric material is preferred because it reacts quickly to applied voltages. While the resulting deflection for a given voltage is small, about 0.1% or less of the length of the piezoelectric, and smaller in other directions, the resulting force is large so that vibration resonance can be achieved.  
      The source  22  can also comprise electrostrictive materials, magneto-restrictive materials (e.g., Terfenol), or other materials that can be used to excite vibrations, according to other embodiments. Preferably, the vibration source  22  comprises materials or devices that convert electrical energy directly into physical motion. For ease of reference, the vibration source  22  will be referred to and described herein as piezoelectric  22 .  
      To avoid confusion between motor  26  and motor assembly  20 , the terminology “vibration element”  26  will be used in most cases to refer to the combination of the piezoelectric element  22  and the resonator  24 .  
      The resonator  24  can have various shapes, but is illustrated as having a rectangular shape with a rectangular cross-section. In order to mount the piezoelectric  22  inside the resonator  24 , it is useful to form a cavity or an opening  28  in the resonator  24  to hold the piezoelectric element  22 . The opening  28  is shown as extending entirely through a portion of the resonator  24  to form a rectangular opening, with sidewalls  29  which define the sides of opening  28 , the sidewalls being located on opposing sides of the longitudinal axis extending through the opening  28 , and with end walls  31  being located on the longitudinal axis extending through the opening  28 . The opening  28  is thus advantageously defined by continuous walls that enclose the opening. Appropriate electrical connections are provided to the piezoelectric  22  and may comprise electrical connections of various types, but which are illustrated as wires  30 .  
      Application of large voltages to an unrestrained piezoelectric  22  can damage the piezoelectric. Thus, the piezoelectric  22  is advantageously placed in compression along at least its longitudinal axis, by end walls  31 . This also causes a preload, which optimizes the piezoelectric lifetime and performance. But a compressive force is not necessarily used if other vibration sources are used that do not require compression, or that do not benefit from compression. Several ways to preload the piezoelectric element  22  are discussed later.  
      In order to make it easier to place the piezoelectric element  22  in compression, the opening  28  is advantageously enclosed on opposing sides, and preferably enclosed on opposing ends of the longitudinal axis of piezoelectric  22 . This arrangement provides opposing surfaces that can be used to provide compression to the piezoelectric  22 . One way to preload the piezoelectric  22  is by movably extending a screw  32  through a threaded opening in the proximal end  35  of resonator  24  so that a distal end of the screw can be moved to compress the piezoelectric  22  against one end of the opening  28  in the resonator  24 . Since the piezoelectric material is brittle, a protective cap  34  is interposed between the distal end of the screw  32  and the adjacent end of the piezoelectric  22 . The cap  34  is made of a protective material that allows the rotation of the screw  32  to compress the piezoelectric while not breaking or cracking the piezoelectric  22 . Metal caps  34  are preferred, but some lubricant or rotational accommodating design is advantageously provided in order to avoid at least some damage to the piezoelectric  22  from rotation of the screw  32 . Other clamping methods of the piezoelectric  22  without a screw and/or a protecting plate can be used, such as expansion or shrinkage of the opening  28 . Additional ways are described below, and other ways will become known to those skilled in the art given the present disclosure.  
      When a voltage is applied to the piezoelectric element  22 , the piezoelectric element extends along longitudinal axis  25 , and that causes the vibration element  26  to also extend in length, in part by elongating the smaller cross-section sidewalls  29 . The vibration of the piezoelectric  22  excites a longitudinal mode in the vibration element  26  which causes the distal end  36  opposite the screw  32  to move back and forth along the longitudinal axis  25 . In addition to that longitudinal motion, bending modes of the vibration element  26  will be excited which are transverse to the longitudinal axis  25 . For the illustrated embodiment, a first preferred bending mode occurs in the direction indicated by arrow  38 , which is perpendicular to the longitudinal axis  25  in the plane of the paper on which the illustration is placed in  FIG. 1 . A second, preferred lateral bending mode occurs along an axis orthogonal to the paper on which the illustration is placed in  FIG. 1 , and is denoted by axis  40 . In practice, the vibration modes are often combinations of various modes involving motion along and rotation about multiple axes.  
      Advantageously, the components of the invention are configured so that the various modes are excited at or very close to their respective resonance frequencies in order to increase the amplitude of motion along the longitudinal axis  25  and preferably only one of the lateral axes  38 ,  40 . As discussed later, the lateral bending can be excited either by asymmetrical placement of the piezoelectric  22  relative to the resonator  24 , or by an asymmetrically placed mass on the vibration element  26 , or by a mounting of the piezoelectric element  22 , or by shaping the resonator  24  to resonate with a desired lateral motion, or by other mechanisms, some of which are discussed later.  
      In the embodiment depicted in  FIGS. 1-2 , the motion along lateral axis  38  is preferably substantially greater than the motion along lateral axis  40 . Substantially greater refers to a difference by a factor of at least 3, and preferably a factor of 10.  
      A driven element  42  is placed in contact with a selected contact portion  44  of the vibration element  26 . As illustrated in  FIGS. 1-2 , the selected contact portion  44  comprises an edge of the vibrating element, although other locations could be used. As used herein, unless otherwise indicated, the term “edge” should be construed to include a comer where multiple surfaces converge, as for example, in the comer of a rectangular cross-sectional rod where three planar surfaces converge (and where three edges converge). Moreover, other shapes of contacting surfaces could be used other than an edge. For example, a beveled surface inclined at an angle selected to place contacting surface  44  into flat engagement with the engaging surface of driven element  42  could be used. Given the present disclosure, many configurations can be derived to ensure that the engaging surface  44  provides the needed engagement to move the driven element  42 .  
      As illustrated, the driven element  42  comprises a rod with a cylindrical cross-section, although other shapes of driven elements can be used. The centerline  25  of the vibration element  26  and a centerline  45  of rod  42  are in the same plane, and separated by an angle α of about 30 degrees as measured in that plane. The orientations of the centerlines  25 ,  45  and the angle α will vary with the particular application. The angle α is difficult to analytically determine, and is preferably adjusted according to the motor design. Typically it is between 10 and 80 degrees, and preferably between 20 and 60 degrees. The driven element  42  is supported so it can move along the longitudinal axis  45  of the driven element  42 . The driven member is supported so that it can move relative to the vibration element  26 , which is effectively held stationary in the illustrated embodiment. The driven element  42  translates along the axis  45 , as explained in greater detail below.  
      As illustrated, the support of the driven element  42  can be achieved by wheels  46 , which provide a low resistance to motion along the axis  45 . This support is achieved here by placing an inclined surface on the wheels  46 , which abut the curved sides of rod-like driven element  42  and rotate as the rod translates along axis  45 . The wheels are located on the side of the driven element  42  opposite the selected contact portion  44 , with the contact portion  44  also being further located between two wheels  46  in a direction along the axis  45 , so that the wheels  46  and selected contact portion  44  restrain motion of the driven element  42  in all directions except along axis  45 . The wheel  46  could also contact the driven element  42  using a flat edge of the wheel concentric with the rotational axis  65 , as illustrated in  FIG. 74 . The wheels  46  could also have contoured peripheries configured to engage mating shapes on adjacent portions of the driven element  42  in order to appropriately support and guide the driven element  42 . Given the present disclosure, a variety of movable support configurations will be apparent to those skilled in the art.  
      The vibration element  26  is advantageously resiliently urged against the driven element  42 , and  FIGS. 1-2  show one of many ways to achieve this. The elliptical motion  100  of the selected contacting portion  44  is preferably an unrestrained motion, one that occurs whether or not the contacting portion  44  engages a driven element, and one that is achieved without relying on any resistance from being urged against the driven element  42 . Nevertheless, the selected contacting portion  44  is advantageously resiliently urged against the driven element  42  in order to enhance the driving engagement of the driving and driven parts.  
      A spring  50  made of flat, elongated spring material is bent into an “L” shape with opposing ends  50   a,    50   b.  A first end  50   a  of the spring is fastened to a base  52 . A second end  50   b  of the spring is fastened to the end of the vibration element  26  through which the screw  32  extends, with a hole in the end  50   b  of spring  50  allowing passage of the screw. A first leg of the spring  50  which contains end  50   a  is generally parallel to the longitudinal axis of vibration element  26 , and the second leg of the spring  50  that contains the end  50   b  is generally parallel to the axis  38 , with the two legs being generally perpendicular to each other. The spring  50  resiliently urges the vibration element  26  against the driven element  42  at the selected contact portion  44 . Variations in the location of the mounting at end  50   a,    50   b  can be used to vary the pre-load with which the vibration element  26  is urged against the driven element  42 . As discussed later, variations in the shape, cross-section, location, and form of the resilient element  50  are possible and can be used to achieve a desired vibration mode.  
      The spring  50  is designed to optimize the vibration characteristics of vibration element  26  as well as provide a sufficient range of flexibility to insure contact between the driven element  42  and vibration element  26 . This contact and a defined range of contact pressure should be maintained throughout the life of the motor assembly  20 . The spring  50  advantageously compensates for manufacturing tolerances and uncertainties and also can compensate for wear that might reduce the size of the vibration element  26  at the selected contact portion  44 .  
      As discussed further below, during operation the vibration element  26  might touch the driven element  42  only part of the time due to the vibration and in such a case the spring  50  preferably is designed to ensure suitable engagement. The spring constant and the location of the spring can be used to adjust the percentage of contact and non-contact time. This allows a designer the ability to configure the motor assembly  20  to ensure the resulting engagement between engaging portion  44  and driven element  42  is with sufficient force to move the driven element  42  with sufficient force to achieve the desired objectives of motor assembly  20 . Moreover, variations in the dimensions affecting the engagement of the selected engaging portion  44  and the driven element  42  will be accommodated by the mounting system, such as spring  50 , that resiliently urges the contacting parts into engagement. This flexibility in manufacturing tolerances allows a reduction in manufacturing costs and in alignment tolerances and costs.  
      In the depicted embodiment the wheels  46  are both rotatably mounted to axles connected to the base  52 . Other schemes of mounting the driven element  42  are possible given the present disclosure. For example, the base  52  could support one or more projections having aligned holes into which linear bearings are preferably placed, with the elongated driven member  42  extending through the holes. This configuration would allow an elongated driven member  42  to translate along an axis, but would restrain other motions. The motor can be as small as 25×25×5 mm 3  or even smaller. Operation:  
      Referring primarily to  FIGS. 1-2 , when an electrical signal of suitable frequency, waveform and voltage is applied to the piezoelectric element  22  the vibration element  26  starts to move the rod  42 . For the embodiment shown, preferred waveforms are sine waves or rectangular shaped waves. The direction of the linear motion generated is determined by the frequency. A motor assembly  20  operating, for example, at about 35 kHz in one direction and at about 60 kHz in the other direction is believed to be suitable for a variety of potential uses. Other frequency pairings are possible and will vary with a variety of factors concerning the design of motor assembly  20 . The operating frequencies can be changed by changing the design of the various components, with the operating frequencies being selected to be inaudible by humans and by most pets in some preferred embodiments. The operating voltage will vary with the type of piezoelectric  22  or other vibration source used. A multi-layered piezoelectric  22  operating at 6 volts peak-to-peak amplitude is believed useful for a variety of applications.  
      The vibration of the piezoelectric element  22  makes the vibration element  26  vibrate in a way so that the selected contact portion  44  performs an elliptic motion relative to the driven element  42 . As discussed below, the vibration of the piezoelectric  22  excited at a first frequency makes the vibration element  26  vibrate in a way so that the selected contact portion  44  performs a first elliptic motion  100   a  relative to the driven element  42 . The elliptical motion is achieved by having the first signal excite two resonant modes of the resonator  24 , resulting in the desired elliptical motion  100   a —preferably without requiring engagement with the driven element  42  to achieve this elliptical motion. That first elliptical motion  100   a  moves the driven element  42  to the right as depicted in  FIG. 1 .  
      Moreover, vibration of the piezoelectric  22  excited at a second frequency makes the vibration element  26  vibrate in a way so that the selected contact portion  44  performs a second elliptic motion  100   b  relative to the driven element  42  in a different direction and orientation than that of elliptical motion  100   a,  and preferably, but optionally, in a direction opposite that of elliptical motion  100   a.  As depicted, the second elliptical motion is clockwise and that will move the driven element  42  in an opposite direction, to the left as depicted in  FIG. 1 . Typically, the elliptical motions  100   a,    100   b  do not overlap, but have different major and minor axes, amplitudes and orientations. Ideally, the elliptical motions  100   a,    100   b  overlap. The elliptical motions  100   a,    100   b  are preferably achieved without requiring that the selected contact portion  44  engage the driven element  42 .  
      This results in the selected contacting portion  44  moving the driven element  42  in a first direction when the source of vibration is driven by the first signal, and moving the driven element in a second direction when the source of vibration is driven by the second signal. But advantageously the selected contacting portion further moves in the first direction when a single sinusoidal signal of a first frequency is applied. Moreover, the selected contacting portion  44  can also move in the first direction when the first frequency is dominant and superimposed with plural sinusoidal signals of different frequencies. In this later instance, the second signal either does not occur simultaneously with the first signal or it is of substantially different amplitude if it occurs simultaneously with the first signal. By different in signal amplitudes a factor of about 10 is considered substantially different, and preferably the amplitudes differ by a factor of 100 or more. The result is that the elliptical motion  100  can be achieved by a simple sinusoidal signal. Alternatively, it can be achieved by complex signals of different frequencies—for example, the complex frequencies that are superimposed to generate sawtooth waves.  
      During driving engagement of the selected contacting portion  44  with the driven element  42 , it is believed that the elliptical motion  100  consists of a phase where the vibration element  26  is pressed against the driven element  42  and a phase where this is not the case. The motion component of the vibration element that has the direction along the longitudinal axis  45  of the driven element is partly transferred to the driven element by the friction between the vibration element  26  and the driven element  42 . In the second phase the vibration element  26  moves in the opposite direction. In this second phase the vibration element does not transfer any motion component parallel to the axis  45  because the vibration element  26  is not pressed against the driven element.  
      In contrast to other vibrating motor designs, the required manufacturing tolerances are believed to be significantly looser so that no precise manufacturing is needed to alternate between the contact and no contact situations. The necessary equilibrium is created by the design, specifically including spring and the mass of the vibration element  26 .  
      Because the high frequencies (over 30 kHz) and small motions make it difficult to actually determine the contact, it is also believed possible that there is always contact between the vibration element  26  and the driven element  42 . In that situation, the motion of the driven element  42  is believed to be caused by the difference in force at the selected contact portion  44  caused by the elliptical motion of the contact portion  44  which provides a resultant force in only one direction, or primarily in only one direction, thus driving the driven element  42  in that direction. Further discussions of this elliptical motion and a number of design aspects are discussed below.  
      Whatever the actual mechanism, the driven element performs a linear motion with the direction of the motion being determined by the motion of the selected contact portion  44  of the vibration element  26 . If the contact portion does a counterclockwise elliptical motion, the driven element  42  will move to the right as depicted in  FIG. 1 . If the motion is clockwise, it will move in opposite direction.  
      It is possible that the vibration element  26  will also be excited to move along axis  40 , which could result in rotation of the cylindrical rod-like driven element  42 . Depending on the relative magnitudes of the motion of the selected contact portion  44  and depending on its orientation and contact with the driven element  42 , and if the bearing supports are properly configured, both translation and rotation could simultaneously occur.  
      Further, referring to  FIG. 2 , it is believed possible to select the axis with the largest motion to be longitudinal axis  25 , but to select the lateral axis  40  as having the next largest and only other significant motion. In that instance the motor assembly  20  would cause a rotation of the rod-like driven element  42  about longitudinal axis  45 . To provide this rotational motion, the selected contact portion  44  would have to provide an elliptic motion having a substantial portion of its motion in a plane generally orthogonal to the axis  45  of the driven element  42  in order to impart rotational motion to the driven element. The direction of rotation would again depend on the direction in which the selected contact portion  44  performs the shape of the ellipse.  
      Moreover, it is believed possible to select the axes with the largest two motions, and only significant motions, to be the lateral axes  38 ,  40 , which could again result in an elliptical motion of the selected contact portion  44  in a manner that engages the driven element  42  to rotate it about longitudinal axis  45  during one portion of movement and to disengage sufficiently to prevent motion or noticeable detrimental motion in the other portion of movement of the selected contact portion  44 .  
      An alternative embodiment is shown in  FIG. 4 , where instead of an elongated driven element  42  a rotatable wheel  60  is mounted to be driven by the vibratory element  26  having a portion placed in contact with an appropriately located driven surface  62  on the wheel  60 . In this embodiment the wheel  60  is mounted to rotate about rotational axis  65  on a bearing. The driven surface  62  is preferably placed on a side  64  of the wheel located in a plane orthogonal to the rotational axis  65  as with driven surface  62   a,  or placed along a surface concentric  62   b  with the axis  65 . The wheel  60  could comprise a variety of elements, including a gear. The selected contact portion  44  of the vibratory element  26  engages the driven surface  62  to cause movement of the wheel  60  about the rotational axis. The wheel  60  will rotate in the opposite direction of the motion of the contact point around the elliptical path traveled by the contact portion  44 . Thus, if the contacting portion  44  of the vibration element  26  moves clockwise the wheel  60  will move counterclockwise, so that the contact portion on the wheel and on the vibration element share the same motion while they are in contact.  
       FIG. 5  shows a further embodiment. The motor assembly  20  has a vibration element  26  that contains a resonator  24  in the shape of a C-clamp  74 . The piezoelectric element  22  is held in the clamp. To transmit the motions, a first electrical signal causes the piezoelectric element  22  to move in the vibration element  26  which causes the contacting portion  44  to move in a first an elliptical motion  100   a.    
      The piezoelectric element  22  is clamped by the screw  32  which extends through leg  73  and presses against insert plate  34  to compress the piezoelectric element  22  between the plate  34  and an opposing leg  75  of the C-clamp resonator  74 . This clamping causes pre-load in the piezoelectric element  22 , which increases and preferably optimizes the lifetime and performance of the piezoelectric element  22 .  
      The legs  73 ,  75  of the C-clamp between which the piezoelectric element  22  is held, could be of similar stiffness, but are advantageously of different stiffness. Advantageously one leg  73  is at least a factor of 10 times stiffer than the opposing leg  73 . The more flexible leg  75  vibrates with larger amplitude than the stiffer leg  73 . The selected contacting portion  44  is preferably located on the less stiff leg  73  in order to achieve a larger amplitude of motion at the selected contacting portion  44 . Moreover, in this configuration the leg  73  is placed in bending stress, with the largest stress being adjacent the interior end of the leg. A notch  77  can be placed adjacent to that location in order to localize the bending so that the leg  75  pivots about the notch  77 .  
      A spring element  50  has a first end  50   a  connected to the base  52  and a second end  52   b  connected to a vibration element  26  to keep the vibration element in contact with the driven element  42 . The second end  50   b  is shown as connected to the head of the screw  32  although other connections to the resonator  74  could be used. In this embodiment the spring  50  is depicted as a tension coil spring. The resonator  74  is loosely pinned by pin  78  extending through hole  80  and into the base  52  so the resonator  74  can pivot about pin  78 . The pin  78  is offset from the line of action of spring  50  so that the contacting portion  44  is resiliently urged against driven element  42 .  
      The spring  50  is under tension in the depicted configuration. The spring  50  provides a sufficient range of flexibility to ensure contact between the driven element  42  and the vibration element  74 . This contact and a defined range of contact pressures are advantageously maintained throughout the life of the motor assembly  20 . The spring  50  advantageously is designed to compensate for manufacturing uncertainties and wear that might reduce the size of the vibration element  26  at the selected contact portion  44 .  
      To prevent the driven element  42  from separating from the vibration element  26 , wheels  46  connected to the base  52  are provided as previously discussed. Alternatively, the base  52  can be equipped with sidewalls  80  having holes through which the driven element  42  extends in order to support the driven element while allowing it to move along its desired translational axis. Advantageously, the holes in the sidewalls  80  are designed to reduce friction, and thus could have linear bearings supporting the driven element  42 . If the holes in the sidewalls  80  are enlarged so that they do not permanently contact the driven element  42 , they function as auxiliary bearings instead and protect the driven element  42  from being forcefully pushed into the vibration element  26  by external forces, which could be damaging to the vibration element  26  as well as its suspension.  
      When an electrical signal of suitable frequency, waveform and voltage is applied to the piezoelectric element  22  the vibration element  46  starts to move the driven element  42 . The direction of the linear motion generated is determined by the frequency. Changing the configuration of various components of the motor assembly  20 , as discussed further below, can change the operating frequencies. In the depicted example, a multi-layered piezoelectric element is used that could operate the motor assembly 20 on 6V peak-to-peak amplitude to drive a cylindrical rod  44 .  
      The vibration of the piezoelectric  22  at a first frequency makes the vibration element  26  vibrate in a way so that the selected contact portion  44  performs a first elliptic motion relative to the driven element  42 . The elliptical motion consists of a phase where the vibration element  26  is pressed against the driven element  42  and a phase where this is not the case, as discussed in further detail below. If the selected contact portion  44  moves in a counterclockwise elliptical path  100   a  as depicted, the driven element  42  will move to the right as depicted in  FIG. 5 .  
      Advantageously, the vibration of the piezoelectric  22  excited at a second frequency makes the vibration element  26  vibrate in a way so that the selected contact portion  44  performs a second elliptic motion  100   b  relative to the driven element  42  in a direction opposite that of elliptical motion  100   a.  As depicted, the second elliptical motion is clockwise and that will move the driven element  42  in an opposite direction, to the left as depicted in  FIG. 5 . Typically, the elliptical motion  100   a,    100   b  do not overlap, but have different major and minor axes, amplitudes and orientations. Ideally, the elliptical motions  100   a,    100   b  overlap. The vibrating element  26  could be configured to cause the second elliptical motion  100   b  to be in a different orientation, as for example, to rotate a driven element  42 .  
      In more detail, vibration of the piezoelectric element  22  causes the vibration element  26  to begin oscillating about the pin  78 , which causes the contact portion  44  to have an up-and-down motion and a back-and forth motion along its elliptical path  100 . The up-down motion and the back-forth motion are out of phase, and the contact portion  44  thus has an elliptical motion along one of paths  100   a,    100   b.  That causes the rod-like driven element  42  to begin motion. The rotation of the vibration element  26  can be caused by interaction of the contacting portion  44  with the driven element  42 , which may be viewed as conserving angular momentum about the pin.  
      The vibratory motor  26  of  FIG. 5  could be used with a rotating driven element  42  as depicted in  FIG. 4 , and could be used in other driving arrangements.  
       FIG. 6  shows a further embodiment in which the vibrating element  26  is mounted in a stationary manner, and the driven element  42  is resiliently urged against the vibrating element. If the driven element  42  is elongated, and especially if it comprises a rod or other structure that is flexible, merely pressing the driven element against the vibrating motor  26  may cause the parts to resiliently urge into contact. That requires the support of the driven element  42  to be such that a resilient support is inherently provided by the flexibility of the driven element. If that is not the case, a resilient support must be provided for the driven element  42 , or a resilient support can be provided in addition to the flexibility of the driven element. Such a resilient support is illustrated schematically by springs  50   a,    50   b,  resiliently urged against the selected contacting portion  44   a,    44   b  of the vibrating element  26 .  
      In this embodiment, the vibration element  26  is configured with a special shape so that there are more than one, and preferably a number of selected contacting portions  44   a,    44   b,  . . .  44   n.  The ability to use different portions of the vibrating element  26  to generate a desired elliptical motion  100  resulting from free vibration modes excited at a specified frequency, offers the ability to have a variety of arrangements. For each of several separate excitation frequencies, a different selected contacting portion  44  can resonate in a predetermined elliptical motion  100 . Alternatively, the same selected contacting portion  44  may resonate at a different excitation frequency to cause an elliptical motion but in a different orientation. Preferably the elliptical motion is in opposite direction to reverse the motion of the driven element, but other motions are possible depending on the needs of the user. As a result, several driven elements  42   a  through  42   n  that are resiliently urged against separate and corresponding selected contacting portions  44   a  through  42   n,  can be individually controlled.  
      For example, it is believed possible to have one driven element  42   a  translate, and another driven element  42   b  rotate, by generating appropriately orientated elliptical paths  100   a,    100   b  respectively, at selected contacting portions  44   a,    44   b,  respectively. The generation of the elliptical paths  100   a,    100   b  is preferably caused by a single excitation frequency to piezoelectric element  22 , which causes a sufficiently resonant vibration to generate the elliptical paths. Alternatively, a first excitation frequency could be required to generate depicted motion  100   a,  and a second excitation frequency used to generate motion  100   b.  Yet other excitation frequencies provided by the piezoelectric  22  could be used to change the direction of the elliptical motion to travel in an opposing direction.  
      Moreover, while the contacting portions  44   a,    44   b  are shown at the distal ends  36  of the vibrating element  26 , the contacting portions  44  could be at various locations and orientations on the vibrating element  26 . This is shown illustratively by engaging portion  44   n  and driven element  42   n,  with the driven element  42   n  rotating (e.g., to drive a gear) or translating along its longitudinal axis.  
      These aspects are further illustrated by the embodiment of  FIG. 62 , which shows that the selected contacting portion  44  need not occur at the distal end  36  of the vibratory element  26 . In  FIG. 62  the vibratory element  26  has one or more selected contacting portions  44   e  located along the periphery of the element along the longitudinal length of the element. A second one or more selected contacting portions  44   f  are located on an opposing surface of the vibratory element. Preferably the selected contacting portions  44  comprise slightly raised areas extending above the surrounding portion of the vibratory element  26 . A driven element  42  such as a cylindrical shaft is placed in contact with the contacting portions  44   e.  In the depicted embodiment, the axes  25 ,  45  of the vibratory element  26  and driven element  42  are aligned and coplanar, but that need not be the case.  
      When the vibratory element  26  is excited at a first frequency, the contacting portions vibrate in an elliptical path  100   a  causing motion of the driven element  42  in a first direction. The contacting portion  44   e  moves in an elliptical path opposite to that of contacting portion  44   f.  To shift the motion of the driven element  42 , the contacting portion  44   f  and driven element  42  are placed into contact. This can be achieved by moving one or both of the vibrating element  26  and driven element  42 . A rotation of the vibratory element  26  would suffice in the illustrated embodiment. Thus, a single excitation frequency could result in opposing directions of movement of the driven element  42 . This embodiment also shows that the contact between the vibration element  26  and the driven element  42  can be a multiple point contact. It is not limited to a single point contact. This also allows, for example, the use of only one bearing pressing a driven rod  42  at two to four points against the vibration element  26 . The increased number of contacting portions  44  can increase the frictional engagement with the driven element  42  and allow a greater power to be exerted on the driven element  42 , and thus allow a greater power to be exerted by the driven element  42 .  
      Alternatively, more than one driven element  42  could be placed in contact with differing portions  44  of the vibratory element  26 , achieving different motions for each driven object. Moreover, the vibratory element  26  could be urged against a stationary surface and by selecting various contacting portions  44  (e.g.,  44   e,  or  44   f ), move the vibratory element and any object connected to the vibratory element in various directions over the surface.  
      Preloaded Motor Configurations  
      It is advantageous in many cases to use multilayer piezoelectric elements  22 . These elements  22  are preferably of rectangular cross-sectional shape, but other shapes could be use such as square, circular, or other shapes. The piezoelectric elements  22  have layers of piezoelectric material with printed electrodes that are stacked on top of each other. Often many piezoelectric components are made at the same time by producing a large stacked plate that is pressed and cut to form many single piezoelectric elements.  
      As a result of this manufacturing method, the mechanical output areas of the piezoelectric are typically parallel to the electrode layers and are also flat. In order to use multilayer piezoelectric elements, a mechanical preload is often applied. This increases the lifetime of the piezoelectric by preventing delamination under dynamic movement of the piezoelectric element, and it also optimizes the contact between the piezoelectric element  22  and the resonator  24  in which it is mounted. As a result, mechanical motion generated by the piezoelectric element  22  is efficiently transferred through the contact zone to the resonator.  
      There are different methods to generate the preload. A resonator  24  can be used that has two parts. A spring is used to generate the preload by inserting a piezoelectric element  22  and the compressed spring between the two parts of the resonator which are then welded or otherwise fastened together. This way, a permanent preload is generated.  
      An alternative way to generate the preload is shown in  FIG. 1 , where the preload is preferably achieved by having the resonator  24  exert a pressure on the piezoelectric element  22 . The compression ensures that the vibrations of the piezoelectric element  22  are transferred to the resonator  26  and selected contacting portion  44 . The compression also avoids at least some damage to the piezoelectric element  22  when high voltages are applied. The pressure is equal to the axial force on the piezoelectric element  22  divided by the area over which the force acts. This area is the contact area between the piezoelectric element  22  and the abutting portions of the resonator  26 . Because the contact area can be difficult to measure, it is more straightforward to use the force rather than the pressure as a characterizing parameter.  
      The force exerted on the piezoelectric element  22  when no current is passed through the piezoelectric element  22  includes: a static pre-load equal to the axial force in the sidewalls  29  counteracting the preload and a load component from the contact force arising from the contact surface  44  being urged against the driven object  42 . All of these forces fluctuate when a fluctuating current is passed through the piezoelectric element  22 .  
      The piezoelectric element  22  can be aligned so that the preload on the piezoelectric element  22  is in the most active direction of the piezoelectric element  22 . While this is not necessary for the vibratory motor  26  to operate, this configuration results in the highest efficiency. Preferably, for the beam-type vibratory element  26  depicted in  FIG. 1 , the greatest motion occurs in the line  3 - 3  direction that is preferably aligned with the longitudinal axis  25 .  
      Methods of producing a preload on the piezoelectric element  22  that are described herein include: (1) clamping the piezoelectric element  22  in the resonator  24  with a threaded fastener or other compressive mechanism; (2) using force to press the piezoelectric element  22  into a hole in the motor body in a manner similar to press fitting of shafts; and (3) combinations thereof. Other preload mechanisms can be used. The following disclosure expands on the threaded fasteners described thus far, and then discusses some press-fit mechanisms and methods. A wedge-based method and some variations on the above methods concludes the discussion.  
      Threaded Preload Device &amp; Method:  FIGS. 1 and 5  illustrate a threaded fastener preload method and apparatus that is further described below. The resonator element  26  is configured so that a hole, cavity or opening  28  is formed to accommodate the piezoelectric element  22 . The resonator can have various shapes, for example cross-sections that are round, square, rectangular, or polyhedral. The opening  28  is larger than the piezoelectric element  22  in all dimensions. A threaded fastener  32  extends through a hole in a stationary object in order to allow the distal end of the fastener  32  to press the piezoelectric element  22  against the resonator  24 . The threaded fastener  32  advantageously passes through a threaded hole in the resonator to directly abut the plate  34  that is urged against the piezoelectric element  22 .  
      Once the parts are assembled, a preload can be achieved in the piezoelectric element  22  by tightening the threaded fastener  32 . The preload can be approximately calculated by tightening the fastener  32  to a known torque. The threaded fastener  32  need not be aligned with the longitudinal axis of the piezoelectric  22 , but can be offset in a variety of ways so that tightening the threaded fastener urges two bodies toward each other to compress the piezoelectric element  22 . A variety of other mechanisms can be used to place the piezoelectric element  22  in compression. Other preloading mechanisms are discussed later.  
      Uni-Axially-Stressed, Press-Fit Preload Device &amp; Method: Several aspects of the press-fit of the piezoelectric element  22  are described with respect to  FIG. 7   a.  The resonator  24  is configured so that a hole or opening  28  for the piezoelectric element  22  is formed in the resonator, with sidewalls  29  defining the sides of the opening. The opening  28  is slightly smaller in the axial direction of the piezoelectric element  22  than the combined length of the piezoelectric element  22  and any other elements to be pressed into the opening  28 . The required interference between the resonator  24  and the parts to be pressed into the opening  28  depends on the geometry and dimensions of all parts and also the elastic strain of the material from which the resonator  24  is made.  
      Referring to  FIG. 7   b,  the piezoelectric element  22  can be pressed directly into the resonator  24 .  
      This press-fit can be made easier by providing a tapered surface  82  which places an inclined contact area between the abutting edges of at least one of the piezoelectric element  22  and an end wail of the resonator  24  at the mating portion of opening  28 . The inclined surface  82  avoids an offset, abutting-type of interference, and provides a sliding interference at the start of the press-fit. An improved way to achieve this press-fit is described later. This preload mechanism and method produces large shear stresses on the contacting surfaces of the piezoelectric element  22 . Because the piezoelectric material is brittle, the stresses can result in cracking of the piezoelectric element  22 . To avoid these shear stresses and protect the piezoelectric element  22 , it is also possible to simultaneously press in a piezoelectric element  22  sandwiched between two strips of a less brittle material  84  ( FIG. 7   a ) such as a metal, preferably steel. The strips of material  84  can have a variety of shapes suitable to the configuration of the piezoelectric element  22  and the vibrational element  26 . The protective cap  84  can also advantageously be used to guide the piezoelectric element  22  into the opening  28 , thereby eliminating the need for tapering of any parts. One of the strips of material  84  can advantageously comprise the end  50   b  of spring  50  that connects the vibrating element  26  to the base  52 .  
      When the piezoelectric element  22  and any end protectors  84  are inserted into the opening  28 , the sidewalls  29  are stretched to accommodate the longer length element  22  and any end protectors  84 . The stretched sidewalls  29  act as springs and maintain the preload on the piezoelectric element  22 . Ideally, the preload on the piezoelectric element  22  could be specified by knowing the cross-sectional dimensions of the sidewalls  29  and fixing an interference that results in an elastic strain in the sidewalls  29  and therefore known stress and preloads in the sidewalls. The preload is then this stress multiplied by the combined cross-sectional area of the sidewalls  29 .  
      Unfortunately, this method may not be practical because the required interference for small vibratory elements  26  of an inch or less in length is likely to be too small, on the order of 0.0001 inches, which is beyond a tolerance currently obtainable by traditional machining processes at a reasonable cost. Larger vibratory elements may have larger preloads that require larger dimensions, but the accuracy needed to achieve those dimensions is likely to require similarly small tolerances and thus also require expensive machining or polishing. This arises in part because small variances in the interference would result in great differences in the preload when the sidewalls  29  are in the elastic portion of the stress-strain curve and act as a spring as the piezoelectric element  22  expands and contracts.  
      Because of these disadvantages, it is desirable to make the interference between the length of the vibratory element  26  and the opening  28  sufficiently large so that the sidewalls  29  forming the opening  28  are stressed beyond their yield strength but below their ultimate tensile strength, and a sufficient amount below their fatigue strength to provide an acceptable product life. When stressed beyond the yield strength, the sidewalls  29  provide a relatively constant preload even though the dimensions of the opening  28 , or the piezoelectric element  22  or the end protectors  84  may vary. This allows looser manufacturing tolerances and results in greatly simplified manufacturing and significantly lower costs.  
      The plastic portion of the stress-strain curve from yield up to the point where necking of the sidewalls  29  begins, can be used to achieve the desired preload. The usable portion of the strain occurring after yield and before necking is at least ten times larger than the elastic portion in strain. This is believed to apply to all non-ferrous metals, which are the preferred material for the resonator  24 , with aluminum being the most preferred non-ferrous metal. Ferrous metals and some non-metallic materials could also be used in other embodiments.  
      This method significantly loosens the required tolerance on the interference fit between the vibratory element  26  and the opening  28 . Further, the slope of the stress-strain curve above yield is much less than that of the elastic portion. Thus, the preload will also not depend so greatly on the amount of interference. Using this method and design, the preload can be estimated as the yield strength multiplied by the combined cross-sectional area of the sidewalls  29  for the depicted configuration. Other configurations will require other calculations, but such calculations are known to one skilled in the art and are thus not described in detail herein.  
      The press-fit method has several advantages over using threaded fasteners to preload the piezoelectric element  22 . The performance of press-fit piezoelectric elements  22  is more repeatable because the preload and contact area are better defined. Furthermore, the preload of a press-fit piezoelectric elements  22  can be easily calculated and does not depend heavily on manufacturing tolerances. The press-fit method also reduces the number of total motor parts, because it does not require the spring  50  to be clamped separately to the vibrating element  50  as the end  50   b  can be used to press-fit the piezoelectric elements  22  into the opening  28 . In addition, assembly of the vibratory element  26  is made easier by eliminating the need for a threaded fastener  32  and uncertainties in its tightening and loosening during vibration. Eliminating the threaded fastener  32  also eliminates the need for a tapped hole and thus reduces manufacturing costs.  
      The vibratory element  26  shown in  FIG. 1  has two straight sidewalls  29  on opposing sides of the opening  28 . The sidewalls could comprise different configurations, such as beams at each comer. But in these configurations the sidewalls  29  are straight and generally parallel to the longitudinal axis of the piezoelectric element  22 . That results in sidewalls that remain primarily in uni-axial tension during preloading and operation of the piezoelectric element  22 .  
      Curved-Beam Configurations For Press-Fit Preloads: Alternative configurations having sidewalls that curve away from the piezoelectric element and from each other can provide a number of advantages. The press-fit operation for these two general types of vibratory elements  26  does not differ. But the resulting advantages of the basic configuration can differ significantly, as discussed below. The source of the problem and some partial solutions are discussed first, and then the advantages of curved sidewalls  29  are discussed relative to  FIG. 8 .  
      Referring to  FIG. 7 , the preload on the piezoelectric element  22  is estimated as the yield strength of the material multiplied by the combined cross-sectional area of the sidewalls  29 , because the sidewalls are stressed in uniaxial tension. This means that the entire cross-section of a sidewall  29  experiences the same stress. If the sidewalls  29  have the same cross-sectional area and the piezoelectric element  22  is pressed so its longitudinal axis coincides with the longitudinal axis  25  of the vibratory element  26 , then the sidewalls  29  also experience the same force and the same stress. If the sidewalls  29  are of constant cross-sectional area, the stress is also constant over the length of the sidewalls measured along the longitudinal axis  25  of the vibratory element.  
      The piezoelectric element  22  must move the resonator  24  and selected driving portion  44  to achieve a sufficient physical displacement to move the driven element  42 . Because the sidewalls  29  act as springs to preload the piezoelectric element  22 , a portion of the preload must be overcome in order to extend the vibratory element  26  and move the selected contacting portion. If the stiffness of the sidewalls  29  is too large, too much of the energy of the piezoelectric element  22  may be expended in pushing against the sidewalls  29  and the amount of vibratory energy that is transferred to movement of the selected contacting portion  44  and driven element  42  thus is reduced.  
      For a small vibratory element  26  of about one inch (2.54 cm) or less in length, the maximum forces on the piezoelectric element  22  and the desire to have the sidewalls  29  in the yield region result in configuring the sidewalls  29  to have a thickness on the order of 0.01 inches (0.25 mm). At such dimensions, or smaller, inaccuracies in manufacturing parts of aluminum can result in significant percentage differences in the thickness of sidewalls  29 . This leads to larger stresses in areas with smaller cross-sections and ultimately a concentration of stresses and strains in the smallest cross-sectional area. This concentration of stresses and strains over a short section of the sidewall  29  increases the chance of necking in this region during the press-fit operation.  
      Necking is undesirable for several reasons. Because all further strain in the sidewalls  29  produced by handling, temperature changes, or operation of the motor assembly  20  will be concentrated in the very short necked region, the large stresses and strains in the necked region can lead to fatigue failure during operation of the motor assembly  20 . Moreover, the necking can result in the geometry and therefore the vibrations of the sidewall  29  and vibratory element  26  to change and alter the performance of the motor assembly  20 .  
      Fatigue failure in vibratory elements  24  with sidewalls  29  in predominantly uniaxial tension is a concern even when necking is not present. Because the sidewalls  29  are put into yield, the fatigue mean stress during motor operation is near the yield strength of the material. The amplitude of the stress is very small because the piezoelectric element  22  produces deflections on the order of hundreds of nanometers as it operates at about 30 kHz-90 kHz. The high frequencies result in very large cycles of operation, but at very small amplitudes. Ferrous metals have a stress endurance limit such that these metals, if operated below this limit, do not suffer from fatigue failure. An endurance limit for aluminum and other nonferrous metals has not been observed (at least not below 100 million cycles). There is a concern that small stress amplitudes eventually may lead to fatigue failure in these materials because the motors  20  are operated at frequencies in the range of tens of kilohertz, and at this rate it does not take more than several hours for a motor to accumulate more than a billion stress amplitude cycles, albeit cycles of low amplitude.  
      Published fatigue data here is not available but fatigue failures in such motors have been observed at more than one billion cycles implying that it is desirable to take steps against fatigue failure. Using a manufacturing process that produces sidewalls  29  with nearly constant cross-sectional dimensions will improve fatigue properties by allowing the entire sidewall  29  to absorb stresses and strains instead of just one small area of the sidewall. Improving the surface finish of the sidewalls  29  also helps by reducing the number of crack initiation sites. Assuring that the sidewalls are equally stressed by giving them the same cross-sectional area and taking care to center the piezoelectric element  22  will also help avoid fatigue failure.  
      Referring now to  FIG. 8 , a vibratory element  26   p  is shown that can be used with any of the motor assemblies  20  described herein. The vibratory element  26   p  has curved sidewalls  29   p,  which are put in a more complicated state of stress, when the piezoelectric element  22  is pressed into opening  28   p  in resonator  26   p.  The opening  28   p  has opposing flat portions  31  to abut the ends of the piezoelectric element  22 , and is configured to produce curved sidewalls  29   p.  Thus the opening  28   p  is generally circular but with two opposing flats located orthogonal to an axis  25   p  corresponding to the longitudinal axis of the piezoelectric element  22 . The remainder of the resonator  24   p  can have various configurations suitable to the desired motion of and location of the selected contacting portion  44 . Here the resonator  24   p  is shown with a rectangular configuration except for the opening  28   p  defined by curved sidewalls  29   p.  The curved sidewalls advantageously have a uniform cross section along the curved length, with the depicted configuration having a rectangular cross-section along the length of the curved sidewalls. The curved sidewalls preferably have a uniform cross section for a substantial portion of the length of the sidewall. As used here, that substantial length advantageously refers to more than half the length of the sidewall  29 , and preferably refers to 75% of the length of the sidewall  29 , and ideally refers to over 90% of the length of the sidewall  29  between the end walls  31 .  
      For curved sidewalls  29   p,  the stress state can still be approximated as uniaxial but the stress in the sidewalls is not uniform and is actually a combination of bending and axial stresses. These stresses can be determined using classical beam theory calculations. Alternatively, the deformations of the sidewalls can be approximated by finite element methods or Castigliano&#39;s theorem.  
      In this embodiment, the sidewalls  29  are also advantageously put into plastic deformation during the press-fit of the piezoelectric element  22  and any protective plates  84  in order to make the preload approximately constant regardless of small differences in the amount of the interference fit. But the vibratory element  26   p  has sidewalls  29   p  that are not uniformly stressed, and are instead stressed like a curved beam in bending. The curved configuration of the sidewalls  29   p  always results in the maximum stress being located on the outside and inside surfaces of the sidewalls  29   p,  at the ends of the curved walls  29   p  joining the main body of the resonator  24 . These stresses basically occur where the curved walls  29   p  join the remainder of the body of the resonator  24 . These stresses occur on the inside of the walls  29   p  forming the opening  28   p,  and also on the outside of the walls  29   p.  The curved walls result in four defined areas of maximum stress  86  on each sidewall  29   p,  two on the inside of the walls and two on the outside of the walls.  
      Significantly, this implies that these areas reach plastic deformation first rather than having the entire cross-section of sidewall  29  reach plastic deformation simultaneously when the piezoelectric element  22  is press-fit into the opening  28   p.  This localized yielding can have advantageous results.  
      The vibratory element  26   p  has several advantages over the vibratory element  26  of  FIG. 7   a.  Because the sidewalls  29   p  are curved, they can be much thicker than straight sidewalls  29  and still achieve the same preload on the piezoelectric element  22 . This is better for manufacturing and better for the fatigue lifetime of the vibratory element  26   p.  Thicker walls increase the fatigue lifetime because small material flaws and manufacturing errors will be proportionally smaller. Such material flaws and manufacturing errors are the most probable locations of crack initiation leading to fatigue failure.  
      Further, in high cycle fatigue, most of the fatigue lifetime is spent in initiating the crack, and the thicker walls help reduce that crack initiation. Moreover, fatigue cracks start in the wall sections that are under the highest stress. In the walls  29   p  the locations of maximum stress are known as explained above, and that allows steps to be taken to reduce stress concentrations. For example, in order to reduce the stress concentrations in these high-stress areas it is preferable that the sidewalls  29   p  have fillets or rounded junctures (at points  86 ) with adjoining walls, on both the inside and outside of the walls  29   p,  as shown in  FIG. 8 . Because the critical stress areas are known and can be either reinforced or have stress-relieving steps applied to them, it is believed unnecessary with the vibratory element  26   p  to require more than a machined surface finish. The expense and effort of a polished surface is not believed necessary.  
      Additionally, necking is also not a severe problem with the vibratory elements  26   p  because of the non-uniform stress distribution across the thickness of the sidewalls  29   p.  The vibratory element  26   p  also has an advantage in that the spring constant of the sidewalls  29   p,  the axial force divided by axial deflection, is lower compared to the sidewalls  26  of  FIG. 7   a.  A lower spring constant allows the piezoelectric element  22  to expend more energy in moving the driven element  42  rather than pushing against the preload spring formed by sidewalls  29 ,  29   p.  For these reasons, it is advantageous to use curved sidewalls  29   p  rather than straight, uni-axial tension sidewalls  29 . The sidewalls  29   p  are preferably of uniform curvature, and symmetric about the portion of the longitudinal centerline  25   p  extending through the opening  28   p.  In comparison to straight sidewalls, curved sidewalls also allow the opening  28  to be dilated by a larger amount (elastically or plastically).  
      Wedging Preload Methods &amp; Designs:  
      Referring to  FIGS. 9-11 , a method and apparatus using a wedging effect is described using a resonator of the configuration of  FIG. 1 . The resonator  24  is thus illustrated as a rectangular body with a rectangular opening  28  both symmetrically aligned along longitudinal axis  25 . Other shapes could be used. The opening  24  is slightly larger than the piezoelectric element  22  and any protective cap  84 , measured along the longitudinal axis as reflected in  FIG. 10 . A slight press-fit is also acceptable. A hole  90  is placed through the resonator  24  at one end of the opening  28 . The hole  90  is shown here as being placed in the end of the resonator  24  opposite the driving end  44  ( FIG. 1 ) and adjacent the end connected to spring  50  ( FIG. 1 ).  
      As shown in  FIGS. 10-11 , a wedge  92  is forced into the hole  90  sufficiently to deform the hole  92  and the adjacent end of the opening  28 . As illustrated, the hole and wedge are both cylindrical, and located adjacent an end of the opening  28 , so as to cause a bulging along the longitudinal axis into the opening  28  sufficient to compress the piezoelectric element  22  within the opening  28 . Basically, the wedge distorts one wall of the opening  28  to place the piezoelectric into compression. The intervening protective plate  34  ( FIG. 1 ) could be used on one or both ends of the piezoelectric element  22 , or omitted.  
      Because the dimensions of the cylindrical hole and wedge can be closely controlled and positioned on the resonator  24 , and because the material properties of the parts are known and predictable, a precise deformation of the opening  28  can be achieved. The distortion must be symmetrically achieved if the forces in sidewalls  29  are to be kept equal. But if an offset compression is desired in order to potentially skew the axis along which the force of the piezoelectric element  22  acts relative to the resonator  24 , then the hole  90  can be offset from the longitudinal axis  25 .  
      Referring to  FIGS. 12 , the hole need not be circular, but could comprise a rectangular slot, with the wedge  92  being correspondingly configured to distort the hole  90  as needed to create the appropriate preload. A wedge  92  with a rectangular shaped cross-section, or with an elliptically shaped cross section, could be used. As the shape of the wedge  92  changes to increase the amount of deformed material, the force needed to insert the wedge  92  into the hole  90  increases.  
      As discussed later, there are advantages in some situations if the piezoelectric element  22  applies its force along an axis either parallel to but offset from the longitudinal axis  25  of the resonator  24 , or at a skew angle relative to that longitudinal axis  25 .  FIGS. 13-16  illustrate several ways to achieve this offset and skewing of the relative longitudinal axes of piezoelectric element  22  and resonator  24 . Another variation is discussed later regarding  FIG. 53 .  
       FIG. 13  shows the piezoelectric element  22  offset within the opening  28  so the centerline  95  of the longitudinal axis of the piezoelectric element  22  is laterally offset from the centerline of the axis  25  of resonator  24 . The offset can be above, below, or to either side of the centerline  25 , depending on the desired motion of the selected contacting portion  44 .  
       FIG. 14  shows a small, hardened insert  94  interposed between one end of the piezoelectric element  22  and the adjacent wall of the opening  28 . A hardened steel ball or a small disk could be used, but it must be sized or shaped relative to the abutting portions of the resonator so that no unacceptable deformation of the insert  94  occurs under driving forces applied by the piezoelectric element  22 . In this embodiment a protective cap  34  is preferably used in order to avoid localized forces on the more brittle piezoelectric element  22  that might damage the piezoelectric. The location of the insert  94  can be above, below, or to either side of the centerline  25 , depending on the desired motion of the selected contacting portion  44 . More than one insert can be used.  
       FIG. 15  shows the opening  28  and piezoelectric element  22  aligned along axis  95  of the piezoelectric element  22 , but both located at a skew angle relative to longitudinal axis  25  of the resonator  24 . This results in an asymmetrical mounting of the piezoelectric element  25  relative to the centerline of the resonator  24 . The amount of skewing of the relative axes of the piezoelectric element  22  and the resonator  24  will depend on the desired motion of the selected contacting portion  44 . This configuration has the disadvantage of creating sidewalls  29  having a varying cross-section. But given the present disclosure, it is possible for one skilled in the art to mount the piezoelectric element  22  at a skew angle to the longitudinal axis  25  of the resonator  24  and the vibratory element  26 . Placing small inserts  94  on opposing ends of the piezoelectric element  22  (see  FIG. 16 ), and on opposing sides of the longitudinal axis  25 , could also achieve a skew axis of the piezoelectric element  22  relative to axis  25 . Various combinations of the above, and later described mounting systems can be used.  
      Mounting Of Vibratory Elements &amp; Driven Elements  
      Given the present disclosure, a variety of mounting configurations are possible for the vibrating element  26  and resilient mounting system  50 . The mounting configuration is often determined by the location of the selected driving portion  44  and the mating engaging portion of the driven element  42 , and the required motion of those elements.  
      Referring to  FIG. 17 , the vibratory element  26  is mounted to a distal end of a rigid beam  102  that is pivotally mounted at a pivoted end to rotate about pivot point  104 . The vibratory element  26  has a selected contacting portion  44  resiliently urged against the driven element  42 . The selected contacting portion  44  is shown inward of the distal end  36  of the vibratory element  26  to reiterate that the location of the selected contacting portion  44  can be at various locations on the vibratory element  26 . The same applies for the other mounting configurations discussed herein.  
      As illustrated in  FIG. 17 , spring  50  resiliently urges the parts to maintain sufficient contact during the desired portion of the motion to move the driven element. The spring  50  may take various forms and be connected in a variety of ways. The driven object  42  can have a variety of shapes, or motions. Useful forms of driven objects  42  comprise one of a rod, a ball or a wheel that is located at a distal end of the vibratory element  26 . The driven element  42  needs to be appropriately supported to allow its intended motion, and that support is not shown here as the motion can vary according to design.  
       FIG. 18  shows an arrangement similar to  FIG. 17 , but with the location of the resilient force altered so that it is exerted on the distal end of the pivoted rigid element  102  and pulls the vibratory element  26  into contact with the driven element  42  rather than pushing it into contact. The spring  50  applying the resilient force advantageously applies its urging force along an axis aligned with the longitudinal axis  25  of the vibratory element  26 , but that is optional.  
       FIG. 19  shows an arrangement similar to  FIGS. 17-18 , but with the location of the resilient force altered so that it is exerted adjacent to the pivoted end of rigid element  102 . The location of the resilient force, such as applied by spring  50 , can affect the displacement of the spring. When the spring  50  is located nearer the pivot point  104 , the spring  50  does not move much because the effective moment arm between the pivot point and the connection to the spring is less.  
      If the vibratory element  26  is rigidly mounted, configurations similar to those described herein can be used to resiliently urge the driven element  42  to maintain sufficient contact with the fixedly mounted vibratory element  26  to achieve the desired movement of the driven element.  
       FIG. 20  shows one advantageous mounting configuration that uses a flat strip of spring metal for the spring  50 . The spring  50  has a first end  50   a  mounted to a base  52 , and an opposing end  50   b  connected to the vibratory element  26 . A first leg of the spring  50  containing end  50   a  is parallel to the longitudinal axis  25  of the vibratory element  26 , with the second leg of the spring being bent at about a right angle. The distal end of the vibratory element  26  is resiliently urged against the driven element  42 . The driven element  42  can in principle have any sufficiently smooth shape, but readily useful forms of driven objects  42  comprise one of a rod, a ball or a wheel that is located at a distal end of the vibratory element  26 . The driven element  42  needs to be appropriately supported to allow its intended motion, and that support is not shown here as the motion can vary according to design.  
       FIG. 21  shows a straight, leaf spring  50   a  having one end rigidly mounted to base  52 , with an opposing distal end  50   b  mounted to the vibratory element  26 . The distal end of the vibratory element  26  is urged by the spring  50  against the driven element  42 . Other shapes of springs  50  are possible.  
       FIG. 22  shows a vibratory element  26  having a first end pivoted about pivot  106  and an opposing distal end resiliently urged by spring  50  against a driven element  42 . The selected contacting portion  44  is intermediate the pivot point  106  and the connection of spring  50  along the longitudinal axis  25 , and is on an opposing side of the axis  25  as is the spring  50  and pivot  106 . But various locations of the selected contacting portion  44  relative along the axis  25  are possible, depending on the desired motion and the configuration of the parts.  
      There is thus provided a method and apparatus for generating at least two components of motion at the selected contacting portion  44 . These two motion components have mutually different directions, with each component oscillating when the piezoelectric element  22  is excited at a predefined frequency, and with the two components having mutually different phases. These two motion components are shaped so as to create an elliptical motion  100  along a desired orientation, by configuring the vibratory element  26 , its suspension  50 , or both. There is also advantageously provided a method and apparatus by which the same or other contacting portions  44  create suitable ellipses  100  at various excitation frequencies of the piezoelectric element  22 , resulting in mutually different macroscopic motions of the driven body  42  engaging one or more of the selected contacting portions(s)  44 .  
      In one embodiment, the vibration element  26  is attached to the base  52  with a spring-like element  50 , wherein the spring can be of the bending, torsion, pneumatic, elastomeric, or any other type. The spring could for example be made from portions of an electronic circuit board, which has advantages in manufacturing. The spring constant or flexibility of the spring can be adjusted to compensate for wear in the contact area between contacting portion  44  and the engaging portion of the driven element  42 , and can also compensate for production inaccuracies. For instance, high compliance of the spring  50  results in small variations in the resilient contact force the contacting portion  44  exerts on the driven element  42  despite relatively large deflections of the contacting portion  44 .  
      The spring  50  can be attached to the vibration element  26  in many different ways. In one embodiment, the vibration element  26  contains an opening  28  with a dimension slightly smaller than the sum of a corresponding dimension of the undeformed piezoelectric element  22  and the thickness of the spring  50 . Inserting the spring  50  and the piezoelectric element  22  into the opening  28  causes the opening to expand, thus creating a press-fit that functionally connects said vibration element to the spring and the piezoelectric element. In another embodiment, the vibration element contains an opening, such as a slot, into which an end of the spring can be pressed, glued, screwed, or otherwise fastened. The slot can be oriented in any preferable way. In one embodiment, the slot is oriented perpendicular to the longitudinal axis  25  of the vibrating element  26 . In another variation, it is oriented parallel thereto.  
      In addition to connecting the vibration element  26  to the housing or base  52 , the spring-like element  50  can also be a functional extension of the vibration element  26 . With proper adjustments, the spring  50  can isolate the vibrations of the vibration element  26  functionally from the housing  52 . Also, the spring  50  can influence the dynamic behavior of the vibration element  26  in order to enhance the performance of the vibration element  26  through amplification or other dynamic effects. For instance, if the spring  50  has axes of symmetry which are different than, or offset to those of, the axes of symmetry of the vibration element  26 , then the assembly of the piezoelectric element  22 , vibration element  26  and spring  50  becomes dynamically asymmetrical resulting in a coupling of what were formerly independent modes of vibration.  
      In a further embodiment, the vibrating element  26  can be attached directly to the housing or base  52 . It is preferable if the housing or base  52  functionally isolates the vibrations of the vibrating element  26 . In this embodiment, the housing itself must exert the resilient force that presses the vibrating element  22  against the moving element  42 . It is preferable if the design of the housing or base  52  provides a mechanism to adjust the contact force and to compensate for motor wear. A screw in a slot to adjustably position the vibratory element  26  relative to the driven element  42  is one example.  
      In yet another embodiment, the vibration element  26  is rigidly connected to the housing  52 , and the driven element  42  is resiliently urged against the vibrating element  26  by the bearings in or on which the moving element is supported. For this purpose, it is preferable if the support contains some sort of spring or other compressible medium to provide a resilient force to urge the parts into contact. In this situation, the motor assembly  20  could also mounted in a position so that gravity acting on the driven element  42  may provide the necessary resilient force.  
      Locating Driving &amp; Driven Elements  
      Referring to  FIGS. 23-36 , various configurations are shown for mounting the vibratory element  26  relative to the driven element  42 . These figures are schematically shown, and omit the mounting systems of the parts that allow the desired motion and that maintain the parts in sufficient contact for the intended use. For illustration the driven object  42  is shown as a rod with a cylindrical cross section, but it could be a ball, a wheel, a rod, a bar, a gear or something else. The vibratory element  26  needs to be urged against the driven object  42  with a certain force and angle to achieve the contact needed to cause motion. This can be achieved through the mounting mechanisms described earlier or apparent to those skilled in the art given the present disclosure. The mechanism causing that resilient contact is not shown. Also not shown is the mounting arrangement that allows the desired movement of the driven part  42  as that will vary with the particular design. The following arrangements are only examples. Others are possible but are not described since it is not possible to cover all of them. Combinations of these arrangements are possible as well.  
      Single Vibratory Element Configurations:  FIGS. 23-26  show configurations using a single vibratory element  26 . In  FIG. 23 , the vibratory element  26  is above the driven element  42 , with at least one of the elements  26 ,  42  being resiliently urged to maintain the selected driving portion  44  in sufficient contact with the selected engaging portion of driven element  42  to achieve the desired motion. The longitudinal axes of the vibratory element  26  and the driven element  42  are perpendicular to each other, but they could be at various intermediate angles. The contact portion  44  is inward of the distal end  36 , but could be at an location along the length of the vibratory element  26  achieving the desired motion at a selected amplitude. The contact portion  44  is thus advantageously selected to occur at a location having the desired elliptical motion  100 . The motion  100  is shown generally aligned with the axis  25  of the vibratory element  26 , which will cause rotation of the driven element  42  about axis  45 . But that need not be the case, as it could be in a plane orthogonal to axis  25  to cause translation along axis  45 , or at orientations in between, depending on the desired motion and the design of the components. As used herein, an alignment of about 0-5 degrees will be considered to be aligned.  
       FIG. 24  shows the longitudinal axis  25  vertically offset from and perpendicular to the axis  45  of the driven element. Various intermediate angles of inclination are possible. The selected contact portion  44  is at the distal end  36 , at a lower peripheral edge of the vibratory element  26 . This arrangement lends itself to producing rotation of the driven element  42  about axis  45 , or translation along that axis, or combinations of those motions.  
      In  FIGS. 25-26 , the longitudinal axes  25 ,  45  are coplanar and inclined relative to each other at an angle α as discussed relative to angle α of  FIG. 1 . The selected contact portion  44  is at the distal end  36 , at a lower peripheral edge of the vibratory element  26 . This arrangement lends itself to producing translation of the driven element  42  along axis  45 , or rotation about that axis, or combinations of those motions. Referring to  FIG. 26 , the axes  25 ,  45  are shown as coplanar, but they need not be so and could intersect at skew angles.  
      Multiple Vibratory Element Configurations: Configurations using multiple vibratory elements  26  that cooperate to move the driven element  42  are shown in  FIGS. 27-42 . The use of multiple vibratory elements  26  has the advantage of providing more locations of support to the driven element  42  so that some of the bearings may be omitted. Thus, cost savings and friction reduction that typically comes with low cost bearings or bushings are achieved. In some applications, it can be enough to suspend the driven element  42  entirely using vibratory elements  26  without the need for additional bearings. The vibration of the contacting portion of the vibratory element  26  can provide a low friction support, and an elliptical motion of the supporting portion of the vibratory element  26  is not necessary for this low friction support application.  
      Further, the use of multiple vibratory elements  26  can accordingly multiply the force, and/or the speed with which the driven element  42  is moved. A single, common excitation signal could be provided to each of the vibratory elements  26  in order to simplify the electrical system, or separate signals could be provided to cause different simultaneous motions of the driven element  42 .  
      In the following, configurations with a specific number of vibratory elements  26  are described. Given the disclosures therein, a variety of other mounting configurations can be configured that use multiple vibratory elements  26  to restrain various degrees of freedom of the driven element  42 .  
      Double Vibratory Element Configurations: Configurations using specifically two vibratory elements  26  and a single driven element  42  are shown in  FIGS. 27-36 . In  FIG. 27  there are two vibratory elements  26  resiliently urged against opposing sides of driven element  42 . The two vibratory elements  26  have axes  25  perpendicular to the longitudinal axis  45  of driven element  42 , and on opposing sides of that axis  45 . The selected contacting portion  44  of each vibratory element  26   a,    26   b  is preferably intermediate the distal ends of the vibratory elements, but that need not be the case as the contacting portion  44  could be at distal end  36 . The axes  25  of the vibratory elements  26  can be parallel and coplanar, but they do not have to be either parallel or coplanar. This arrangement lends itself to producing translation of the driven element  42  along its longitudinal axis  45 , or rotation about that axis, or combinations of those motions.  
       FIG. 28  shows two vibratory elements  26  resiliently urged against a common side of driven element  42 . The two vibratory elements  26  have axes  25  perpendicular to the longitudinal axis  45  of driven element  42 , but the axes  25  could be inclined to axis  45 . The axes  25  of the vibratory elements  26  can be coplanar, but need not be so. The contact portions  45  are at distal edges of each face  36 . The contact portion  44  is at an angle 45 degrees from the horizontal plane in which the axis  45  is shows as being located, but on opposing sides of that plane. This configuration lends itself to producing translation of the driven element  42  along its longitudinal axis  45 , or rotation about that axis, or combinations of those motions.  
       FIG. 29  shows a configuration similar to  FIG. 28  except that the vibratory elements  26  face each other and are located on opposing sides of the driven element  42  relative to the vertical axis.  
       FIG. 30  shows a configuration similar to  FIG. 24 , except there are two vibratory elements  26  on opposing sides of the driven element  26 , on a common axis  25 . The longitudinal axes  25  of each vibration element  26  need not coincide, but could be coplanar and skewed relative to each other.  
       FIG. 31  has two vibratory elements  26  on opposing sides of the driven element  42 , with the elements  26  facing each other but oriented at inclined angles α, β, respectively, relative to a plane through the longitudinal axis  45  of driven element  42 . The angles α, β are shown so the axes  25  of each vibratory element  26  are parallel, but they need not be parallel. The angles preferably cause the longitudinal axes  25  to intersect the longitudinal axis  45  of the driven element  42 , but need not do so. The selected contact portion  44  is at the distal end  36  of each vibratory element  26 .  
       FIG. 32  has two vibratory elements  26  on opposing sides of the driven element  42 , with the elements  26  facing the same direction and oriented at inclined angles α, β, respectively, relative to a plane through the longitudinal axis  45  of driven element  42 . The angles α, β are such that the longitudinal axes  25  preferably intersect longitudinal axis  45  of the driven element  42 , but need not do so. The selected contact portion  44  is at the distal end  36  of each vibratory element  26 .  
       FIG. 33  has two vibratory elements  26  on the same side of the driven element  42 , with the elements  26  facing the same direction and orientated at inclined angles α, β, respectively, relative to a plane through the longitudinal axis  45  of driven element  42 . The angles α, β are such that the longitudinal axes  25  preferably intersect longitudinal axis  45  of the driven element  42 , but need not do so. The axes  25  need not lie in the same plane, but preferably do so. The selected contact portion  44  is at the distal end  36  of each vibratory element  26 .  
       FIG. 34  has two vibratory elements  26  on the same side of the driven element  42 , with the elements  26  facing each other and oriented at inclined angles α, β, respectively, relative to a plane through the longitudinal axis  45  of driven element  42 . The angles α, β are such that the longitudinal axes  25  preferably intersect longitudinal axis  45  of the driven element  42 , but need not do so. The axes  25  need not lie in the same plane, but preferably do so. The selected contact portion  44  is at the distal end  36  of each vibratory element  26 .  
       FIGS. 35-36  show a configuration with two vibratory elements  26  on opposing sides of the driven element  42 , with the elements  26  facing the same direction and orientated at inclined angles α, β, respectively, relative to a plane through the longitudinal axis  45  of driven element  42 . The angles α, β are such that the longitudinal axes  25  preferably intersect longitudinal axis  45  of the driven element  42 , but need not do so. The axes  25  need not lie in the same plane, but preferably do so. Advantageously, the axes  25  intersect at a common location on axis  45 , with the engaging portions  44  being in the same plane orthogonal to axis  45 .  
      In this configuration, the selected contact portion  44  is at the distal end  36  of each vibratory element  26 . The selected contact portions  44  of each element  26  are configured to have a shape mating with the shape of the engaged portion of driven element  42 . Here, the circular cross-section of rod  42  results in a convex curved surface for the selected contacting portions  44 . This curved engagement results in the vibratory elements  26  providing a support for the driven element  42  that restrains motion except for translation along axis  45 . If the contacting portion  44  has a small engaging surface along the length of axis  45 , then the driven element  42  will rock about the engaging portions  44 . If the contacting portion  44  has an engaging surface with a sufficient length along the length of axis  45 , then the driven element  42  can be supported without rocking about the engaging portions  44 . This configuration can simplify the mounting of the driven element  42  by allowing the vibratory elements  26  to also act as bearings by clamping the rod between the tips of two vibratory elements  26 .  
      Triple Vibratory Element Configurations:  
       FIGS. 37-40  show configurations using three vibratory elements  26   a,    26   b  and  26   c,  with the letters a, b, and c being associated with the various corresponding parts of the first, second and third vibratory elements, respectively.  FIG. 37  shows two vibratory elements  26   a,    26   b  as described in  FIG. 27 , each of which is resiliently urged against opposing sides of driven element  42 . The two vibratory elements  26   a,    26   b  each have axes  25  perpendicular to the longitudinal axis of driven element  45 , and on opposing sides of that axis  45 . The selected contacting portion  44   a,    44   b  of each vibratory element  26   a,    26   b  is preferably intermediate the distal ends of the vibratory elements, but that need not be the case as the contacting portion  44  could be at distal end  36 . A third vibratory element  26   c  is located on an opposing side of the driven element  42  with its selected contacting portion  44   c  intermediate, and preferably equally between, the contacting portions  44   a,    44   b  along an axis between  44   a  and  44   b.  The driven element  42  has its longitudinal axis along the z axial direction. Preferably, the first and second vibratory elements  26   a,    26   b  contact the driven element  42  at the 12 and 6 o&#39;clock positions, with the third vibratory element  26   c  contacting the driven element  42  at the 3 o&#39;clock position. Other contact locations are possible. The contacting portion  44   c  is preferably at a distal edge of the vibratory element  26   c,  with the third vibratory element  26   c  being oriented at an angle α parallel to the plane containing axes  25   a,    25   b.  The axes  25  of the vibratory elements  26   a,    26   b  are preferably parallel and the axes  25   a,    25   b  and  25   c  are preferably coplanar, but the various axes do not have to be either parallel or coplanar. This configuration provides for translation and rotation of the driven element  42  along and about its longitudinal axis  45 , with the vibratory elements restraining translation in both directions along the y-axis, and in the +x direction, but allowing motion along the −x direction.  
       FIG. 38  shows the vibratory elements  26  with their longitudinal axes  25  perpendicular to a radial axis extending in a plane orthogonal to the axis  45  of the driven element  42 . The contacting portions  44  are illustrated as offset from distal ends  36 , but that need not be the case. The vibrating elements  26  are shown as equally spaced with angles β, γ, and α each being about 60 degrees, but the angles can vary. The axes  25   a,    25   b,    25   c  are shown as coplanar, but they need not be so. The driven element  42  has its longitudinal axis along the z axial direction. This arrangement allows the vibrating elements  26  to restrain translation of the driven element  42  in both directions along the x-axis and y-axis.  
       FIG. 39  places two of the vibratory elements  26   a,    26   b  on one side of the driven element with axes  25   a,    25   b  parallel to the x-axis, and with their respective contacting portions  44   a,    44   b  engaging the peripheral portion of the driven element at corresponding locations along an axis parallel to the vertical y-axis. The contacting portions are located at edges of the distal ends  36   a,    36   b.  The axes  25   a,    25   b  are parallel and coplanar, but need not be coplanar or parallel. The third vibratory element  26   c  is on the opposing side of the driven element  42 , with axis  25   c  parallel to the y-axis. The axis  25   c  is preferably coplanar with axes  25   a,    25   b,  but need not be so. The driven element  42  has its longitudinal axis along the z axial direction. This arrangement allows the vibratory elements  26  to restrain translation of the driven element  42  in both directions along the x-axis and y-axis.  
       FIG. 40  places two of the vibratory elements  26   a,    26   b  on one side of the driven element with axes  25   a,    25   b  parallel to the x-axis, and with their respective contacting portions  44   a,    44   b  engaging the peripheral portion of the driven element at corresponding locations along an axis parallel to the vertical y-axis. The axes  25   a,    25   b  are parallel and coplanar, but need not be coplanar or parallel. The driven element  42  has its longitudinal axis along the z axial direction. The third vibratory element  26   c  is on the opposing side of the driven element  42 , with axis  25   c  parallel to the x-axis, and coaxial with axis  25   b.  The contacting portions are located at edges of the distal ends  36   a,    36   b,    36   c.  This arrangement allows the vibratory elements  26  to restrain translation of the driven element  42  in both directions along the x-axis and y-axis, but does permit motion along one direction of a skew axis at 45 degrees from the horizontal as shown in  FIG. 40 .  
      In the above configurations using multiple vibratory elements  26 , each vibratory element is preferably activated at the same time as the other vibratory elements so that the vibratory elements cooperate to produce the desired motion of the driven element  42 . But the vibratory elements  26  could be separately activated at different times or in different combinations or in different sequences in order to achieve separate motions of the driven element.  
      Six Vibratory Elements:  
       FIGS. 41-42  show a configuration in which six vibratory elements  26   a  through  26 f are used to support a driven element  42  that can rotate and translate about its longitudinal axis  45 . The vibratory elements  26  each have one end attached to a ring  110  that encircles the driven element  42 , preferably in a plane orthogonal to the longitudinal axis  45  of the driven element. The opposing distal end  36   a  through  36 f of the vibratory elements  26  is pressed against driven element  42 . Three of the vibratory elements  26  extend toward the driven element  42  in directions opposite to the other three vibratory elements as best seen in  FIG. 42 . The relative position of each vibratory element  26  viewed in the x-y plane orthogonal to the axis  45  of driven element  42  ( FIG. 41 ), is determined through the angles α, β, γ, σ, ε, and ρ. These angles are preferably 60 degrees in order to equally distribute the support and driving forces, but the angles can be different from that. The vibratory elements  26  advantageously have their longitudinal axes  25  intersecting the longitudinal axis  45  of the driven element  42 , but the axes  25  could be skewed so they do not intersect the axis  45 . The angles between the vibratory elements  26  and the driven element  42  are defined by δ and φ as shown in the drawing, and will vary depending on the dimensions of the various parts and on the orientation of the vibratory elements  26 . The flexibility of the ring  110  helps to ensure that the vibration elements  26  are pressed against the driven element  42 . As a result the rod is suspended at six points.  
      This configuration allows the vibratory elements  26  to support the driven element  42  so as to allow translation only along the longitudinal axis  45  of the driven element  45  and to allow rotation about that axis.  
      Motor Operating Principles  
      The following description helps understand the operation of the above-described embodiments, and helps understand the variety of ways to implement these embodiments and variations thereon.  
      The present motor uses only one piezoelectric element  22  with one electrical excitation signal to excite various modes of vibration of the vibration element  26 . The motion of the contact portion  44  is determined by these modes of vibration. In particular, the present motor achieves an elliptical movement of the contact portion  44  in a first direction for a sinusoidal electrical excitation signal at a first frequency, and an elliptical movement of the contact portion  44  in a second direction for a sinusoidal excitation signal at a second frequency, providing a required force or amplitude of motion or speed at the contact portion  44 . Elliptical movements of the contact portion  44  in a third and more directions for sinusoidal excitation signals at third and more frequencies are known to be possible.  
      The motor assembly  20  is advantageously configured so that the contact portion  44  traces the elliptical motion several tens of thousand times per second to make motor operation inaudible for humans and most pet animals. During a selected segment of each elliptical cycle, the contact portion  44  comes into contact with the engaging surface of the driven object  42  where it exerts a frictional contact force that transports the driven object  42  by a small amount into a corresponding direction. The observed macroscopic motion of the driven object  42  is the accumulation of all individual transportation steps.  
      While the bulk of this disclosure refers to a contact portion  44  located at a distal end  36  of vibration element  26  and moving in a first elliptical path  100   a  causing the driven object  42  to be transported in direction of the driven object&#39;s longitudinal axis  45 , and the same selected contact portion  44  moving in a second elliptical path  100   b  causing transportation in an opposing direction (as in  FIGS. 2 and 5 ), the first and second selected contacting portions  44  need not be the same, need not be adjacent, and need not be located at a distal end  36 . They need only be located on the same vibratory element  26 . Further, the number of selected contacting portions  44  and the directions and orientations of respective elliptical paths  100  at each contacting portion can vary according to the particular design. There could be three, or there could be more. There can thus be a plurality of selected contacting portions  44  on the vibratory element  26  moving in a plurality of elliptical paths  100  in a plurality of directions.  
      Advantageously the desired motion of a selected contact portion  44  is identified, whether it is in a single direction or multiple directions, and whether there is a single selected contact portion  44  or several contact portions  44 , or combinations thereof. The motor assembly  20  is then designed to achieve that motion. As often occurs, the design does not achieve perfection but instead achieves an acceptable approximation of the desired motion. A number of the factors that can be used to configure the components of motor assembly  20  to achieve that desired motion are discussed below.  
      Generation of elliptical motion: If the piezoelectric element  22  is excited with a sinusoidal electrical signal, it generates a sinusoidal force and a sinusoidal displacement principally along its longitudinal axis  95 , shown in  FIGS. 1-3  as being alongside longitudinal axis  25  of the vibration element  26 , or shown in  FIG. 15  as being at an oblique angle to the longitudinal axis  25 . Said force and displacement are then used to excite modes of vibration of the vibration element  26 . The vibration element  26  is preferably configured so that at a predetermined excitation frequency at least two of its modes of vibration are substantially excited. If a mode has only a uniform motion component in the direction of the longitudinal axis  25 , it is considered to be a longitudinal mode. If the motion components of a mode lie in a direction perpendicular to the longitudinal axis  25 , the mode is considered to be a bending mode. Further well known modes include torsion and shear modes. A mixed mode is neither of these modes but can have components of motion in, or rotating around, any of the directions  25 ,  38  or  40 . Each mode that is excited adds a sinusoidal motion component to the motion of the contact portion  44 . If at least two of these components of motion are non-parallel and mutually out of phase, the resulting motion of the contact portion  44  is known to be elliptical.  
      The bulk of this disclosure refers to a piezoelectric element  22  that generates force and displacement principally along its longitudinal axis  95 , but a piezoelectric element having a different principal direction, or a force and displacement-generating element other than a piezoelectric element, could be used.  
      Making use of elliptical motion: It is an advantage of the present invention over prior art motors that elliptical motions do not have to be achieved exclusively with mutually perpendicular longitudinal and bending modes that are excited 90 degrees out of phase, but instead that the elliptical motion can be generated with at least two excited modes that can be mutually oblique and have a phase difference that can be substantially different from 90 degrees. In this case, the contact portion  44  traces an ellipse  100  whose semi-axes are not necessarily aligned with any of the directions  25 ,  38  or  40 , thus making it advantageous for the vibration element  26  to be mounted at an oblique angle to the driven object  42 . That is, the longitudinal axis  25  is preferably inclined to the vibration element  26  at an angle α ( FIG. 1 ), which will vary with the particular design and components involved. Oblique mounting of the vibration element  26  rotates the ellipse  100  with respect to the driven object  42 . Associated with this rotation, a coordinate transformation is formulated elsewhere that exposes the beneficial and enhancing effects of this rotation on relative phase shifts between components of motions that generate the elliptical path  100 .  
      While the elliptical motion  100  of the selected contacting portion  44  is achieved even if the selected contacting portion  44  is not engaged with the driven element  42 , in order to achieve useful motion the contact portion  44  of vibration element  26  is placed in physical contact with the engaging surface of the driven object  42  during a certain portion of each elliptical cycle  100 . This portion preferably remains the same for each subsequent cycle. During each engagement, the vibration element  26  exerts frictional forces on the driven object  42 . These forces can vary over the period of an engagement but their accumulative effect transports the driven object  42  relative to vibration element  26 . It is believed that this transport is most efficient if the direction of transport coincides with the direction of motion of the contact portion  44  at the point of the ellipse that is closest to the driven object  42 .  
      The speed of contacting portion  44  tangential to the elliptical path  100  is largest where the minor-axis of the ellipse intersects the elliptical path, and smallest where the major axis of the ellipse intersects the elliptical path. An ellipse whose major axis is tangential to the engaging surface of driven object  42  is therefore expected to provide an efficient transportation mechanism. It can be beneficial to use an ellipse whose major axis is inclined with respect to the engaging surface of driven object  42 . In this situation, the contact portion  44  moves towards the driven object  42  at a different rate than the rate at which it moves away from it after having passed the point closest to driven object  42 . Inherent to the elliptical shape, a faster approach typically results in a slower retreat and vice versa, so that the process of engagement with the driven element  42  can be selected to be more gradual or more abrupt. At its extreme, such a motion is known as a saw-tooth motion. Motors that generate exact saw-tooth motion are in the prior art. Purposefully employing an inclined ellipse in the present disclosure provides therefore some of the advantages only seen in those saw-tooth motors.  
      To ensure efficiency of transport, it is preferable that the frictional engagement is sufficiently large, and that the contacting portion  44  moves against the direction of desired transport of driven object  42  only while the friction forces are reduced or vanish, which occurs when some or all of the contact portion  44  has lost contact with the engaging surface of the driven object  42 .  
      The amount of friction and wear depends also on the friction parameters and the material combination used for the contact portion  44  and the size of contact portion  44 . These parameters also influence the strength of the motor  20 . More friction typically results in stronger force but may also result in more wear. Material combinations that are believed suitable for use include steel, aluminum and glass on one side of the contact, and glass, fiberglass, PMMA, PVC, ABS or steel on the other side of the contact. The friction parameters of glass surfaces are modifiable chemically or physically by adding particles or etching a texture.  
      It is an advantage of the motor assembly  20  that the dimensions of the engaging surface of the driven object  42  do not have to be precise and that variations are accommodated by the resilient mounting system of the motor  20 , which is discussed later. Also, it has been shown that wear due to the vibrations can modify the contact portion  44  of vibration element  26  and create a larger contact area. This effect is especially strong at the beginning of the lifetime of the motor. The effect fades quickly, resulting in better motor performance. This wear can be used to advantage since the resilient mount  50  urges the selected contacting portion  44  against the driven element  42 , allowing for wear-in between harder and softer materials, that can reduce initial manufacturing tolerances. As desired, the wear-in can also be used to increase the selected contacting area  44 .  
      Achieving desired elliptical motion: The size and orientation of the elliptical trajectory  100  depend on the amplitudes and phases used to generate the ellipse. The ability to maintain a useful elliptical trajectory  100  of contact portion  44  over a sufficiently large frequency range depends on the vibration design properties of the motor assembly  20 .  
      It is known that a mode of the vibration element  26  undergoes a smooth phase change of −180 degrees with respect to the excitation signal applied to the piezoelectric element  22  if the frequency of excitation is increased across the resonance frequency of the mode. The width of the frequency range within which this transition occurs increases with the amount of mechanical damping in the system. It is desirable that such a frequency range is sufficiently large in order to assure that the phase difference between two excited modes can remain sufficiently different from 0 or 180 degrees over an extended frequency range and can potentially sustain a desired elliptical motion of the contact portion  44 . This renders the motor  20  in principle less sensitive to variations in manufacturing and operating conditions. In order to achieve a desired amount of damping at a particular location, a separate dampening element could be added to any part of the vibration element  26  or the portion of the suspension that participates in the mechanical vibrations. But preferably the damping that is inherent in the system design and materials is used.  
      To achieve a stronger motor, it is also desirable if the excited modes of vibration show significant amplitudes at the contact portion  44  near the desired frequency of excitation and thus it is preferable to have a frequency of excitation that is close to a resonance frequency of a selected mode. Since the amplitude of a mode at the contact portion  44  also depends on the amplitude of its excitation, it is preferable that the vibration element  26  is designed to appropriately distribute the mechanical vibrations generated by the piezoelectric element  22  to the various modes. This distribution can be achieved in a controlled fashion in a number of ways using combinations of damping, geometric and material properties of the vibration element  26 , and the forces that are generated between the vibration element  26  and the driven object  42  at the contact portion  44 . Conceptually, methods and modifications that affect the force distribution are different from methods that affect the shape of a mode and its resonance frequency. In reality however, a modification that affects force distribution very often modifies also a mode shape and its resonance frequency. For example, it is known for a rod-like vibration element  26  that some modifications that would distribute mechanical energy forces to a pure longitudinal and a pure bending mode would typically also couple the two modes together to create new modes of mixed type.  
      Distributing mechanical vibrations: Internal damping forces can couple one mode to another so that the piezoelectric element  22  can potentially drive a first mode, which in turn excites a second mode indirectly by way of damping. This effect is particularly strong if the respective resonance frequencies and the frequency of excitation lie close together.  
      A first mode that is excited by the piezoelectric element  22  at a certain frequency can excite a second mode also by way of the contact forces generated in the contact portion  44 . Specifically, the elliptical motion of the contact portion  44  can produce a force that is sinusoidal or a force that is intermittent with the same frequency that drives the first mode. This force then excites other modes in the vibration element  26 , as well as vibration modes of the driven object  42  discussed elsewhere. This form of excitation can be mutual, and this effect can be deliberately used, so that formerly independent modes can be coupled together to form new modes. The orientation of the ellipse  100  at the contact portion  44  and the portion of the ellipse during which contact forces are generated determine the phase with which a second mode is excited relative to the first mode. This phase is preferably not a multiple of 180 degrees.  
      Which modes of vibration element  26  are excited by way of contact, and by how much they are excited, also depends on the position and orientation relative to vibration element  26  of contact portion  44  and engaging surface of driven object  42 . The contact portion  44  can be chosen to lie in a plane of symmetry of the assembly  20 , vibration element  26 , resilient mounting system  50  or driven object  42 , or not. Non-symmetric positioning can be used to excite modes that otherwise would be harder to excite by the piezoelectric element  22  alone, for example certain bending modes or torsion modes of vibration element  26 . To the same end, the orientation of the engaging surface relative to vibration element  26  can be chosen to be perpendicular or parallel to certain planes of symmetry, or not.  
      Location and orientation of piezoelectric element: Referring to  FIG. 1 , the vibrating element  26  preferably has an elongated, rod-like shape with an opening  28  perpendicular to the longitudinal axis  25  of the rod. The opening  28  has dimensions slightly smaller than corresponding dimensions of the piezoelectric element  22 , so that the piezoelectric element can be inserted into the opening  28  in a press-fit manner. If the vibrating element  26  has a symmetric shape, and if the piezoelectric element  22  is inserted symmetrical with respect to the longitudinal axis  25 , and if the contact between the piezoelectric element  22  and the vibration element  26  is nearly perfect, then it is expected that primarily only longitudinal vibrations in direction of axis  25  are generated. These vibrations can be transformed into bending or other vibrations by way of the previously discussed contact forces at contact portion  44 , or they can be transformed by the action of the resilient mounting system  50  discussed below, which urges vibration element  26  against the driven object  42 .  
      The piezoelectric element  22  can directly generate other than longitudinal vibrations in the vibration element  26  if element  22  is non-symmetrically inserted into the opening  28 , e.g., if the longitudinal axis  95  of the piezoelectric element  22  is offset (cf.  FIG. 14 ) or inclined (cf.  FIG. 15 ) with respect to the longitudinal axis  25  of vibration element  26 , or if at least one of the contact areas between the piezoelectric element  22  and the resonator  24  is made non-symmetrical. For example, the vibration element  76  in  FIG. 5  has the longitudinal axis of the piezoelectric element  22  offset from a principal longitudinal axis of the resonator  74 . This offset couples various modes of the resonator  74  and the vibration element  76 . Moreover, the resonator  74  rotates about pin  78 , and that may further modify vibrating modes of the vibrating element  76 .  
      The more pronounced the modifications are that let the piezoelectric element  22  be inserted in a asymmetric fashion, the more bending and other vibrations are typically excited. Also, such modifications typically couple formerly independent longitudinal and bending modes together to create new modes of mixed type. Torsion modes in a rod-like vibration element could also be excited.  
      In a preferred embodiment, the piezoelectric element  22  is inserted into the opening  28  in the resonator  24  such that the resonator and the piezoelectric element  22  do not enter into perfect contact along the entire area where contact could be possible. To achieve such a purposefully partial contact during the insertion process, the sidewalls  29  of the opening  28  could for example be deformed by the inserted piezoelectric element  22  such that contact is lost in certain portions of the potential contact area. Alternatively, partial contact can be achieved by making the potential contact surface of the piezoelectric  22  non-even, for example by removing material in parts of the contact area of the resonator  24  before inserting the piezoelectric into the opening  28 . In  FIGS. 10 and 11 , this could also be achieved by inserting a pin  92  at a location offset from the depicted longitudinal axis  25 . Also, inserts  94  ( FIG. 16 ) could be used to provide localized contact areas at the location of the insert. Moreover, combinations of the above methods may be used to achieve a desired partial contact and to induce a desired combination of lateral and longitudinal motion components at a desired contacting portion  44 .  
      Shape of vibration element: The resonance frequencies of the various vibration modes typically decrease if the vibration element  26  is made longer, and vice versa. Also, the shape and size of the cross-sections of the resonator  24  affect the resonance frequencies and modes involving bending and torsion. For example, referring to  FIG. 73 , the cross-section of the resonator  24 , or at least of a portion of the distal end of the resonator  24  could be I-shaped, which can be used to vary the relative stiffness and resonance frequencies of modes involving longitudinal motion and lateral bending since the I beam cross-section can have the stiffness along one lateral axis much different than the stiffness along the other lateral axis. It also produces a lower lateral bending stiffness without having to greatly increase the length of the resonator  24 .  FIG. 73  also shows a T-shaped cross section, which could introduce a twisting mode if the T was made non-symmetric about its vertical axis. C-shaped cross-sections and variety of other cross-sectional shapes can be used to vary the resonance modes of the resonator  24  and of the vibrating element  26 . Other non-symmetric, cross-sectional shapes can be used.  
      To purposefully achieve modes of vibration that can generate a desired elliptical motion  100  at the contact portion  44  of vibration element  26  so that the ellipse  100  is inclined with respect to the longitudinal axis  25  of vibration element  26  and/or the engaging surface of driven object  42 , it can be advantageous to have a non-symmetric design of the vibration element  26 . For example, the resonator  24  could be made helical, or it could have an arched or an L-shape. Other shapes are possible. The asymmetric mass distribution that is achieved this way results in modes of vibration that are neither purely longitudinal nor purely transversal in nature, which is beneficial for generating inclined elliptical motion  100 .  
      Moreover, referring to  FIG. 77 a  further embodiment is shown that has advantageous design features. This embodiment illustrates a resonator  24  that is not straight. Further, it illustrates the location of the piezoelectric element  22  along an axis that does not intersect the driven element. Moreover, it illustrates a different alignment and orientation of the piezoelectric element  22  and the resonator  24 . The axes are inclined relative to each other, with the axis of piezoelectric element  22  generally parallel with the axis  45  of the driven element  42 . The axis  25  of the resonator  24  is inclined to bridge the gap between the two axes  95 ,  45 . The selected contacting surface  44  comprises a curved surface conforming the shape of the abutting contact area on the rod-like driven element  42 . The curved surface may be manufactured or it may be generated by natural wear during operation of the motor. The resilient mounting system accommodates motion of the selected contact surface  44  that moves the rod  42 .  
      Advantageously, the resilient mounting system comprises one or more springs  50 . In the illustrated embodiment, if the rod  42  is in a horizontal plane, then a spring  50   a  aligned in the horizontal plane through piezoelectric axis  95  and perpendicular to (but offset from) the longitudinal axis  45  provides the resilient mounting system. Advantageously, there are two springs  50   a  extending on opposing sides of the resonator  24  to provide a symmetric resilient mounting, although only one spring  50   a  could be used. The springs  50   a  are shown connected to the resonator  24  by interposing distal ends of the springs  50   a  between the piezoelectric element  22  and the opening  28  in resonator  24 . Instead of separate springs  50   a,  a single leaf spring element with its middle abutting piezoelectric element  22  could be used.  
      Alternatively to spring  50   a,  or in addition to spring  50   a,  a spring  50   b  connects to the resonator  24  adjacent the end  35  in an axis orthogonal to the horizontal plane. Depending on the relative stiffness of the springs  50   a,    50   b,  and the relative location of those springs, various motions of the driven element  42  can be achieved. Preferably, the motion is a combined rotation about, and translation along axis  45 , but a pure rotation or a pure translation of the driven object  42  could also be achieved.  
      Suspension: A resilient mounting system  50 , called the suspension, is connected to the vibration element  26  to ensure that the selected contact portion  44  is consistently urged against the engaging portion of the driven element  42  so that the elliptic motion  100  of the contact portion  44  can transport the driven element  42 . Similar principles apply if the driven element  42  is resiliently suspended and urged against the contacting portion  44  instead. This consistent resilient force is preferably maintained even if the driven element has a varying surface smoothness or configuration and if the contact portion  44  shows signs of wear. For a small resilient force, these motors have been shown to transport the driven object  42  quickly, but provide small force. For a larger resilient force, the transportation speed decreases, but the transportation force increases. If the resilient force is selected too large, the driven object typically stops.  
      Depending on the location of the selected contacting portion  44  and the configuration in which one or more vibratory elements  26  are arranged (e.g.,  FIGS. 23-42 , different suspension systems will be needed. A variety of suspension systems are illustrated in  FIGS. 1, 2 ,  5  and  17 - 22 , and portions of the suspension system are discussed in the section on Mounting Of Vibratory Elements &amp; Driven Elements. The suspension system described here is primarily a spring-based suspension system, but need not be so limited. The suspension could include leaf springs, coil springs and other types of springs; it could include resilient materials such as elastomers or compressed gas springs, to name a few. The effect of the suspension system on the vibration modes of the vibratory element  26  will vary with the specific type of suspension system used and its arrangement.  
      For example,  FIG. 74  shows a suspension system using a curved, flat spring  188  having a first end  188   a  connected to base  52  and an opposing end  188   b  connected to the vibratory element  26 . In the depicted embodiment, the spring  188  is interposed between one end of the piezoelectric element  22  and the adjacent wall, which defines an opening  28 . The vibratory element  26  is inclined at an angle α relative to the engaging surface of the driven object  42 . The curved spring  188  offers the possibility of providing a smaller motor assembly  20  because the curved spring can reduce the needed space for the suspension. The wheel  46  could contacts the driven element  42  using a flat edge of the wheel concentric with the rotational axis  65 , as illustrated in  FIG. 74 . The wheels  46  could also have contoured peripheries configured to engage mating shapes on adjacent portions of the driven element  42  in order to appropriately support and guide the driven element  42 . Given the present disclosure, a variety of movable support configurations will be apparent to those skilled in the art.  
      Another example is shown in  FIG. 1  where the vibrating element  26  is mounted to and moves about the location where end  50   a  is mounted to base  52 . The selected contact portion  44  is located relative to the mounting of spring end  50   a  to the base  52  so that a generally vertical axis passes through both the mounting point  50   a  and the contacting portion  44 .  
      In contrast, the C-clamp configuration of  FIG. 5  has the vibrating element  76  rotating about pin  78 . A vertical axes passing through the contacting portion  44  is offset from a vertical axis passing through the pivot pin  78 . The offset, combined with an asymmetric location of the piezoelectric element  22 , results in a different suspension system that can have different characteristics.  
      Portions of the resilient suspension system typically participate in the vibrations of the vibration element  26  and therefore affect the vibration modes. The design of the suspension system is advantageously such that it enhances the desired motion of the selected contact portion  44 .  
      If a resilient suspension system, such as spring  50  is connected at a node of vibration at an operational frequency of the vibrator element  26 , then it does not participate in the vibration. But if the resilient suspension system is connected at a location other than a node of vibration at selected operating frequencies, then it creates an asymmetry that can couple various otherwise independent modes of vibration of vibration element  26  together. This can result in elliptical motion  100  at the selected contacting portion  44  that is especially useful if the engaging surface of the driven object  42  is inclined with respect to the vibration element  42 .  
      For example, in the embodiment of  FIG. 5 , the vibration element  76  oscillates about the pin  78 , which can cause the contact portion  44  to have an up-and-down motion along its elliptical path  100 . The mounting of the vibrating element  46 ,  76  can result in a variety of vibration modes of the motor assembly  20  and various movement of the contacting portion  44 .  
      Moreover, referring to  FIG. 77  the further embodiment is shown that is suitable for use in a torsional motion or rotational motion of the driven element. In this embodiment, the driven element  42  rotates about its longitudinal axis  45 . The longitudinal axis  95  of the piezoelectric element  22  is not aligned with the longitudinal axis  25  of the resonator  24 . The axes are inclined relative to each other, with the axis of piezoelectric element  22  generally parallel with the axis  45  of the driven element  42 . The axis  25  of the resonator  24  is inclined to bridge the gap between the two axes  95 ,  45 . The selected contacting surface  44  comprises a curved surface conforming the shape of the abutting contact area on the rod-like driven element  42 . The resilient mounting system accommodates motion of the selected contact surface  44  that rotates the rod  42  about its longitudinal axis  45 .  
      Advantageously, the resilient mounting system comprises one or more springs  50 . In the illustrated embodiment, if the rod  42  is in a horizontal plane, then a spring  50   a  aligned in the horizontal plane through piezoelectric axis  22  and perpendicular to (but offset from) the longitudinal axis  45  allows the rotational motion of rod  42 . Advantageously, there are two springs  50   a  extending on opposing sides of the resonator  24  to provide a symmetric resilient mounting, although only one spring  50   a  could be used. The springs  50   a  are shown connected to the resonator  24  by interposing distal ends of the springs  50   a  between the piezoelectric element  22  and the opening  28  in resonator  24 . Instead of separate springs  50   a,  a single leaf spring element with its middle abutting piezoelectric element  22  could be used.  
      Advantageously, but optionally, a spring  50   b  connects to the resonator  24  adjacent the end  35  in an axis orthogonal to the horizontal plane. Depending on the relative stiffness of the springs  50   a,    50   b,  and the relative location of those springs, various motions of the driven element  42  can be achieved. Preferably, the motion is predominantly or purely rotation about longitudinal axis  45 , although a combined rotation about, and translation along axis  45  could also be achieved.  
       FIG. 77  also illustrates that the vibration element  26  and resonator  24  can be non-symmetric. It also shows that the spring  50  can have various locations, configurations and orientations. Indeed, the spring  50  can be a bending or torsion spring, each of which can affect the suspension and resonant vibration modes of the system or of the vibratory element  26 .  FIG. 77  also shows that the spring  50  need not be connected to the piezoelectric element  22 . Moreover, the axis  25  of the predominant vibrating portion of resonator  24  need not be parallel to the axis  95  through the piezoelectric. Further, the contacting portion  44  can be molded to conform to the abutting surface of the driven element. The molding can be preformed in the resonator  24 , cut or otherwise formed into the resonator  24 , or it can be formed by wear and run-in.  
      A Mode of Operation to Reduce Friction: It is an additional feature of the motors of this disclosure that when excited at certain frequencies, which are not the operational frequencies, they produce a varying contact force at the contact portion  44 , and possibly liftoff, which can reduce the effective frictional holding force on the driven object  44 . In other words, it is easier to pull the driven object through the motor when operated at those frequencies, then when the motor is turned off. This property of selectively reduced friction can be beneficial in certain applications.  
      Theoretical Design Aspects  
      The piezoelectric  22  and resonator  26  are configured to achieve a desired motion of the selected contact portion  44  that moves the driven element  42 . The contact portion  44  preferably moves in an elliptical path  100  as shown in  FIG. 1 . Changes in phase and amplitudes of two rectangular components of motion of the resonator  26  and their superposition to achieve that elliptical motion are described here (similar results can be derived for oblique angles). By modifying the phase and amplitudes several properties of the ellipse useful to the present application in motor assembly  20  are better understood. These properties include the orientation and lengths of the short and long semi-axes of the ellipse that is the path preferably traveled by the selected contact portion  44 . Other relevant properties could also include the speed by which the ellipse is traversed, which correlates to the speed of the contact portion  44  and thus the speed with which the driven element  42  moves. The design may require the direction of the semi-axis of the ellipse to be aligned with certain dimensional tolerances within the piezoelectric motor assembly  20 . The design may also require that the lengths of the semi-axes of the ellipse  100  do not exceed certain predefined limits. Moreover, the ratio of the semi-axes of the ellipse  100  can be advantageously selected to provide greater motion, or faster movement, with the ratio of the axes advantageously being 5:1, preferably 10:1, and ideally from about 10-50:1.  
      Referring to  FIG. 43 , the ellipse  100  represents the potential motion of contact portion  44  of the vibrating element  26  as shown in  FIGS. 1 and 5 , among others. The ellipse  100  is generated by two components of motion, the first acting in the E x -direction (which corresponds to motion along the longitudinal axis  45  of the driven element  42  in  FIG. 1 ). The second component of motion acts in the E y -direction, which is perpendicular to the E x -direction. The two components of motion E x , E y  are generated at the selected contact portion  44  of the motor assembly  20 . The mechanism used to generate the components of motion do not affect the following disclosure. Localized major and minor axes e x , e y , respectively, of the ellipse  100  are also shown.  
      For illustration, the first and second components of motion E x , E y  are assumed to be sinusoidal with amplitudes A and B, respectively, and to have a phase difference of φ=π/2+Δφ[rad]. But other waveforms could be used. The position vector r of the selected contact portion  44  located at the edge of the resonator  24  as depicted in  FIG. 1 , as a function of time, is: 
 
 r=A  cos(ω t +φ) E   x   +B  sin(ω t ) E   y . 
 
      In this equation, ω is the frequency of the oscillation.  FIG. 44  shows an example of the partial components of motion for A=1, B=0.5, ω=1 and φ=π/6 [rad]. The ellipse  100  of  FIG. 43  is traversed counterclockwise for |Δφ|&lt;90°, and clockwise for 90°&lt;|Δφ|&lt;270°.  
      The lengths  2   a  and  2   b  of the long and short semi-axes are then computed from, respectively: 
 
2 a   2   =A   2   +B   2   +{square root}{square root over (A     4     +B     4     −2A     2     B     2      cos(2Δφ))},  
 
2 b   2   =A   2   +B   2   −{square root}{square root over (A     4     +B     4     −2A     2     B     2      cos(2Δφ))}.  
 
       FIG. 45  depicts how b/B depends on A(p and the ratio B/A.  FIG. 46  depicts the dependence of a/A. It is important to notice that the dependence of b/B does not change substantially for ratios of B/A&lt;=0.3. A good approximation of this dependence for |Δφ|&lt;500 and B/A&lt;=0.3 is given by the function  
         b   B     =     1   -           (     Δφ   ⁡     [   rad   ]       )     2     2     .             
      The orientation angle α ( FIG. 43 ) cannot exceed the value atan(B/A) (see  FIG. 47 ). As design rule, one has, for B/A&lt;0.5  
         atan   ⁡     (     B   A     )       ≈       B   A     .         
 
      The angle α can for sufficiently small ratios B/A be approximated by (see  FIG. 48 ):  
         tan   ⁡     (   α   )       =       -     B   A       ⁢       sin   ⁡     (   Δφ   )       .           
 
      The following example illustrates the usage of the previous material. Assuming that B/A=0.3. From  FIG. 47  we find that atan(B/A)≅15°. It follows then from  FIG. 48  that for Δφ=45°, α≅0.8*15°=12°.  FIGS. 45-46  indicate that b/B≅0.7 and a/A≅1.025.  
      This information illustrates how to change A, B and Δφ together in a way that preserves or achieves various properties of the ellipse  100 . In the previous example, the changes can be made such that the angle of inclination α ( FIG. 1 ) between the longitudinal axis  25  of the vibratory element  26  remains close to 12 degrees in order to achieve a large translation of the driven element  42 . The changes may also be made to ensure that 2b, the length of the minor axis of the ellipse  100  ( FIGS. 1, 43 ) remains larger than a given value in order to ensure the vibratory element  26  causes the selected contact portion  44  to disengage from the driven element  42  sufficiently to not only avoid undesired movement of the driven element  42 , but to avoid unacceptable wear of the driven element  42 . Over a relatively wide parameter range a desired ellipse  100  can be achieved that is particularly useful for moving a driven element  42  in the present invention. In the example above, the driven element  42  would be preferably oriented in an angle of 12 degrees to the E x -direction. But it should apparent to one skilled in the art that the optimal angle is in general not restricted to this value.  
      Referring to  FIGS. 1, 43  and  49 - 51 , it is also advantageous to consider the influences of a coordinate transformation from the coordinate system having an axis aligned with the longitudinal axis  25  of the vibratory element  26 , to the coordinate system corresponding to the elliptical motion of the selected driving portion  44 . This can illustrate useful affects on the frequency response curves and therefore on the performance and design of the motor assembly  20 .  FIG. 43  illustrates the motor coordinate system defined by axes E x  and E y , where the E x  axis corresponds to the longitudinal axis  45  of the driven element  42  ( FIG. 1 ). The ellipse  100  is believed to be generated by a first and a second motion component of the selected driving portion  44  on the vibratory element  26  of  FIG. 1 . The localized axes of the ellipse  100  are represented by axes e x  and e y .  
      For example, we assume the first component of motion lies in the E x -direction and has a transfer function that in the vicinity of a selected frequency can be approximated by a constant amplification factor g 1 (s)=A. The second component of motion lies in the E y -direction and has a transfer function that in the vicinity of a selected frequency can be approximated by a second order resonator given by its Laplace transform:  
           g   2     ⁡     (   s   )       =       k       s   2     +     2   ⁢     ⅇω   o     ⁢   s     +     ω   0   2         .         
 
      Here coo is the (undamped) resonance frequency, and e is a dimensionless damping parameter arising inherently from damping in the mechanical system, i.e., the motor assembly  20  in this case.  
      The superposition of g 1 (s) and g 2 (s) yields transfer functions G 1 (s) and G 2 (s) in the e x  and e y  directions, respectively. For illustration, examples are given in which A=1 and ω 0 =1.  FIGS. 49-51  depict G 1 (s) and G 2 (s) for k=0.01 and α=25 degrees. The parameter e increases from  FIG. 49  to  FIG. 50  to  FIG. 51 . The combination of these two signals results in a behavior where the phase difference Δφ between G 1 (s) and G 2 (s) undergoes an intermittent change that becomes more rounded as the damping in the system increases. This effect results in an expanded frequency range where the relative phase difference lies between 0 and 180 degrees, which makes it easier for a resonant frequency to be found that results in a useful, elliptically shaped motion. This frequency range is considerably wider than what would be achieve with the transfer function of a simple second order oscillator. Such a particularly widespread phase range can be used in conjunction with other design aspects to help select the shape and orientation of the resulting ellipse  100  as the selected driving frequency is changed.  
      The influence of the above coordinate transformation becomes more involved as G 1 (s) and G 2 (s) are replaced by higher order, and more realistic, transfer functions as they arise from the piezoelectric motor assembly  20 . Such transfer functions can create relative phase shifts Δφ between G 1 (s) and G 2 (s) that fluctuate between 0 and 180 degrees in even wider frequency ranges, thus rendering the motor assembly  20  even less dependent on production tolerances, material properties, temperature variations, and other manufacturing factors and criteria.  
      This phase shift between the longitudinal and lateral motion is used to achieve the desired elliptical motion. Phase shifts of between 3 and 177 degrees are believed well suitable to achieve useful motion at the selected contacting portion  44 . A 90 degree phase shift results in a circular motion if amplitudes are equal. Preferably, but optionally, the phase shift results in non-circular motion of the selected contacting portion  44  in order to obtain greater movement along the major axis of the elliptical motion.  
      The portion of the ellipse  100  below the E x  axis can be thought of as reflecting the engagement of the driving portion  44  with the driven element  42 . By altering the shape of the ellipse  100  (i.e., 2a, 2b measured along e x  and e y ) the duration of the engagement can be varied and to some extent the pressure of that engagement can be varied. Further, by altering the orientation of the ellipse  100  (i.e., the angle of inclination α between the axis  45  of the driven element and the major axis of the ellipse) the duration of the engagement can be varied. As the angle of inclination α a comes closer to aligning the e x  axis with the E x  axis, the duration of the contact between the driving portion  44  and driven element  42  increases.  
      For practical reasons, the longitudinal axis of the driven element  42  may often be placed between the two axes E x  and e x . But the more important aspect is that these equations show that as the excitation frequency of the piezoelectric  22  changes, the amplitude and phase of the selected driving portion  44  (i.e., ellipse  100 ) change. This shows the ability to alter the amplitude and orientation of the ellipse  100  and thus alter the characteristics of the motion driving the driven element  42 . Moreover, the equations reflect an ability to offer these variations over a wide range of amplitudes and frequencies which offers a flexibility in functional design characteristics of the piezoelectric  22  not previously available. Further, the equations reflect the ability to vary the engagement criteria to a sufficient extent that the manufacturing tolerances can be less, and potentially significantly less than with many of the existing motors using piezoelectric drives.  
      Historically, these various manufacturing criteria have been so precise that they result in costly manufacturing of piezoelectric vibratory elements  26 , and the motors have narrow operating ranges and criteria. Thus, the ability to use more liberal criteria offers the possibility of significant cost savings in producing the motors while offering wider operating parameters.  
      The direction of the motion of the driven element  42  depends on the relative orientation of the driven element  42  and the direction of the selection contacting portion  44  as it moves around its elliptical path of travel  100 . Different points of the vibration element  26  can show different vibration shapes. Typically areas with clockwise and counterclockwise motion around elliptical paths  100 , alternate along the length of the vibration element  26 . The driving direction of a rod shaped vibration element  26  can typically be reversed by turning the vibration element by 180 degrees about longitudinal axis  25 .  
      The shape of the motion of the contact point  44  is important to this invention. This shape must achieve more driving force in one direction than in the other. This is typically achieved by increasing the contact pressure while the selected contact portion  44  moves in the direction the driven element  42  gets moved. When the contact portion  44  moves in the opposite direction the contact pressure is reduced or the contacting portion  44  even looses contact with the driven element  42 . One important aspect is how to generate the appropriate motion.  
      Because of mechanical noise and unwanted vibrations, the shape of the ellipse does not always follow the ideal theoretical path. This may result in the selected contacting portion  44  sometimes performing motions that are undesired, such as figure-eight shaped motion. But these motions may nonetheless regularly appear with the vibrator element  26 . They are, however, not used to drive the driven elements  42 . This is clarified in the discussion of the three-dimensional vibration shapes of the contacting portion  44 .  
      In the description only the two-dimensional shape of the vibration will be addressed. In actuality the contacting portion  44  will have some slight motion in the third dimension, the direction perpendicular to both directions of the driving force along axis  25  and the direction of the contact force between vibration element  26  and driven element  42  which is generally along axis  45 . These vibrations might also contain higher frequency components. As a result the motion of the contact portion  44  could look like a figure-eight motion if projected into certain planes. Although this figure-eight motion can be observed, it is not relevant for the operation of the vibratory element  26  driving the driven element  42  and is merely a side effect of unused motion.  
      Ideally, the major axis of the elliptical motion  100  is perfectly aligned with the direction in which the driven element  42  moves in order to optimize performance. Perfect alignment is difficult to achieve for many reasons, including manufacturing tolerances and performance variations. Further, even the elliptical path  100  is not perfectly elliptical and may vary over time. Variations in voltage, current, power disruptions or fluctuations, degradation over time, electrical noise, mechanical noise, electromagnetic interference, to name a few, can affect the shape and smoothness of the elliptical paths  100 . Thus, it is desirable to be able to configure a system that can accommodate a practical range of variations in order to reduce manufacturing costs and assembly costs, and to produce a system that can accommodate environmental variations and other variations that arise during use of the system. Because of such variations, an alignment of about 0-5 degrees will be considered to be aligned, in part because in most instances this variation from perfect alignment does not substantially affect the performance of the systems disclosed herein.  
      The vibrator element  26  does not rely on traveling waves for the movement of the selected contacting portion  44 . But any mechanical wave existing in the material also travels through it. In the present invention such waves get reflected at some part of the vibration element  26  causing another traveling wave that superimposes with the first one. This results in a standing wave, and in some instances this standing wave can be used in connection with a selected contacting portion  44 . Several prior art motors require a wave that is not standing, but rather traveling—with the driven object moving with or being moved by the traveling wave. The traveling wave is different from the standing wave.  
      Practical Design Aspects:  
      The contacting portion  44  is the point of the vibrating element  26  that comes in contact with the driven object  42  in order to move the driven object. That contacting portion is typically a portion of the resonator  26 , and is preferably on the distal end  36  of the resonator. The power of the motor assembly  20  to move heavier driven elements  42  and the efficiency of the motor assembly  20  are functions of the periodic motion of the contacting portion  44  and of the force between the contacting portion  44  and the driven element  42 .  
      The spatial motion of the selected contacting portion  44  is the result of the superposition of several vibration modes of the motor. These modes are all excited, to varying amplitudes and relative phases, at the same frequency generated by the piezoelectric element  22 . Their contributions to the desired motion of contacting portion  44  and forces applied by contacting portion  44  are a function of the relative magnitudes and the relative phase angles of each of these several vibration modes. These vibration modes in turn are functions of the motor geometry, constitutive relations, and the material properties.  
      In order to increase the performance of the motor assembly  20 , the following guidelines may be used. Preferably all of the following guidelines are simultaneously satisfied at the selected contact portion  44  in order to optimize the performance of the motor assembly  20 , but compromises of one or more of these guidelines can occur if the resulting motor performs satisfactorily.  
      The motion of the selected contacting portion  44  is elliptical with major and minor axes of lengths a and b, respectively. As used here, and unless specified otherwise, the reference to elliptical motion or to an ellipse includes ellipses with the major and minor axes are equal, which forms a circle. The reference to elliptical motion or to an ellipse also includes ellipses in which either of the major or minor axes are small relative to the other axis, which results in a very elongated ellipse approaching a straight line.  
      The major axis of the ellipse is preferably aligned with the driving direction of the driven element  42 . The length of the major and minor axes, a and b, are both large enough to achieve their desired uses, and preferably large enough to provide optimum performance for the selected application. The generally preferred elliptical shape has an elongated major axis “a” relative to the minor axis “b” in order to increase speed, and has a minor axis “a” sufficient to disengage the contacting portion  44  from the driven member  42  during the return portion of the ellipse, as discussed next. As discussed above, ratios of about 3:1 up to 150:1 or even greater are believed usable, although the higher ratio&#39;s provide more linear motion and result in more impact motion with the driven element.  
      The force at the selected contact portion  44  normal to the contact surface on driven element  42  is large when the contacting portion  44  moves in the driving direction, and small (or zero), when the contact portion  44  moves against it. If the force is zero, the contacting portion  44  has lost contact with the driven object  42 . In that lost-contact case, the backward motion of the vibratory element  26  tip is very efficient, but the motor assembly  20  also loses traction during that period of time. This loss of traction should be considered when evaluating motor efficiency and strength. If the normal force is too large when the contact portion  44  moves against the driving direction, the driven element may not be properly transported in the driving direction, which results in a loss of performance.  
      Moreover, the normal contact force between the selected contacting portion  44  and the driven element  42  is a measure of the friction force between the contacting portion  44  and the driven object  42 . Larger normal forces provide the motor assembly  20  with stronger thrust. But the wear occurring over the repeated contact from the many thousands of cycles of elliptical travel must also be considered. Larger contact areas on the contacting portion  44  have the advantage of tolerating more defects in the surface of the driven element  42  that engages the contacting portion  44 .  
      In the embodiments thus far disclosed, the selected contacting portion  44  is often illustrated as being located on one edge of the distal end  36  of the vibration element  26 , in part because the desired elliptical motion can be readily achieved at that location. Moreover, the edge location provides a narrow area of contact and good frictional engagement. But it is not necessary that the selected contacting portion has to be located on an edge. Moreover, typically some material wear will wear out the edge and provide a flat or flattened contact surface  44  after some period of use. This wear typically does not affect the operation or use of the motor assembly  20 . As discussed elsewhere, the contact portion  44  can also be located at other places on the vibration element  26 . For example, the contact portion  44  could be located on the side of the vibration element  26  as in  FIG. 62 . The selected contacting portion  44  does not have to be a point contact. The particular applications will thus influence the size and location of the selected contacting portions  44 .  
      The displacement of the contacting portion  44  in the driving direction and the normal contact force are not in phase. These two quantities form an ellipse when plotted in a displacement/force diagram. The orientation of the major axis of this ellipse with respect to the displacement axis provides another design parameter. Depending on this orientation, the maximal contact force is generated earlier or later during the forward motion of the tip. In a certain sense this could be interpreted as somewhat analogous to a saw-tooth-like movement. Because useful motion can be achieved when one semi-elliptical axis of the elliptical path  100  is about 5, 10 or more times greater than each other axis, even relatively small motions can be of potential use for one of the semi-axes.  
      The motion of the selected contacting portion  44  is the result of the vibrations of the entire motor assembly  20  and its components. Large motions of the selected contacting portion  44  are achieved if the excitation frequency lies close to a resonance frequency of the system, and if the selected contacting portion  44  is located where a large amplitude occurs. In order for the motion of the selected contacting portion  44  to be multi-directionally large, the motor assembly  20  is advantageously designed to have several resonance vibrations clustered in a selected frequency range. For example, if the natural frequency of a bending mode is close to that of a longitudinal mode, and the excitation frequency lies in between the frequencies that excite these bending and longitudinal modes, then the resulting motion of the selected contacting portion  44  will have moderately large amplitudes. The elliptical nature of the motion of the selected contacting portion  44  is generated by the phase difference of the respective motions. The phase difference is generated in part by the damping in the system. Various combinations of these factors can be used to achieve the desired motion of contacting portion  44  and to achieve other criteria of the motor assembly  20 , such as power, reliability, wear, etc.  
      The absolute and relative locations of the resonance frequencies and vibration modes of the motor assembly  20  are affected by a multitude of parameters. The following factors can be used to configure an acceptable design of the motor assembly  20 .  
      Lower vibration modes are generally stronger than higher vibration modes because the lower vibration modes store relatively less elastic energy, leaving more energy for driving the object  42  through the selected contacting portion  44 .  
      The location of the longitudinal resonance of the vibrating element  26  in a frequency diagram is affected mainly by the length of the piezoelectric  22  and resonator  24  and by the material properties of the parts. The first longitudinal mode is by far the strongest and therefore the more desirable mode to use.  
      The location of the longitudinal resonance of the vibratory element  26  in a frequency diagram can further be affected by the motor suspension, i.e., by the spring steel support  50  ( FIG. 1 ) or other mechanisms that connect the vibratory element  26  to its housing. If a natural (resonance) frequency of the support such as spring  50  is brought close to the longitudinal resonance frequency of the vibratory element  26 , it has the effect of splitting the longitudinal frequency into two frequencies which are close to each other. The phases of the modes fluctuate strongly between 0 and 180 degrees in these resonance areas. Resonance splitting can therefore be used to spread the working region of a motor over a wider frequency range, making the motor therefore more robust.  
      Phase differences other than 0 and 180 degrees are induced by damping mechanisms. In order to expand this effect over wider frequency areas, additional damping elements such as damping layers can be added to the vibratory element  26 , or to various portions of the motor assembly  20 . Also, internal damping is affected by the material properties of the piezoelectric  22  and resonator  24  and by the way in which they are assembled. These factors in turn can be affected by the material&#39;s history, i.e., its manufacturing process.  
      Moreover, whether the damping is inherent in the system materials or added by design components, the damping can be used so that a primary resonance mode is used to excite a secondary vibrational mode that results in the desired elliptical motion of a selected contacting portion  44  along path  100 . Recall that the elliptical semi-axes can have amplitude ratios of 5, 10 or more, such that a vibration mode excited by damping need only have an amplitude of ⅕, {fraction (1/10)} or so of the amplitude resulting from the primary vibrational mode. Because damping can couple vibration modes, the damping can be used to achieve the desired elliptical motion of the selected contacting portion.  
      Bending resonance vibration modes are affected mainly by the length and cross-sectional areas and shapes of the piezoelectric  22  and resonator  24  and also by the material properties of those parts. Lower resonance vibration modes are stronger than higher ones. Guidelines for placing and splitting of resonance longitudinal vibration modes also apply to bending modes.  
      Shearing resonance vibration modes can contribute to the longitudinal motion of the selected contacting portion  44 , especially if the contacting portion  44  is located at a distal end  36  of the vibratory element  26  and on an edge of the distal end. The shape of the cross-sections of the resonator  24  affects these resonance vibration modes, as does the placement of the piezoelectric  22  relative to the resonator  24 . Further, as an example, see  FIG. 2 . If the longitudinal axis of the piezoelectric  22  is appropriately offset from the longitudinal axis of the resonator  24 , an edge of the distal end  36  can have a shearing resonance that causes opposing edges at distal end  36  to pivot about axis  40 . Removing material close to the centerline of the motor can have an especially strong effect on this resonance mode. One configuration with material removed along the centerline is shown in  FIG. 52 , and described later.  
      Torsion resonance vibration modes can be used to support selected, and preferably vertical motion of the selected contact portion  44  if the portion  44  is close to a side of the vibratory element  26 . The torsion resonant vibration modes are usually of smaller magnitude than other vibration modes, but they offer the possibility of using various portions along the length of the vibratory element  26  to drive various objects. Torsion resonant vibration modes could be used to rotate the driven element  42  in the embodiments of  FIGS. 23, 25 ,  27 ,  28 ,  29 ,  30 ,  32  and others. Torsion resonant vibration modes could be used to translate the driven element  42  in the embodiments of  FIGS. 38-40 .  
      Resonant vibration modes arising from cross-sectional contraction are of little benefit when the driven element is elongated, such as the rod-like driven element  42  depicted in  FIG. 1 . The cross-sectional contractions appear at frequencies that are too high to produce readily usable amplitudes. Cross-sectional contraction is governed by the Poisson-effect. This effect is strongest where the longitudinal strains in the piezoelectric element  22  or resonator  24  motors are the highest, i.e., where the stresses are highest. Cross-sectional contraction can therefore be large where the piezoelectric element  22  is connected to the resonator or whatever frame is holding the piezoelectric element and the portion of that connection in which the forces are high. This contraction can drive the bending vibrations of the thin sidewalls  29  ( FIG. 1 ) of the resonator  24 . If the bending resonant vibration modes of the sidewalls  29  are tuned to the longitudinal vibration mode of the vibratory element  26 , yet another splitting of natural vibration frequencies can occur with similar benefits as mentioned above.  
      The piezoelectric element  22  generates predominantly longitudinal forces in the resonator  24  within which it is mounted. Coupling of these longitudinal forces from the vibratory element  26  into directions other than along longitudinal axis  25  creates a number of other possible vibration modes within the vibration element  26 , such as bending, shear and torsion. The intensity of the coupling of the longitudinal motion with other vibratory motions within the vibratory element  26  can determine the relative amplitudes of the various modes of the vibratory element  26  and therefore their relative contributions to the motion of the selected contact portion  44 . Coupling can be generated by material properties, geometric imperfections and asymmetries within the components of the vibratory element  26 , primarily the piezoelectric  22  and the resonator  24 .  
      Some of these coupling effects are often poorly defined, difficult to analyze, and hard to measure or design. Well-defined mechanisms are therefore preferable. These mechanisms include mounting the piezoelectric element  22  off-center of the longitudinal axis  25 , or at an angle to the longitudinal axis  25  of the vibratory element  26 , or using flexible mountings for the vibratory element  26  such as a spring  50  or similar elements. In the case of a spring  50 , the longitudinal motion of the vibratory element  26  generates bending in the spring  50 . The end  50   b  of the spring that is clamped to the vibratory element  26  is forced to bend or possibly to twist. This bending or twisting causes bending moments to be generated in the vibrational element  26 . The configuration of the spring  40  could be used to vary the vibrational mode, as for example by introducing bends, edges and similar modifications into a flat metal spring. Furthermore, the spring  50  can be made more flexible at specified locations to better define an axis of rotation about the flexible portion, if that is useful to the design. Coupling of vibration modes within the vibratory element  26  can also be achieved if the piezoelectric element  26  is selected or configured or excited to perform other than pure longitudinal motions.  
      Several additional factors are preferably considered in configuring the vibratory element  26  and the motor  22 . These factors include: the orientation of ellipse  100  in which the selected contact portion  44  moves when it is not in contact with anything; the orientation of the force-displacement ellipse of the contact portion  44  when it is in contact with the driven element  42 ; and an estimate of mechanical power generated at the selected contact portion  44  when it is in contact with the driven element  42 .  
      Reversing Direction  
      If a principle of operation of the vibration element  26  is known to transport the driven object  42  in one direction at a first frequency, it is desirable to use the same principle of operation at a second frequency to transport the driven object in the opposite direction. Such a design is not only useful for vibration elements that operate using elliptical motion, but also for vibration elements that operate on other principles. The vibration modes of the vibration element  26  that produce the transporting motion in the contacting portion  44  at the first frequency are not necessarily the same as those that produce the transporting motion at the second frequency, nor are they necessarily of the same type.  
      It is an advantage of such a multi-directional design that—provided the vibration element  26  is appropriately designed—the same mechanical components that are necessary to achieve unidirectional movement can be used to achieve bi-directional movement at two distinct operational frequencies. In particular, a single vibration source  20 , e.g., a piezoelectric element, is sufficient.  
      The realization of a multi-directional design is simplified if the axis  25  of vibration element  26  is oblique to the direction of transport of driven element  42 . Also, in many cases the shape of the motion of the contacting portion  44  at either operational frequency may not be optimal to achieve maximal force or speed of transport, but only a compromise to achieve suitable bi-directional performance. Furthermore, the frequency range within which the vibration element transports in one direction is not necessarily as large as the range within which it transports in the other. Testing has shown that a frequency range of 5 kHz at a first frequency and at least 300 Hz at the second frequency are possible to move or transport a driven element in opposing directions.  
      Illustrative Designs  
      Various modifications on the design of the resonator  24  holding the piezoelectric element  22  are possible to enhance the performance of the vibratory element  22 . The following implementations are some possibilities. Combinations of these following embodiments, and of the prior embodiments, are possible. All combinations of methods for clamping the piezoelectric element  22  and of the various mounting methods are also believed possible.  
       FIGS. 52-55  show a vibratory element  26  having a resonator  24  with a slot  112  extending from adjacent the cavity  28  to adjacent the distal end  36 , and extending through the resonator, along the direction of longitudinal axis  25 . The slot  112  preferably has rounded ends and parallel sides. But the slot could have rectangular shaped ends. There are advantages to using longer, narrower  112  compared to wide slots as shown in  FIG. 54 . The narrower slots  112  result in beams  114  with larger dimensions, so that manufacturing tolerances have less effect on the resulting vibration. If the slots  112  are large, the walls  114  are usually smaller in dimension so that errors in manufacturing have a larger effect on the vibrational performance.  
      The slot  112  preferably opens onto the same surfaces of the resonator  24  as does the opening  28 . But this need not be so, as the slot could open onto other surfaces of the resonator  24  depending on the vibrational modes and configurations that are desired.  FIG. 55  shows the slot  112  opening onto a lateral surface turned 180 degrees from the orientation of the opening  28 . Various angular orientations are possible, especially if the resonator  24  has a cylindrical body shape. The slot  112  creates a resonator with two beam segments  114   a,    114   b,  on opposing sides of the slot, each of which forms a portion of resonator  24 .  
      In  FIGS. 52-54 , the slot  112  is illustrated as fairly symmetrically located in order to produce side-beams  114  of approximately equal dimension with close vibrational modes and frequencies. But the slot  112  need not be symmetrically located as reflected in  FIG. 55 , and can be located to produce beams  114   a,    114   b  of very different dimension and with different resonance frequencies. Moreover, more than one slot  112  can be used.  
      The slot  112  in the resonator  24  can thus create an increased number of beams  114  in the resonator, with each beam vibrating at its own eigenfrequencies and selected for that very reason. The increased eigenfrequencies leads to an increased number of phase shifts of the vibrations in the resonator  24 . By having two almost identical beams  114   a,    114   b  with eigenfrequencies very close together, it is also possible to get a wider frequency range with high amplitudes.  
      The slot  112  also changes the mass distribution of the resonator, the bending of the resonator, and the shear stiffness of the resonator  24 . Each of these changes has an influence on the resonant frequencies and resonant vibration modes of the resonator  24  and of the vibratory element  26 . This gives a flexibility of design that allows a broader range of frequencies to excite the requisite vibration modes of vibratory element  26  while allowing lower manufacturing tolerances.  
      In  FIG. 53 , the opening  28  for the piezoelectric element  22  has rounded ends rather than flat ends over the portion that abuts the piezoelectric element  22 . The contact area between piezoelectric element  22  and the end of the opening  28  comprises two lines when the piezoelectric element has a square or rectangular cross-sectional area. This can provide a more defined contact. If the opening  28  is formed by a wall abutting the piezoelectric element  22 , the wall is typically not perfectly flat and not perfectly orthogonal to the longitudinal axis  25 . Moreover, the end of the piezoelectric element  22  is not perfectly flat and not perfectly orthogonal to the centerline (e.g., longitudinal axis  25 ). Thus, when the end of the piezoelectric element  22  abuts the walls (e.g., end walls  31 ) defining the opening  28 , it is possible that the piezoelectric will not be compressed along its centerline, with the result that the piezoelectric will be compressed along an offset axis or a skewed axis. The offset axis or skew axis can result in a variation of vibrational modes. Alternative ways of resolving this contact location are discussed relative to  FIGS. 9-16 .  
       FIG. 56  shows an embodiment with two slots, on each side of the opening  28 , along the piezoelectric element  122 . The slots  112  open into the opening  28  to form an “H” shaped configuration with the piezoelectric element  122  mounted at the center of the “H”. This configuration makes it easier to press-fit the piezoelectric element  122  into the resonator  24  since the sidewalls  29  can take more deformation before necking begins.  
       FIG. 57  shows an embodiment in which the opening  28  is formed in one leg  114  defined by centrally located slot  112 , resulting in the leg  1   14   a  being divided for a portion of its length into further legs  114   c.  Configurations such as this can have a high shear contribution to the motion at the selected contacting portion  44 , which is illustrated as being aligned along the axis of leg  114   a.  A different selected contacting portion  44   b  on leg  114   b  could be used to drive a different element at a frequency other than that used to activate the driving mode of leg  114   a.  A third potential contacting portion  44   c  on the leg  114  could represent yet another frequency to yet another driven element when activated. This is another illustration that the selected contacting portion  44  need not always be at the same location on the vibratory element  26 , as it will depend on a variety of factors, including the number, configuration and arrangement of the vibratory element(s)  26  and the configuration of the driven element or elements motor assembly  20 .  
       FIG. 58  shows an embodiment having a hole  116  in the resonator  24 . The hole is shown extending along the longitudinal axis  25  of the vibratory element  26 , but it could be located off-axis, or skewed relative to that axis  25 . The hole  116  is shown as opening onto the distal end  36 , but it could be formed on any of the surfaces of the resonator  26 . The hole  116  is preferably cylindrical and results from drilling of the hole as close tolerances can be maintained at low cost with such holes. But other shapes could be used, as a drilled hole can be broached to achieve various cross-sectional shapes. The diameter of the hole  116  can vary depending on the desired effect, as the hole changes the mass distribution by removing material, and it changes the stiffness of the material remaining after the hole is formed.  
       FIG. 59  shows an embodiment with a larger mass behind the piezoelectric element  122 , located between the piezoelectric element  122  and the proximal end  35  of the resonator  24  that is opposite the distal end  36 . This extra mass enhances the vibration of the distal end  36  of the vibratory element  26  and is useful when the selected contacting portion  44  is on the distal end  36 .  
       FIG. 60  illustrates an embodiment with multiple sidewalls  29 . It is possible to not only have solid sidewalls  29  next to the piezoelectric element  22 , but it is also have a more complex sidewall configurations.  
       FIG. 61  shows a further embodiment in which the piezoelectric is substantially enclosed and surrounded by the frame. This configuration is akin to inserting batteries into a flashlight. The opening  28  comprises a close-ended hole, with the end  120  of the hole having either a conical shape or a flat shape depending on the drill used to create the hole. A cap  122  threadingly engages corresponding threads on the end of the hole  28  to compress the piezoelectric element  22  placed in the hole. The cap  122  is shown as having a curved end  124  to abut the cap  34  on the abutting end of piezoelectric element  22  and create a point contact. Preferably, one or more small holes  126  are formed in the sidewalls  29  defining the opening  28  so that the electrical wires  30  can be connected to the piezoelectric element  22 . But other ways of providing electrical connections can be devised. The end  120  against which the piezoelectric element  22  abuts forms an area contact if the bottom  120  is flat; it forms a four point contact if the cross-section of the piezoelectric or any protective cap  34  (not show) is square; and it forms a line contact if the cross sectional area of the piezoelectric or any protective cap  34  (not shown) is round.  
      Preferably the resonator  24  is machined or cast of non-ferrous metal, preferably aluminum. The resonator could be sintered of appropriate materials. Moreover, it is believed possible that the resonator could comprise two separate sections joined by an appropriate adhesive to opposing sides of the piezoelectric element  22 . Further, the resonator  24  could be formed of a suitable ceramic material. If formed of a ceramic material that is sintered, the resonator could be sintered directly to the piezoelectric during the sintering of the resonator  24 .  
      Suspension Of The Driven Element: The driven element  42  is preferably suspended so that it can move relative to the vibratory element  26  and support or move a desired load. Usually the load is moved by pressing a portion of the driven element  42  against the load, as for example a fiberglass rod connected to a CD tray that is moved reciprocally in and out of a housing by a linear motor assembly  20 . But in some situations the driven element  42  itself may be the desired load. The driven element  42  can be suspended on bearings. Less expensive methods are to suspend the driven element on small wheels, or to use bushings as linear bearings. The bushings are believed to work well with rod-like driven elements  42 . A low friction and stiction coefficient between the bushings and a glass or fiberglass rod reduces the performance loss of the motor assembly  20  due to friction. Self-lubricating bearings are desirable to further reduce friction losses. Other methods are possible. Other driven objects like a wheel or a ball also easily be suspended on an axle.  
      When the driven element  42  comprises a rod, it can also be suspended on at least four balls such that the rod can move linearly. The stiction of such a mounting using four Delrin balls is believed to be less than with four ball bearings. The balls preferably need to run in grooves in order to transfer radial loads applied to the balls by the rod. Thus, the rod could be grooved to provide a driven element  42  with longitudinal grooves in it when the configuration of the motor assembly  20  is arranged to translate the rod. The orientation of the grooves would change depending on the desired movement of the rod or driven element  42 . Further, the length of the grooves could limit the motion of the rod.  
      A plate driven by the vibratory element  26  could also be suspended on at least three balls. This would give the motion of the plate three degrees of freedom. Other methods are possible.  
      The driven element  42  could be suspended in a manner that resiliently urges it against the selected contacting portion. 44 , using principles discussed above for mounting the vibratory element  26 . One resilient support is discussed regarding  FIG. 6  above.  
      Electronics  
      A number of different electronic circuits can be used to drive the piezoelectric element  22  of vibratory element  26  since the motor  26  is functional with a variety of different signal shapes applied to the piezoelectric element  22 , as long as the power spectrum of the input signal provides a substantial amount of vibratory energy at the desired driving frequency sufficient to achieve the desired motion of the selected contacting portion  44 . This ability is an advantage over those prior art motors that require specialized, more expensive electronics to generate special waveforms, such as, for example, saw-tooth waveforms. Some specific examples of driver circuits are shown in FIGS.  63  to  66 .  
       FIG. 63  shows an example of a driver circuit, preferably using a halfbridge  152  such as a NDS8858H halfbridge available from Fairchild Semiconductor. A discrete halfbridge is also believed suitable, but is not as preferred for size reasons. A rectangular input timer-signal  150  of specified frequency can be used to repeatedly switch between the inputs of the integrated halfbridge  152 . This process generates an oscillatory waveform in the capacitor  154 , which represents the piezoelectric element  22 . It is however not necessary for the signal  150  to be rectangular as long as it reaches the necessary thresholds that can switch the halfbridge  152 . The signal  150  can thus be generated by a microcontroller, or by other suitable signal generators such as a LM555 timer circuit available from National Semiconductor. The input timer-signal  150  is used to switch between the inputs of the halfbridge  153 . The period during which one of the said inputs is connected to the output of the halfbridge is determined by the input signal  150  and can be appropriately chosen. Typically, the cycle during which the supply voltage (VCC) is connected through to the output of the halfbridge accounts for about 50% or less of the time in order to achieve the best energy efficiency in the circuit and in the piezoelectric element  22 . If the signal  150  is high, the n-channel transistor  153   a  in the halfbridge  152  becomes conductive and discharges the capacitor  154 . After this discharge, it is preferable for the signal  150  to change to a lower level so that the p-channel transistor  153   b  in the halfbridge becomes conductive instead and charges the capacitor  154 . This process can be repeated indefinitely and, since the capacitor  154  represents the piezoelectric element  22 , it results in a vibratory motion of the piezoelectric element  22  and therefore of the vibratory element  26  ( FIG. 1 ).  
      As an alternative, one of the transistors  153   a,    153   b  in the driver circuit can be replaced with a component  156 , e.g. a passive component like a resistor, or an active component such as a constant current diode. Such an alternative embodiment is shown in  FIG. 64 , where the transistor  153   b  has been replaced with a component  156  such as a resistor.  
      In accordance with specific embodiments, the driver circuits of  FIGS. 63, 64  have the advantage that they can be implemented within an integrated circuit, e.g. as part of a microcontroller.  
       FIG. 65  shows an alternative driver circuit for the piezoelectric element  22  that uses a switched resonance circuit having a capacitor  154  (piezoelectric element  22 ), an electromagnetic storage device, such as inductive coil  158 , and optional resistor  156  connected in parallel. One advantage of using a resonance circuit to drive the piezoelectric element  22  is the ability to lower the supply voltage (VCC) to battery level (e.g. 3V) while maintaining the higher voltages necessary to operate the piezoelectric element  22 . Moreover, the entire circuit consists of only three electronic parts besides the capacitor  154 , which represents the piezoelectric element  22 .  
      In  FIG. 65 , an input signal  150  (like the one previously described in the halfbridge driver circuit of  FIG. 63 ) is used to switch a control element  153 , such as transistor, on and off in a well-determined fashion. Typically the cycle during which the transistor  153  is conductive is chosen to be about 50% or less in order to achieve the best energy efficiency in the piezoelectric element  22 . When the input signal  150  is high, the transistor  153  becomes conductive and reverses the charge of the capacitor  154  while increasing the current through the coil  158 . The current in the coil  158  reaches its maximum when the capacitor  154  is fully charged. At that point in time, the coil  158  stores a maximal amount of energy in its electromagnetic field, and it is preferable if the input signal  150  is set to low so that the transistor  153  is no longer conductive. The energy stored in the coil  158  sustains the flow of current, which in turn reverses the charge of the capacitor  154  resulting in an increased voltage across the capacitor  154  and therefore the piezoelectric element  22 .  
      When the capacitor  154  has fully reversed its charge and if the circuit adjustments are correct, the energy in the coil  158  increases the voltage across the capacitor  154  beyond the supply voltage (VCC). When the coil  158  has relinquished its energy, the voltage across the capacitor  154  reaches a maximum, and the capacitor  154  now stores the entire electric energy of the system. Next, the current flow through the coil  158  reverses which in turn causes another reversal of the capacitor charge. At this point or shortly thereafter, it is preferable if the input signal  150  is switched to high again so that the cycle can be repeated.  
      The resistor  156  is not necessary for the operation of the circuit of  FIG. 65 , but it provides a method to shape the waveform at the capacitor  154  and to cut off possible voltage peaks that can originate from the fast switching of the current through the transistor  153 , hence reducing potential electromagnetic interference as well as leakage of vibratory energy into undesired frequency spectra. Alternatively, the resistor can also be put in series with the inductor  158 . As a further alternative, it can be beneficial to place the inductive coil  158  in series with the capacitor  154  to form another type of electric resonance circuit. If the resonance frequency of this circuit is chosen sufficiently close to an operation frequency of the motor, higher voltages at the piezoelectric element  22  can be achieved while maintaining relatively low electric power consumption. As mentioned earlier, the inductor  158  can advantageously also be a wire coil made from the same wire that connects the capacitor  154 .  
      Further, referring to  FIGS. 78-80 , it is possible to integrate the coil  158 , and even the resistor  156 , directly into the vibratory element  26  by, for example, wrapping an insulated wire around the vibratory element  26  to form an inductive coil as shown in  FIG. 78 . In such an embodiment, the two ends of the wire coil  158  can concurrently be used as electrical leads to the piezoelectric element, as shown in  FIG. 79 . The wire coil  158  can be wound around the resonator  24  as in  FIG. 30 , or separate as in  FIG. 80 . These configuration place inductive coil  158  in parallel to the capacitor  154  and save additional wiring, although the coil  185  could be placed in series, with or without the damping resistors, half-bridge or single transistors.  
      Moreover, the inductive coil  158  can be mounted close to the piezoelectric element  22  with which it can form the electric resonator circuit. The physical proximity of the piezoelectric element  22  and coil  158  can reduce the inherent electrical resistance in the electrical connections of those parts and can make the circuit more effective, especially since most of the current used to drive the motor oscillates in this electric resonator. As a result, the wires leading from the electric resonator consisting of the coil  158  and piezoelectric element  22  to the signal generating unit can be reduced in diameter and increased in length and may result in lower electrical interference.  
      A source of electrical signals, such as a signal generator, is electrically connected to the vibratory element  26 , and the source of vibratory motion,  22 , through various ways, e.g., a pair of wires  30 . In order to move the driven object  42  in a first direction, the signal generator produces an electrical signal with a spectrum whose dominant frequency is the corresponding operating frequency. Typical and usable signals include, but are not restricted to, pure sinusoidal, triangular and rectangular waveforms. Similarly, a signal with a spectrum whose dominant frequency is a second or third operating frequency, can cause the driven object  42  to move in a second or third direction.  
      The capability of the various vibratory motors described herein to reliably operate with a variety of waveforms is an advantage over those prior art motors that require special waveforms other than sinusoidal waves to function, e.g., saw-tooth waveforms, and that would not function reliably with a purely sinusoidal waveform. Therefore, since the quality of the signal applied to the piezoelectric element  22  can be less than compared to some prior art motors, the signal generator can have a simpler construction, which results in a reduced cost of the entire motor system.  
      Furthermore, it is desirable to have all electrical signals produced by the signal generator communicated through the same set of electrical connections to the vibratory element  26 , and particularly to the piezoelectric element  22 , e.g., by wires  30 . When all signals are communicated through the same electrical connections, there is no need for a unit that switches between various selected connections. This further simplifies the vibratory motor compared to prior art motors. Further, some prior art devices generate a phase shift between two electrical signals and then communicate the signals individually through separate electric connectors to at least one piezoelectric element, and the present, more simplified electrical connection can avoid that more complex design. This can further reduce the cost of the motor in comparison to some prior art motors.  
      As illustrated in  FIGS. 78-80 , the piezoelectric element  22  can be sized to extend beyond the portions that engage the walls forming the opening  28 . Thus, the piezoelectric element  22  is shown as extending beyond the end walls  31 . This variation in the dimensions of the piezoelectric element can be used to vary the value of capacitor  154 , and thus the performance of the control circuits, such as the circuit depicted in  FIG. 65 .  
      One potential disadvantage of the driver circuit of  FIGS. 65 and 78 - 80  is due to negative voltages that can appear across the capacitor  154 . Negative voltages can be damaging to the piezoelectric element  22 , which is a polarized electrical component. In order to amend the situation for piezoelectric elements susceptible to negative voltages, a modification of the circuit can be provided as discussed relative to  FIG. 66 .  
       FIG. 66  shows a driver circuit suitable for use with a piezoelectric element  22  that may be more sensitive to a negative voltage. In this circuit, a second physical capacitor  154   b  is added to the piezoelectric element  22  (represented as capacitor  154   a ), or multiplayer piezoelectric element  22 , if it has multiple piezoelectric layers that are electrically split as shown, can be represented as two capacitors  154   a  and  154   b.  Also, another resistor  156   b  is included in the circuit in addition to the existing resistor  156   a.  Parallel to the resistors  156  and the capacitors  154 , two diodes  160   a,    160   b  are added.  
      The orientation of the diode  156   a  prevents the voltage of the node between the two resistors  156   a,    156   b  from falling below the supply voltage (VCC). The voltage across the capacitor  154   a  therefore cannot become more negative than the typical voltage drop across a conductive diode (about 0.5-0.7 Volts). This small negative voltage can be sustained by most piezoelectric elements.  
      If, in the same manner as before, the circuit is excited by the input signal  150  to resonate, the amplitude of the oscillating voltage across the piezoelectric element  22  (represented by capacitor  154   a ) can be made larger than the supply voltage (VCC), but the voltage cannot become negative. A similar statement holds true for the physical capacitor  154   b  so that a polar electrical component may be chosen there as well. Further, if the piezoelectric element has multiple piezoelectric layers and is electrically split so as to be represented as two capacitors  154   a  and  154   b,  the driver circuit of  FIG. 66  advantageously requires only a single control signal  150  to drive the piezoelectric element.  
      It has been observed that for a given voltage amplitude of the electric input signal to the piezoelectric element  22 , the electrical current consumption of the piezoelectric element increases sharply for excitation frequencies just below certain resonance frequencies of the vibration element  26 , and drops sharply just above those resonance frequencies. For rod-like vibration elements  26 , these frequencies typically correspond to longitudinal modes. This electrical effect can be used to cheaply and quickly determine a particular vibration mode without using specialized measuring equipment such as a laser vibrometer. The sharp decrease in current just above a certain resonance frequency can be used to reduce the electrical power necessary to drive the vibratory unit  26  if the motor assembly  20  can be operated at these frequencies. Also, the electronics could be configured to automatically detect the drop in current and track the frequency at which the drop occurs, hence advantageously providing feedback. This feedback can be used to adapt the optimal operating frequency to changing external influences, such as temperature and humidity. Also, this kind of feedback can be used to detect the mechanical load that the motor must move.  
      The electric current consumption of the piezoelectric motor  26  can also be called a draw of electric current. It refers to the electric current flowing in and out of the piezoelectric element  22  during operation of the piezoelectric motor  26 . The electric current changes with time depending on the frequency of excitation and the voltage of the excitation signal to the piezoelectric element  22 . For practical purposes, the electric current draw is defined as the peak or the mean amplitude of the oscillating electric current signal as measured over one or preferably over at least five vibration periods, or, alternatively, is defined as the mean of the squares of the values of the electric current computed over one or preferably over at least five vibration periods.  
      Specially Configured Piezoelectric Elements  
      In some embodiments where the piezoelectric element  22  is press-fit into the opening  28  in the resonator, the walls defining the opening  28  deform elastically and/or plastically during the press-fit process in order to accept the larger piezoelectric element  22  and generate the preload. One way to prevent the piezoelectric from experiencing sheer forces during the press-fit and to prevent the piezoelectric from breaking is to put additional metal layers on the mechanical contact sides of the piezoelectric. But this time and labor to do so increases costs.  
       FIGS. 67-69  show a piezoelectric element  22  with a specially shaped end  170  that is configured for press-fitting into recess  28 . The end  170  can eliminate the need for additional metal layers and result not only in cost savings, but also in fewer mechanical contact surfaces and therefore in better performance. The new piezoelectric shape can also generate a more defined contact area.  
      The shaped end  170  has at least on one flat  172  adjoining an edge of the piezoelectric, and preferably two flats  172  on opposing edges of the shaped end  170 . The interior end of the flat  172  joins an incline or taper  174  that helps widen the hole  28  that the piezoelectric element  22  gets pressed into. The taper  174  joins a flat, central contact area  176 .  
      The shaped end  170  is advantageously placed on two opposing ends of the piezoelectric element  22 , the ends that will abut the walls defining the opening  28  and cause the preload on the piezoelectric element. The flats  172  on opposing ends  170  are spaced a distance apart selected to allow the piezoelectric element  22  to be inserted into an undeformed opening  28 . That helps position the piezoelectric. The inclines  174  make it easier to press the piezoelectric into the opening  28 . The inclined surface  174  is of sufficient length and inclination to allow insertion without unacceptably damaging the piezoelectric element  22 . The specific length and inclination angle will vary with the particular application. The central contact area  176  defines the final dimension of the opening  28  and sets the preload, it also provides a localized contact area to reduce the area in driving engagement with the resonator  24  in which the opening  28  is formed. That helps locate the contact area and axis of engagement and excitation, and it helps improves the engagement. Advantageously, the shape of the end  170  is symmetric about central axis  25  so the piezoelectric can be press-fit from two directions, but that need not be so.  
      The piezoelectric element  22  could be ground after the sintering process that produces the piezoelectric elements, in order to shape the taper(s)  174  and flat(s)  172 . Alternatively, the taper could be produced during the pressing process by which the piezoelectric elements are formed. The pressing process typically occurs after the layer stacking process in the piezoelectric production sequence. In this way no additional process step is necessary. This method also has an advantage over grinding in that no electrode surfaces are in danger of being ground, resulting in lower piezoelectric efficiency.  
      Referring to  FIG. 69 , the following process is believed suitable to produce this piezoelectric element, although someone skilled in the art can devise other methods given the present disclosure. The layer-stacking machine starts with the bottom die and places the first piezoelectric layer on top. All other layers follow just as in the normal lay-up process. Finally the top die is placed onto the stack and the whole stack is then pressed. During the pressing process, the piezoelectric elements  22  are forced to accept the shape of the die.  
      The die  178  has a shape configured to produce the depicted surface contour. The die  178  thus has flats  172 , inclined surfaces  174 , and central flats  176  located so as to form those surfaces on the pressed and sintered piezoelectric elements produced by the die. The contours of the die  178  are modified as needed to account for shrinkage and deformation that might occur during formation of the piezoelectric elements.  
      The combined surface contour is repeated for as many piezoelectric elements  22  as are placed in the die. It is important that the relative position of the electrodes  180  matches the position of the die. The stacking machine can ensure proper alignment. The stacked elements are pressed and produce the group of piezoelectric elements depicted in  FIGS. 67-69 .  
      Following the pressing process, the piezoelectric elements will be cut and processed as usual. During the cutting, it may be beneficial to leave the die attached to the stack for stability and alignment. The result is a piezoelectric element with said advantages.  
      If the piezoelectric is shaped as shown in  FIG. 67-68 , an additional advantage arises. Typically the electrodes  180  are printed onto the sides of the piezoelectric element  22 . If machines typically used to make multilayer capacitors are used, the electrodes  180  partially cover the edges of the adjacent sides, and here that includes a portion of the electrode over the recessed flat surface  172 . Because this surface  172  is sized to fit in the opening  28  without deformation the slight thickness of the added electrode layer does not affect installation. But if that electrode layer  180  were on a normal, square-ended wall of the piezoelectric element being press-fit against the walls defining opening  28 , then the edges of the piezoelectric would be larger than the center which lacks the electrode layer, and that would render a press-fit more difficult. The piezoelectric shape of  FIGS. 67-68  thus eliminates the need for removing the excess electrode material.  
      It is also possible to shape the die producing the piezoelectric elements such that the deformation that is caused by the polarization of the piezoelectric is accounted for. When polarized, the flat contact area  176  bulges slightly outward, convex to the piezoelectric element  22 . To offset this polarizing bulge, the die  178  is advantageously formed with a slightly convex surface at the contact surface  176  so that the resulting piezoelectric element  22  has a slightly concave surface in it at the contact surface  176 . The amount of curvature is selected so that after the piezoelectric element  22  is polarized, the contact surface  176  is flat. The amount of curvature will vary with the specific design of the piezoelectric element involved.  
       FIG. 75  shows a potential press-fit insertion sequence. Optionally, by first inserting a tapered plug  182  into the opening  28 , the insertion edges of the opening  28  are preferably slightly plastically deformed, which widens portions of the mating edge of the opening. For the illustrated embodiment the end walls  31  engage the piezoelectric element  22  and place it in compression. To avoid overstretching and breaking the sidewalls  29  during formation of the taper, the insertion edges on the end walls  31  can be shaped individually in two separate steps, or the entire frame can be constrained against axial deformation.  
      When the plug  182  is removed, the piezoelectric element  22  with shaped ends  170  is aligned with the opening  28 . The flats  172  preferably are able to enter the opening  28 , with or without the widened edge produced by plug  182 . The inclined edge formed on the end walls  31  defining opening  28  mate with the inclined surface  174  on the piezoelectric element  22  to provide a sliding insertion to position the piezoelectric element in the opening  28 . The tapered end walls benefit, but are not necessary for the press-fit to occur. They do, however, have the added benefit of inducing asymmetry in the resonator if so desired. If the sidewalls  29  were to engage the piezoelectric element  22  and apply a compressive force, then an inclined surface could be formed on the sidewalls  29  or on the corresponding edges of the piezoelectric element  22 .  
      The above discussion described the piezoelectric element  22  as comprising a plurality of piezoelectric layers. This need not be the case as a single piezoelectric crystal or ceramic block could be formed having the specially configured ends  170 .  
       FIG. 76  shows another advantageous method to press-fit the piezoelectric element  22  into the opening  28  of resonator  24 . To put this in perspective, a short discussion is given of the objectives, the problem, and then the solution.  
      Repeatability in the performance of the vibratory motors  26  requires a consistent preload be applied to the piezoelectric elements  22 . In order to accommodate variations in the dimension of the piezoelectric element  22 , while achieving the same preload, the sidewalls  29  can be placed in plastic deformation. The slope of the stress-strain-curve is very small in the plastic region, which leads to very small changes in preload when the length of the piezoelectric element  22  changes. This allows combinations of shortest piezoelectric element  22  with the largest opening  28 , and the longest piezoelectric element  22  with the smallest opening  28 , to result in essentially the same preload on the piezoelectric element.  
      But when the piezoelectric element  22  is press-fit into the opening  28 , it is subjected to frictional forces that lead to high shear forces on the piezoelectric element. Because the piezoelectric material is brittle, the shear forces can act to delaminate adjacent layers of the piezoelectric material. To prevent shear forces from acting on the piezoelectric the protective plates  34 ,  84  can be added to take the shear forces. This not only alleviates the stress on the piezoelectric element  22 , but also helps the press-fit as the plates  34 ,  84  act to guide the piezoelectric into the opening  28 .  
      To reduce the cost of the vibratory motor  26  and to also improve mechanical coupling between the piezoelectric element  22  and the resonator  24 , it is desirable to press-fit the piezoelectric element  22  without any protective layers of steel such as plates  34 ,  84 . The following process allows this by reducing the forces acting on the piezoelectric during the press-fit operation to a constant and low level and by making the press-fit process more controllable and therefore easier to automate.  
      The objective is to have most of the elongation of the sidewalls  29  done not by the piezoelectric being forced into the opening  28 , but to have the elongation done by another machine. This machine pulls the resonator  24  with force P as shown in  FIG. 76  to stretch the sidewalls  29 . The piezoelectric with tapered edges  82  sits on top of the opening  28  in the resonator  24  and is pressed into the opening  28  with a force, F, that is preferably constant, and that is not strong enough to push the piezo into the hole by itself. The force F is also not strong enough to cause damage to the piezoelectric element  22 , and is especially not strong enough to cause shear forces that delaminate the piezoelectric material.  
      At some point during the elongation of the sidewalls  29  by increasing force, P, the piezoelectric  22  starts to slide into the hole under force, F. By setting the force, F, to a specified value, shear forces between the piezoelectric element  22  and the resonator  24  are limited to the resulting normal force multiplied by the coefficient of friction. This resulting normal force equals the desired preload force minus the force, P.  
      Once the piezoelectric element  22  starts sliding into the opening  28 , it is necessary to stop the elongation of the sidewalls  29  by the machine because otherwise the resultant preload on the piezoelectric will be reduced.  
      The pulling machine applying the force P can be controlled by one of two principles, load control or displacement control. Load control refers to controlling the applied load and measuring the resultant displacement. Displacement control is just the opposite: controlling the displacement and measuring the resultant load. To prevent overstretching the sidewalls  29 , it is preferable to use displacement control for this application.  
      The sidewalls  29  could be curved toward, or away from the longitudinal axis  25  extending through opening  28 . If the sidewalls  29  are curved away from the longitudinal axis  25  extending through the opening  28 , then by applying opposing forces to opposing sidewalls  29 , the end walls  31  can be forced apart, allowing the piezoelectric element  22  to be inserted into the opening  28 . Upon removal of the force pressing the curved sidewalls  29  toward each other, the piezoelectric element  22  is placed in compression. Advantageously, in pressing the curved sidewalls  29  together in order to enlarge the space between end walls  31 , the sidewalls  29  are stressed beyond their elastic limit so as to achieve the advantages discussed herein.  
      Similarly, if the sidewalls  29  are curved toward each other, then by applying a force to the sidewalls that urges them apart, the end walls  31  are moved away from each other, allowing the piezoelectric element  22  to be inserted into the opening  28 . Upon removal of the force pressing the curved sidewalls  29  away from each other, the piezoelectric element  22  is placed in compression. Advantageously, in pressing the curved sidewalls  29  apart in order to enlarge the space between end walls  31 , the sidewalls  29  are stressed beyond their elastic limit so as to achieve the advantages discussed herein.  
      Instead of tapering the piezoelectric element  22  by applying inclined surfaces  82  (or  174  ( FIGS. 67-69 ) it is also possible to taper the edges of the opening  28 . It is also possible to have both parts tapered. If neither the piezoelectric element  22  nor the opening  28  are tapered, the piezoelectric element  22  starts to slide in at the point where the piezoelectric element and the opening are the same size. This presents alignment problems and requires very precise control to avoid overstretching of the sidewalls  29 . Therefore, it is desirable to have at least one mating part tapered.  
      The press-fit method described here is also adaptable to all other press-fits. The vibratory motor  26  with the piezoelectric element  22  and the resonator  24  or frame, is used as an example.  
      Stepper Motor Approximations  
      Referring to  FIG. 70 , the vibratory element  26  can be operated at a selected excitation frequency that does not coincide with any frequency being used for regular operation of the motor assembly  22  and that therefore does not transport the driven element  42  in a specified direction, but rather excites a mode of vibration in the driven element  42  itself. This is illustrated in  FIG. 70 , where the induced mode of a rod-shaped driven element  42  has nodes  190 . Similar nodes are observed if the driven element is a rotational object such as in  FIG. 4 .  
      In this situation, there is a tendency for the rod  42  to shift its position so that the contacting portion  44  becomes centered at the node  190 . Depending on what node is closest to the contacting portion  44 , this results in a forward or backward motion of the rod  42 . Thus, by seeking out a specific position along the driven element  42 , the motor assembly  22  can provide the incremental movement and locating aspects of a stepper motor. The step sizes are determined naturally by the particular mode of vibration that is being excited in the driven element  42 , and will vary with the mode that is being excited.  
      Centering the driven element  42  at known nodes  190  can be exploited to move the driven element  42  into a pre-defined position. This eliminates positioning errors that may have accumulated during regular operation of the vibration motor  26  and can be used to increase the accuracy and repeatability of the motor without the need of position feedback. This mode of operation requires that the actual position of the driven element  42  be within a certain distance to a desired node  190 , so that the resonant vibration causes movement toward the desired node.  
      The suspension of the driven element  42  affects the frequencies and node locations of the natural vibration modes of the driven element  42 . For the purpose of stepping the driven element  42 , the influence of the suspension must therefore be considered in choosing appropriate excitation frequencies to achieve this locating activity achieved by the nodes  190 . Conversely, the design of the suspension may be influenced by the need for particular excitation frequency or frequencies and designed to achieve those frequencies. There is thus provided a method and apparatus for using the vibration nodes of the driven object to transport the driven object to a known position for calibration.  
      Position Sensing  
      There are situations where it is desirable to exactly know the position of the driven element  42  relative to the vibration element  26 . Referring to  FIG. 71 , an illustrative implementation is described that uses the characteristic travel duration of a vibration or acoustic pulse from the piezoelectric element  22  to monitor the position of the driven element. The position of the driven element  42  relative to the vibration element  26  can be determined by measuring the time it takes for a mechanical vibratory pulse to travel from the vibration element  26  into and through the driven element  42 , and/or vice versa. The vibratory pulse can be generated in the vibration element  26  by the piezoelectric element  22  or in the driven element  42  by some other generating mechanisms  198 , such as a solenoid, a spring driven impact mechanism, or other mechanical or electronic mechanisms.  
      A receiver  196 , e.g., a piezoceramic microphone, that is mounted adjacent a distal end of the driven element  42  at a known location on the driven element can be used to sense the pulse generated by the piezoelectric element  22 , the pulse being sufficient to cause an impact vibration at the selected contacting portion  44 . Alternatively, the piezoelectric element  22  can be used to sense the vibratory pulse generated by the generator  198  by exploiting the piezoelectric material&#39;s inherent ability to convert a mechanical movement (e.g., of selected contacting portion  44 ) back into an electrical signal.  
      It is also possible for the piezoelectric element  22  to sense a vibratory pulse that it has generated earlier. This requires that the pulse travel through the vibratory element  26  and the driven element  42 , be reflected at a location on driven element  42 , such as the distal end of the driven element  42 , and return to the piezoelectric element  22  where it can be sensed. In a similar fashion, by way of reflection, it is possible for a sensor  196  to sense a pulse generated by a generator  198 .  
      The vibratory pulse can be chosen such that its power spectrum does not contain significant vibratory energy at frequencies that could cause the driven element  42  to move. Alternatively, the vibratory pulse can be incorporated into the operational input signal to the piezoelectric element  22 , for example in form of a brief pause. Because the geometries and material properties of the vibratory element  26  and of the driven element  42  are known, and because the position of the contacting portion  44  on the vibratory element  26  is known, the monitored time difference between pulse generation and sensing is representative of or characteristic of the distance between the piezoelectric element  22  and the receiver  196  or a distal end of the driven element  42 .  
      In some of the various position-sensing embodiments, it is desirable that the piezoelectric element  22  is temporarily deactivated prior to the position sensing so that undesirable vibrations are allowed to dampen out. Then the specified signals can be emitted and detected, with the operation of the vibrator element  26  resuming thereafter. It is an advantage of some of these embodiments that, if the piezoelectric element  22  is used as a sensor as well as actuator, only a single piezoelectric element is needed to move the driven element  42  as well as provide position feedback.  
      The pulse generated in either the piezoelectric element  22  or in a generator  198  is reflected at any surface where the mechanical impedance changes abruptly. These surfaces include the opposing ends of the vibrating element  26  and the opposing ends  200  of the driven element  42 . Some of these reflections are undesirable and must either be masked or be otherwise distinguishable from the position-determining pulse. Ways to achieve this include, but are not restricted to, degrading undesired reflected signals by inclining, damping or roughening certain reflecting surfaces such as the distal end  200   b  of the driven element  42 . Given the present disclosure, other ways of altering the ends  200  could be devised to allow signals reflected from the ends to be distinguished for use in the position sensing method and system.  
       FIG. 72  shows a different position-sensing embodiment that uses a resistive position measurement method that uses characteristic resistance of a resistive driven object to monitor the position of the driven object in a way that is analogous to how an integrated potentiometer operates. The position of the selected contact portion along the length of driven element  42  varies a resistance that is detected and used to define a relative position of the elements.  
      For illustration, the driven element  42  comprises a cylindrical rod. The driven element  42  is either made of an electrically resistive material, or it is made of an electrically insulating material that has been entirely or partially coated with an electrically resistive material  204 . A carbonated plastic material is believed suitable for either use. Since the electrical resistance between the selected contacting portion  44  of vibratory element  26  and either of the opposing ends of the rod  42  depends on the position of the contacting portion  44  relative to the rod  42 , the position can be determined by measuring the electrical conductivity, or the electrical resistance, between one of the opposing ends of the rod  42  and the vibration element  26 .  
      The voltage that is necessary for measuring the electric resistance can be small and it can be applied between the vibrating element  26  and a distal end of the driven element  42 . Preferably though, one end of the driven object  42  is connected to a positive supply voltage and the other end to a negative supply voltage. By measuring the voltage at the vibration element  26 , e.g., by a voltmeter  204 , accurate position information can be obtained.  
      Instead of applying the necessary voltage directly to ends  200   a,    200   b  of the driven object  42 , this voltage can also be applied to the wheels or bearings  46  supporting the driven element  42  provided that the wheels or bearings  46  are made from an electrically conductive material and that they are electrically insulated from each other and from the vibrating element  26 . In such an embodiment, the electrical contact resistance between the bearings  46  and the driven element  26  may have to be accounted for.  
      A suitably coated electrically nonconductive driven object  42  for use in the described embodiment can be cut from a sufficiently thick sheet of plastic that has previously been dipped into conductive paint and let dry. This forms a conductive layer on the exterior surface of the plate. The plate is then cut into strips creating two opposing edges that have no conductive layer. Thus, except for the strips or rods formed from the sheet edges, the cutting process advantageously exposes the nonconductive plastic on two elongated sides of the driven object (the strips or rods), which results in the conductive paint forming an elongated resistor  202  that wraps around the longitudinal axis of the driven object. This embodiment can be modified by further removing the conductive layer from one of the ends  200   a,    200   b  of the driven element. The position can be determined from a measurement of the resistance between the vibration element  26  and one contact point, e.g., bearing  46   a  or  46   b.    
      Electrically conductive driven objects  42  can also be used in position sensing embodiments if appropriate portions of them are first coated with an insulating layer and then with an electrically resistive layer. For example, an insulating layer may be applied to that side of a metallic rod-like driven object  42  that faces the bearings. On top of this layer, an electrically resistive layer  202  is applied so that it contacts the underlying metal only at the ends of rod  42 . The position dependent resistance lies now between a bearing and the end of the rod  42 .  
      Other Variations &amp; Advantages  
      In comparison to prior art piezoelectric motors, the present motor  26  requires only one piezoelectric element  22  and only one electrical excitation to generate motion. Due to the use of resonant vibration modes to generate elliptical motion  100  with a single excitation frequency, the piezoelectric element  22  can be smaller than those of other bidirectional piezoelectric motors can, and the overall motor  26  can also be smaller.  
      The present invention works very well to provide linear motion of driven elements  42 . The traditional solution is to use a motor with a gearbox driving a rack and pinion arrangement. The present motor assembly  20  provides a simpler arrangement, and less costly arrangement than prior art motors. Because the selected contacting portion  44  engages the driven element  42  by friction, the motor  20  is not damaged if the driven element is externally pushed so as to back-drive the motor. In contrast, such motion would strip the gears of conventional electric motors.  
      The present invention is especially suited for low cost applications. The simple design can avoid the need for precision manufacturing requirements and the attendant cost. It allows low cost manufacturing methods and inexpensive piezoelectric elements. In return, the design provides less power and efficiency than some other piezoelectric motors. But the available power and low cost make the embodiments especially suitable for many traditional markets, such as toys, office equipment, and home automation. Some illustrative examples of the uses for the vibratory motor  26  are given below. But one significant advantage of this motor is its size and simplicity, which can result in low cost.  
      Vibratory motor assemblies  20  are believed possible that are as small as 0.4×0.4×0.8 inches (1×1×2 cm) in size, moving driven elements  42  at 0.5-10 inches per second (1.3-25 cm/sec), with a force of 0.1-3 N. Rotational drive units are believed possible with sizes as small as 0.6×0.8×0.8 inches (1.5×0.2×2 cm) with torques and revolutions per minute (RPMs) depending on the diameter of the rotationally driven object  42 . The voltage of the vibratory motor assemblies can be varied depending on the circuit design and the power needed, but can range from 3V, 6V, 12V, 24V 48V, 110V or 220V. Other custom voltages can be used. There is thus a wide range of operating voltages available for the vibratory motors  26 .  
      The size of the vibratory element  26  can be very small, with elements as small as 2×3×10 mm 3  believed possible. The cost of the vibratory motors  20  is believed to be half that of competing electric motors. These motors  26  can produce linear motion, rotary motion, both linear and rotational motion, and blocking force when un-powered. They start and stop without delay in as little as 0.6 milliseconds, have no backlash because there are no gears, and can provide fast motion yet also provide slow motion without using gears. They are inaudible because they are driven in the ultrasonic range. The motors require no lubricants and use no controlling the times during which they are powered, they can achieve various speeds and distances. They generate no magnetic fields, have no brush discharge, and no inductive voltage peaks.  
      The advantages of the vibratory motors  26  make them very suitable for use in CD-ROMs as tray actuators, in scanners to move the light bar or rotary elements, in printers and copiers to transport and guide paper. In home automation applications, the vibratory motors  26  could actuate air conditioning elements, automatic blinds, lighting controls and switches, dust protection doors on dust sensitive appliances, automatic locks, or elements in motion detectors. The vibratory motors  26  could also be used to position, pan, tilt or zoom remotely operated cameras, e.g., security cameras.  
      The ability to directly engage and drive glass offers advantages for using the vibratory motor  26  to control the position and focus of accent lighting in homes, retail stores, theaters, galleries, museums, hotels and restaurants. In automotive applications, the vibratory motors  26  can be used to position mirrors, headlights and air conditioning vents, and to actuate automatic locks. The stepper-like operation of the motor assembly  20  allows the storing and retrieving of mechanical settings, such as mirror position, for each of several drivers under the direction of a computer.  
      The combination of a computer that stores position information in connection with a positionable motor  20  finds many possibilities in sensors that automatically adjust their position. These include optical sensors, lens cleaning mechanisms for such sensors, protective covers that open and close by the use of vibratory motors  26 , automatic alignment mechanisms, proximity lasers, and adjustments of a variety of products that require movement of small parts by simple motors.  
      The vibratory motor  26  is especially useful for toys due to advantageously low cost, small size and low noise. Dolls could have limbs moved and eye lids actuated by the motors  26 . Remotely controlled vehicles could have steering controlled by the motors  26 . Animated toy books could be provided. Railway models could have moving forks, cranes, signals, railway gates and other actuated components. Various other applications of the motor  26  can be made. Further, the vibratory motor  26  can be made resistant to liquids, such as water, with appropriate modifications and coatings, and can provide multiple motions. The low force output reduces the possibility of injury. Also, it is possible to mix the electrical operating signal that is supplied to the piezoelectric element  22  with an electrical signal that contains a non-operational yet audible frequency spectrum. In such an embodiment, a piezoelectric motor  20  can also serve as a simple device for generating sound and music.  
      There is thus advantageously provided a motor assembly  20  that costs less to produce than traditional motors of comparable power and speed. The size and weight of the motor assembly  20  is less, and the invention allows for exceptional miniaturization of the motor. The motor can achieve stepper-like motion of the driven elements  42 , and positioning of the driven elements is possible without using positioning sensors on the driven part. The motor assembly  20  allows the use of fast, or slow driving speeds, and does not need a gearbox. Because the motor does not use gears, there is no backlash as associated with gear trains. The motor assembly  20  allows the driven element  42  to be translated, or rotated, or both. The positioning of the driven element  42  has positioning accuracy of 1 μm. The operating frequency can be selected to be inaudible to humans so that motor operation is silent. Due to the absence of magnetic fields or spark discharges, the motor assembly  20  and its vibratory element  26  are suitable for use in environments that are sensitive to electromagnetic interference or sparks. Quick reaction times of the motor assembly  20  permit control with binary state controllers, which are easier to implement and less expensive than PID controllers.  
      The invention further advantageously provides a vibratory element  26  having a piezoelectric driving element  22  and a resonator  24  that advantageously holds the driving element  22  in compression. This vibratory element combined with a resilient suspension system such as spring  10  can advantageously be provided to users who apply the components to a variety of driven elements at the discretion of the user. These parts are advantageously designed and configured to cooperate to generate an elliptical path  100  at a selected driving portion  44  for one or more predetermined applications or for one or more generic applications. This combination can be provided as a unit, and could be provided with or without the spring  50 . A user could thus use these components to drive a variety of driven elements  42 .  
      Alternatively, a user could be provided with a complete motor assembly  20  containing not only the vibratory element  26  and resilient mount such as spring  50 , but also driven elements  42  supported in a predetermined relation to the vibratory element  26 . In this alternative situation, the motor assembly  20  is preferably encased in a housing along with a suspended driven object  42 , as for example a rod for a linear motor. In this alternative situation, the motor assembly  20  and driven element  42  are ready for installation and use as the user sees fit. The assembly can be used with a driven element used in other motors, or it could be used as a part of a product configured for use with the components. Providing the driving elements and suspension elements allows the user to acquire a low cost driving mechanism having great flexibility in its application.  
      The driven object  42  preferably has a smooth and hard surface located to engage the selected driving portion  44 . The driven element  42  can have a variety of shapes, for example it can be a disc, a rod, a wheel, a gear, a beam, a ball, etc, as long as a fairly constant contact force can be maintained between the selected contact portion  44  and the driven element  42 . This gives designers a wide range of possible implementation methods for the motor assembly  20 .  
      The motor assembly  20  is advantageously encased in a housing to protect it from dirt and other extraneous contact and potential damage. The housing can be manufactured out of plastic through an injection molding process, or made of sheet metal. It is preferably designed such that it can be assembled through snap joints. This assembly avoids the use of more expensive methods including threaded fasteners and is good for a fully automated assembly.  
      This provides the possibility for the end user to have an inexpensive and small motor unit, which is easy to implement into a design. In order to increase the flexibility of use, the base  10  or the housing can have clamping holes or other clamping mechanisms to make it easier to attach to the end user&#39;s product. If the volume of a specific designed base  52  or housing is sufficient, the base  52  and/or housing unit could be specially configured to meet the mounting needs of the user.  
      There is thus provided a mechanism and method for generating an ellipse  100  that has a first semi-axis and a second semi-axis, with the length of the first semi-axis being useful to generate a difference in friction force between the selected contact portion  44  and an engaging surface of a driven element  42 , during the motion components in the direction or directions of travel along the elliptical path  100 . This elliptical motion is advantageously provided by a single excitation frequency provided to a piezoelectric element  22  that results in at least two vibrational modes generating the elliptical path  100 . This elliptical motion  100  is achieved by exciting at least two vibrational modes at least one of which, and preferably both of which, are not pure longitudinal or pure bending modes, and superimposing those modes to generate the elliptical motion at the selected contacting portion. This elliptical motion  100  is advantageously achieved without having to place the selected contacting portion  44  into contact with any driven element  42 .  
      The practical result of having modes other than purely longitudinal and purely bending, is that the major axis defining the elliptical path  100  of the selected contacting portion  44  is angled relative to the longitudinal axis  25  of the vibratory element  26 . The major and minor axes of the elliptical path  100  are not aligned with the longitudinal axis  25  of the resonator as is common with prior art vibratory devices. The angle of the major axis of the elliptical path  100  relative to longitudinal axis  25  is advantageously not near 0 degrees or multiples of 90 degrees. For ease of description, the angle will be described relative to the orientation of parts in  FIG. 1  in the first quadrant, but one skilled in the art will appreciate that the parts can be rotated through other quadrants or that the orientation of parts can be changed—without altering the relative angles discussed here.  
      Because the greatest motion and fastest rate of travel is achieved when the longitudinal axis of the elliptical path  100  is aligned with the travel path of the driven element  42 , the vibratory element  26  is preferably angled relative to the driven element  42  in order to align those axes. If the major axis of the elliptical path  100  aligns with the longitudinal axis of the driven element  42 , then this above-discussed angle can be considered to be the angle α, discussed above. The perfect alignment of the major axis of the elliptical path  100  with the longitudinal axis of the driven element  42  is often compromised for practical reasons.  
      Because the elliptical motion  100  is angled relative to the longitudinal axis  25  of the resonator  24 , elliptical motions with large aspect ratios can be used. Ratios of the major to minor axes of the elliptical path  100  are advantageously over 5, more advantageously over 10, and preferably over 20 to  1 . But when the semi-axis becomes too small, the selected contacting portion may not adequately disengage from the driven element when the ellipse is aligned with the driven element and thus ratios of 30:1 or more are difficult to achieve, especially at low cost. Further, as the ratios increase, the performance approaches that of an impact drive vibrator element. Thus, ratios of over 150:1, and even 30:1 are difficult to achieve and use. While most useful sized elliptical paths  100  are believed to have aspect ratios of about 3:1 to 150:1, preferably the ratios are from about 4:1 to 30:1, and ideally from about 5:1 to 15:1. If aspect ratios are used up to and over 150:1, then the resulting applications find use primarily in impact drive types of devices.  
      The amplitudes needed to achieve elliptical path  100  at the selected contacting portion  44  are preferably obtained by large magnification of small input signals. That requires selecting or creating resonance modes of vibration sufficiently close to the selected input signal to achieve a usable amplitude. Advantageously, for each volt input to the piezoelectric element  22 , the selected contacting portion  44  can achieve 0.3-0.5 microns of motion along the major axis of the elliptical path  100 . Preferably, for each volt input, the motion along the major axis of the elliptical path  100  is 1 micrometer or greater. These motions are the result of resonant vibration mode amplifications that increase the motion by factors of at least 100, and typically by factors of 1000 or more.  
      It is possible, but less desirable, to use a small resonance magnification and instead provide a larger input signal in order to achieve the needed amplitude to generate an acceptable elliptical path  100  at the selected contacting portion  44 . If one of the vibration modes that results in the usable elliptical path  100  is off-resonance, then the electric input signal to the piezoelectric element  22  can be increased sufficiently to result in a usable elliptical motion, one sufficient to moves the driven element  42 . Thus, it is believed suitable in some applications to have one volt input to the piezoelectric element  22  result in motion along the major axis of the elliptical path of 20-50 nanometers, but with movements of  100  nanometers or more being desirable.  
      Thus, the selected contacting portion  44  moves in a first elliptical path having a major axis and minor axis when the vibration source, such as piezoelectric element  22  is excited by a first electrical signal that causes at least two vibration modes that are superimposed to create the first elliptical path  100 . Preferably, at least one of the vibration modes is other than a pure longitudinal mode and other than a pure bending mode. When at least one of the two vibration modes is off-resonance, the first electrical signal is amplified sufficiently to cause the at least one off-resonance vibration mode to produce a motion of the selected contacting portion  44  having sufficient amplitude that the resulting elliptical path  100  can move the driven element  42  during use. As used here, the reference to an off-resonance vibration mode refers to a vibration mode that is sufficiently away from the resonance mode that the resulting motion does not generate a usable elliptical motion, motion insufficient to drive the driven element  42 .  
      The desired elliptical motion  100  is advantageously achieved without requiring the selected contacting portion  44  to engage the driven element  42 . Depending on the angle of engagement, reflected by angle α, the engagement can cause impact or bending that may affect the elliptical path  100  or the resulting motion of the driven element  42 , and appropriate compensation can be made for those effects.  
      As mentioned above, the generation of the elliptical path  100  at the selected contacting portion is most easily determined in a localized coordinate system that does not align with the longitudinal axis  25  of the vibratory element  26 . A coordinate transformation to align the motion so that one axis of the elliptical path  100  aligns with the vibratory element  26  or preferably with the driven element  42  allows the practical use of the elliptical path  100  to be evaluated.  
      If multiple motions of a driven element  42  are desired to be produced from a single vibratory element  26 , then the selected elliptical path  100  is likely to be a compromise among several potential elliptical paths at various frequencies, and if desired, at several selected contacting portions  44 . If multiple motions are desired to be produced by a single piezoelectric element  22 , it is preferable that the frequency used to achieve the different elliptical motions be sufficiently different to clearly separate the frequencies and their resulting motions. The frequencies for the separate motions are preferably separated by at least the same margin as the frequency range over which the substantially uniform elliptical motion  100  is achieved. Thus, for example, if a first elliptical motion  100  is achieved over a frequency range of 2.5 kHz on either side of a first frequency, for a total range of 5 kHz, then the second frequency is advantageously at least 5 kHz from the first frequency, and preferably more.  
      Ideally, the major axis of the elliptical path  100  is aligned with the axis along which the driven part  42  moves. As shown in  FIG. 1 , that alignment angle corresponds to the angle α between the longitudinal axis  25  of the vibratory element  26  and the axis  45  of a rod-like driven element  42 . This alignment may be achievable if the driven element  42  is moved in only one direction. But when the same vibratory element  26  is used to move the driven element  42  in opposing directions, relative alignment is difficult or impossible to achieve, especially in both directions. Further, the alignment considerations for bidirectional motion as discussed below are advantageously used even when only a single direction of motion of the driven element  42  is used.  
       FIG. 81  will be used to illustrate the considerations in this alignment.  FIG. 81  illustrates a first elliptical path  100   a  having a major axis e x1  for moving the driven element  42  in a first direction, and a second elliptical path  100   b  having a major axis e x2  for moving the driven element in a second, opposing direction. The major axis e x1  is inclined at an angle β 1  relative to axis  45  of the driven element  42  and the major axis e x2  is inclined at an angle β 2  relative to that axis  45 . The axis  45  can be viewed as parallel to a tangent to the driven element  42  in the direction of motion of the driven element  42  at the selected contacting portion  44 . The motion along the first direction, the motion resulting from ellipse  100   a  is believed to typically be the easiest to achieve and will typically have the major axis e x1  of ellipse  100   a  most closely aligned with the axis  45  of the driven element  42 , while the major axis e x2  is not as closely aligned with that axis  45 . Thus, β 1  is typically smaller than β 2  when β 1  is selected first. But that need not always be the case as the ultimate selection of elliptical paths  100   a,    100   b  is a result of compromising several factors as discussed herein.  
      While it is ideal for β 1  and β 2  to be 0, so that the major or minor axes of the elliptical paths  100  to align as closely as possible with the desired motion of the driven element  42 , that is difficult to achieve for multi-direction motion. For bi-directional motion where the same motion is used to move a driven part, but move the same driven part in different directions, it is believed that β 1  and β 2  will range from 5 to 40 degrees with respect to a tangent to the driven element  42 , along the direction of motion of the driven element  42 , at the selected contacting portion  44 . It is believed possible, but less desirable to have the angles go from 40 to 45 degrees. It is very desirable to have the angles β 1  and β 2  perfectly align the major axis of the elliptical paths  100   a,    100   b  with the direction of motion of the driven element, and preferably align them within 0 to 5 degrees. As used herein, because the orientation of parts can increment the angles through various multiples of 90 degrees relative to a horizontal axis, the angles given should be construed as relative angles rather than as absolute numbers. Thus, for example, the reference to aligning the major axes and the driven path within 0 to 5 degrees includes angles on opposing sides of the horizontal X-axis as shown in the drawings. That thus includes an absolute angle of 355-365 degrees relative to a common axis of measurement.  
      As shown in  FIG. 81  the angle is relative to the longitudinal axis  45  of the translating rod  42 . But the driven element  42  could comprise a rotating disk (e.g.,  FIG. 4 ). Usable but sometimes undesirable performance is believed to be achieved if β 1  and β 2  range from about 5 to 85 degrees from the tangent to the driven element at the location of the selected contacting portion  44 . Preferred performance levels are believed to be achieved if β 1  and β 2  range from about 5-35 degrees and 55-85 degrees, and the best performance is believed to be achieved when β 1  and β 2  range from about 15 to 25 degrees and 65 degrees to 75 degrees.  
      As stated or implied above, because of symmetry considerations relative to the 0 and 90 degree axes, the above range can vary in 90 degree increments in absolute value relative to a common axis of origin. The above discussions and angle ranges are believed to apply to multi-direction motion.  
      In order to achieve the desired angles β 1  and β 2 , it is believed that the angle α should be maintained within the previously discussed ranges. The particular combination of β 1  and β 2  that is used is typically chosen so that the major axis of elliptical paths  100  aligns as close as possible with the axis of the driven element  42 . The closer the alignment, the more efficient the transfer of motion from the vibratory element  26  to the driven element  42  along the selected axis of motion  45 .  
      But from the various angles discussed, it can be observed that the selected vibration mode(s) of the resonator  24  that result in usable vibratory motions along elliptical paths  100  orientated at angles β 1  and β 2 , are neither purely longitudinal nor purely bending modes. Thus, in producing the elliptical motion  100  at the selected contacting portion  44 , the angles β 1  and/or β 2  are such that the major and minor axes of the elliptical paths  100   a,    100   b  do not align with the longitudinal axis of resonator  24  of the vibratory element  26 . Further, the angles β 1  and/or β 2  are such that the major and minor axes of the elliptical paths  100   a,    100   b  do not align with a pure bending mode of that vibratory element  26 , e.g., along axes  38 ,  40  in  FIG. 1 . The angle ∝ between the driving element such as vibrating element  26  and the driven element  42  is varied in order to allow the advantageous alignment of the major and minor axes with the direction of motion desired for the driven element  42 .  
      This also means that the vibrational modes of the vibratory element  26  that generate elliptical paths  100   a,    100   b  at the selected contacting portion  44 , have at least one vibration mode that is not a purely longitudinal vibration mode along axis  25 , and not a pure bending mode (e.g., along the axes  38 ,  40  for the configuration depicted in  FIG. 1 ). Thus, for example, the two vibration modes being selected to generate elliptical path  100   a  preferably do not include a pure longitudinal or pure bending mode of the resonator  24  in order to produce the first elliptical motion  100   a  of the selected contacting portion  44 , and the same is true with the vibration modes to generate the second elliptical path  100   b.  If a pure longitudinal or pure bending mode is used to generate the first elliptical path  100   a,  then the vibration modes used for the second elliptical path  100   b  do not necessarily include a pure longitudinal or pure bending mode of the resonator  24  in order to produce the elliptical path  100   b.  Further, if a vibration mode is used that includes a pure longitudinal vibration mode along axis  25 , then desirably the axis  25  is inclined to the driven element  42  at an angle α that is other than 0 and 90 degrees or multiples thereof, and that is preferably between about 5-85 degrees and multiples thereof.  
      As the angles β 1  and β 2  become greater relative to the driven element  42 , the contact results in greater impact between the selected contacting portion  44  and the driven element  42 . When the aspect ratio of one or both elliptical paths  100   a,    100   b  becomes large, so that one axis is much larger than the other axis, the contact approaches that of an impact drive. It is believed possible to have one of the elliptical paths  100   a,    100   b  have a high aspect ratio, sufficiently high that the resulting motion can effectively be considered an impact drive, and have the other elliptical path with a lower aspect ratio to produce a non-impact drive. Advantageously, aspect ratios of the elliptical paths  100  that produce a pure impact type drive, are avoided.  
      Further, it is believed possible that the teachings of this disclosure can be used to configure a vibratory element  26  having very high aspect ratio elliptical motions  100   a,    100   b  in opposing directions. When the aspect ratio of the major to minor elliptical axes become great enough, the particular direction of motion around the elliptical path is not determinative of the direction in which the driven element moves. Instead, the angle of inclination β of the major axis relative to the driven element becomes the determinative factor in deciding the direction of motion. Thus, it is believed possible to use two high aspect ratio elliptical paths  100   a,    100   b,  at the same (or different) selected contacting portions  44 , to create an impact drive moving the driven element  42  in the same direction. Indeed, the principles of this disclosure could be used to have a single piezoelectric element  24  generate two longitudinal resonance modes at two different frequencies, each of which is used in an impact drive.  
      Whether high aspect elliptical motion is used to approximate pure impact drive, or whether a pure linear motion is achieved to implement an impact drive, the motion of the driven element  42  can be achieved at two separate frequencies. But the use of two frequencies can result in different rates of travel of the driven element. The differences in the rate of travel by using different frequencies can be enhanced if a high aspect elliptical motion is used in which the direction of travel of the ellipse (e.g., clockwise vs. counter-clockwise) changes, or in which the angle β, changes. Further, the teachings herein can be used even if more than a single piezoelectric element  22  is used to cause the multiple frequencies for the impact-type motion or to cause actual impact motion using mainly longitudinal resonance of the vibratory element  26 .  
      It is desirable to have the angles β 1  and β 2  be reasonably constant over as wide a range of excitation frequencies to vibratory driving element  22 , as possible. For example, if any excitation frequency signal to piezoelectric element  22  over a 2 kHz range results in elliptical motion  100  at the selected contacting portion where the angle β 1  does not vary by more than 5 degrees, then it becomes easier to design the vibratory system, and it becomes easier to allow the use of components with larger tolerances in performance parameters. It is desirable to have the angles β 1  or β 2  vary less than 10 degrees, and preferably less than 5 degrees, and ideally less than 3 degrees, over as large range of excitation frequencies as possible. This allows the angle α of inclination between the predominant axis  25  of the vibrating element  26  and the motion axis  45  of the driven element  42  to be set with reasonable tolerances, and to use components with tolerances obtainable at affordable prices, and produce acceptable motion. This especially allows the use of low cost motors in a wide variety of commercial applications, as discussed herein.  
      It is thus desirable to have the selected contacting portion  44  move in approximately the same elliptical path  100  when the frequency of the driving signal input to the piezoelectric element  22  varies by as little as 200 Hz on either side of the selected frequency. Advantageously, approximately the same elliptical path  100  is achieved when the frequency of the driving signal varies as much as 2.5 kHz, or more, from the selected frequency. It is thus desirable that the excitation frequency to the source of vibration  22  can vary by as much as 2.5 kHz on either side of the selected frequency, and preferably greater, while still producing suitable amplitudes for elliptical paths  100  at the inclination angles β 1  and β 2 . In relative terms, it is desirable to have a range of 5-10% of the selected excitation frequency achieve suitable elliptical paths  100 , with the inclination angles β 1  and β 2  varying less than 25 degrees, and preferably less than 10 degrees, and ideally by about 5 degrees or less, over that frequency range. The ability to do so will vary with the particular design criteria and performance requirements.  
      One way to help maintain the inclination angles β 1  and β 2  reasonably constant over a reasonably wide range of excitation frequencies is to vary the various design parameters of the motor as discussed herein. The above discussed angles of 25 degrees, preferably 10 degrees and ideally about 5 degrees or less are each considered to be reasonably constant, with angles of about 5 degrees or less being the most preferred and most reasonably constant. Maintaining these inclination angles reasonably constant is most easily achieved by having the effect of the relative phase change on the angles β 1  and β 2  compensate for the effect of the amplitude change on the angle. To achieve this it is useful to select a set of vibration modes that have suitable directions of motion and frequency response curves for phase and amplitude. Further, using a coordinate transformation to view and analyze the elliptical motion  100  in a localized orientation also makes the design easier.  
      As used herein, the predominant axis is used to indicate an angle of inclination between the vibratory element  26  and the elliptical path  100  of the selected contacting portion  44 . The predominant axis will vary with the geometry and shape of the vibratory element  26 , and the location and orientation of the selected contacting portion  44  on the vibratory element  26 . For elongated vibratory elements  26  with the selected contacting portion  44  located at a distal end, as in  FIG. 1 , the predominant axis is likely to be the longitudinal axis  25 , or an axis orthogonal thereto, or a rotation about such axes. For non-straight vibratory elements  26  as shown in  FIG. 77 , with the selected contacting portion  44  located on a distal end, the predominant axis is the axis  25  through the distal end, or an axis orthogonal thereto, or a rotation about such axes. For selected contacting portions  44   n  located along the length or on intermediate portions of vibratory elements  26  as shown in  FIG. 6 , the predominant axis is again the longitudinal axis through the distal end  36   a,  or an axis orthogonal thereto, or a rotation about such axes. The particular predominant axis will vary in part with the motion of the selected contacting portion  44  and an adjacent axis of the vibratory element  26  that can be readily used for orientating the vibratory element to achieve alignment of the elliptical path  100  at the selected contacting portion  44  with the driven element  42 .  
      To test the quality of a motor  20  after it has been assembled, it is advantageous and cost-effective to measure a few electromechanical characteristics of the motor using its piezoelectric element  22 . The characteristics include, but are not limited to, the current that is drawn by the piezoelectric element  22  for a predetermined input signal, and the electrical signal that is generated by the piezoelectric element when it is turned off after having appropriately excited vibrations in the vibration element  26 . It is also possible to track these characteristics during the lifetime of a motor  20 , and in doing so to monitor motor efficiency without the need of special equipment such as a laser vibrometer.  
      The above disclosure focuses on using a single electrical signal to excite a single piezoelectric element  22  to produce an elliptical motion  100  at the selected contacting portion  44  that is inclined to the predominant driving axis (e.g., longitudinal axis  25 ) of the resonator  24 . That elliptical motion  100  is an unrestrained motion of the selected contacting portion  44  and occurs whether or not the contacting portion  44  engages the driven element  42 . But that inclined elliptical motion  100  can be produced by using more than a single piezoelectric element  22  on the resonator  24 . This invention thus has broader applicability to vibratory elements  26  that use plural piezoelectric elements  22  to achieve the elliptical motion  100  inclined to predominant driving axis (e.g., the longitudinal axis  25 ) of the resonator  24 . Thus, for example, as shown in  FIG. 81 , first and second piezoelectric elements  22   a,    22   b  could be on different portions or sides of resonator  26  (or contacting differently located walls defining one or more openings  28  in the resonator  26  as in  FIG. 2 ), in order to produce an inclined elliptical motion  100   a  at the selected contacting portion  44 . A third piezoelectric element  22   c  could be on yet another portion of the resonator in order to produce a different elliptical motion  100   b  at the selected contacting portion  44 . This use of multiple piezoelectric elements  22   a - 22   c  requires more complex electronics and thus has disadvantages, and it may limit the applicability of some aspects of the present disclosure. But it illustrates that some aspects of this disclosure have applicability beyond use with the single piezoelectric element  22  as described herein.  
       FIGS. 82   a -  82   e  show various views and sections of a piezoelectric motor  26  comprising a resonator  24  of length LA and a piezoelectric element  22 . The resonator  24  comprises two sidewalls  29  that at least partially have the shape of a circular section. Said sidewalls  29  have a maximal width LF. The piezoelectric element  22  has width LD, height LC and length LB, and it is clamped into the opening  28  in the resonator  24  that is formed by the sidewalls  29  such that a partial height LE of the piezoelectric element  22  extends above the plane that is spanned by the resonator  24  surface. Furthermore, a resilient element  50  joins the resonator  24  to the base  52 . Preferably the resilient element  50  comprises a form-fit spring coil  507  joined to the resonator  24 .  
       FIGS. 83   a -  83   g  show a resonator  24  having overall length LA and height LG. The long leg of the resonator  24  that is proximal to the driven element  42  has length L 9  and width L 5 . Furthermore, in the contact portion  44  to the driven element  42  there is provided a circular depression having width L 8  whose shape essentially conforms to the surface of the driven element  42  (not shown). Opposite to the side where the resonator  24  clamps the piezoelectric element  22  (not shown), the resonator opening  28  comprises chamfered edges  550 . Under the rear side of the resonator  24  leg that is distal to the driven element  42  (not shown), there are provided lateral recesses  551 , each for receiving one connecting portion  50   b  of the spring element  507 . These recesses  551  have a depth L 13  and width L 17 ; their center is located at a distance L 14  from the end face of the resonator  24 . Directly next to the recesses  551  there are provided material bulges  552  with radius R 7  on the sides of the resonator  24 , the material bulges  552  having a maximal height L 16  as measured from the deepest point of the recess  551 . When the connecting portions  50   b  of the resilient element  50  are pressed into the recesses  551 , the material bulges  552  support the form-fit of the joint.  
       FIG. 84  shows a piezoelectric motor  26  that is fastened to a base  52  by way of a spring coil  507  and a mounting screw  509 , its resonator  24  abutting a driven element  42 . During operation of the piezoelectric motor  26 , the contact portion  44  of the resonator  24  executes preferably elliptical or linear vibrations, which cause the driven element  42  to move in a preferably rotary or linear motion. In the example, the contact element  506  is implemented as a contact spring that closes an electrical circuit with the resonator  24  when the wear of the contact portion  44  becomes too big. In effect, when the contact portion  44  wears a sufficient amount the resonator  24  abuts the contact element  506  to activate the electrical circuit indicative of a predetermined amount of wear. For example, activation of the circuit by contact between spring  506  and resonator  24  could shut off the motor and activate an audio or visual signal (or both) indicating the motor is off and needs replacement.  
       FIG. 84  also provides an indication of what happens to the rate of wear as the contact portion  44  is worn. The resilient element  50  resiliently urges the contact portion  44  against the driven element  42  to cause a static contact force. The contact pressure is a function of that contact force and a function of the size of the portion of the contact portion  44  that contacts the driven element  42 . The contact pressure further affects the rate of wear. Therefore, the rate of wear changes if the size of the contact area changes due to wear, or if the resilient contact force changes because the resilient element  50  seeks to maintain the contact portion  44  in contact with the driven element while wear drives the contact surface of the contact portion  44  further into the resonator. The rate of wear can further be influenced by the material selection. For example, the surface of the driven element may comprise an abrasive material (for high rates of wear) that is worn off (to subsequently reduce the rate of wear).  
      The run-in phase of a piezoelectric motor  26  is discussed with reference to  FIGS. 84   a,    89   a,  and  89   b.  It is possible that a newly manufactured piezoelectric motor  26  exhibits less than acceptable performance due to unsatisfying contact between the contact portion  44  and the driven element  42 . That contact could be improved by a finishing step during assembly where a contact surface  520  is cut, ground or otherwise manufactured into the resonator. In this process, however, no use is made of the actual shape of the driven element, which may be subject to tolerances of its own. Better results having better matched contact surfaces between the resonator  24  and the driven element  42  are achieved by moving the selected contacting portion  44  against the driven element  42 , or vice versa, to cause deliberate, shape-matching wear. This relative motion causes the run-in of the motor and it is especially preferable that the piezoelectric motor  26  and/or the resilient element  50  are configured to increase the motor performance during this run-in phase. The relative motion may be caused by piezoelectric motor  26  operation when an electric signal is applied to the piezoelectric element  22 . The driven element  42  during run-in is preferably the same driven element that later will be in the piezoelectric motor assembly  20  during the service life of the piezoelectric motor  26 . Other driven elements may, however, also be used specifically for the run-in phase. Such driven elements  42  preferably have surface properties that cause the contact portion  44  to quickly attain the desire shape. To that end it may also be preferable to cause relative motion between the driven element  42  and the resonator  24  during run-in using an external source such as an electric DC motor, any other type of motor or actuator, or even another piezoelectric motor  26  to drive the driven element  42  against the resonator  24  in a linear or rotary fashion as indicated in  FIGS. 89   a  and  89   b  by double-headed arrows  540  for rotary external driving and by double-headed arrows  541  for linear external driving. In this process, the piezoelectric motor  26  may or may not be powered. In this context, a contact element  506  may be used to indicate when a predetermined amount of contact portion  44  wear is reached.  
      During run-in, the relative motion may be in only one direction or it may occur in two opposing or more directions, e.g., also in an oscillating manner that is sufficient to produce wear. In this manner one achieves a run-in piezoelectric motor  26  that has been prepared for driving a driven element  42  in two or more directions, wherein the two directions are preferably opposite. It may be that this run-in creates a contact surface  521  on the selected contact portion  44  that is equally suitable for driving the driven element  42  in two or more directions but with less than optimal performance that could be achieved if the piezoelectric motor  26  had been run in using only one direction of motion of the driven element  42 . In this sense it is possible to achieve a compromise between the performances of directions of motion of the driven element  42 .  
      In a preferred embodiment, the material of the selected contacting portion  44  and of the engaged portion of the driven element  42  are selected to wear during driving engagement. Also, in a preferred embodiment, the motion of the driven element  42  relative to the selected contact portion  44  is selected to cause wear of the selected contact portion  44  to cause the shape of the selected contact portion  44  to conform to the shape of the engaged portion of the driven element  42 .  
      During the run-in phase, the resonator and the driven element are preferably and resiliently urged toward each other by a force sufficient to allow movement of the driven element and to cause wear while not causing excessive wear. In this method, the resonator may be urged against the driven element or vice versa. In a further preferred embodiment, an electronic circuit is activated when a predetermined amount of wear occurs on one of the contacting portion  44  or the driven element  42 . Also, the motion of the contact portion  44  is preferably changed with the amount of wear, the change in motion being selected to adjust the movement of the driven element  42  to at least partially offset any degradation in the movement of the driven element  42  caused by the wear.  
      In  FIGS. 85   a -  85   c,  various views of a coil spring  507  are shown. The connecting portions  50   b  of the coil spring  507  that are joined to the resonator  24  ( FIG. 90 ) comprise a modified, preferably not circular, here rectangular (cf.  FIG. 87   a ) cross-section at the connecting portion  50   b  by which a twist-resistant joint to the resonator  24  can be assured. The windings  515  of the coil spring  507  undergo almost two complete circular revolutions during the transition from one leg of the coils spring  507  to the other. Any desired number of windings could be used to help vary the desired spring constant of spring  50 , and the cross-sectional shape could also be varied. Both legs of the coil spring  507  are preferably connected through two angled leg pieces  514  and a straight cross bar  513 .  
       FIG. 86  shows a further coil spring  507 , which differs from those shown in  FIGS. 85   a -  85   c  in that the respective legs comprise three almost complete windings  515  and that the cross bar  513  has essentially the shape of a circular section.  
      In  FIGS. 87   a -  87   d,  a third coil spring  507  is shown in several views. This third coil spring  507  also comprises almost three complete windings  515 , which, however, are tilted with respect to the plane of the circles. The two legs of the spring are connected through a cross bar  513  that has essentially the shape of the letter Omega.  
      The diagram in  FIG. 88   a  illustrates the vibrations of various regions of the resonator  24  having a clamped piezoelectric element  22  during operation of the piezoelectric motor  26  with a first signal at a first frequency. One can clearly see the distinct local variation of the directions of motion in the region of the contact portion  44 . Thus, a first point  520  on the exposed surface  522  of the contact portion  44  moves in a first direction, while a second point  521  interior to the resonator  24  but adjacent the contact portion  44 , moves in a different direction. By selecting the first point  520  and second point  521  to be sufficiently close that the second point  521  is exposed by wear of the resonator  24 , various advantages can be achieved as discussed herein. Alternatively, the second point  521  can be at a different location on the resonator  24  and can come into engagement with an element  42  to be driven by the second point  521  when the first point  520  experiences a predetermined amount of wear.  
      In one illustrative embodiment the resonator  24  comprises an elongated aluminum member having a length of  20  mm, a width of  3  mm and a height or thickness of  3  mm, with an internally mounted piezoelectric element  22 . A contacting portion  44  initially comprises the center of a square comer of the resonator (e.g.,  FIG. 1 ) and after wear-in the contacting portion the shape of the contacting portion  44  comprises a curved surface conforming to the shape of a driven rod with a circular cross-section (e.g.,  FIG. 77 ). The center of the curved surface extends inward into the resonator by about 0.5 mm or about 10-20% of the resonator height/thickness, and preferably 12-18% of that height or thickness. After operational wear, the center of the curved contacting surface  44  is about 1.5 mm from the original, square comer, and about 1 mm from the initial operational contacting surface  44 . That change is thus about 50% of the thickness or height of the distal end or contacting end of the resonator  24  for this particular illustrative embodiment. Changes of 10-60% are believed suitable, and changes of 20-35% are believed preferable, but larger and smaller changes are possible, depending on the uses and materials involved.  
      In a preferred embodiment of the piezoelectric motor assembly  20 , the driven element  42  moves in a first direction while the first point  520  is exposed to contact the driven element  42  when an electric signal is applied to the piezoelectric element  22 , and further moves in the first direction while the second point  521  is exposed to contact the driven element  42  when the same electric signal is applied to the piezoelectric element  22 .  
      Wear of the selected contact portion  44  is caused by the friction that occurs during the relative movements between the resonator and the driven element. In a preferred embodiment of the motor assembly  20 , the contact portion  44  and driven element  42  comprise materials that in combination cause the contact portion  44  to wear during use of the piezoelectric motor  26 .  
      The first point  520  on the resonator  24  is located on a first, virgin contact surface  522 , which in the cross-section of the embodiment shown in  FIG. 88   a  is located at an edge of the resonator  24 . Other shapes and locations of virgin contact surfaces  522  are possible.  FIG. 90   a,  for example, exhibits a cylindrical first, virgin contact surface  522  that is configured to closely, but not necessarily exactly, match the shape of a driven element  42 .  
       FIG. 88   b  illustrates the vibrations of the embodiment of  FIG. 88   a  after wear of the contact portion  44  due to regular motor  26  operation or due to a running-in procedure (discussed later). Wear of the contact portion  44  exposes a second, worn surface  524  in the interior of the resonator  24 , the second surface  524  comprising the second point  521  to contact the driven element  42 . In the cross-section of  FIG. 88   b,  the second, worn surface  524  appears as a chamfered edge. The second, worn surface  524  preferably is worn to closely match the shape of the driven element  42  to create a snug fit for the driving engagement of the contact portion  44  with the driven element  42 . Accordingly, the second surface  524  could comprise a cylindrical surface if it engages a cylindrical portion of a driven element  42 .  
       FIG. 88   b  shows the vibrations being the same as in  FIG. 88   a.  This need not be the case. In particular as the amount of wear increases, the vibration pattern may change in a preferably predetermined way, usually by changing the magnitude or the direction of the motion of a particular point on the resonator  24 . It is believed preferable to have the magnitude and/or direction change by 5-25%, and preferably 10-25%, but the change could be much greater, and in some cases could completely reverse direction.  
      In a preferred embodiment of the piezoelectric motor  26 , the shapes and/or the surface areas of the first and second surfaces  522 ,  524  are different In particular, the first surface  522  may indeed be only an edge of the resonator  24 .  
      In a preferred embodiment, the piezoelectric motor  26  comprises a resilient element  50  for urging the selected contact portion  44  against the driven element  42  during use of the system. The resilient element  50  is preferably configured to compensate for at least a portion of the difference between the oscillation orientations of the first and second points  520 ,  521  so that the performance of the piezoelectric motor  26  changes in a predetermined manner as the contact portion  44  wears. The resilient element  50  preferably compensates for at least a portion of the difference between the oscillation orientations of the first and second points  520 ,  521  by resiliently changing the orientation and/or the location of the resonator  24  and/or the resonator contact portion  44  with respect to the driven element  42 . In the process, the motor performance preferably remains the same for the service life of the piezoelectric motor  26  or improves for at least a portion of that service life. Performance preferably does not drop below a predetermined threshold.  
      As used herein, performance is one or more or combinations of relative speed between the driven element  42  and the resonator  24 , the driving force of the piezoelectric motor assembly  20 , the mechanical motor power, the motor efficiency, or the electric current draw by the piezoelectric element  22 .  
      Thus, in a preferred embodiment of the piezoelectric motor  26 , the resilient element  50  compensation causes the piezoelectric motor  26  to move the driven element  42  with substantially similar performance when the first point  520  is exposed to the driven element  42  than when the second point  521  is exposed to the driven element  42  after wear. In a further preferred embodiment of the piezoelectric motor  26 , the resilient element  50  compensation causes the piezoelectric motor  26  to move the driven element with better performance when the first point  520  is exposed to the driven element  42  than when the second point  521  is exposed to the driven element  42  after wear.  
      As used herein, a substantially similar or equal performance is one that is within 50% of a nominal or predetermined performance, and is preferably within 25%, and most preferably within 10% of a nominal performance.  
      In a further preferred embodiment, the resilient element  50  is configured to always hold the resonator in an orientation with respect to the driven element that causes a snug contact between the contact portion  44  and the driven element  42 . In order to achieve this, it is preferable if the resilient element  50  resiliently changes the orientation and/or the location of the resonator  24  with respect to the driven element  42  as the contact portion  44  wears. In particular, the orientation of the resonator  24  when the first contact surface  522  is in contact with the driven element  42  is different from the orientation of the resonator  24  when the second contact surface  524  is in contact with the driven element  42 . It is believed preferable to have the angle or inclination between the resonator and the driven element  42  change from 0.1° to 1° and more preferably from 1° to 10°, but the change could be greater depending on the particular application and the materials used.  
      The present description typically refers to the selected contacting portion  44  of the resonator  24  moving the driven element  42 , but the driven element  42  could be stationary and movement of the contacting portion  42  could cause the resonator  24  to move. Thus, which element moves is not necessarily dependent upon whether it is referred to as the driving element or the driven element.  
       FIG. 89   a  and  89   b  show a piezoelectric motor  26  in driving engagement with a linear and a rotary driven element  42 , respectively. In both cases, the resonator  24  is connected by a coil spring  507  to a base  52  so the spring  507  resiliently urges the contacting portion  44  toward engagement with the driven element  42 . Connecting portions  50   b  of the spring  507  are fit into recesses  551  on opposing sides of the resonator  24 . Spring portions  513 ,  514  (hidden from view) are fastened to the base  52  by mounting screws  509 . The contact portion  44  of the resonator  24  is shown to comprise a cylindrical depression with an edge portion that conforms to the toroidal or curved surface of the driven element  42 .  
       FIGS. 90   a - 90   b  show two views of a resonator  24  that comprises protrusions  607  on the end walls  31  of the resonator opening  28  in between which the piezoelectric element  22  gets installed. In the present case, the protrusions  607  are created by removing the material surrounding the protrusions  607 . But the protrusions could be formed by any suitable means, including adding material to the resonator  24  or displacing material from the resonator to form the protrusions  607 .  
       FIG. 90   a  also shows a first contact surface  522  of the contact portion  44  that has a cylindrical shape. It is thus not flat but has a spatial curvature. Other shapes comprising spatial curvatures are, e.g., saddles. A second contact surface  524  in the interior of the resonator  24  also preferably is not flat, preferably due to wear-interaction with a driven element  42 .  
       FIGS. 91   a - 91   c  show three further configurations of the resonator opening end walls  31  that are believed suitable. In  FIG. 91   a,  the end walls  31  each have an inclination  609  that facilitates the installation of the piezoelectric element  22 . The inclined walls  609  form a wider portion of the opening  28  and a narrower portion of the opening so the piezoelectric element  22  can be inserted into the wider portion of the opening  28  and pressed into the narrower portion of the opening  28 . For assembly, the piezoelectric element  22  is pushed up into the resonator  24  from below, which is indicated by the arrow V. As illustrated, the piezoelectric element  22  is preferably located so a central longitudinal axis of the piezoelectric element intersects the aligned protrusions  607 , which in this embodiment comprise the edges of the opening  28  at the narrow portion of the opening. Thus, half the piezoelectric element  22  is inside the resonator  24  and half is outside the resonator  24 . Different amounts or volumes of the piezoelectric element  22  can be inside or outside the resonator body  24 . Offsetting the piezoelectric element  22  relative to the resonator  24  provides a means for varying the vibration of the resonator  24  and of the selected contact portions  44 . The edges of the piezoelectric element  22  may be slightly tapered to make it easier to insert the piezoelectric element  22  into the narrow portion of the opening  28 .  
       FIG. 91   b  shows a resonator  24  comprising two protrusions  607 . The protrusions  607  are mutually offset in the vertical direction so that the protrusion  607  on the left is located higher than the protrusion  607  on the right. Correspondingly, different points of contact result on the piezoelectric element  22 . However, the piezoelectric element  22  is preferably horizontal and having vertical walls on the engaging portions of the protrusions  607  helps maintain the horizontal orientation of the piezoelectric element  22 .  
       FIG. 91   c  corresponds essentially to the embodiment according to  FIG. 91   b  with the exception that in this example the piezoelectric element  22  is not horizontal. This is achieved by having an inclined engaging surface on the protrusion  607  abut the opposing and parallel walls of the piezoelectric element  22 . It is believed possible to have the walls of the piezoelectric element  22  be inclined to achieve the angled orientation of the piezoelectric element  22  relative to the portions  607 , but not as desirable.  
      The protrusions  607  allow the piezoelectric element  22  to exert vibrations along axes which are not parallel to the longitudinal axis  25  of the resonator  24 . Preferably the protrusions  607  and piezoelectric element  22  are aligned so the longitudinal axis of the piezoelectric element  22  intersects the protrusions  607  and further intersects the longitudinal axis  25  of the resonator  24 , but that need not be so. The protrusions  607 , alone or in combination with an offset piezoelectric element  22 , provide means for changing the vibration orientation of the piezoelectric element relative to the resonator  24 , and preferably relative to the longitudinal axis of the resonator  24 . The protrusions  607 , alone or in combination with an offset piezoelectric element  22 , also provide means for changing the direction and/or magnitude of vibration of various portions of the resonator  24 , and of the selected contacting portions  44 .  
       FIG. 92  shows an example of a coil-type spring  507 , which can serve as a resilient element  50 . As shown, the windings  515  of the spring  507  comprise a foamed material  519  that have been pressed into the opening formed by the coil windings or has been applied to the windings  515  to completely cover them. It is preferable that the foamed material gets in between the gaps  518  that exist between the windings  515 . The foamed material  519  preferably covers at least a portion of the windings  519  and is generally located in the region of the windings  515 . The foamed material adds damping to the resilient element  50 , which is illustrated by the schematics in  FIG. 93 , where a dashpot element represents the foamed material  519  and a spring element represents the coil spring  507 . The foamed material  519  preferably dampens relative vibrations between the windings  519  and particularly prevent the windings from impacting each other or from creating friction between each other that could result in acoustical noise. Furthermore, acoustical noise can be damped by the foamed material  519  once it has been generated.  
      The above description is given by way of example, and not limitation. Given the above disclosure, one skilled in the art could devise variations that are within the scope and spirit of the invention. Further, the various features disclosed herein can be used alone, or in varying combinations with each other and are not intended to be limited to the specific combination described herein. Thus, the invention is not to be limited by the illustrated embodiments but is to be defined by the following claims when read in the broadest reasonable manner to preserve the validity of the claims.  
      Partial List of Reference Numbers:  
     
         
           22  piezoelectric element  
           24  resonator  
           26  piezoelectric motor  
           28  opening in the resonator  24   
           29  sidewalls  
           42  driven element  
           44  contact portion  
           50  resilient element  
           50   a  first end of the resilient element  50  for fastened to a base  52   
           50   b  second end of the resilient element  50  fastened to the resonator  24   
           52  base  
           506  contact element  
           507  coil spring  
           509  mounting screw  
           513  cross bar of coil spring  507   
           514  leg pieces of coil spring  507   
           515  coil windings of coil spring  507   
           520  first point on first, virgin contact surface  522   
           521  second point on second, worn surface  524   
           522  first, virgin contact surface of contact portion  44   
           524  second, worn surface of contact portion  44   
           550  chamfered, or tapered edge of the resonator opening  28   
           551  recess in the resonator  24   
           552  material bulge at recess  551   
          LF resonator  24  width in the region of the sidewalls  29  that have the shape of a circular section  
          LD width of the piezoelectric element  22   
          LC height of the piezoelectric element  22   
          LB length of the piezoelectric element  22   
          LE partial height of the piezoelectric element  22   
          LA overall length of the resonator  24   
          L 9  leg length of the resonator  24   
          L 5  leg width of the resonator  24   
          L 8  width of the recess  551   
          L 13  depth of the recess  551   
          L 17  width of the recess  551   
          L 14  distance from end face of the resonator  24  to the center of the recess  551   
          R 7  material bulge  552   
          L 16  height of the material bulge  552