Abstract:
A piezoelectric actuator comprises a plurality of piezoelectric elements stacked in a thickness direction thereof for undergoing expansion/contraction movement to vibrationally drive the piezoelectric actuator in accordance with a driving signal applied to the piezoelectric elements. Each of the piezoelectric elements has a length extending in a direction generally perpendicular to the stacking direction. The length of each of at least two of the piezoelectric elements being different from the length of at least one other of the piezoelectric elements.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to piezoelectric actuators represented by ultrasonic motors and bimorph type actuators used in clocks, cameras, printers, storage devices and the like and, more particularly, to a piezoelectric actuator whose output is improved from that available in the prior art. 
     2. Description of the Related Art 
     Piezoelectric actuators that utilize a vibration of a piezoelectric element in response to the application of a driving signal such as an AC voltage as a motive force to move a movable body are attracting attention especially in the field of micromechanics because of their high electromechanical energy conversion efficiency. 
     A description will now be made which references to FIGS. 13A,  13 B and  13 C on a piezoelectric actuator  100  which is an example of conventional piezoelectric actuators. 
     The configuration of the piezoelectric actuator  100  will now be described. 
     As shown in FIG. 13A, the piezoelectric actuator  100  is substantially comprised of a rectangular elastic plate  101  made of metal, a piezoelectric element  102  integrally stacked on one of the surfaces of the elastic plate  101  and a piezoelectric element  103  formed on the other surface of the elastic plate  101 . 
     The piezoelectric elements  102  and  103  are polarized in the direction of the thickness thereof. Referring to the polarizing direction, for example, surfaces  102   a  and  103   a  in contact with the elastic plate  101  are polarized to be negative and positive respectively, whereas surfaces  102   b  and  103   b  opposite thereto are polarized to be positive and negative respectively. That is, the piezoelectric elements  102  and  103  are polarized in opposite directions. 
     An electrode is provided on each of the surfaces  102   b  and  103   b  to substantially cover the entire surface. The elastic plate  101  serves as an electrode for the surfaces  102   a  and  103   a.    
     An operation of the piezoelectric actuator  100  will now be described. 
     As shown in FIG. 13A, a voltage is first applied with the electrodes on the surfaces  102   b  and  103   b  serving as the negative pole and the elastic plate  101  serving as the positive pole. 
     The piezoelectric element  102  expands in the longitudinal direction because the voltage is applied in the direction opposite to the polarizing direction of the surfaces  102   b  and  102   a.    
     The piezoelectric element  103  contracts in the longitudinal direction because the voltage is applied in the same direction as the polarizing direction of the surfaces  103   a  and  102   b.    
     As a result, the piezoelectric actuator  100  is bent in the direction indicated by the arrow X in FIG. 13B, which generates a driving force to move the movable body (not shown) in the bending direction. 
     When a voltage is applied with the surfaces  102   b  and  103   b  as the positive pole and the surfaces  102   a  and  103   a  as the negative pole, the piezoelectric actuator  100  is bent in the direction opposite to the arrow X, which generates a driving force to move the movable body in the direction opposite to the direction shown in FIG.  13 B. 
     However, upper limits have existed for the output and displacement of the piezoelectric actuator  100  because it is formed by simply forming one each piezoelectric element  102 ,  103  on both sides of the elastic plate  101  integrally. 
     As a technique to improve the piezoelectric actuator  100 , a piezoelectric actuator  110  as shown in FIG. 13C has been provided in which piezoelectric elements  104  and  105  identical in configuration to the piezoelectric elements  102  and  103  are formed on the piezoelectric elements  102  and  103 , respectively. However, increases in output or displacement was smaller than expected from the magnitude of the voltage, i.e., electric power input thereto. The output or displacement of the piezoelectric actuator  110  could be smaller than the output of the piezoelectric actuator  100  depending on the conditions. 
     The inventors identified a cause for the above-mentioned problem with the piezoelectric actuator  110  as follows. The same piezoelectric element as the piezoelectric element  102  is used as the piezoelectric element  104 , which results in the same amount of expansion in spite of the fact that the expansion of the piezoelectric element  104  must be greater than the expansion of the piezoelectric element  102  because it is located further than the elastic plate  101  having a distortion-neutral plane. The same piezoelectric element as the piezoelectric element  103  is used as the piezoelectric element  105 , which results in the same amount of contraction in spite of the fact that the contraction of the piezoelectric element  105  must be greater than the contraction of the piezoelectric element  103  because it is located further than the elastic plate  101 . 
     That is, the piezoelectric element  104  has hindered the expansion of the piezoelectric element  102 , and the piezoelectric element  105  has hindered the contraction of the piezoelectric element  103 . 
     The invention has been conceived based on the above-described idea, and it is an object of the invention to provide a piezoelectric actuator which transmits a driving force of a plurality piezoelectric elements to the outside without loss. 
     SUMMARY OF THE INVENTION 
     In order to solve the above problem, according to one aspect of the invention, there is provided a piezoelectric actuator which is distorted according to an input driving signal to generate a driving force, characterized in that it is formed by integrally stacking a plurality of piezoelectric elements such that they do not hinder the operation of each other. 
     In the above-described aspect of the invention, for example, the piezoelectric actuator is a bimorph type actuator or an ultrasonic motor. 
     There is no limitation on the material of the piezoelectric elements. 
     Further, the thickness of each of the plurality of piezoelectric elements is appropriately adjusted in accordance with the operation and position of the piezoelectric element. Basically, a piezoelectric element is made thinner, the greater the distortion it must undergo. All of the plurality of piezoelectric elements may be different in thickness and, alternatively, some of them may have the same thickness. 
     In this aspect of the invention, by adjusting the thickness of the piezoelectric elements depending on the operations and positions of the piezoelectric elements, all of the piezoelectric elements contribute to the operation of the piezoelectric actuator without interfering with each other. It is therefore possible to fabricate a piezoelectric actuator which provides output greater than that available in the prior art with the same power consumption, which can be made smaller in size than that in the prior art having the same output and which consumes less power. 
     According to the invention, there is provided a piezoelectric actuator as described above, characterized in that the thickness of piezoelectric elements located on the side of the actuator with smaller distortion is larger than the thickness of piezoelectric elements located on the side thereof with greater distortion. 
     In this aspect of the invention, the thickness of each piezoelectric element is smaller, the further the piezoelectric element from a distortion-neutral plane of the actuator. Therefore, all of the piezoelectric elements contribute to the operation of the piezoelectric actuator without interfering with each other. It is therefore possible to fabricate a piezoelectric actuator which provides output greater than that available in the prior art with the same power consumption, which can be made smaller in size than that in the prior art having the same output and which consumes less power. 
     According to the invention, there is provided a piezoelectric actuator as described above, characterized in that at least two of the plurality of piezoelectric elements undergo identical vibrations. 
     All of the plurality of piezoelectric elements may undergo identical vibrations. 
     When the piezoelectric actuator is an ultrasonic motor, for example, the identical vibrations may be longitudinal vibrations, bending vibrations or torsional vibrations. 
     In this aspect of the invention, since at least part of the plurality of piezoelectric elements undergo identical vibrations, the vibrations are greater in magnitude than those in the prior art. It is therefore possible to fabricate a piezoelectric actuator which provides output greater than that available in the prior art with the same power consumption, which can be made smaller in size than that in the prior art having the same output and which consumes less power. 
     According to the invention, there is provided a piezoelectric actuator as described above, characterized in that the plurality of piezoelectric elements are stacked in a direction in parallel with a driving force extracting portion of the piezoelectric actuator. 
     In this aspect of the invention, the same effect as that described above is achieved. 
     According to the invention, there is provided a piezoelectric actuator as described above, characterized in that it is an ultrasonic motor which utilizes a composite vibration resulting from two different kinds of vibrations generated at the piezoelectric elements as a driving force and in that the two different kinds of vibrations are excited by separate piezoelectric elements. 
     The two different kinds of vibrations are, for example, a torsional vibration and an expansion vibration, although not limited to them. 
     Further, there are normally a plurality of piezoelectric elements for exciting each kind of vibration, and the thickness of them is adjusted such that each vibration does not interfere with the vibration, i.e., distortion of other piezoelectric elements. 
     In this aspect of the invention, in addition to the effect as described above, the adjustment of the thicknesses of the plurality of piezoelectric elements makes it possible to optimize the ratio of the magnitudes of the two different kinds of vibrations. 
     According to the invention, there is provided a piezoelectric actuator as described above, characterized in that it includes a piezoelectric element for detecting vibrations and in that the piezoelectric element for detecting vibrations is different in thickness from the other piezoelectric elements. Since driving piezoelectric elements are provided in a region which is greatly distorted, the detecting capability is higher, the smaller the thickness of the detecting piezoelectric element which is provided in a region having smaller distortion is. 
     In this aspect of the invention, the piezoelectric element for detection vibrations does not hinder the distortion of piezoelectric elements used as a source of a driving force and has higher detecting capability. This improves the accuracy of control over a piezoelectric actuator. 
     According to the invention, there is provided a piezoelectric actuator as described above, characterized in that it is an ultrasonic motor in which the thickness of the plurality of piezoelectric elements is equal to the thickness of a vibrating element integrally stacked on the plurality of piezoelectric elements. 
     In this aspect of the invention, since a driving force generated at the plurality of piezoelectric elements is transmitted to the vibrating element with highest efficiency, it is possible to fabricate an ultrasonic motor which provides output greater than that available in the prior art with the same power consumption, which can be made smaller in size than that in the prior art having the same output and which consumes less power. 
     According to the invention, there is provided an electronic apparatus having a piezoelectric actuator as described above. 
     For example, the electronic apparatus is an electronic clock, measuring apparatus, camera, printer, machine tool, robot, transfer apparatus, storage apparatus or the like. 
     In this aspect of the invention, an ultrasonic motor as described above is used which provides greater output with low power compared to conventional ultrasonic motors. Since this makes it possible to make an ultrasonic motor compact, an electronic apparatus with an ultrasonic motor can be provided which is compact and consumes less power. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a sectional view showing a configuration of a piezoelectric actuator  1  which is a first embodiment of the invention. 
     FIG. 2 is a sectional view showing a configuration of a piezoelectric actuator  2  which is a second embodiment of the invention. 
     FIGS. 3A through 3D are schematic views illustrating a configuration and operation of a piezoelectric actuator  3  which is a third embodiment of the invention. 
     FIGS. 4A and 4B are schematic views illustrating a configuration of an ultrasonic motor  4  which is a fourth embodiment of the invention. 
     FIG. 5 is a schematic view illustrating an operation of the ultrasonic motor  4 . 
     FIGS. 6A and 6B are schematic views respectively illustrating configurations of ultrasonic motor motors  5  and  6  which are modifications of the ultrasonic motor  4 . 
     FIGS. 7A through 4D are schematic views illustrating a configuration of an ultrasonic motor  7  which is a fifth embodiment of the invention. 
     FIG. 8 is a schematic view illustrating an operation of the ultrasonic motor  7 . 
     FIG. 9 is a schematic view illustrating a configuration of an ultrasonic motor  7   a  which is a modification of the ultrasonic motor  7 . 
     FIGS. 10A through 10E are schematic views illustrating a configuration of an ultrasonic motor  8  which is a sixth embodiment of the invention. 
     FIG. 11A through 11C are schematic views illustrating an operation of the ultrasonic motor  8 . 
     FIG. 12 is a block diagram illustrating a configuration of an electronic apparatus  9  with a piezoelectric actuator which is a seventh embodiment of the invention. 
     FIGS. 13A through 13C are schematic views illustrating a configuration and operation of a piezoelectric actuator  100  which is an example of a conventional piezoelectric actuator. 
    
    
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
     Preferred embodiments of the invention will now be described in detail with reference to FIGS. 1 through 12. 
     FIG. 1 is a sectional view showing a configuration of a piezoelectric actuator  1  which is a first embodiment of the invention. 
     First, the configuration of the piezoelectric actuator  1  will be described. 
     The piezoelectric actuator  1  is substantially comprised of a rectangular piezoelectric element  11 , a rectangular piezoelectric element  12  integrally stacked on the top surface of the piezoelectric element  11 , a rectangular piezoelectric element  13  integrally stacked on the top surface of the piezoelectric element  12 , a rectangular piezoelectric element  14  integrally stacked on the bottom surface of the piezoelectric element  11 , a rectangular piezoelectric element  15  integrally stacked on the bottom surface of the piezoelectric element  14  and a rectangular piezoelectric element  16  integrally formed on the bottom surface of the piezoelectric element  15 . One of the ends of the piezoelectric actuator  1  is a fixed end and the other is a free end. 
     An electrode  17   a  is provided at the interface between the piezoelectric elements  11  and  12 . An electrode  17   b  is provided at the interface between the piezoelectric elements  12  and  13 . An electrode  17   c  is provided on surface of the piezoelectric element  13  opposing the electrode  17   b . An electrode  17   d  is provided at the interface (a distortion-neutral plane) between the piezoelectric elements  11  and  14 . An electrode  17   e  is provided at the interface between the piezoelectric elements  14  and  15 . An electrode  17   f  is provided at the interface between the piezoelectric elements  15  and  16 . An electrode  17   g  is provided on surface of the piezoelectric element  15  opposing the electrode  17   b . The electrodes are provided so as to cover the respective surfaces excluding the peripheral regions. 
     The electrodes  17   a ,  17   c ,  17   e  and  17   g  are in conduction to each other, and the electrodes  17   b ,  17   d  and  17   f  are in conduction to each other. 
     For example, the piezoelectric element  11  is made of barium titanate and is polarized in the stacking direction to provide the positive polarity on the surface thereof in contact with the electrode  17   a  and the negative polarity on the surface thereof in contact with the electrode  17   d.    
     The piezoelectric element  12  is made of the same material as that of the piezoelectric element  11 , identical to the piezoelectric element  11  in the configuration of the stacking surface and is thinner than the piezoelectric element  11 . It is polarized in the stacking direction to provide the positive polarity on the surface thereof in contact with the electrode  17   a  and the negative polarity on the surface thereof in contact with the electrode  17   b.    
     The piezoelectric element  13  is made of the same material as that of the piezoelectric element  11 , identical to the piezoelectric element  12  in the configuration of the stacking surface and is thinner than the piezoelectric element  12 . It is polarized in the stacking direction to provide negative polarity on the surface thereof in contact with the electrode  17   b  and the positive polarity on the surface thereof in contact with the electrode  17   c.    
     The piezoelectric element  14  has the substantially same configuration as that of the piezoelectric element  11  and is made of the same material as that of the piezoelectric element  11 . It is polarized in the stacking direction to provide the positive polarity on the surface thereof in contact with the electrode  17   d  and the negative polarity on the surface thereof in contact with the electrode  17   e.    
     The piezoelectric element  15  has the substantially same configuration as that of the piezoelectric element  12  and is made of the same material as that of the piezoelectric element  11 . It is polarized in the stacking direction to provide the negative polarity on the surface thereof in contact with the electrode  17   e  and the positive polarity on the surface thereof in contact with the electrode  17   f.    
     The piezoelectric element  16  has the substantially same configuration as that of the piezoelectric element  13  and is made of the same material as that of the piezoelectric element  11 . It is polarized in the stacking direction to provide the positive polarity on the surface thereof in contact with the electrode  17   f  and the negative polarity on the surface thereof in contact with the electrode  17   g.    
     An operation of the piezoelectric actuator  1  will now be described. 
     The discussion will refer to a case in which a voltage is applied to the piezoelectric actuator  1  with the electrodes  17   a ,  17   c ,  17   e  and  17   g  serving as the positive pole and the electrodes  17   b ,  17   d  and  17   f  serving as the negative pole. 
     The piezoelectric element  11  contracts in the longitudinal direction because the surface thereof having the positive polarity is in contact with the electrode  17   a , i.e., the positive pole and the surface thereof having the negative polarity is in contact with the electrode  17   d , i.e., the negative pole. 
     Similarly, the piezoelectric element  12  contracts in the longitudinal direction because the surface thereof having the positive polarity is in contact with the electrode  17   a , i.e., the positive pole and the surface thereof having the negative polarity is in contact with the electrode  17   b , i.e., the negative pole. 
     The piezoelectric element  12  contracts more than the piezoelectric element  11  in spite of the fact that it is applied with the same voltage as that of the piezoelectric element  11 , because it is thinner than the piezoelectric element  11 . 
     Similarly, the piezoelectric element  13  contracts in the longitudinal direction because the surface thereof having the negative polarity is in contact with the electrode  17   b , i.e., the negative pole and the surface thereof having the positive polarity is in contact with the electrode  17   c , i.e., the positive pole. 
     The piezoelectric element  13  contracts more than the piezoelectric element  12  in spite of the fact that it is applied with the same voltage as that of the piezoelectric element  12  because it is thinner than the piezoelectric element  12 . 
     The piezoelectric element  14  expands in the longitudinal direction because the surface thereof having the positive polarity is in contact with the electrode  17   d , i.e., the negative pole and the surface thereof having the negative polarity is in contact with the electrode  17   e , i.e., the positive pole. 
     Similarly, the piezoelectric element  15  expands in the longitudinal direction because the surface thereof having the positive polarity is in contact with the electrode  17   e , i.e., the negative pole and the surface thereof having the negative polarity is in contact with the electrode  17   f , i.e., the positive pole. 
     The piezoelectric element  15  expands more than the piezoelectric element  14  in spite of the fact that it is applied with the same voltage as that of the piezoelectric element  14  because it is thinner than the piezoelectric element  14 . 
     Similarly, the piezoelectric element  16  expands in the longitudinal direction because the surface thereof having the negative polarity is in contact with the electrode  17   f , i.e., the positive pole and the surface thereof having the positive polarity is in contact with the electrode  17   g , i.e., the negative pole. 
     The piezoelectric element  16  expands more than the piezoelectric element  15  in spite of the fact that it is applied with the same voltage as that of the piezoelectric element  15  because it is thinner than the piezoelectric element  15 . 
     As a result, the piezoelectric elements  11 ,  12  and  13  of the piezoelectric actuator  1  contract, and the piezoelectric elements  14 ,  15  and  16  expand with the electrode  17   d  being a distortion neutral plane. This results in a driving force in the direction of the arrow X shown in FIG.  1 . Among the piezoelectric elements  11 ,  12  and  13 , the piezoelectric element  13  located furthest from the electrode  17   d  which is a distortion-neutral plane undergoes the greatest contraction, and the piezoelectric element  11  located directly above the electrode  17   d  undergoes the smallest contraction. As a result, the piezoelectric element  12  does not interfere with the contraction of the piezoelectric element  11  and thus increases the driving force of the piezoelectric actuator  1 , and the piezoelectric element  13  does not interfere with the contraction of the piezoelectric elements  11  and  12  and thus increases the driving force of the piezoelectric actuator  1 . 
     Similarly, among the piezoelectric elements  14 ,  15  and  16 , the piezoelectric element  16  located furthest from the electrode  17   d  undergoes the greatest expansion, and the piezoelectric element  14  located directly under the electrode  17   d  undergoes the smallest expansion. As a result, the piezoelectric element  15  does not interfere with the expansion of the piezoelectric element  14  and thus increases the driving force of the piezoelectric actuator  1 , and the piezoelectric element  16  does not interfere with the expansion of the piezoelectric elements  14  and  15  and thus increases the driving force and displacement of the piezoelectric actuator  1 . 
     When a voltage is applied to the piezoelectric actuator  1  with the electrodes  17   a ,  17   c ,  17   e  and  17   g  conversely serving as the negative pole and the electrodes  17   b ,  17   d  and  17   f  conversely serving as the positive pole, the piezoelectric elements  11 ,  12  and  13  expand and the piezoelectric elements  14 ,  15  and  16  contract with the electrode  17   d  being a distortion-neutral plane. This results in a driving force in the direction opposite to the arrow X. 
     Among the piezoelectric elements  11 ,  12  and  13 , the piezoelectric element  13  located furthest from the electrode  17   d  undergoes the greatest expansion, and the piezoelectric element  11  located directly above the electrode  17   d  undergoes the smallest expansion. As a result, the piezoelectric element  12  does not interfere with the expansion of the piezoelectric element  11  and thus increases the driving force of the piezoelectric actuator  1 , and the piezoelectric element  13  does not interfere with the expansion of the piezoelectric elements  11  and  12  and thus increases the driving force and displacement of the piezoelectric actuator  1 . 
     Similarly, among the piezoelectric elements  14 ,  15  and  16 , the piezoelectric element  16  located furthest from the electrode  17   d  undergoes the greatest contraction, and the piezoelectric element  14  located directly under the electrode  17   d  undergoes the smallest contraction. As a result, the piezoelectric element  15  does not interfere with the contraction of the piezoelectric element  14  and thus increases the driving force of the piezoelectric actuator  1 , and the piezoelectric element  16  does not interfere with the contraction of the piezoelectric elements  14  and  15  and thus increases the driving force of the piezoelectric actuator  1 . 
     As described above, in the piezoelectric actuator  1  which is an embodiment of the invention, integrally stacked on the top side of the piezoelectric element  11  are the piezoelectric element  12  which is thinner than the piezoelectric element  11  and which expands and contracts in the same direction as that of the piezoelectric element  11  at the same voltage and the piezoelectric element  13  which is thinner than the piezoelectric element  12  and which expands and contracts in the same direction as that of the piezoelectric element  11  at the same voltage. Integrally stacked on the bottom side of the piezoelectric element  11  are the piezoelectric element  14  which expands and contracts oppositely to the piezoelectric element  11  in direction at the same voltage, the piezoelectric element  15  which is thinner than the piezoelectric element  14  and which expands and contracts in the same direction as that of the piezoelectric element  14  at the same voltage and the piezoelectric element  16  which is thinner than the piezoelectric element  15  and which expands and contracts in the same direction as the piezoelectric element  14  at the same voltage. Therefore, the expansion and contraction of each of the piezoelectric elements  11 ,  12 ,  13 ,  14 ,  15  and  16  contributes to the driving force without interfering with the expansion and contraction of other piezoelectric elements. 
     Therefore, the piezoelectric actuator  1  has a simple structure and has higher output and efficiency than that available in the prior art, so that its size and power consumption can be smaller compared to those of conventional devices having the same output. 
     Any modification may be made on the present embodiment as long as it does not depart from the principle of the invention. 
     For example, any piezoelectric material may be used for the piezoelectric elements  11  through  16 . 
     The optimum ratio between the thicknesses of the piezoelectric elements  11  through  16  is not uniquely determined, and it is rather determined by a plurality of factors such as the electromechanical coupling coefficient of the piezoelectric material and the surface area of the stacking surfaces of the piezoelectric elements  11  through  16 . 
     The polarizing direction of each of the piezoelectric elements and the structure of the electrodes are not limited to the present embodiment, and any modification is possible as long as they undergo expansion and contraction in the respective same directions which are separated at the distortion-neutral plane. 
     It is not necessary to stack the same number of piezoelectric elements on both sides of the neutral plane as long as a plurality of piezoelectric elements are provided on each side. Especially, when the number of stacked layers is three or more, it is not necessary that all of the piezoelectric elements are different in thickness, and the same effect can be achieved even if some of them have the same thickness. For example, the reverse of the above-described effect can be achieved by attaching a weight to the free end, to provide an acceleration sensor or force sensor that outputs signals with a great magnitude. 
     FIG. 2 is a sectional view showing a configuration of a piezoelectric actuator  2  which is a second embodiment of the invention. 
     The configuration of the piezoelectric actuator  2  will now be described. 
     The piezoelectric actuator  2  is substantially comprised of a first group of six rectangular piezoelectric elements  21   a,    21   b,    21   c,    21   d,    21   e  and  21   f  integrally arranged in the longitudinal direction to form one rectangular element, a second group of twelve rectangular piezoelectric elements  22   a,    22   b,    22   c,    22   d,    22   e,    22   f,    22   g,    22   h,    22   i,    22   j,    22   k  and  22   l  integrally stacked on the top surfaces of the six piezoelectric elements  21   a  through  21   f  having a length in the longitudinal direction which is one half of that of the piezoelectric elements  21   a  through  21   f,  a third group of six piezoelectric elements  23   a,    23   b,    12   c,    23   d,    23   e  and  23   f  having the same configuration as the piezoelectric elements  21   a  through  21   f  integrally stacked on the bottom surfaces of the piezoelectric elements  21   a  through  21   f,  and a fourth group of twelve rectangular piezoelectric elements  24   a,    24   b,    24   c,    24   d,    24   e,    24   f,    24   g,    24   h,    24   i,    24   j,    24   k  and  24   l  having the same configuration as the piezoelectric elements  22   a  through  22   l  integrally stacked on the bottom surfaces of the six piezoelectric elements  23   a  through  23   f.  One end of the element is a fixed end, and the other is a free end. 
     In addition, the bottom surfaces of the piezoelectric elements  21   a  through  21   f  act as a distortion-neutral plane in the context of the present invention. 
     For example, the piezoelectric elements  21   a  through  21   f ,  22   a  through  22   l ,  23   a  through  23   f  and  24   a  through  24   l  are made of barium titanate or lead zirconate titanate and are polarized in the longitudinal direction. The piezoelectric elements have the same thickness in the stacking direction. 
     The piezoelectric elements  21   a ,  21   c  and  21   e  are polarized to have the positive polarity at the fixed end and the negative polarity at the free end. The piezoelectric elements  21   b ,  21   d  and  21   f  are polarized to have the negative polarity at the fixed end and the positive polarity at the free end. 
     The piezoelectric elements  22   a ,  22   b ,  22   e ,  22   f ,  22   i  and  22   j  are polarized to have the positive polarity at the fixed end and the negative polarity at the free end. The piezoelectric elements  22   c ,  22   d ,  22   g ,  22   h ,  22   k  and  22   l  are polarized to have the negative polarity at the fixed end and the positive polarity at the free end. 
     The piezoelectric elements  23   a ,  23   c  and  23   e  are polarized to have the negative polarity at the fixed end and the positive polarity at the free end. The piezoelectric elements  23   b ,  23   d  and  23   f  are polarized to have the positive polarity at the fixed end and the negative polarity at the free end. 
     The piezoelectric elements  24   a ,  24   b ,  24   e ,  24   f ,  24   i  and  24   j  are polarized to have the negative polarity at the fixed end and the positive polarity at the free end. The piezoelectric elements  24   c ,  24   d ,  24   g ,  24   h ,  24   k  and  24   l  are polarized to have the positive polarity at the fixed end and the negative polarity at the free end. 
     Further, at the interfaces between the adjoining piezoelectric elements  21   a  through  21   f , electrodes  25   a ,  25   b ,  25   c ,  25   d  and  25   e  (listed in the order of closeness to the fixed end) are sequentially provided such that they extend to substantially cover the entity of respective interfaces between the piezoelectric elements  22   a  through  22   l , piezoelectric elements  23   a  through  23   f  and piezoelectric elements  24   a  through  24   l.    
     Electrodes  25   f  and  25   g  are provided to substantially cover the entirety of the fixed end faces of the piezoelectric elements  21   a ,  22   a ,  23   a  and  24   a  and the free end faces  21   f ,  22   l ,  23   f  and  24   l , respectively. 
     Electrodes  26   a ,  26   c ,  26   e ,  26   g ,  26   i  and  26   k  are respectively provided on the sides of the piezoelectric elements  22   a ,  22   c ,  22   e ,  22   g ,  22   i  and  22   k  toward the free end. Electrodes  26   b ,  26   d ,  26   f ,  26   h ,  26   j  and  26   l  are respectively provided on the sides of the piezoelectric elements  22   b ,  22   d ,  22   f ,  22   h ,  22   j  and  22   l  toward the fixed end. 
     Electrodes  27   a ,  27   c ,  27   e ,  27   g ,  27   i  and  27   k  are respectively provided on the sides of the piezoelectric elements  24   a ,  24   c ,  24   e ,  24   g ,  24   i  and  24   k  toward the free end. Electrodes  27   b ,  27   d ,  27   f ,  27   h ,  27   j  and  27   l  are respectively provided on the sides of the piezoelectric elements  24   b ,  24   d ,  24   f ,  24   h ,  24   j  and  24   l  toward the fixed end. 
     The electrodes  25   a ,  25   c ,  25   e ,  26   a ,  26   d ,  26   e ,  26   h ,  26   i ,  26   l ,  27   a ,  27   d ,  27   e ,  27   h ,  27   i  and  27   l  are in conduction to each other, and the electrodes  25   b ,  25   d ,  25   f ,  25   g ,  26   b ,  26   c ,  26   f ,  26   g ,  26   j ,  26   k ,  27   b ,  27   c ,  27   f ,  27   g ,  27   j  and  27   k  are in conduction to each other. 
     An operation of the piezoelectric actuator  2  will now be described. 
     The discussion will refer to a case in which a voltage is applied to the piezoelectric actuator  2  with the electrodes  25   a ,  25   c ,  25   e ,  26   a ,  26   d ,  26   e ,  26   h ,  26   i ,  26   l ,  27   a ,  27   d ,  27   e ,  27   h ,  27   i  and  27   l  serving as the negative pole and the electrodes  25   b ,  25   d ,  25   f ,  25   g ,  26   b ,  26   c ,  26   f ,  26   g ,  26   j ,  26   k ,  27   b ,  27   c ,  27   f ,  27   g ,  27   j  and  27   k  serving as the positive pole. 
     The piezoelectric elements  21   a ,  21   b ,  21   c ,  21   d ,  21   e  and  21   f  contract in the longitudinal direction because the positively polarized surfaces thereof are respectively in contact with the electrodes  25   f ,  25   b ,  25   d  and  25   g , i.e., the positive poles and the negatively polarized surfaces thereof are respectively in contact with the electrodes  25   a ,  25   c  and  25   e , i.e., the negative poles. 
     The piezoelectric elements  22   a ,  22   b ,  22   c ,  22   d ,  22   e ,  22   f ,  22   g ,  22   h ,  22   i ,  22   j ,  22   k  and  22   l  contract in the longitudinal direction because the positively polarized surfaces thereof are respectively in contact with the electrodes  25   f ,  26   b ,  26   c ,  25   b ,  25   b ,  26   f ,  26   g ,  25   d ,  25   d .  26   j ,  26   k , and  25   g , i.e., the positive poles and the negatively polarized surfaces thereof are respectively in contact with the electrodes  26   a ,  25   a ,  25   a ,  26   d ,  26   e ,  25   c ,  25   c ,  26   h ,  26   i ,  25   e ,  25   e  and  26   l , i.e., the negative poles. 
     Since the length of the piezoelectric elements  22   a  through  22   l  in the longitudinal direction, i.e., the direction in which the voltage is applied is one half of that of the piezoelectric elements  21   a  through  21   f,  they contract in an amount which is about twice the contraction of the piezoelectric elements  21   a  through  21   f  in spite of the fact that the same voltage is applied to them. 
     The piezoelectric elements  23   a,    23   b,    23   c,    23   d,    23   e  and  23   f  expand in the longitudinal direction because the positively polarized surfaces thereof are respectively in contact with the electrodes  28   a,    28   c  and  28   e,  i.e., the negative poles and the negatively polarized surfaces thereof are respectively in contact with the electrodes  28   f,    28   b,    28   d,    28   g,  i.e., the positive poles. 
     The piezoelectric elements  24   a ,  24   b ,  24   c ,  24   d ,  24   e ,  24   f ,  24   g ,  24   h ,  24   i ,  24   j ,  24   k  and  24   l  expand in the longitudinal direction because the positively polarized surfaces thereof are respectively in contact with the electrodes  27   a ,  28   a ,  28   a ,  27   d ,  27   e ,  28   c ,  28   c ,  27   h ,  27   i ,  28   e ,  28   e  and  27   l , i.e., the negative poles and the negatively polarized surfaces thereof are respectively in contact with the electrodes  28   f ,  27   b ,  27   c ,  28   b ,  28   b ,  27   f ,  27   g ,  28   d ,  28   d ,  27   j ,  27   k  and  28   g , i.e., the positive poles. 
     Since the length of the piezoelectric elements  24   a  through  24   l  in the longitudinal direction, i.e., the direction in which the voltage is applied is one half of that of the piezoelectric elements  23   a  through  23   f , they expand in an amount which is about twice the expansion of the piezoelectric elements  23   a  through  23   f  in spite of the fact that the same voltage is applied to them. 
     As a result, the piezoelectric elements  21   a  through  21   f  and  22   a  through  22   l  of the piezoelectric actuator  2  contract with the bottom surfaces of the piezoelectric elements  21   a  through  21   f  being a distortion-neutral plane, and the piezoelectric elements  23   a  through  23   f  and  24   a  through  24   l  expand with the bottom surfaces of the piezoelectric elements  23   a  through  23   f  being a distortion-neutral plane. This results in a driving force in the direction of the arrow X shown in FIG.  2 . 
     Since the piezoelectric elements  22   a  through  22   l  located further from the bottom surfaces of the piezoelectric elements  21   a  through  21   f  undergo contraction greater than the contraction of the piezoelectric elements  21   a  through  21   f , the piezoelectric elements  22   a  through  22   l  do not interfere with the contraction of the piezoelectric elements  21   a  through  21   f  and thus increase the driving force and displacement of the piezoelectric actuator  2 . 
     Similarly, since the piezoelectric elements  24   a  through  24   l  located further from the upper surfaces of the piezoelectric elements  23   a  through  23   f  undergo expansion greater than the expansion of the piezoelectric elements  23   a  through  23   f , the piezoelectric elements  24   a  through  24   l  do not interfere with the expansion of the piezoelectric elements  23   a  through  23   f  and thus increase the driving force and displacement of the piezoelectric actuator  2 . 
     When a voltage is applied to the piezoelectric actuator  2  with the electrodes  25   a ,  25   c ,  25   e ,  26   a ,  26   d ,  26   e ,  26   h ,  26   i ,  26   l ,  27   a ,  27   d ,  27   e ,  27   h ,  27   i  and  27   l  conversely serving as the positive pole and the electrodes  25   b ,  25   d ,  25   f ,  25   g ,  26   b ,  26   c ,  26   f ,  26   g ,  26   j ,  26   k ,  27   b ,  27   c ,  27   f ,  27   g ,  27   j  and  27   k  conversely serving as the negative pole, the piezoelectric elements  21   a  through  21   f  and  22   a  through  22   l  expand with the bottom surfaces of the piezoelectric elements  21   a  through  21   f  being as a distortion-neutral plane, and the piezoelectric elements  23   a  through  23   f  and  24   a  through  24   l  contract with the upper surfaces of the piezoelectric elements  23   a  through  23   f  being a distortion-neutral plane. This results in a driving force in the direction opposite to the arrow X. 
     Since the piezoelectric elements  22   a  through  22   l  located further from the bottom surfaces of the piezoelectric elements  21   a  through  21   f  undergo expansion greater than the expansion of the piezoelectric elements  21   a  through  21   f , the piezoelectric elements  22   a  through  22   l  do not interfere with the expansion of the piezoelectric elements  21   a  through  21   f  and thus increase the driving force and displacement of the piezoelectric actuator  2 . 
     Similarly, since the piezoelectric elements  24   a  through  24   l  located further from the bottom surfaces of the piezoelectric elements  23   a  through  23   f  undergo contraction greater than the contraction of the piezoelectric elements  23   a  through  23   f , the piezoelectric elements  24   a  through  24   l  do not interfere with the contraction of the piezoelectric elements  23   a  through  23   f  and thus increase the driving force and displacement of the piezoelectric actuator  2 . 
     As described above, on the top side of the six piezoelectric elements  21   a  through  21   f  integrally arranged in the longitudinal direction of the piezoelectric actuator  2  which is an embodiment of the invention, the twelve piezoelectric elements  22   a  through  22   l  are integrally stacked which are one half of the piezoelectric elements  21   a  through  21   f  in the longitudinal length and which expand and contract in the same direction as that of the piezoelectric elements  21   a  through  21   f  at the same voltage. Integrally stacked on the bottom side of the piezoelectric elements  21   a  through  21   f  are the six piezoelectric elements  23   a  through  23   f  which contract and expand oppositely to the piezoelectric elements  21   a  through  21   f  in direction at the same voltage and the twelve piezoelectric elements  24   a  through  24   l  which are one half of the piezoelectric elements  23   a  through  23   f  in the longitudinal length and which contract and expand in the same direction as that of the piezoelectric elements  23   a  through  23   f  at the same voltage. Therefore, the expansion and contraction of each of the piezoelectric elements  21   a  through  21   f ,  22   a  through  22   l ,  23   a  through  23   f  and  24   a  through  24   l  contributes to the driving force and displacement without interfering with the expansion and contraction of other piezoelectric elements. 
     Therefore, the piezoelectric actuator  2  has a simple structure and has higher output and efficiency than that available in the prior art, its size and power consumption can be smaller compared to those of conventional devices having the same output. 
     Any modification may be made on the present embodiment as long as it does not depart from the principle of the invention. 
     For example, any piezoelectric material may be used for the piezoelectric elements  21   a  through  21   f ,  22   a  through  22   l ,  23   a  through  23   f  and  24   a  through  24   l.    
     It is not essential that the length of the piezoelectric elements  22   a  through  22   l  in the longitudinal direction is one half of the length of the piezoelectric elements  21   a  through  21   f , and they are only required to be shorter than the piezoelectric elements  21   a  through  21   f . It is not essential that all of the piezoelectric elements  21   a  through  21   f  have the same length and that all of the piezoelectric elements  22   a  through  22   l  have the same length. 
     Similarly, it is not essential that the length of the piezoelectric elements  24   a  through  24   l  in the longitudinal direction is one half of the length of the piezoelectric elements  23   a  through  23   f , and they are only required to be shorter than the piezoelectric elements  23   a  through  23   f . It is not essential that all of the piezoelectric elements  23   a  through  23   f  have the same length and that all of the piezoelectric elements  24   a  through  24   l  have the same length. 
     The optimum ratio between the length of the piezoelectric elements is not uniquely determined, and it is rather determined by a plurality of factors such as the electromechanical coupling coefficient of the piezoelectric material and the surface area of the stacking surface of each piezoelectric element. 
     It is not essential that all of the piezoelectric elements  21   a  through  21   f ,  22   a  through  22   l ,  23   a  through  23   f  and  24   a  through  24   l  have the same thickness, and the piezoelectric elements  22   a  through  22   l  and the piezoelectric elements  23   a  through  23   f  may be thinner than the piezoelectric elements  21   a  through  21   f  and the piezoelectric elements  24   a  through  24   l , respectively. 
     The polarizing direction of each of the piezoelectric elements and the structure of the electrodes are not limited to the present embodiment, and any modification is possible as long as they undergo expansion and contraction in the respective same directions which are separated at the distortion-neutral plane. 
     It is not necessary to stack the same number of piezoelectric elements on both sides of the neutral plane as long as a plurality of piezoelectric elements are provided on each side. Especially, when the number of stacked layers is three or more, it is not necessary that the lengths of the piezoelectric elements in the longitudinal direction are all different depending of the stacking positions, and the same effect can be achieved even if some of them have the same thickness. For example, the reverse of the above-described effect can be achieved by attaching a weight to the free end to provide an acceleration sensor or force sensor that outputs signals with a great magnitude. 
     FIGS. 3A through 3D are schematic views showing a structure and operation of an ultrasonic motor  3  having a piezoelectric actuator according to a third embodiment of the invention. FIG. 3A is a schematic front view of the ultrasonic motor  3 . FIG. 3B is a schematic view showing a stacking structure of the ultrasonic motor  3 . FIG. 3C is a schematic view illustrating an operation of the ultrasonic motor  3 . 
     The structure of the ultrasonic motor  3  will now be described. 
     As shown in FIG. 3B, the ultrasonic motor  3  is substantially comprised of a disc-shaped piezoelectric element  31 , a disc-shaped piezoelectric element  32  integrally stacked on the piezoelectric element  31  and a vibrator  33  integrally stacked on the piezoelectric element  32 , and a rotor  34  provided on the vibrator  33  in contact therewith is moved about a shaft  36 . 
     That is, the ultrasonic motor  3  has a structure in which the piezoelectric element  31  thinner than the piezoelectric element  32  is integrally stacked on the bottom surface of the piezoelectric element  32  which corresponds to a piezoelectric element of a conventional rotary ultrasonic motor. 
     The sum of the thickness of the piezoelectric elements  31  and  32  is equal to the thickness of the vibrator  33 . As a result, the interface between the piezoelectric element  32  and vibrator  33  acts as a distortion-neutral plane in the context of the present invention. 
     For example, the piezoelectric element  31  is made of barium titanate or lead zirconate titanate and has a structure formed by alternately providing six pairs of sector regions  31   a ,  31   a  adjacent to each other having a center angle of 30° and six pairs of sector regions  31   b ,  31   b  adjacent to each other having a center angle of 30°. 
     The regions  31   a  and  31   b  are polarized to have opposite polarities in the direction of the thickness thereof. 
     A hole is provided in the center of the piezoelectric element  31  to insert a shaft  36 . 
     The piezoelectric element  32  is made of the same material as that of the piezoelectric element  31  and is fabricated to have the same diameter as that of the piezoelectric element  31  and to have a thickness greater than that of the piezoelectric element  31 . It has a structure formed by alternately providing six pairs of sector regions  32   a ,  32   a  adjacent to each other having a center angle of 30° and six pairs of sector regions  32   b ,  32   b  adjacent to each other having a center angle of 30°. 
     All of the regions  32   a  and  31   a  overlap with the regions  32   b  and  31   b , respectively. 
     The regions  32   a  and  32   b  are polarized oppositely to the regions  31   a  and  31   b  respectively in the direction of the thickness thereof. 
     A hole is provided in the center of the piezoelectric element  32  to insert the shaft  36 . 
     Electrodes  35   a  and  35   b  in conduction to the regions  31   a  and  32   a  are alternately provided on substantially the entirety of the six interfaces between the regions  31   a  and  32   a  and the six interfaces between the regions  31   b  and  32   b.    
     All of the six electrodes  35   a  are in conduction to each other, and all of the six electrodes  35   b  are in conduction to each other. 
     Further, an electrode  35   c  is provided to substantially cover the entire bottom surface of the piezoelectric element  31 , and an electrode  35   d  is provided to substantially cover the entire top surface of the piezoelectric element  32 . The electrodes  35   c  and  35   d  are grounded. 
     The vibrator  33  is a disc-shaped elastic element having the same diameter as those of the piezoelectric elements  31  and  32  and is formed with a hole for receiving the shaft  36  in the center thereof. 
     The vibrator  33  has six projections  33   a  in total which are in contact with the rotor  34  and which are provided in locations substantially corresponding to the centers of the boundaries between the adjoining regions  32   a  and locations substantially corresponding to the centers of the boundaries between the adjoining regions  32   b.    
     An operation of the ultrasonic motor  3  will now be described. 
     First, let us assume that an AC voltage oscillating in a sinusoidal manner is input to the electrodes  35   a  of the ultrasonic motor  3  as a driving signal. 
     When the electrodes  35   a  have a negative potential, the three regions  31   a  and three regions  32   a  in contact with the electrodes  35   a  among six each regions  31   a  and  32   a  expand in the direction of the thickness because the negative potential is applied to the positively polarized surfaces of them. The regions  31   a  undergo expansion greater than the expansion of the regions  32   a  in spite of the fact that the same voltage is applied, because they are thinner than the regions  32   a , and the expansion of the regions  32   a  is not therefore interfered. 
     The three regions  31   b  and three regions  32   b  in contact with the electrodes  35   a  among six each regions  31   b  and  32   b  contract in the direction of the thickness because the negative potential is applied to the negatively polarized surfaces of them. The regions  31   b  undergo contraction greater than the contraction of the regions  32   b  in spite of the fact that the same voltage is applied because they are thinner than the regions  32   b , and the contraction of the regions  32   b  is not therefore interfered. 
     Therefore, when the potential of the electrodes  35   a  is increased in the negative direction, as indicated by ( 1 ) in FIG. 3C, an interaction between the expansion of the regions  31   a  and the expansion of the regions  32   a  causes the projections  33   a  provided between the regions  32   a  to incline in the direction of the arrow in FIG. 3C more than those in a conventional rotary ultrasonic motor to be urged against the rotor  34  more strongly than in the conventional rotary ultrasonic motor. As a result, the rotor  34  is moved in the direction of the arrow in FIG. 3C by a force greater than that in the conventional rotary ultrasonic motor. 
     When the electrodes  35   a  conversely have a positive potential, the three regions  31   a  and three regions  32   a  in contact with the electrodes  35   a  among six each regions  31   a  and  32   a  contract in the direction of the thickness, because the positive potential is applied to the positively polarized surfaces of them. The regions  31   a  undergo contraction greater than the contraction of the regions  32   a  in spite of the fact that the same voltage is applied, because they are thinner than the regions  32   a , and the expansion of the regions  32   a  is not therefore interfered. 
     The three regions  31   b  and three regions  32   b  in contact with the electrodes  35   a  among six each regions  31   b  and  32   b  expand in the direction of the thickness because the positive potential is applied to the negatively polarized surfaces of them. The regions  31   b  undergo expansion greater than the expansion of the regions  32   b  in spite of the fact that the same voltage is applied, because they are thinner than the regions  32   b , and the expansion of the regions  32   b  is not therefore interfered. 
     Therefore, when the potential of the electrodes  35   a  is increased in the positive direction, as indicated by ( 2 ) in FIG. 3C, an interaction between the expansion of the regions  31   b  and the expansion of the regions  32   b  causes the projections  33   a  provided between the regions  32   b  to incline in the direction of the arrow in FIG. 3C more than those in a conventional rotary ultrasonic motor to be urged against the rotor  34  more strongly than in the conventional rotary ultrasonic motor. As a result, the rotor  34  is moved in the direction of the arrow in FIG. 3C by a force greater than that in the conventional rotary ultrasonic motor. 
     The ultrasonic motor  3  thus causes the rotor  34  to be smoothly moved in the direction indicated by the arrow in FIG. 3C with a force greater than that in the conventional rotary ultrasonic motor. 
     Let us now assume that an AC voltage oscillating in a sinusoidal manner is conversely input to the electrodes  35   b  of the ultrasonic motor  3  as a driving signal. 
     When the electrodes  35   b  have a negative potential, the three regions  31   a  and three regions  32   a  in contact with the electrodes  35   b  among six each regions  31   a  and  32   a  expand in the direction of the thickness, because the negative potential is applied to the positively polarized surfaces of them. The regions  31   a  undergo expansion greater than the expansion of the regions  32   a  in spite of the fact that the same voltage is applied, because they are thinner than the regions  32   a , and the expansion of the regions  32   a  is not therefore interfered. 
     The three regions  31   b  and three regions  32   b  in contact with the electrodes  35   b  among six each regions  31   b  and  32   b  contract in the direction of the thickness because the negative potential is applied to the negatively polarized surfaces of them. The regions  31   b  undergo contraction greater than the contraction of the regions  32   b  in spite of the fact that the same voltage is applied, because they are thinner than the regions  32   b , and the contraction of the regions  32   b  is not therefore interfered. 
     Therefore, when the potential of the electrodes  35   b  is increased in the negative direction, as indicated by ( 1 ) in FIG. 3D, an interaction between the expansion of the regions  31   a  and the expansion of the regions  32   a  causes the projections  33   a  provided between the regions  32   a  to incline in the direction of the arrow in FIG. 3D, i.e., in the direction opposite to the direction in FIG. 3C more than those in the conventional rotary ultrasonic motor to be urged against the rotor  34  more strongly than in the conventional rotary ultrasonic motor. As a result, the rotor  34  is moved in the direction of the arrow in FIG. 3D by a force greater than that in the conventional rotary ultrasonic motor. 
     When the electrodes  35   b  conversely have a positive potential, the three regions  31   a  and three regions  32   a  in contact with the electrodes  35   b  among six each regions  31   a  and  32   a  contract in the direction of the thickness, because the positive potential is applied to the positively polarized surfaces of them. The regions  31   a  undergo contraction greater than the contraction of the regions  32   a  in spite of the fact that the same voltage is applied, because they are thinner than the regions  32   a , and the expansion of the regions  32   a  is not therefore interfered. 
     The three regions  31   b  and three regions  32   b  in contact with the electrodes  35   b  among six each regions  31   b  and  32   b  expand in the direction of the thickness, because the positive potential is applied to the negatively polarized surfaces of them. The regions  31   b  undergo expansion greater than the expansion of the regions  32   b  in spite of the fact that the same voltage is applied, because they are thinner than the regions  32   b , the expansion of the regions  32   b  is not therefore interfered. 
     Therefore, when the potential of the electrodes  35   b  is increased in the positive direction, as indicated by ( 2 ) in FIG. 3D, an interaction between the expansion of the regions  31   b  and the expansion of the regions  32   b  causes the projections  33   a  provided between the regions  32   b  to incline in the direction of the arrow in FIG. 3D more than those in a conventional rotary ultrasonic motor to be urged against the rotor  34  more strongly than in the conventional rotary ultrasonic motor. As a result, the rotor  34  is moved in the direction of the arrow in FIG. 3D by a force greater than that in the conventional rotary ultrasonic motor. 
     The ultrasonic motor  3  thus causes the rotor  34  to be smoothly moved in the direction indicated by the arrow in FIG. 3D with a force greater than that in the conventional rotary ultrasonic motor. 
     As described above, in the ultrasonic motor  3  which is an embodiment of the invention, on the bottom surface of the piezoelectric element  32  corresponding to a piezoelectric element in a conventional rotary ultrasonic motor, the piezoelectric element  31  thinner than the piezoelectric element  32  is stacked such that it operates in the same direction as the piezoelectric element  32  in response to the same driving signal The piezoelectric element  31  therefore expands and contracts more than the piezoelectric element  32  to increase the output of the ultrasonic motor  3 . While the current consumed by an ultrasonic motor increases with the capacity of the piezoelectric elements thereof, the current consumed can be reduced to improve efficiency by increasing the thickness of piezoelectric elements in a region whose factor of contribution to distortion is small. 
     Since the ultrasonic motor  3  thus makes the same operation as that of a conventional rotary ultrasonic motor with a greater force, the size and power consumption of the same can be smaller than those of the conventional rotary ultrasonic motor when they provide the same output. 
     Since the sum of the thickness of the piezoelectric elements  31  and  32  is equal to the thickness of the vibrator  33  and the interface between the piezoelectric element  32  and vibrator  33  acts as a distortion-neutral plane, a driving force originating from the expansion and contraction of the piezoelectric elements  31  and  32  is transmitted to the vibrator  33  most efficiently. 
     Any modification may be made on the present embodiment as long as it does not depart from the principle of the invention. 
     For example, any piezoelectric material may be used for the piezoelectric elements  31  and  32 . 
     The optimum ratio between the thicknesses of the piezoelectric elements  31  and  32  is not uniquely determined, and it is rather determined by a plurality of factors such as the electromechanical coupling coefficient of the piezoelectric material and the surface area of the stacking surfaces of the piezoelectric elements. 
     The polarizing direction of each of the piezoelectric elements and the structure of the electrodes are not limited to the present embodiment, and any modification is possible as long as a plurality of stacked piezoelectric elements expand and contract in the same direction. 
     It is not essential to stack two piezoelectric elements, and what is required is that a plurality of piezoelectric elements are provided with a total thickness which is equal to the thickness of the vibrator. Especially, when three or more piezoelectric elements are stacked, it is not essential that they are all different in thickness depending on the stacking positions, and the same effect can be achieved even if some of them have the same thickness. 
     A description will now be made with reference to FIGS. 4A and 4B and FIG. 5 on an ultrasonic motor  4  which is a fourth embodiment of the invention. 
     FIGS. 4A and 4B are schematic views illustrating a configuration of the ultrasonic motor  4 . FIG. 5 is a schematic view illustrating an operation of the ultrasonic motor  4 . 
     The configuration of the ultrasonic motor  4  will be first described with reference to FIGS. 4A and 4B. 
     As shown in FIG. 4A, the ultrasonic motor  4  is substantially comprised of a disc-shaped piezoelectric element  41 , a disc-shaped piezoelectric element  42  stacked on the piezoelectric element  41 , a disc-shaped piezoelectric element  43  stacked on the piezoelectric element  42 , a disc-shaped piezoelectric element  44  stacked under the piezoelectric element  41 , a disc-shaped piezoelectric element  45  stacked under the piezoelectric element  44 , a disc-shaped piezoelectric element  46  stacked under the piezoelectric element  45 , a disc-shaped piezoelectric element  47  stacked on the piezoelectric element  43 , a disc-shaped piezoelectric element  48  stacked under the piezoelectric element  46 , a vibrator  49  integrally stacked on the piezoelectric element  47  and a vibrator  40  integrally stacked under the piezoelectric element  48 . It performs an operation to be detailed later to rotate a rotor  4   a  (shown in FIG. 5) which is located slightly above the vibrator  49  when not operated. 
     The piezoelectric elements  41  through  48  are all stacked integrally and are all equal in the diameter. The piezoelectric elements  41  through  46  are piezoelectric elements to cause expansion and contraction in the stacking direction, i.e., a longitudinal vibration, and the piezoelectric elements  47  and  48  are piezoelectric elements to cause a torsional vibration in the circumferential direction. 
     For example, the piezoelectric element  41  is made of barium titanate or lead zirconate titanate and is polarized in the direction of the thickness to have the positive polarity on the top surface thereof and the negative polarity on the bottom surface thereof. 
     The piezoelectric elements  42  and  43  are made of the same material as that of the piezoelectric element  41 . The piezoelectric element  42  is thicker than the piezoelectric element  41 , and the piezoelectric element  43  is thicker than the piezoelectric element  42 . 
     Both of the piezoelectric elements  42  and  43  are polarized in the direction of the thickness thereof. The piezoelectric element  43  is polarized similarly to the piezoelectric element  41 , and the piezoelectric element  42  is polarized to have the negative polarity on the top surface thereof and the positive polarity on the bottom surface thereof oppositely to the polarization of the piezoelectric element  41 . 
     The piezoelectric elements  44 ,  45  and  46  are made of the same material as that of the piezoelectric element  41 , and are equal in thickness to the piezoelectric elements  41 ,  42  and  43 , respectively. 
     Each of the piezoelectric elements  44 ,  45  and  46  is polarized in the direction of the thickness thereof. The piezoelectric elements  44  and  46  are polarized similarly to the piezoelectric element  42 , and the piezoelectric element  45  is polarized similarly to the piezoelectric element  41 . 
     The piezoelectric elements  47  and  48  are made of the same material as that of the piezoelectric element  41  and, as shown in FIG. 4B, are respectively split into four equal sector regions  47   a  and  48   a  having a center angle of, for example, 90°. 
     The regions  47   a  and  48   a  are polarized in the circumferential direction thereof to have the polarity that changes from positive to negative counterclockwise when viewed from above, as shown in FIG.  4 B. 
     An electrode  41   b  is provided to substantially cover the entire top surface of the piezoelectric element  41  to apply the same voltage to both of the top surface of the piezoelectric element  41  and the bottom surface of the piezoelectric element  42 . 
     Similarly, an electrode  42   b  is provided to substantially cover the entire top surface of the piezoelectric element  42  to apply the same voltage to both of the top surface of the piezoelectric element  42  and the bottom surface of the piezoelectric element  43 . 
     Similarly, an electrode  43   b  is provided to substantially cover the entire top surface of the piezoelectric element  43  to apply a voltage only to the top surface of the piezoelectric element  43 . 
     An electrode  41   c  is provided to substantially cover the entire bottom surface of the piezoelectric element  41  to apply the same voltage to both of the bottom surface of the piezoelectric element  41  and the top surface of the piezoelectric element  44 . 
     Similarly, an electrode  44   b  is provided to substantially cover the entire bottom surface of the piezoelectric element  44  to apply the same voltage to both of the bottom surface of the piezoelectric element  44  and the top surface of the piezoelectric element  45 . 
     Similarly, an electrode  45   b  is provided to substantially cover the entire bottom surface of the piezoelectric element  45  to apply the same voltage to both of the bottom surface of the piezoelectric element  45  and the top surface of the piezoelectric element  46 . 
     Similarly, an electrode  46   b  is provided to substantially cover the entire bottom surface of the piezoelectric element  46  to apply a voltage only to the bottom surface of the piezoelectric element  46 . 
     Electrodes  47   b ,  47   c ,  48   b  and  48   c  are provided to substantially cover the entirety of the top surface of the piezoelectric element  47 , the bottom surface of the same, the top surface of the piezoelectric element  48  and the bottom surface of the same, respectively. 
     The electrodes  41   b ,  43   b ,  44   b  and  46   b  are in conduction to each other, and the electrodes  41   c ,  42   b  and  45   b  are in conduction to each other. 
     The electrodes  47   b  and  48   c  are in conduction to each other, and the electrodes  47   c  and  48   b  are in conduction to each other. 
     An operation of the ultrasonic motor  4  will now be described with reference to FIG.  5 . 
     Lest us first assume that an AC voltage is applied to the electrodes  41   b ,  43   b ,  44   b  and  46   b  as a driving signal with the electrodes  41   c ,  42   b  and  45   b  serving as reference electrodes which are grounded and that an AC voltage at a phase lag of 90° from the AC voltage applied to the electrodes  41   b ,  43   b ,  44   b  and  46   b  is applied to the electrodes  47   b  and  48   c  with the electrodes  47   c  and  48   b  serving as reference electrodes which are grounded. 
     A discussion will follow on a case wherein the electrodes  41   b ,  43   b ,  44   b  and  46   b  have a negative potential. 
     The piezoelectric element  41  expands in the direction of the thickness thereof, because the negative potential is applied by the electrode  41   b  to the positively polarized surface of the same and the negatively polarized surface thereof is grounded through the electrode  41   c.    
     Similarly, since the negative potential is applied by the electrodes  41   b ,  43   b ,  44   b ,  44   b  and  46   b  to the respective positively polarized surfaces of the piezoelectric elements  42 ,  43 ,  44 ,  45  and  46  respectively and the negatively polarized surfaces thereof are grounded through the electrodes  42   b ,  41   c , and  45   b  respectively, all of the piezoelectric elements expand in the direction of the thickness thereof. 
     The greater the distortion of a location in the mode of vibration to be achieved, the smaller the thickness of the piezoelectric element provided in the location. The piezoelectric elements  42  and  45  undergo expansion smaller than that of the piezoelectric elements  41  and  44  in spite of the fact that they are applied with the same voltage as those of the piezoelectric elements  41  and  44 , because they are thicker than the piezoelectric elements  41  and  44 . Similarly, the piezoelectric elements  43  and  46  undergo expansion smaller than that of the piezoelectric elements  42  and  45  in spite of the fact that they are applied with the same voltage as those the piezoelectric elements  42  and  45 , because they are thicker than the piezoelectric elements  42  and  45 . 
     Since the potential at the top surfaces of the piezoelectric elements  47  and  48  becomes positive relative to that at the bottom surfaces at a lag of 90° from the piezoelectric elements  42  through  46 , contraction occurs at the upper part of a side thereof having the positive polarity while expansion occurs at the upper part of a side thereof having the negative polarity. Therefore, the top surface of the piezoelectric element  47  is twisted counterclockwise at a lag of 90° from the expansion of the piezoelectric elements  42  through  46 . 
     Therefore, as the voltage applied to the piezoelectric elements  41  through  46  increases in the negative direction, the ultrasonic motor  4  expands upward, recovers from clockwise twist and further twists counterclockwise as indicated by ( 1 ) and ( 2 ) in FIG.  5 . As a result, it urges the rotor  4   a  and moves it counterclockwise. 
     Since the piezoelectric elements  41  through  46  having different thicknesses are integrally stacked, the expansion is greater than that in the prior art and the rotor  4   a  is therefore moved counterclockwise with a greater force. 
     When the voltage applied to the piezoelectric elements  41  through  46  has a different phase, the rotor  4   a  is not contacted because the piezoelectric elements  41  through  46  contract. Therefore, the rotor  4   a  is not moved as indicated by ( 3 ) and ( 4 ) in FIG.  5 . 
     Then, the state indicated by ( 1 ) in FIG. 5 recurs, and the same operation is repeated. 
     A discussion will follow on a case wherein each of the electrodes  41   b ,  43   b ,  44   b  and  46   b  has a positive potential. 
     The piezoelectric element  41  contracts in the direction of the thickness thereof, because the positive potential is applied by the electrode  41   b  to the positively polarized surface of the same and the negatively polarized surface thereof is grounded through the electrode  41   c.    
     Similarly, since the positive potential is applied by the electrodes  41   b ,  43   b ,  44   b ,  44   b  and  46   b  to the respective positively polarized surfaces of the piezoelectric elements  42 ,  43 ,  44 ,  45  and  46  respectively and the negatively polarized surfaces thereof are grounded through the electrodes  42   b ,  42   b ,  41   c ,  45   b  and  45   b  respectively, all of the piezoelectric elements contract in the direction of the thickness thereof. 
     The greater the distortion of a location in the mode of vibration to be achieved, the smaller the thickness of the piezoelectric element provided in the location. The piezoelectric elements  42  and  45  undergo contraction smaller than that of the piezoelectric elements  41  and  44  in spite of the fact that they are applied with the same voltage as those of the piezoelectric elements  41  and  44 , because they are thicker than the piezoelectric elements  41  and  44 . Similarly, the piezoelectric elements  43  and  46  undergo contraction smaller than that of the piezoelectric elements  42  and  45  in spite of the fact that they are applied with the same voltage as those of the piezoelectric elements  42  and  45 , because they are thicker than the piezoelectric elements  42  and  45 . 
     Since the potential at the top surfaces of the piezoelectric elements  47  and  48  becomes negative relative to that of the bottom surfaces at a lag of 90° from the piezoelectric elements  42  through  46 , expansion occurs at the upper part of a side thereof having the positive polarity while contraction occurs at the upper part of a side thereof having the negative polarity. Therefore, the top surface of the piezoelectric element  47  is twisted clockwise at a lag of 90° from the expansion of the piezoelectric elements  42  through  46 . 
     Therefore, as the voltage applied to the piezoelectric elements  41  through  46  increases in the negative direction, the ultrasonic motor  4  expands upward, recovers from counterclockwise twist and further twists clockwise, although not shown. As a result, it urges the rotor  4   a  and moves it clockwise. 
     Since the piezoelectric elements  41  through  46  having different thicknesses are integrally stacked, the expansion is greater than that in the prior art and the rotor  4   a  is therefore moved clockwise with a greater force. 
     When the voltage applied to the piezoelectric elements  41  through  46  has a different phase, the rotor  4   a  is not contacted because the piezoelectric elements  41  through  46  contract. Therefore, the rotor  4   a  is not moved. 
     As described above, in the ultrasonic motor  4  which is an embodiment of the invention, the piezoelectric elements  41  through  46  for a longitudinal vibration are integrally stacked, and the piezoelectric elements  47  and  48  for a torsional vibration are integrally stacked above and under the same, to which a driving signal separate from that to the piezoelectric elements  41  through  46  is input. Therefore, an adjustment of the driving signal allows the ultrasonic motor  4  to be urged against the rotor  4   a  only when the piezoelectric elements  41  through  46  expand, i.e., when the piezoelectric elements  47  and  48  are twisted in one direction. Further, the piezoelectric elements  41  through  46  have different thicknesses depending on the distribution of distortion. Therefore, the ultrasonic motor  4  causes the rotor  4   a  to rotate in a predetermined direction with a force greater than that in the prior art and with reduced power consumption. 
     The ratio between the strengths of longitudinal and torsional vibrations can be adjusted to an optimum value by adjusting the ratio between the thicknesses of the piezoelectric elements  41  through  46  and the thicknesses of the piezoelectric elements  47  and  48  appropriately. 
     Any modification may be made on the present embodiment as long as it does not depart from the principle of the invention. 
     For example, any piezoelectric material may be used for the piezoelectric elements  41  through  48 . 
     The optimum ratio between the thicknesses of the piezoelectric elements  41  through  46  is not uniquely determined, and it is rather determined by a plurality of factors such as the modes of vibration, the electromechanical coupling coefficient of the piezoelectric material and the surface area of the stacking surfaces of the piezoelectric elements  41  through  46 . 
     The method and direction of stacking the piezoelectric elements for longitudinal and torsional vibrations are not limited to the present embodiment. 
     For example, the same effect can be achieved by an ultrasonic motor  5  as shown in FIG. 6A in which piezoelectric elements  51  and  52  for a torsional vibration having the same polarizing structure as that of the piezoelectric element  47  are stacked and piezoelectric elements  53  and  54  for a longitudinal vibration having the same polarizing structure as that of the piezoelectric element  41  are stacked on both of the end faces of the piezoelectric elements  51  and  52 . For example, when both of the longitudinal and torsional vibrations are first-order modes of vibration, the distortion is greatest in the central region and decreases toward both ends. Therefore, the balance between those vibrations is maintained by making the piezoelectric elements for the longitudinal vibration provided at the periphery thicker than the piezoelectric elements for the torsional vibration provided in the middle. 
     Further, the same effect can be achieved by an ultrasonic motor  6  as shown in FIG. 6B in which a piezoelectric element  62  for a longitudinal vibration thicker than a piezoelectric element  61  and a piezoelectric element  63  for a longitudinal vibration thicker than the piezoelectric element  62  are stacked on the piezoelectric element  61  for a longitudinal vibration having the same polarizing structure as that of the piezoelectric element  41  and in which a piezoelectric element  64  for a torsional vibration having the same polarizing structure as that of the piezoelectric element  47 , a piezoelectric element  65  for a torsional vibration thicker than the piezoelectric element  64  and a piezoelectric element  66  for a torsional vibration thicker than the piezoelectric element  65  are stacked under the piezoelectric element  61 . 
     While both of the longitudinal and torsional vibrations used here are first-order modes of vibration, the invention is not limited thereto. Higher-order modes of vibration may be used, and it is not essential that the two modes of vibration are of the same order. Such alternative arrangements will provide the same effect as that described above as long as the piezoelectric elements have different thicknesses depending on the distribution of distortion. 
     A description will now be made with reference to FIGS. 7A through 7D and FIG. 8 on an ultrasonic motor  7  which is a fifth embodiment of the invention. 
     FIGS. 7A through 7D are schematic views illustrating a configuration of an ultrasonic motor  7  which is a fifth embodiment of the invention, and FIG. 8 is a schematic view illustrating an operation of the ultrasonic motor  7 . 
     The configuration of the ultrasonic motor  7  will be first described. 
     As shown in FIG. 7A, the ultrasonic motor  7  is substantially comprised of a disc-shaped piezoelectric element  71 , a disc-shaped piezoelectric element  72  stacked on the piezoelectric element  71 , a disc-shaped piezoelectric element  73  stacked on the piezoelectric element  72 , a disc-shaped piezoelectric element  74  stacked under the piezoelectric element  71 , a disc-shaped piezoelectric element  75  stacked under the piezoelectric element  74 , a disc-shaped piezoelectric element  76  stacked under the piezoelectric element  75 , a vibrator  77  integrally stacked on the piezoelectric element  73  and a vibrator  78  integrally stacked under the piezoelectric element  76 . It performs an operation to be detailed later to rotate a rotor  79  which is located slightly above the vibrator  78 . 
     The piezoelectric elements  71  through  76  are all stacked integrally and are all equal in the radius. The interface between the piezoelectric elements  71  and  74  constitutes a distortion-neutral plane in the context of the invention. 
     For example, the piezoelectric element  71  is made of barium titanate or lead zirconate titanate. As shown in the plan view in FIG. 7C, the piezoelectric element  71  is split into two semicircles which are polarized oppositely in the direction of the thickness thereof. 
     The piezoelectric element  72  is fabricated thicker than the piezoelectric element  71  using the same material as that of the piezoelectric element  71 . The piezoelectric element  72  is split into two semicircles in the same direction as that of the piezoelectric element  71  and is polarized oppositely to the piezoelectric element  71 . 
     The piezoelectric element  73  is fabricated thicker than the piezoelectric element  72  using the same material as that of the piezoelectric element  71 . The piezoelectric element  73  is split into two semicircles in the same direction as that of the piezoelectric element  71  and is polarized similarly to the piezoelectric element  71 . 
     The piezoelectric element  74  is fabricated with the same thickness as that of the piezoelectric element  71  using the same material as that of the piezoelectric element  71 . As shown in the plan view in FIG. 7D, the piezoelectric element  74  is split into two semicircles which are polarized oppositely in the direction of the thickness thereof. The piezoelectric element  74  is split in a direction orthogonal to the direction in which the piezoelectric element  71  is split. 
     The piezoelectric element  75  is fabricated with the same thickness as that of the piezoelectric element  72  using the same material as that of the piezoelectric element  71 . The piezoelectric element  75  is split into two semicircles in the same direction as that of the piezoelectric element  74  and is polarized oppositely to the piezoelectric element  74 . 
     The piezoelectric element  76  is fabricated with the same thickness as that of the piezoelectric element  73  using the same material as that of the piezoelectric element  71 . The piezoelectric element  76  is split into two semicircles in the same direction as that of the piezoelectric element  74  and is polarized similarly to the piezoelectric element  74 . 
     An electrode  71   a  is provided to substantially cover the entire top surface of the piezoelectric element  71  to apply the same voltage to both of the top surface of the piezoelectric element  71  and the bottom surface of the piezoelectric element  72 . 
     Similarly, an electrode  72   a  is provided to substantially cover the entire top surface of the piezoelectric element  72  to apply the same voltage to both of the top surface of the piezoelectric element  72  and the bottom surface of the piezoelectric element  73 . 
     An electrode  73   a  is provided to substantially cover the entire top surface of the piezoelectric element  73  to apply a voltage to the top surface of the piezoelectric element  73 . 
     An electrode  71   b  is provided to substantially cover the entire bottom surface of the piezoelectric element  71  to apply a voltage to the bottom surface of the piezoelectric element  71 . 
     An electrode  74   a  is provided to substantially cover the entire top surface of the piezoelectric element  74  to apply a voltage to the top surface of the piezoelectric element  74 . 
     An electrode  74   b  is provided to substantially cover the entire bottom surface of the piezoelectric element  74  to apply the same voltage to both of the bottom surface of the piezoelectric element  74  and the top surface of the piezoelectric element  75 . 
     Similarly, an electrode  75   a  is provided to substantially cover the entire bottom surface of the piezoelectric element  75  to apply the same voltage to both of the bottom surface of the piezoelectric element  75  and the top surface of the piezoelectric element  76 . 
     An electrode  76   a  is provided to substantially cover the entire bottom surface of the piezoelectric element  76  to apply a voltage to the bottom surface of the piezoelectric element  76 . 
     The electrodes  71   a  and  73   a  are in conduction to each other, and the electrodes  71   b  and  72   a  are in conduction to each other. 
     The electrodes  74   a  and  75   a  are in conduction to each other, and the electrodes  74   b  and  76   a  are in conduction to each other, 
     Flexible substrates, metal plates and the like may be used as those electrodes for applying signals without any restriction. 
     Methods for bonding the piezoelectric elements, electrodes and vibrators include the use of an adhesive or a structure in which they are bored in the middle to be fastened and secured together with a bolt and nuts or the like. 
     An operation of the ultrasonic motor  7  will now be described. 
     Lest us first assume that an AC voltage is applied to the electrodes  71   a  and  73   a  as a driving signal with the electrodes  71   b  and  72   a  serving as reference electrodes and that an AC voltage at a phase lag of 90° from the AC voltage applied to the electrodes  71   a  and  73   a  is applied to the electrodes  74   a  and  75   a  with the electrodes  74   b  and  76   a  serving as reference electrodes. 
     When the voltage applied to the electrodes  71   a  and  73   a  is increased in the negative direction, the left half of the piezoelectric elements  71 ,  72  and  73  as viewed in FIG. 7C expands in the direction of the thickness thereof because a negative voltage is applied to the positively polarized surfaces thereof through the electrodes  71   a  and  73   a , and the right half contracts in the direction of the thickness because a negative voltage is applied to the negatively polarized surfaces thereof. 
     Since the voltage applied to the electrodes  74   a  and  75   a  approaches to zero from a negative value, the piezoelectric elements  74 ,  75  and  76  undergo smaller distortion. 
     Therefore, as shown in FIG.  7 B and (B) in FIG. 8, the ultrasonic motor  7  expands at the left side and contracts at the right side and undergoes smaller distortion this side and the further side of the plane of the drawings when viewed as a whole. As a result, the rotor  79  is rotated in the direction indicated by the arrow at (B) in FIG.  8 . 
     Since the piezoelectric elements  71 ,  72  and  73  have thicknesses that increase in the same order as they are listed, a piezoelectric element undergoes greater distortion, the greater the distortion of the location of the same in the mode of vibration to be achieved. Therefore, it allows the amount of distortion or driving force of the ultrasonic motor  7  to be increased without interfering with the expansion or contraction of other piezoelectric elements. In addition, a piezoelectric element in a location with a small factor of contribution to driving has a great thickness, which allows reductions in the capacity and power consumption. 
     When the voltage applied to the electrodes  71   a  and  73   a  approaches to zero from a negative value, the piezoelectric elements  71 ,  72  and  73  undergo smaller distortion. 
     At this time, the voltage applied to the electrodes  74   a  and  75   a  increases in the positive direction. Therefore, the upper half of the piezoelectric elements  74 ,  75  and  76  as viewed in FIG. 7D contracts in the direction of the thickness thereof because a positive voltage is applied to the positively polarized surfaces thereof through the electrodes  74   a  and  75   a , and the lower half expands in the direction of the thickness because a positive voltage is applied to the negatively polarized surfaces thereof. 
     Therefore, as indicated by (C) in FIG. 8, the ultrasonic motor  7  expands at this side of the plane of the drawing and contracts at its further side, and undergoes smaller distortion at the left and right side thereof when viewed as a whole. As a result, the rotor  79  is rotated in the direction indicated by the arrow at (B) in FIG.  8 . 
     Since the piezoelectric elements  74 ,  75  and  76  have thicknesses that increase in the same order as they are listed, a piezoelectric element undergoes greater distortion, the greater the distortion of the location of the same in the mode of vibration to be achieved. Therefore, it allows the amount of distortion or driving force of the ultrasonic motor  7  to be increased without interfering with the expansion or contraction of other piezoelectric elements. In addition, a piezoelectric element in a location with a small factor of contribution to driving has a great thickness, which allows reductions in the capacity and power consumption. 
     When the voltage applied to the electrodes  71   a  and  73   a  is increased in the positive direction, the left half of the piezoelectric elements  71 ,  72  and  73  as viewed in FIG. 7C contracts in the direction of the thickness thereof because a positive voltage is applied to the positively polarized surfaces thereof through the electrodes  71   a  and  73   a , while the right half expands in the direction of the thickness because a positive voltage is applied to the negatively polarized surfaces thereof. 
     Since the voltage applied to the electrodes  74   a  and  75   a  approaches to zero from a positive value, the piezoelectric elements  74 ,  75  and  76  undergo smaller distortion. 
     Therefore, as indicated by (D) in FIG. 8, the ultrasonic motor  7  contracts at the left side and expands at the right side, and undergoes smaller distortion this side and the further side of the plane of the drawing when viewed as a whole. As a result, the rotor  79  is rotated in the direction indicated by the arrow at (B) in FIG.  8 . 
     Since the piezoelectric elements  71 ,  72  and  73  have thicknesses that increase in the same order as they are listed, a piezoelectric element undergoes greater distortion, the greater the distortion of the location of the same in the mode of vibration to be achieved. Therefore, it allows the amount of distortion or driving force of the ultrasonic motor  7  to be increased without interfering with the expansion or contraction of other piezoelectric elements. In addition, a piezoelectric element in a location with a small factor of contribution to driving has a great thickness, which allows reductions in the capacity and power consumption. 
     When the voltage applied to the electrodes  71   a  and  73   a  approaches to zero from a positive value, the piezoelectric elements  71 ,  72  and  72  undergo smaller distortion. 
     At this time, the voltage applied to the electrodes  74   a  and  75   a  increases in the negative direction. Therefore, the upper half of the piezoelectric elements  74 ,  75  and  76  as viewed in FIG. 7D expands in the direction of the thickness thereof because a negative voltage is applied to the positively polarized surfaces thereof through the electrodes  74   a  and  75   a , and the lower half contracts in the direction of the thickness because a negative voltage is applied to the negatively polarized surfaces thereof. 
     Therefore, as indicated by (A) in FIG. 8, the ultrasonic motor  7  contracts at this side of the plane of the drawing and expands at its further side, and undergoes smaller distortion at the left and right side thereof when viewed as a whole. As a result, the rotor  79  is rotated in the direction indicated by the arrow at (B) in FIG.  8 . 
     Since the piezoelectric elements  74 ,  75  and  76  have thicknesses that increase in the same order as they are listed, a piezoelectric element undergoes greater distortion, the greater the distortion of the location of the same in the mode of vibration to be achieved. Therefore, it allows the amount of distortion or driving force of the ultrasonic motor  7  to be increased without interfering with the expansion or contraction of other piezoelectric elements. In addition, a piezoelectric element in a location with a small factor of contribution to driving has a great thickness, which allows reductions in the capacity and power consumption. 
     That is, the expansion of the ultrasonic motor  7  occurs in the order of the left side, this side, right side and further side of the plane of FIG. 8 to rotate the rotor  79  clockwise. 
     When an AC voltage at a phase lag of 90° from the AC voltage applied to the electrodes  71   a  and  73   a  is applied to the electrodes  74   a  and  75   a  with the electrodes  74   b  and  76   a  serving as reference electrodes, the expansion occurs, in contrast to FIG. 8, in the order of the left side, further side, right side and this side to rotate the rotor  79  counterclockwise. 
     Since the piezoelectric elements  71 ,  72  and  73  and the piezoelectric elements  74 ,  75  and  76  have thicknesses that increase in the same order as they are listed, a piezoelectric element undergoes greater distortion, the greater the distortion of the location of the same in the mode of vibration to be achieved. Therefore, it allows the amount of distortion or driving force of the ultrasonic motor  7  to be increased without interfering with the expansion or contraction of other piezoelectric elements. In addition, a piezoelectric element in a location with a small factor of contribution to driving has a great thickness, which allows reductions in the capacity and power consumption and also leads to the improvement in efficiency. 
     As described above, in the ultrasonic motor  7  which is an embodiment of the invention, the piezoelectric element  72  thicker than the piezoelectric element  71  and the piezoelectric element  73  thicker than the piezoelectric element  72  are integrally stacked on the piezoelectric element  71  as a source of a driving force, and the piezoelectric element  74  having the same thickness as that of the piezoelectric element  71 , the piezoelectric element  75  having the same thickness as that of the piezoelectric element  72  and the piezoelectric element  76  having the same thickness as that of the piezoelectric element  73  are integrally stacked under the piezoelectric element  71 . As a result, the piezoelectric elements  71  through  76  increase the amount of distortion of the ultrasonic motor  7  without interfering with the expansion and contraction of other piezoelectric elements. The driving force of the ultrasonic motor  7  is thus increased, and the capacity and power consumption can be reduced because a piezoelectric element in a location that has a small factor of contribution to driving has a great thickness. 
     Any modification may be made on the present embodiment as long as it does not depart from the principle of the invention. 
     For example, any piezoelectric material may be used for the piezoelectric elements  71  through  76 . 
     The optimum ratio between the thicknesses of the piezoelectric elements  71  through  76  is not uniquely determined, and it is rather determined by a plurality of factors such as the electromechanical coupling coefficient of the piezoelectric material and the surface area of the stacking surfaces of the piezoelectric elements  71  through  76 . 
     Further, the method and direction of stacking the piezoelectric elements  71  through  76  are not limited to the present embodiment. For example, the same effect can be achieved even when the polarizing structure of the piezoelectric element  71  is adopted for all of the piezoelectric elements  71  through  73  and the polarizing structure of the piezoelectric element  74  is adopted for all of the piezoelectric elements  74  through  76 . There is no limitation on the structure of the electrodes. For example, in a structure wherein stacked piezoelectric elements are integrally sintered, the electrode on each layer used at the time of polarization may be shorted at the inner or outer circumference of the piezoelectric element or shorted with a through hole to excite a desired vibration in response to an input signal. 
     Further, although a first-order mode of bending is used here, the invention is not limited to the same and a higher-order mode may be used. The same effect as described above can be achieved also in such a case only by changing the thickness of the piezoelectric elements depending on the distribution of distortion. 
     FIG. 9 is a schematic view illustrating a configuration of an ultrasonic motor  7   a  which is a modification of the ultrasonic motor  7 . 
     The ultrasonic motor  7   a  has a structure in which piezoelectric elements  74  are integrally stacked under piezoelectric elements  72  which are integrally stacked, and a piezoelectric element  79  for detecting a signal used for self-excited vibration or driving and control based on separate excitation is integrally stacked under the same. 
     For example, the piezoelectric element  79  is fabricated with a thickness smaller than those of the piezoelectric elements  72  and  74  using barium titanate or lead zirconate titanate and is polarized in the direction of the thickness to have the positive polarity on the top surface thereof and the negative polarity on the bottom surface thereof. 
     The piezoelectric element  79  has four electrodes provided, for example, at every 90° for detecting the state of distortion of the ultrasonic motor  7   a  as an electrical signal. 
     In the ultrasonic motor  7   a  having the above-described configuration, although the piezoelectric element  79  is provided in a region which undergoes distortion smaller than that of the region of the piezoelectric elements  72 , it exhibits a high detecting capability without interfering with the distortion of the piezoelectric elements  72  and  74  as a source of a driving force, because it is thinner than the piezoelectric elements  72  and  74 . 
     There is no need for providing the piezoelectric elements  72  and  74  with electrodes for detecting a vibration, and the piezoelectric elements  72  and  74  as a whole are used as a source of a driving force. It is therefore possible to obtain a driving force greater than that available in the prior art. 
     Especially, when the ultrasonic motor  7   a  is a high power ultrasonic motor, although the piezoelectric elements  72  and  74  must be thick enough to maintain the strength of the ultrasonic motor  7   a , there is no reduction in the vibration detecting capability of the ultrasonic motor  7   a , because the piezoelectric element  79  for detecting a vibration is provided separately from the piezoelectric elements  72  and  74 . 
     The present embodiment is not limiting the invention, and the same effect can be achieved in piezoelectric actuators including any type of ultrasonic motor by providing a separate piezoelectric element for detecting a vibration with a thickness smaller than that of a piezoelectric element for driving in addition to piezoelectric elements for driving. 
     A detailed description will be made with reference to FIGS. 10A through 10B and FIGS. 11A through 11C on an ultrasonic motor  8  which is the sixth embodiment of the invention. 
     FIGS. 10A through 10E are schematic views illustrating a configuration of the ultrasonic motor  8 , and FIGS. 11A through 11C are schematic views illustrating an operation of the ultrasonic motor  8 . 
     The configuration of the ultrasonic motor  8  will now be described. 
     The ultrasonic motor  8  is substantially comprised of four rectangular piezoelectric elements  81  which are integrally stacked, three rectangular piezoelectric elements  82 ,  83  and  84  integrally stacked on the piezoelectric elements  81  and three rectangular piezoelectric elements  85 ,  86  and  87  integrally stacked under the piezoelectric elements  81 . 
     The piezoelectric elements  81  are piezoelectric elements for generating a longitudinal vibration, and the piezoelectric elements  82  through  87  are piezoelectric elements for generating a bending vibration. That is, the ultrasonic motor  8  is an ultrasonic motor which moves a movable element with an elliptical vibration generated on end faces and sides thereof as a result of synthesis of a longitudinal vibration and a bending vibration. 
     All of the piezoelectric elements  81  through  87  have a same surface configuration. 
     The bottom surface of the second piezoelectric element  81  from the top constitutes a distortion-neutral plane in the context of the invention. 
     For example, the piezoelectric element  81  is fabricated from barium titanate or lead zirconate titanate and is polarized in the direction of the thickness to have the positive polarity on the top surface as shown in the plan view of FIG.  10 C. 
     Electrodes  81   a  and  81   b  are respectively provided on the top and bottom surfaces of the piezoelectric element  81 . 
     The piezoelectric element  82  is fabricated using the same material as that of the piezoelectric element  81 . As shown in the plan view of FIG. 10B, the piezoelectric element  82  has two rectangular polarized regions which are oppositely polarized in the direction of the thickness thereof, for example, to have the positive polarity on the top surface of the region on the left side of FIG.  10 B and the negative polarity on the top surface of the region on the right side of FIG.  10 B. Electrodes  82   a  are provided on the upper surfaces of the two polarized regions, and a single continuous electrode  82   b  is provided on the bottom surfaces of the same. 
     The piezoelectric element  83  is made with a thickness smaller than that of the piezoelectric element  82  using the same material as that of the piezoelectric element  81 . 
     The piezoelectric element  84  is made with a thickness smaller than that of the piezoelectric element  83  using the same material as that of the piezoelectric element  81 . 
     The polarizing structure of the piezoelectric elements  83  and  84  is the same as that of the piezoelectric element  82 . Similarly to the piezoelectric element  82 , electrodes  83   a  and electrodes  84   a  are provided on the upper surfaces of the respective polarized regions, and two single continuous electrodes  83   b  and  84   b  are provided on the respective bottom surfaces. 
     The piezoelectric element  85  is fabricated to have the same thickness as that of the piezoelectric element  82  using the same material as that of the piezoelectric element  81 . As shown in the plan view of FIG. 10D, the piezoelectric element  85  has two rectangular polarized regions with the same configuration as that of the piezoelectric element  82 , which are polarized oppositely to the piezoelectric element  82  in the direction of the thickness thereof, for example, to have the negative polarity on the top surface of the region on the left side of FIG.  10 D and the negative polarity on the top surface of the region on the right side of FIG.  10 D. Electrodes  85   a  are provided on the upper surfaces of the two polarized regions, and a single continuous electrode  85   b  is provided on the bottom surfaces of the same. 
     The piezoelectric element  86  is made with the same thickness as that of the piezoelectric element  83  using the same material as that of the piezoelectric element  81 . 
     The piezoelectric element  87  is made with the same thickness as that of the piezoelectric element  84  using the same material as that of the piezoelectric element  81 . 
     The polarizing structure of the piezoelectric elements  86  and  87  is the same as that of the piezoelectric element  85 . Similarly to the piezoelectric element  85 , electrodes  86   a  and electrodes  87   a  are provided on the upper surfaces of the respective polarized regions, and two single continuous electrodes  86   b  and  87   b  are provided on the respective bottom surfaces. 
     All of the electrodes  82   a ,  83   a ,  84   a ,  85   a ,  86   a  and  87   a  are in conduction to each other. Further, all of the electrodes  81   b ,  82   b ,  83   b ,  84   b ,  85   b ,  86   b  and  87   b  are grounded. 
     An operation of the ultrasonic motor  8  will now be described. 
     First, a discussion follows on a case wherein an AC voltage is applied to all electrodes  81   a  as a driving signal with the electrodes  81   b  serving as reference electrodes and wherein an AC voltage at a phase lead of 90° from the AC voltage applied to the electrodes  81   a  is applied to the electrodes  82   a  through  87   a  with the electrodes  82   b  through  87   b  serving as reference electrodes. 
     When the voltage applied to the electrodes  81   a  increases in the negative direction, the piezoelectric elements  81  expand in the longitudinal direction as shown in FIG. 11A because a negative voltage is applied to the positively polarized surfaces thereof. Therefore, an end face of the ultrasonic motor  8  is brought into contact with the movable element (not shown). 
     At this time, since the voltage applied to the electrodes  82   a  through  87   a  increases in the positive direction, a positive voltage is applied to the top surfaces of the piezoelectric elements  82  through  87 . Therefore, the left half of the piezoelectric elements  82  through  84  contracts and the right half of the same expands, while the left half of the piezoelectric elements  85  through  87  expands and the right half thereof contracts. 
     As a result, the ultrasonic motor  8  is distorted as indicated by the sectional view of FIG.  11 B. 
     The piezoelectric elements  82  through  84  have thicknesses that decrease as the distance from the distortion-neutral plane increases and therefore increase the amount of distortion of the ultrasonic motor  8  without interfering with the expansion and contraction of other piezoelectric elements. 
     Similarly, the piezoelectric elements  85  through  87  have thicknesses that decrease as the distance from the distortion-neutral plane increases and therefore increase the amount of distortion of the ultrasonic motor  8  without interfering with the expansion and contraction of other piezoelectric elements. 
     As a result, the ultrasonic motor  8  moves the movable body which is in contact with the end face of the ultrasonic motor  8  in the direction indicated by the arrow in FIG.  11 C. 
     When the voltage applied to the electrodes  81   a  increases in the positive direction, the piezoelectric elements  81  contract in the longitudinal direction because a positive voltage is applied to the positively polarized surfaces thereof. Therefore, the end face of the ultrasonic motor  8  is not brought into contact with the movable element (not shown) and hence no driving force is transmitted to the movable element. 
     Next, a discussion follows on a case wherein an AC voltage is applied to all electrodes  81   a  as a driving signal with the electrodes  81   b  serving as reference electrodes and wherein an AC voltage having the same phase as the AC voltage applied to the electrodes  81   a  is applied to the electrodes  82   a  through  87   a  with the electrodes  82   b  through  87   b  serving as reference electrodes. 
     When the voltage applied to the electrodes  81   a  increases in the negative direction, the piezoelectric elements  81  expand in the longitudinal direction as shown in FIG. 11A because a negative voltage is applied to the positively polarized surfaces thereof. Therefore, the end face of the ultrasonic motor  8  is brought into contact with the movable element (not shown). 
     At this time, since the voltage applied to the electrodes  82   a  through  87   a  increases in the negative direction, a negative voltage is applied to the top surfaces of the piezoelectric elements  82  through  87 . Therefore, the left half of the piezoelectric elements  82  through  84  expands and the right half of the same contracts, while the left half of the piezoelectric elements  85  through  87  contracts and the right half thereof expands. 
     As a result, the ultrasonic motor  8  undergoes distortion opposite to that indicated by the sectional view of FIG.  11 B. 
     The piezoelectric elements  82  through  84  have thicknesses that decrease as the distance from the distortion-neutral plane increases and therefore increase the amount of distortion of the ultrasonic motor  8  without interfering with the expansion and contraction of other piezoelectric elements. 
     Similarly, the piezoelectric elements  85  through  87  have thicknesses that decrease as the distance from the distortion-neutral plane increases and therefore increase the amount of distortion of the ultrasonic motor  8  without interfering with the expansion and contraction of other piezoelectric elements. 
     As a result, the ultrasonic motor  8  moves the movable body which is in contact with the end face of the ultrasonic motor  8  in a direction opposite to the direction indicated by the arrow in FIG.  11 C. 
     When the voltage applied to the electrodes  81   a  increases in the positive direction, the piezoelectric elements  81  contract in the longitudinal direction because a positive voltage is applied to the positively polarized surfaces thereof Therefore, the end face of the ultrasonic motor  8  is not brought into contact with the movable element (not shown) and hence no driving force is transmitted to the movable element. 
     As described above, in the ultrasonic motor  8  which is an embodiment of the invention, the piezoelectric elements  82 ,  83  and  84  and the piezoelectric elements  85 ,  86  and  87  for a bending vibration having thicknesses that decrease as the distance from distortion-neutral plane increases are respectively stacked above and under the four piezoelectric elements  81  for a source of a longitudinal vibration. Therefore, the piezoelectric elements  82  through  87  increase the amount of the bending vibration of the ultrasonic motor  8  without interfering with the expansion and contraction of other piezoelectric elements. This increases the driving force of the ultrasonic motor  8 , reduces the power consumption and improves the efficiency of the same. 
     Any modification may be made on the present embodiment as long as it does not depart from the principle of the invention. 
     For example, any piezoelectric material may be used for the piezoelectric elements  81  through  87 . 
     The optimum ratio between the thicknesses of the piezoelectric elements  82  through  87  is not uniquely determined, and it is rather determined by a plurality of factors such as the electromechanical coupling coefficient of the piezoelectric material and the surface area of the stacking surfaces of the piezoelectric elements  82  through  87 . 
     The polarizing structure of the piezoelectric elements  82  through  87  is not limited to the present embodiment, and any polarizing structure that generates a bending vibration may be used. For example, electrodes E may be provided between electrodes B, C and D as a common electrode with no insulating layer interposed therebetween. The applied electrical signals are not limited to have a phase of 90°, and they may be in the same phase. 
     The stacking structure and number of the piezoelectric elements  81  through  87  are not limited to those of the present embodiment as long as the thickness of a piezoelectric element is made smaller as the distance of the same from the distortion-neutral plane increases. 
     FIG. 12 is a block diagram showing a configuration of an electronic apparatus  9  with a piezoelectric actuator which is an application of a piezoelectric actuator according to the invention to an electronic apparatus. 
     The electronic apparatus  9  with a piezoelectric actuator is comprised of a piezoelectric actuator  91  having piezoelectric elements polarized in a predetermined manner, a movable element  92  moved by the piezoelectric actuator  91 , an urging mechanism  93  for urging the piezoelectric element  91  and movable element  92 , a transmission mechanism  94  which moves in cooperation with the movable element  92  and an output mechanism  95  moving in accordance with the operation of the transmission mechanism  94 . For example, a spring is used as the urging mechanism  93 . 
     The electronic apparatus  9  with a piezoelectric actuator includes, for example, an electronic clock, a measuring instrument, a camera, a printer, a machine tool, a robot, a transfer apparatus, a storage apparatus and the like. 
     For example, the piezoelectric actuator  91  is the piezoelectric actuator  1  or  2  or the ultrasonic motor  3 ,  4 ,  5 ,  6 ,  7 ,  7   a  or  8 . For example, a transmission wheel such as a toothed wheel or frictional wheel is used as the transmission mechanism  94 . For example, the output mechanism  95  is a shutter driving mechanism or a lens driving mechanism in a camera, a needle driving mechanism or calendar driving mechanism in an electronic clock, a head driving mechanism in a storage apparatus to drive a head for writing and reading information to and from a storage medium in the information storage apparatus, a blade feeding mechanism or a workpiece feeding mechanism in a machine tool, or the like. 
     The size and power consumption of the piezoelectric actuator of the electronic apparatus  9  can be small because it is a piezoelectric actuator according to the invention which provides greater output in comparison to that of conventional piezoelectric actuaters. Therefore, it can be made smaller than conventional electronic apparatuses with a piezoelectric actuator. 
     An ultrasonic motor alone serves as a driving mechanism when an output shaft is attached to the movable element  92  and a power transmission mechanism is provided to transmit torque from the output shaft.