Abstract:
In a drive apparatus having a vibration-generator including a plurality of electromechanical conversion elements, a plurality of weights located between each of the electromechanical conversion elements, and a rod fixed to one end of an electromechanical conversion element, a mass and a spring constant of each component included in the vibration-generator are determined, so that a relationship of integral multiple is satisfied. Thus, highly efficient driving with taking advantage of the resonance frequencies can be realized, and thus the energy consumption can be saved.

Description:
This application is based on applications No. 133293/2004 filed in Japan, the contents of which are hereby incorporated by reference. 
   BACKGROUND OF THE INVENTION 
   1. Field of the Invention 
   The present invention relates to a driving device, and particularly to a driving device which comprises an electromechanical conversion element such as a piezoelectric element or the like. The driving device of the present invention is suitably used as a lens-driving mechanism of a camera or a driving mechanism of a precision stage. 
   2. Description of the Related Art 
   The Japanese patent publication of JP-A-11-98865 discloses a driving device as shown in the exploded perspective view in  FIG. 1A  and the assembly diagram thereof in  FIG. 1B . This driving device comprises an electromechanical conversion element such as a piezoelectric element, of which the length changes (expands or contracts) under application of a voltage. 
   In this driving device, the slider  10  can be moved relative to the base  1 . For example, this driving device can be used as a lens-driving device for a camera. That is, a lens frame of the camera is connected with the slider  10 , such that the associated lens is driven together with the slider  10 . 
   The piezoelectric element  4  is composed of a plurality of laminated piezoelectric plates. One end  4   a  of the piezoelectric element  4  in its expanding or contracting direction is fixed to the base  1 , and the other end  4   b  is fixed to one end  5   a  of the rod  5 . The rod  5  is slidably supported by the support walls  2  and  3  which are integrally formed with the base  1 . 
   The slider  10  is frictionally engaged with the outer surface of the rod  5 , wherein the rod  5  is sandwiched between the body  11  and the cap  12  of the slider  10 , and a leaf spring  13  imparts an urging force to the body  11  and the cap  12  in approaching direction. 
   The piezoelectric element  4  is connected to a voltage-controlling circuit or drive-pulse generator (not shown). When a predetermined drive voltage with sawtooth-like waveform is applied to the piezoelectric element  4 , the piezoelectric element  4  vibrates while drawing substantially the same sawtooth-like displacement ( FIG. 2 ). With this vibration, the rod  5  also vibrates in its longitudinal direction, while drawing sawtooth-like displacement. That is, the graph in  FIG. 2  shows the displacement of the rod  5  due to the vibration, as well as the displacement of the piezoelectric element  4  due to the vibration. 
   This is described in more detail. During the interval “A” of the first waveform  100 , a gentle rising inclination  101  is shown, and during this interval, the piezoelectric element  4  expands at a relatively slow rate, and the rod  5  slowly moves in the arrow direction I in  FIG. 1B . Next, during the interval “B”, the piezoelectric element  4  abruptly contracts to its initial length (the portion of the waveform indicated by the falling inclination  102 ), and the rod  5  rapidly moves in the arrow direction II. 
   The same is repeated thereafter, and consequently, the rod  5  vibrates while alternately repeating the slow motion in the direction I and the rapid motion in the direction II. In this way, the rod  5  vibrates while drawing sawtooth-like vibration waveforms, with gentle and acute alternations. 
   The force of the leaf spring  13  of the slider  10  (i.e., the frictional engaging force of the slider  10  to the rod  5 ) is adjusted, such that when the rod  5  slowly moves (in other words, when the rod  5  moves at a rate smaller than a predetermined value), the slider  10  moves together with the rod  5 , and when the rod  5  rapidly moves (in other words, when the rod  5  moves at a rate larger than the above predetermined value), the slider  10  stops there due to its inertia, or moves a little compared with that of the rod  5 , as shown in  FIG. 3 . As a result, the slider  10  moves relative to the base  1  in the direction I during the rod  5  is vibrating. 
   When the slider  10  is to be moved in the arrow direction II on  FIG. 1B , the vibration waveform of the piezoelectric element  4  and the rod  5  would be a reverse of the waveform shown in  FIG. 2 , and such the reversed waveform has alternately an acute rising portion and a gentle falling portion. The principle for moving the slider  10  is the same as described above. 
   As mentioned above, it is needed to vibrate the rod  5 , with which the slider  10  is frictionally engaged, with drawing a sawtooth-like waveform. One of the methods for causing such the vibration is disclosed in the Japanese patent publication of JP-A-11-41953. 
   According to the Japanese patent publication of JP-A-11-41953, a plurality of piezoelectric elements are disposed in series. To each of the piezoelectric elements, a waving voltage expressed by a laterally symmetric pulse waveform (such as a sine wave) is applied. But, the waving voltages are of different amplitudes and cycles from each other. Thus, in a total observation during the rod&#39;s vibration, the moving speed of the rod in one direction and the moving speed in the other direction are made different from each other. 
   According to the Japanese patent publication of JP-A-11-41953, the rod can be vibrated drawing sawtooth-like waveforms, by applying a voltage with simple sine waveform to each of the piezoelectric elements, without the need of generating a voltage with sawtooth-like waveform. Therefore, this method has an advantage in that the constitution of the drive circuit can be simplified. 
   In general, the gain can be the local maximum when a resonance frequency is input to a physical vibration system. That is, in the case where the driving is caused by inputting alternating signals to a plurality of piezoelectric elements as in the Japanese patent publication of JP-A-11-41953, highly efficient driving would be realized if a resonance frequency of the vibration system is input. 
   Accordingly, in the case of using a plurality of piezoelectric elements, it becomes possible to realize highly efficient driving and the energy consumption can be saved by the followings: that is, a first resonance frequency is input to a first piezoelectric element, a second resonance frequency is input to a second piezoelectric element, and the same is repeated, so that an “n”th resonance frequency is input to an “n”th piezoelectric element. 
   If the resonance frequencies are in a relationship of integral multiple (that is, f 1 =f 2 /2= . . . =fn/n, where the first resonance frequency is f 1 , the second resonance frequency is f 2 , . . . , and the “n”th resonance frequency is fn), inputting each of the resonance frequencies to each of the piezoelectric elements respectively would realize a regular sawtooth-like vibration waving with a constant cycle. 
   However, in case that the resonance frequencies are not in the above-mentioned relationship of integral multiple (in general, in a certain physical vibration system, it is not always that the resonance frequencies are in the relationship of integral multiple), the local maximum value of the gain is not regularly obtained even if each of the resonance frequencies is input to each of the piezoelectric elements. Therefore, a regular sawtooth-like vibration waveform consisting of regularly repeated same wave-sections can not be obtained. In other words, highly efficient driving with taking advantage of resonance can not be realized. 
   SUMMERY OF THE INVENTION 
   An object of the present invention is therefore to realize a physical structure in which the resonance frequencies thereof are of a relationship of integral multiple, in a driving device comprising a plurality of electromechanical conversion elements disposed in series (a physical vibration system), and thereby to realize a highly efficient driving with taking advantage of resonance frequencies. 
   In order to achieve the objects of the present invention, there is provided a driving device comprising:
         a vibration-generator comprised of a first weight fixed to a predetermined steady surface; a first electromechanical conversion element, one end of which in its expanding or contracting direction is fixed to the first weight; a second weight fixed to the other end of the first electromechanical conversion element in the expanding or contracting direction; a second electromechanical conversion element, one end of which in its expanding or contracting direction is fixed to the second weight; and a rod fixed to the other end of the second electromechanical conversion element in the expanding or contracting direction,   a slider which is frictionally engaged with the outer surface of the rod of the vibration-generator, and a degree of the frictional engagement is set, so that the slider can move together with the rod when the rod is moved at a speed lower than a predetermined speed, and a moving amount of the slider becomes smaller than that of the rod when the rod is moved at a speed higher than the predetermined speed, and   a drive circuit which applies voltages to the first electromechanical conversion element and the second electromechanical conversion element to cause the rod to vibrate, such that a moving speed of the rod in one direction is lower than the predetermined speed and a moving speed of the rod in the other direction is higher than the predetermined speed, so as to move the slider along the rod in the one direction,   characterized in that a second resonance frequency of the vibration-generator is set to be twice a first resonance frequency thereof by satisfying the following equation, where a mass of the first weight is m w1 ; a mass of the first electromechanical conversion element is m p1 ; a spring constant of the first electromechanical conversion element is k p1 ; a mass of the second weight is m w2 ; a mass of the second electromechanical conversion element is m p2 ; a spring constant of the second electromechanical conversion element is k p2 ; and a mass of the rod is m r .       

   
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             1 
                             
                               
                                 K 
                                 p1 
                               
                               ⁢ 
                               
                                 K 
                                 p2 
                               
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 m 
                                 w1 
                               
                               + 
                               
                                 
                                   2 
                                   
                                     π 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   m 
                                   p1 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   2 
                                   
                                     π 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   m 
                                   p1 
                                 
                               
                               + 
                               
                                 m 
                                 w2 
                               
                               + 
                               
                                 
                                   2 
                                   
                                     π 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   m 
                                   p2 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           
                             ( 
                             
                               
                                 
                                   2 
                                   
                                     π 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   m 
                                   p2 
                                 
                               
                               + 
                               
                                 m 
                                 r 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 m 
                                 w1 
                               
                               + 
                               
                                 m 
                                 w2 
                               
                               + 
                               
                                 m 
                                 r 
                               
                               + 
                               
                                 
                                   4 
                                   
                                     π 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   m 
                                   p1 
                                 
                               
                               + 
                               
                                 
                                   4 
                                   
                                     π 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   m 
                                   p2 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           ( 
                           
                             
                               
                                 1 
                                 
                                   K 
                                   p2 
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     
                                       2 
                                       
                                         π 
                                         2 
                                       
                                     
                                     ⁢ 
                                     
                                       m 
                                       p1 
                                     
                                   
                                   + 
                                   
                                     m 
                                     w2 
                                   
                                   + 
                                   
                                     
                                       2 
                                       
                                         π 
                                         2 
                                       
                                     
                                     ⁢ 
                                     
                                       m 
                                       p2 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     
                                       2 
                                       
                                         π 
                                         2 
                                       
                                     
                                     ⁢ 
                                     
                                       m 
                                       p2 
                                     
                                   
                                   + 
                                   
                                     m 
                                     r 
                                   
                                 
                                 ) 
                               
                             
                             + 
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               ( 
                               
                                 
                                   1 
                                   
                                     K 
                                     p1 
                                   
                                 
                                 + 
                                 
                                   1 
                                   
                                     K 
                                     p2 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   m 
                                   w1 
                                 
                                 + 
                                 
                                   
                                     2 
                                     
                                       π 
                                       2 
                                     
                                   
                                   ⁢ 
                                   
                                     m 
                                     p1 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     2 
                                     
                                       π 
                                       2 
                                     
                                   
                                   ⁢ 
                                   
                                     m 
                                     p2 
                                   
                                 
                                 + 
                                 
                                   m 
                                   r 
                                 
                               
                               ) 
                             
                           
                           + 
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 1 
                                 
                                   K 
                                   p1 
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     m 
                                     w1 
                                   
                                   + 
                                   
                                     
                                       2 
                                       
                                         π 
                                         2 
                                       
                                     
                                     ⁢ 
                                     
                                       m 
                                       p1 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     
                                       2 
                                       
                                         π 
                                         2 
                                       
                                     
                                     ⁢ 
                                     
                                       m 
                                       p1 
                                     
                                   
                                   + 
                                   
                                     m 
                                     w2 
                                   
                                   + 
                                   
                                     
                                       2 
                                       
                                         π 
                                         2 
                                       
                                     
                                     ⁢ 
                                     
                                       m 
                                       p2 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
                 = 
                 
                   4 
                   25 
                 
               
             
             
               
                 [ 
                 Equation 
                 ] 
               
             
           
         
       
     
   
   The above equation is derived by modeling the physical system of the above vibration-generator into an electric circuit in accordance with the Force-Voltage Method. 
   The number of the electromechanical conversion elements is not always limited to two as above. Even in the case where three or more (“n”) electromechanical conversion elements are used, the resonance frequencies of such the vibration-generator can be set in accordance with the same idea. That is, it is possible to determine the masses of the weights, and the masses and the spring constants of the electromechanical conversion elements of the vibration-generator, so that the relationship of f 1 =f 2 /2, . . . , =fn/n can be satisfied, where the first resonance frequency is f 1 , the second resonance frequency is f 2 , . . . , and the “n”th resonance frequency is fn, in the vibration-generator. 
   In the driving device of the present invention constructed as above, the resonance frequencies of the vibration-generator have the relationship of integral multiple. Accordingly, by inputting each of the resonance frequencies to each of the electromechanical conversion elements, a repeatable sawtooth-like waveform of the rod vibration can be realized, which waves at a predetermined cycle. In other words, highly efficient driving with taking advantage of the resonance frequencies can be realized, and thus the energy consumption can be saved. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     These and other objects and features of the present invention will become apparent from the following description taken in conjunction with the preferred embodiments thereof with reference to the accompanying drawings. 
       FIG. 1A  shows an exploded perspective view of a conventional driving device, and  FIG. 1B  shows an assembly diagram thereof. 
       FIG. 2  shows a graph illustrating the principle for driving of the driving device shown in  FIG. 1 . 
       FIG. 3  shows a diagram illustrating the principle for driving of the driving device shown in  FIG. 1 . 
       FIG. 4A  shows an exploded perspective view of a driving device according to the first embodiment of the present invention, and  FIG. 4B  shows an assembly diagram thereof. 
       FIG. 5  shows a model of the vibration-generator of the driving device shown in  FIG. 4 . 
       FIG. 6  shows an electric circuit diagram corresponding to the model shown in  FIG. 5 . 
       FIG. 7  shows a graph illustrating the vibration characteristics of the vibration-generator of the driving device shown in  FIG. 4 . 
       FIG. 8  shows a graph illustrating the waveforms of voltages input to the driving device shown in  FIG. 4 . 
       FIG. 9  shows a graph illustrating the waveform of the rod vibration when the voltages shown in  FIG. 8  are applied. 
       FIG. 10  shows a graph illustrating the waveforms of voltages input to a driving device according to the second embodiment of the present invention. 
       FIG. 11  shows a graph illustrating the waveform of the rod vibration when the voltages shown in  FIG. 10  are applied. 
       FIG. 12  shows a graph illustrating the waveforms of voltages input to a driving device according to the third embodiment of the present invention. 
       FIG. 13  shows a graph illustrating the waveform of the rod vibration when the voltages shown in  FIG. 12  are applied. 
       FIG. 14  shows a perspective view of the essential portion of a driving device according to the fourth embodiment of the present invention. 
       FIG. 15  shows a perspective view of the essential portion of a driving device according to a modification to the fourth embodiment shown in  FIG. 14 . 
       FIG. 16  shows a perspective view of the essential portion of a driving device according to the fifth embodiment of the present invention. 
       FIG. 17  shows a graph illustrating the waveform of the rod vibration in the fifth embodiment. 
       FIG. 18  shows another embodiment, wherein “n” piezoelectric elements are used. 
   

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
   &lt;First Embodiment ( FIGS. 4 to 9  )&gt; 
     FIG. 4A  shows an exploded perspective view of a driving device according to the first embodiment of the present invention, and  FIG. 4B  shows an assembly diagram thereof. 
   In this driving device, the first piezoelectric element  31  and the second piezoelectric element  32  are connected in series in the expanding or contracting direction, via the second weight  22 . As shown in  FIG. 4B , the piezoelectric elements  31  and  32  are driven by the drive circuits (driving-pulse generator)  1  and  2 , respectively. 
   The first weight  21  corresponds to the base  1  in the structure shown in  FIGS. 1A and 1B . For example, when the first weight  21  is fixed on an inner steady surface of a lens barrel, lenses connected to the slider  10  can be driven in a direction of an optical axis. The first piezoelectric element  31  is disposed between the first weight  21  and the second weight  22 , and the second piezoelectric element  32  is disposed between the rod  25  and the second weight  22 . 
   The slider  10  is frictionally engaged with the outer surface of the rod  25 . Herein, the same reference numerals corresponding to those in  FIGS. 1A and 1B  are assigned and the explanation of the slider  10  is omitted, since the structure of the slider  10  is the same as that of the conventional slider shown in  FIGS. 1A and 1B . 
   As will be described later, predetermined driving voltages are respectively applied to the two piezoelectric elements  31  and  32  from the drive circuits  1  and  2 , such that the rod  25  is vibrated with drawing sawtooth-like waveforms, to thereby drive the slider  10  in a desired direction. In the present invention, the manner for realizing the sawtooth-like waveform vibration of the rod is different from the conventional one. But, the principle for moving the slider  10  along the vibrating rod  25  is the same as that in the prior art described with reference to  FIGS. 1A to 3 . 
   Preferably, the rod  25  or the slider  10  is vibrated or reciprocated while being supported by a suitable guiding means, although not shown herein. 
   In the driving device shown in  FIGS. 4A and 4B , the weights  21  and  22 , the piezoelectric elements  31  and  32 , and the rod  25  compose a vibration-generator  40 . The masses and the spring constants of the respective members of the vibration-generator  40  are set to specified values to thereby satisfy the relationship of f 1 =f 2 /2 in the vibration-generator  40  (a physical vibration system), where fl is the first resonance frequency of this system, and f 2  is the second resonance frequency of this system. 
   To determine the specified values of the masses and the spring constants, the following modeling is carried out. 
   The physical structure of the vibration-generator  40  shown in  FIGS. 4A and 4B  can be modeled as shown in  FIG. 5 , with taking into account the spring constants and damping forces of the piezoelectric elements. 
   According to the Force-Voltage Method, the model shown in  FIG. 5  is equivalent to an electric circuit shown in  FIG. 6 . With determining the first resonance frequency and the second resonance frequency of this electric circuit, the first resonance frequency and the second resonance frequency of the vibration-generator  40  (a physical vibration system) shown in  FIG. 4  can be determined. 
   In such modeling, the relationship of the mass, spring constant and damping constant of the respective members in the physical vibration system shown in  FIG. 5 , with the capacity of the capacitor, the reactance of the coil and the electrical resistance of the electric circuit shown in  FIG. 6  are as summarized in Table 1 below. 
   
     
       
             
             
           
         
             
                 
               TABLE 1 
             
             
                 
                 
             
           
           
             
                 
               The relationship of the spring constant of the 
             
             
                 
               piezoelectric element with the capacity of the capacitor: 
             
             
                 
               C 1  = 1/k p1   
             
             
                 
               C 2  = 1/k p2   
             
             
                 
               The relationship of the masses of the weight, the 
             
             
                 
               piezoelectric element and the rod with the reactance of the 
             
             
                 
               coil: 
             
             
                 
               L 1  = m w1  + 2/π 2 m p1   
             
             
                 
               L 2  = 2/π 2  m p1  + m w2  + 2/π 2 m p2   
             
             
                 
               L 3  = 2/π 2 m p2  + m r   
             
             
                 
               The relationship of the damping constant of the 
             
             
                 
               piezoelectric element with the electrical resistance: 
             
             
                 
               R 1  = h p1   
             
             
                 
               R 2  = h p2   
             
             
                 
               where 
             
             
                 
               m w1  = the mass of the weight 21 
             
             
                 
               m w2  = the mass of the weight 22 
             
             
                 
               m p1  = the mass of the piezoelectric element 31 
             
             
                 
               m p2  = the mass of the piezoelectric element 32 
             
             
                 
               m r  = the mass of the rod 25 
             
             
                 
               k p1  = the spring constant of the piezoelectric element 31 
             
             
                 
               k p2  = the spring constant of the piezoelectric element 32 
             
             
                 
               h p1  = the damping constant of the piezoelectric element 31 
             
             
                 
               h p2  = the damping constant of the piezoelectric element 32 
             
             
                 
                 
             
           
        
       
     
   
   The first resonance frequency f 1  and the second resonance frequency f 2  in the electric circuit shown in FIG.  6  can be expressed as the following equations (1) and (2). 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         [Equation  (1)] 
                       
                       ⁢ 
                       
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         f 
                         1 
                       
                       = 
                       
                         
                           1 
                           
                             2 
                             ⁢ 
                             π 
                           
                         
                         ⁢ 
                         
                           
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             C 
                                             2 
                                           
                                           ⁢ 
                                           
                                             L 
                                             2 
                                           
                                           ⁢ 
                                           
                                             L 
                                             3 
                                           
                                         
                                         + 
                                         
                                           
                                             C 
                                             2 
                                           
                                           ⁢ 
                                           
                                             L 
                                             1 
                                           
                                           ⁢ 
                                           
                                             L 
                                             3 
                                           
                                         
                                         + 
                                         
                                           
                                             C 
                                             1 
                                           
                                           ⁢ 
                                           
                                             L 
                                             1 
                                           
                                           ⁢ 
                                           
                                             L 
                                             3 
                                           
                                         
                                         + 
                                         
                                           
                                             C 
                                             1 
                                           
                                           ⁢ 
                                           
                                             L 
                                             1 
                                           
                                           ⁢ 
                                           
                                             L 
                                             2 
                                           
                                         
                                       
                                       
                                         
                                           C 
                                           1 
                                         
                                         ⁢ 
                                         
                                           C 
                                           2 
                                         
                                       
                                     
                                     - 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             
                                               
                                                 ( 
                                                 
                                                   
                                                     
                                                       
                                                         C 
                                                         2 
                                                       
                                                       ⁢ 
                                                       
                                                         L 
                                                         2 
                                                       
                                                       ⁢ 
                                                       
                                                         L 
                                                         3 
                                                       
                                                     
                                                     + 
                                                     
                                                       
                                                         C 
                                                         2 
                                                       
                                                       ⁢ 
                                                       
                                                         L 
                                                         1 
                                                       
                                                       ⁢ 
                                                       
                                                         L 
                                                         3 
                                                       
                                                     
                                                     + 
                                                     
                                                       
                                                         C 
                                                         1 
                                                       
                                                       ⁢ 
                                                       
                                                         L 
                                                         1 
                                                       
                                                       ⁢ 
                                                       
                                                         L 
                                                         3 
                                                       
                                                     
                                                     + 
                                                     
                                                       
                                                         C 
                                                         1 
                                                       
                                                       ⁢ 
                                                       
                                                         L 
                                                         1 
                                                       
                                                       ⁢ 
                                                       
                                                         L 
                                                         2 
                                                       
                                                     
                                                   
                                                   
                                                     
                                                       C 
                                                       1 
                                                     
                                                     ⁢ 
                                                     
                                                       C 
                                                       2 
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                               2 
                                             
                                             - 
                                           
                                         
                                       
                                       
                                         
                                           
                                             
                                               
                                                 L 
                                                 1 
                                               
                                               ⁢ 
                                               
                                                 L 
                                                 2 
                                               
                                               ⁢ 
                                               
                                                 
                                                   L 
                                                   3 
                                                 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   
                                                     
                                                       L 
                                                       1 
                                                     
                                                     + 
                                                     
                                                       L 
                                                       2 
                                                     
                                                     + 
                                                     
                                                       L 
                                                       3 
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                             
                                               
                                                 C 
                                                 1 
                                               
                                               ⁢ 
                                               
                                                 C 
                                                 2 
                                               
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                             
                               2 
                               ⁢ 
                               
                                 L 
                                 1 
                               
                               ⁢ 
                               
                                 L 
                                 2 
                               
                               ⁢ 
                               
                                 L 
                                 3 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 1 
                 ) 
               
             
           
           
             
               
                 
                   
                     
                       
                         [Equation  (2)] 
                       
                       ⁢ 
                       
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         f 
                         2 
                       
                       = 
                       
                         
                           1 
                           
                             2 
                             ⁢ 
                             π 
                           
                         
                         ⁢ 
                         
                           
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             C 
                                             2 
                                           
                                           ⁢ 
                                           
                                             L 
                                             2 
                                           
                                           ⁢ 
                                           
                                             L 
                                             3 
                                           
                                         
                                         + 
                                         
                                           
                                             C 
                                             2 
                                           
                                           ⁢ 
                                           
                                             L 
                                             1 
                                           
                                           ⁢ 
                                           
                                             L 
                                             3 
                                           
                                         
                                         + 
                                         
                                           
                                             C 
                                             1 
                                           
                                           ⁢ 
                                           
                                             L 
                                             1 
                                           
                                           ⁢ 
                                           
                                             L 
                                             3 
                                           
                                         
                                         + 
                                         
                                           
                                             C 
                                             1 
                                           
                                           ⁢ 
                                           
                                             L 
                                             1 
                                           
                                           ⁢ 
                                           
                                             L 
                                             2 
                                           
                                         
                                       
                                       
                                         
                                           C 
                                           1 
                                         
                                         ⁢ 
                                         
                                           C 
                                           2 
                                         
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             
                                               
                                                 ( 
                                                 
                                                   
                                                     
                                                       
                                                         C 
                                                         2 
                                                       
                                                       ⁢ 
                                                       
                                                         L 
                                                         2 
                                                       
                                                       ⁢ 
                                                       
                                                         L 
                                                         3 
                                                       
                                                     
                                                     + 
                                                     
                                                       
                                                         C 
                                                         2 
                                                       
                                                       ⁢ 
                                                       
                                                         L 
                                                         1 
                                                       
                                                       ⁢ 
                                                       
                                                         L 
                                                         3 
                                                       
                                                     
                                                     + 
                                                     
                                                       
                                                         C 
                                                         1 
                                                       
                                                       ⁢ 
                                                       
                                                         L 
                                                         1 
                                                       
                                                       ⁢ 
                                                       
                                                         L 
                                                         3 
                                                       
                                                     
                                                     + 
                                                     
                                                       
                                                         C 
                                                         1 
                                                       
                                                       ⁢ 
                                                       
                                                         L 
                                                         1 
                                                       
                                                       ⁢ 
                                                       
                                                         L 
                                                         2 
                                                       
                                                     
                                                   
                                                   
                                                     
                                                       C 
                                                       1 
                                                     
                                                     ⁢ 
                                                     
                                                       C 
                                                       2 
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                               2 
                                             
                                             - 
                                           
                                         
                                       
                                       
                                         
                                           
                                             
                                               
                                                 L 
                                                 1 
                                               
                                               ⁢ 
                                               
                                                 L 
                                                 2 
                                               
                                               ⁢ 
                                               
                                                 
                                                   L 
                                                   3 
                                                 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   
                                                     
                                                       L 
                                                       1 
                                                     
                                                     + 
                                                     
                                                       L 
                                                       2 
                                                     
                                                     + 
                                                     
                                                       L 
                                                       3 
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                             
                                               
                                                 C 
                                                 1 
                                               
                                               ⁢ 
                                               
                                                 C 
                                                 2 
                                               
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                             
                               2 
                               ⁢ 
                               
                                 L 
                                 1 
                               
                               ⁢ 
                               
                                 L 
                                 2 
                               
                               ⁢ 
                               
                                 L 
                                 3 
                               
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 2 
                 ) 
               
             
           
         
       
     
   
   From the equations (1) and (2), the necessary conditions for satisfying that f 1 =f 2 /2 is expressed as the equation (3) below, where the electrical resistances R 1  and R 2  (i.e., the damping constants h p1  and h p2 ) are ignored, because they have no relationship with the resonance frequencies. 
   
     
       
         
           
             
               
                 
                   
                     
                       [Equation  (3)] 
                     
                   
                 
                 
                   
                     
                       
                         
                           1 
                           
                             2 
                             ⁢ 
                             π 
                           
                         
                         ⁢ 
                         
                           
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             C 
                                             2 
                                           
                                           ⁢ 
                                           
                                             L 
                                             2 
                                           
                                           ⁢ 
                                           
                                             L 
                                             3 
                                           
                                         
                                         + 
                                         
                                           
                                             C 
                                             2 
                                           
                                           ⁢ 
                                           
                                             L 
                                             1 
                                           
                                           ⁢ 
                                           
                                             L 
                                             3 
                                           
                                         
                                         + 
                                         
                                           
                                             C 
                                             1 
                                           
                                           ⁢ 
                                           
                                             L 
                                             1 
                                           
                                           ⁢ 
                                           
                                             L 
                                             3 
                                           
                                         
                                         + 
                                         
                                           
                                             C 
                                             1 
                                           
                                           ⁢ 
                                           
                                             L 
                                             1 
                                           
                                           ⁢ 
                                           
                                             L 
                                             2 
                                           
                                         
                                       
                                       
                                         
                                           C 
                                           1 
                                         
                                         ⁢ 
                                         
                                           C 
                                           2 
                                         
                                       
                                     
                                     - 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             
                                               
                                                 ( 
                                                 
                                                   
                                                     
                                                       
                                                         C 
                                                         2 
                                                       
                                                       ⁢ 
                                                       
                                                         L 
                                                         2 
                                                       
                                                       ⁢ 
                                                       
                                                         L 
                                                         3 
                                                       
                                                     
                                                     + 
                                                     
                                                       
                                                         C 
                                                         2 
                                                       
                                                       ⁢ 
                                                       
                                                         L 
                                                         1 
                                                       
                                                       ⁢ 
                                                       
                                                         L 
                                                         3 
                                                       
                                                     
                                                     + 
                                                     
                                                       
                                                         C 
                                                         1 
                                                       
                                                       ⁢ 
                                                       
                                                         L 
                                                         1 
                                                       
                                                       ⁢ 
                                                       
                                                         L 
                                                         3 
                                                       
                                                     
                                                     + 
                                                     
                                                       
                                                         C 
                                                         1 
                                                       
                                                       ⁢ 
                                                       
                                                         L 
                                                         1 
                                                       
                                                       ⁢ 
                                                       
                                                         L 
                                                         2 
                                                       
                                                     
                                                   
                                                   
                                                     
                                                       C 
                                                       1 
                                                     
                                                     ⁢ 
                                                     
                                                       C 
                                                       2 
                                                     
                                                   
                                                 
                                                 ) 
                                               
                                               2 
                                             
                                             - 
                                           
                                         
                                       
                                       
                                         
                                           
                                             
                                               
                                                 L 
                                                 1 
                                               
                                               ⁢ 
                                               
                                                 L 
                                                 2 
                                               
                                               ⁢ 
                                               
                                                 
                                                   L 
                                                   3 
                                                 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   
                                                     
                                                       L 
                                                       1 
                                                     
                                                     + 
                                                     
                                                       L 
                                                       2 
                                                     
                                                     + 
                                                     
                                                       L 
                                                       3 
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                             
                                               
                                                 C 
                                                 1 
                                               
                                               ⁢ 
                                               
                                                 C 
                                                 2 
                                               
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                             
                               2 
                               ⁢ 
                               
                                 L 
                                 1 
                               
                               ⁢ 
                               
                                 L 
                                 2 
                               
                               ⁢ 
                               
                                 L 
                                 3 
                               
                             
                           
                         
                       
                       = 
                       
                         
                           1 
                           2 
                         
                         × 
                       
                     
                   
                 
                 
                   
                     
                       
                         1 
                         
                           2 
                           ⁢ 
                           π 
                         
                       
                       ⁢ 
                       
                         
                           
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           C 
                                           2 
                                         
                                         ⁢ 
                                         
                                           L 
                                           2 
                                         
                                         ⁢ 
                                         
                                           L 
                                           3 
                                         
                                       
                                       + 
                                       
                                         
                                           C 
                                           2 
                                         
                                         ⁢ 
                                         
                                           L 
                                           1 
                                         
                                         ⁢ 
                                         
                                           L 
                                           3 
                                         
                                       
                                       + 
                                       
                                         
                                           C 
                                           1 
                                         
                                         ⁢ 
                                         
                                           L 
                                           1 
                                         
                                         ⁢ 
                                         
                                           L 
                                           3 
                                         
                                       
                                       + 
                                       
                                         
                                           C 
                                           1 
                                         
                                         ⁢ 
                                         
                                           L 
                                           1 
                                         
                                         ⁢ 
                                         
                                           L 
                                           2 
                                         
                                       
                                     
                                     
                                       
                                         C 
                                         1 
                                       
                                       ⁢ 
                                       
                                         C 
                                         2 
                                       
                                     
                                   
                                   - 
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             
                                               ( 
                                               
                                                 
                                                   
                                                     
                                                       C 
                                                       2 
                                                     
                                                     ⁢ 
                                                     
                                                       L 
                                                       2 
                                                     
                                                     ⁢ 
                                                     
                                                       L 
                                                       3 
                                                     
                                                   
                                                   + 
                                                   
                                                     
                                                       C 
                                                       2 
                                                     
                                                     ⁢ 
                                                     
                                                       L 
                                                       1 
                                                     
                                                     ⁢ 
                                                     
                                                       L 
                                                       3 
                                                     
                                                   
                                                   + 
                                                   
                                                     
                                                       C 
                                                       1 
                                                     
                                                     ⁢ 
                                                     
                                                       L 
                                                       1 
                                                     
                                                     ⁢ 
                                                     
                                                       L 
                                                       3 
                                                     
                                                   
                                                   + 
                                                   
                                                     
                                                       C 
                                                       1 
                                                     
                                                     ⁢ 
                                                     
                                                       L 
                                                       1 
                                                     
                                                     ⁢ 
                                                     
                                                       L 
                                                       2 
                                                     
                                                   
                                                 
                                                 
                                                   
                                                     C 
                                                     1 
                                                   
                                                   ⁢ 
                                                   
                                                     C 
                                                     2 
                                                   
                                                 
                                               
                                               ) 
                                             
                                             2 
                                           
                                           - 
                                         
                                       
                                     
                                     
                                       
                                         
                                           
                                             
                                               L 
                                               1 
                                             
                                             ⁢ 
                                             
                                               L 
                                               2 
                                             
                                             ⁢ 
                                             
                                               
                                                 L 
                                                 3 
                                               
                                               ⁡ 
                                               
                                                 ( 
                                                 
                                                   
                                                     L 
                                                     1 
                                                   
                                                   + 
                                                   
                                                     L 
                                                     2 
                                                   
                                                   + 
                                                   
                                                     L 
                                                     3 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                           
                                             
                                               C 
                                               1 
                                             
                                             ⁢ 
                                             
                                               C 
                                               2 
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                           
                             2 
                             ⁢ 
                             
                               L 
                               1 
                             
                             ⁢ 
                             
                               L 
                               2 
                             
                             ⁢ 
                             
                               L 
                               3 
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 3 
                 ) 
               
             
           
         
       
     
   
   The equation (4) is derived from the equation (3). 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         [Equation  (4)] 
                       
                       ⁢ 
                       
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             C 
                             1 
                           
                           ⁢ 
                           
                             C 
                             2 
                           
                           ⁢ 
                           
                             L 
                             1 
                           
                           ⁢ 
                           
                             L 
                             2 
                           
                           ⁢ 
                           
                             
                               L 
                               3 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   L 
                                   1 
                                 
                                 + 
                                 
                                   L 
                                   2 
                                 
                                 + 
                                 
                                   L 
                                   3 
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             ( 
                             
                               
                                 
                                   C 
                                   2 
                                 
                                 ⁢ 
                                 
                                   L 
                                   2 
                                 
                                 ⁢ 
                                 
                                   L 
                                   3 
                                 
                               
                               + 
                               
                                 
                                   C 
                                   2 
                                 
                                 ⁢ 
                                 
                                   L 
                                   1 
                                 
                                 ⁢ 
                                 
                                   L 
                                   3 
                                 
                               
                               + 
                               
                                 
                                   C 
                                   1 
                                 
                                 ⁢ 
                                 
                                   L 
                                   1 
                                 
                                 ⁢ 
                                 
                                   L 
                                   3 
                                 
                               
                               + 
                               
                                 
                                   C 
                                   1 
                                 
                                 ⁢ 
                                 
                                   L 
                                   1 
                                 
                                 ⁢ 
                                 
                                   L 
                                   2 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                       
                       = 
                       
                         4 
                         25 
                       
                     
                   
                 
               
             
             
               
                 ( 
                 4 
                 ) 
               
             
           
         
       
     
   
   Based on the relationships shown in Table 1, the equation (4) can be transformed using the masses and the spring constants, and the following equation (5) is derived. 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         [Equation  (5)] 
                       
                       ⁢ 
                       
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   1 
                                   
                                     
                                       K 
                                       p1 
                                     
                                     ⁢ 
                                     
                                       K 
                                       p2 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       m 
                                       w1 
                                     
                                     + 
                                     
                                       
                                         2 
                                         
                                           π 
                                           2 
                                         
                                       
                                       ⁢ 
                                       
                                         m 
                                         p1 
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       
                                         2 
                                         
                                           π 
                                           2 
                                         
                                       
                                       ⁢ 
                                       
                                         m 
                                         p1 
                                       
                                     
                                     + 
                                     
                                       m 
                                       w2 
                                     
                                     + 
                                     
                                       
                                         2 
                                         
                                           π 
                                           2 
                                         
                                       
                                       ⁢ 
                                       
                                         m 
                                         p2 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         2 
                                         
                                           π 
                                           2 
                                         
                                       
                                       ⁢ 
                                       
                                         m 
                                         p2 
                                       
                                     
                                     + 
                                     
                                       m 
                                       r 
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       m 
                                       w1 
                                     
                                     + 
                                     
                                       m 
                                       w2 
                                     
                                     + 
                                     
                                       m 
                                       r 
                                     
                                     + 
                                     
                                       
                                         4 
                                         
                                           π 
                                           2 
                                         
                                       
                                       ⁢ 
                                       
                                         m 
                                         p1 
                                       
                                     
                                     + 
                                     
                                       
                                         4 
                                         
                                           π 
                                           2 
                                         
                                       
                                       ⁢ 
                                       
                                         m 
                                         p2 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       1 
                                       
                                         K 
                                         p2 
                                       
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         
                                           
                                             2 
                                             
                                               π 
                                               2 
                                             
                                           
                                           ⁢ 
                                           
                                             m 
                                             p1 
                                           
                                         
                                         + 
                                         
                                           m 
                                           w2 
                                         
                                         + 
                                         
                                           
                                             2 
                                             
                                               π 
                                               2 
                                             
                                           
                                           ⁢ 
                                           
                                             m 
                                             p2 
                                           
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         
                                           
                                             2 
                                             
                                               π 
                                               2 
                                             
                                           
                                           ⁢ 
                                           
                                             m 
                                             p2 
                                           
                                         
                                         + 
                                         
                                           m 
                                           r 
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         1 
                                         
                                           K 
                                           p1 
                                         
                                       
                                       + 
                                       
                                         1 
                                         
                                           K 
                                           p2 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         m 
                                         w1 
                                       
                                       + 
                                       
                                         
                                           2 
                                           
                                             π 
                                             2 
                                           
                                         
                                         ⁢ 
                                         
                                           m 
                                           p1 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         
                                           2 
                                           
                                             π 
                                             2 
                                           
                                         
                                         ⁢ 
                                         
                                           m 
                                           p2 
                                         
                                       
                                       + 
                                       
                                         m 
                                         r 
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       1 
                                       
                                         K 
                                         p1 
                                       
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         
                                           m 
                                           w1 
                                         
                                         + 
                                         
                                           
                                             2 
                                             
                                               π 
                                               2 
                                             
                                           
                                           ⁢ 
                                           
                                             m 
                                             p1 
                                           
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       ( 
                                       
                                         
                                           
                                             2 
                                             
                                               π 
                                               2 
                                             
                                           
                                           ⁢ 
                                           
                                             m 
                                             p1 
                                           
                                         
                                         + 
                                         
                                           m 
                                           w2 
                                         
                                         + 
                                         
                                           
                                             2 
                                             
                                               π 
                                               2 
                                             
                                           
                                           ⁢ 
                                           
                                             m 
                                             p2 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                         
                       
                       = 
                       
                         4 
                         25 
                       
                     
                   
                 
               
             
             
               
                 ( 
                 5 
                 ) 
               
             
           
         
       
     
   
   Thus, if the vibration-generator  40  is assembled from the members which have the masses and the spring constants satisfying the equation (5), the physical resonance frequencies of the vibration-generator  40  will satisfy the relationship of f 1 =f 2 /2. 
   The masses and the spring constants of the respective members in the embodiment shown in  FIGS. 4A and 4B  are shown in Table 2. 
   
     
       
             
             
             
             
           
             
             
             
             
             
           
         
             
                 
               TABLE 2 
             
             
                 
                 
             
             
                 
                 
               Mass 
               Spring constant 
             
             
                 
                 
             
           
           
             
                 
             
           
        
         
             
                 
               Weight 21 
               148 
               mg 
               — 
             
             
                 
               Piezoelectric element 31 
               55 
               mg 
               41250000 N/m 
             
             
                 
               Weight 22 
               65 
               mg 
               — 
             
             
                 
               Piezoelectric element 32 
               55 
               mg 
               41250000 N/m 
             
             
                 
               Rod 25 
               12 
               mg 
               — 
             
             
                 
                 
             
           
        
       
     
   
     FIG. 7  is a graph showing the characteristics of the frequencies of the vibration-generator  40  (a physical vibration system) assembled as above. The first resonance frequency f 1  of this system is 123 kHz, and the second resonance frequency f 2  thereof is 246 kHz. Thus, the relationship of f 1 =f 2 /2 is satisfied. 
   The relationships of the applied voltages with the vibration waveform of the rod are shown in  FIGS. 8 and 9 , where a voltage of sine waveform pulses having an amplitude of 2 V and a frequency of 123 kHz is applied to the first piezoelectric element  31  of the vibration-generator  40 , and a voltage of sine waveform pulses having an amplitude of 1V and a frequency of 246 kHz is applied to the second piezoelectric element  32  thereof, with a phase difference of 90 deg therebetween.  FIG. 8  shows the waveforms of the applied voltages, and  FIG. 9  shows the vibration waveform of the rod. 
   A regular sawtooth-like vibration waveform of the rod is realized. Thus, the slider  10  can be driven in a predetermined direction. 
   &lt;Second Embodiment ( FIGS. 10 and 11  )&gt; 
   In the second embodiment, instead of the voltages of sine waveform pulses, voltages of rectangular waveform pulses having duty ratios of 1/2 are applied to the driving device, which is assembled as above-mentioned to satisfy the relationship of f 1 =f 2 /2.  FIG. 10  shows the voltages of rectangular waveform pulses applied to the first piezoelectric element  31  and the second piezoelectric element  32 , wherein a voltage of rectangular waveform pulses, having an amplitude of 2V, a frequency of 123 kHz and a duty ratio of 1/2 is applied to the first piezoelectric element  31 , and a voltage of rectangular waveform pulses, having an amplitude of 1V, a frequency of 246 kHz and a duty ratio of 1/2 is applied to the second piezoelectric element  32 . 
   In general, a rectangular waveform can be expressed as the sum of a plurality of sine waveforms, according to the Fourier expansion. The general formula thereof is expressed by the following equation (6). 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         [Equation  (6)] 
                       
                       ⁢ 
                       
                           
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             
                               V 
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             = 
                               
                             ⁢ 
                             
                               
                                 
                                   V 
                                   d 
                                 
                                 · 
                                 
                                   4 
                                   π 
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         sin 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             π 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             Du 
                                           
                                           ) 
                                         
                                       
                                       1 
                                     
                                     ⁢ 
                                     
                                       sin 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             ω 
                                             d 
                                           
                                           ⁢ 
                                           t 
                                         
                                         ) 
                                       
                                     
                                   
                                   + 
                                   
                                     
                                       
                                         sin 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         2 
                                         ⁢ 
                                         
                                           ( 
                                           
                                             π 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             Du 
                                           
                                           ) 
                                         
                                       
                                       2 
                                     
                                     ⁢ 
                                     sin 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         
                                           2 
                                           ⁢ 
                                           
                                             ω 
                                             d 
                                           
                                           ⁢ 
                                           t 
                                         
                                         + 
                                         
                                           π 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                               
                             ⁢ 
                             
                               
                                 
                                   
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     3 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         π 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         Du 
                                       
                                       ) 
                                     
                                   
                                   3 
                                 
                                 ⁢ 
                                 
                                   sin 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         3 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           ω 
                                           d 
                                         
                                         ⁢ 
                                         t 
                                       
                                       + 
                                       π 
                                     
                                     ) 
                                   
                                 
                               
                               + 
                               
                                 
                                   
                                     sin 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     4 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         π 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         Du 
                                       
                                       ) 
                                     
                                   
                                   4 
                                 
                                 ⁢ 
                                 
                                   sin 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         4 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           ω 
                                           d 
                                         
                                         ⁢ 
                                         t 
                                       
                                       + 
                                       
                                         
                                           3 
                                           ⁢ 
                                           π 
                                         
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               + 
                               ⋯ 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 6 
                 ) 
               
             
           
         
       
     
   
   In the equation (6), Vd is amplitude of a rectangular waveform pulse; ωd is angular velocity; Du is duty ratio; and t is time. Further, in the large bracket in the equation (6), the first term, the second term, . . . , and the Nth term, respectively, represents the first component, the second component, . . . , and the “n”th component of the plurality of sine waveforms. 
   In the equation (6), if Du equals 1/2 (50%), the second term in the bracket is zero. In other words, the voltage of rectangular waveform pulses having a Du of 1/2 does not contain the second sine wave component. Thus, if the frequency of the rectangular waveform signal input to the first piezoelectric element  31  is 123 kHz (i.e., the first resonance frequency of the system), the first resonance component alone is enlarged by this inputting, where the higher terms equal or more than the third order is ignored, because such higher terms are reduced due to the transfer characteristics of the piezoelectric element. 
   Likewise, if the frequency of the rectangular waveform signal input to the second piezoelectric element  32  is 246 kHz (i.e., the second resonance frequency of the system), the second resonance component alone is enlarged by this inputting. Thus, a regular sawtooth-like vibration of the rod, as is obtained when two voltages of sine waveform pulses as shown in  FIG. 8  are input, can be obtained (see  FIGS. 10 and 11 ). 
   &lt;Third Embodiment ( FIGS. 12 and 13  )&gt; 
   In the third embodiment, the same voltages of rectangular waveform pulses as shown in  FIG. 12  are applied to both of the first piezoelectric element  31  and the second piezoelectric element  32  of the driving device, which is assembled as above-mentioned to satisfy the relationship of f 1 =f 2 /2. The voltage of rectangular waveform pulses have an amplitude of 2V, a frequency of 123 kHz (which equals the first resonance frequency of the system), and a duty ratio of 2/3. 
   In the case where the same voltages of rectangular waveform pulses are applied to two piezoelectric elements, it is necessary that the rectangular waveform contains the second sine wave component. Therefore, the duty ratio should not be 1/2. This is because the voltage of rectangular waveform pulses having a duty ratio of 1/2 does not contain the second sine wave component, as mentioned above. 
   For this reason, the duty ratios of the voltages of rectangular waveform pulses in the third embodiment are set at 2/3. Note that the duty ratio of 2/3 is not always needed, and a duty ratio not equal to 1/2 would be enough. The frequencies of these voltages of rectangular waveform pulses are set at 123 kHz, which equals the first resonance frequency of the system. 
   With the voltages of rectangular waveform pulses as above being applied to the piezoelectric elements, the first resonance component and the second resonance component are enlarged, so that a regular sawtooth-like vibration of the rod as shown in  FIG. 13  can be obtained. 
   &lt;Fourth Embodiment ( FIGS. 14 and 15  )&gt; 
   In the fourth embodiment, a roll type piezoelectric element is used, in which two weight-sections and two piezoelectric-element-sections are integrally composed. As shown in  FIG. 14 , electrodes  51  and  52  with an interval are applied on a piezoelectric sheet  50 . This piezoelectric sheet  50  is rolled up, and a rod (not shown) is fixed to the resultant roll to provide a vibration-generator. The fields  56  and  57  of the piezoelectric sheet on which electrodes are not applied can not be expanded or contracted, and therefore can serve as weights. 
   In the embodiment shown in  FIG. 14 , the weight-section  56  corresponds to the weight  21  in  FIG. 4 , and the weight-section  57  corresponds to the weight  22  in  FIG. 4 . The fields  51  and  52  of the piezoelectric sheet on which the electrodes are applied function as piezoelectric elements, and these sections (after rolled up) correspond to the piezoelectric elements  31  and  32  in  FIG. 4 , respectively. The rod is fixed to one end  52   a  of the piezoelectric-element-section  52 . 
     FIG. 15  shows a modification to the embodiment shown in  FIG. 14 . The modification shown in  FIG. 15  differs from the embodiment shown in  FIG. 14  in that electrodes are applied also to the weight-sections  56 ′ and  57 ′. With no voltages being applied thereto, the sections would not expand or contract even if the electrodes are applied thereto. Thus, such the sections  56 ′ and  57 ′ can serve as weight-sections. The electrodes on the weight-sections  56 ′ and  57 ′ are applied with slit-like spaces between themselves and the electrodes  51  and  52 , respectively, such that the voltages are not to be applied thereto. In the case where the fields on which the electrodes are applied function as weight-sections, it is possible to reduce the volumes of such weight-sections, because the specific gravity of such weight-section is larger than that of weight-sections having no electrodes thereon. As a result, the driving device can be advantageously made compact. 
   In the illustrated embodiment, the weight-sections  56 ′ and  57 ′ are insulated by providing the slits so that voltages can not be applied to the electrodes thereon. But, not only slits but also other appropriate means may be used to insulate the weight-sections  56 ′ and  57 ′. 
   Also in the constitutions shown in  FIGS. 14 and 15  (a physical vibration system), the relationship of f 1 =f 2 /2 can be obtained, by determining the masses and the spring constants of the respective members of the systems in accordance with the modeling as described with reference to  FIGS. 5 and 6 . 
   In the fourth embodiment, the two piezoelectric-element-sections are formed of the same material, and therefore C 1  equals C 2  (C 1 =C 2 ). Accordingly, the equations (4) and (5) are expressed as the following equations (4′) and (5′). 
                         [Equation       (     4   ′       )]     ⁢                             L   1     ⁢     L   2     ⁢       L   3     ⁡     (       L   1     +     L   2     +     L   3       )             (         L   2     ⁢     L   3       +       L   1     ⁢     L   3       +       L   1     ⁢     L   3       +       L   1     ⁢     L   2         )     2       =     4   25                   (     4   ′     )                       [Equation       (     5   ′       )]     ⁢                                   (       m   w1     +       2     π   2       ⁢     m   p1         )     ⁢     (         2     π   2       ⁢     m   p1       +     m   w2     +       2     π   2       ⁢     m   p2         )                   (         2     π   2       ⁢     m   p2       +     m   r       )     ⁢     (       m   w1     +     m   w2     +     m   r     +       4     π   2       ⁢     m   p1       +       4     π   2       ⁢     m   p2         )                     (         (         2     π   2       ⁢     m   p1       +     m   w2     +       2     π   2       ⁢     m   p2         )     ⁢     (         2     π   2       ⁢     m   p2       +     m   r       )       +                   2   ⁢     (       m   w1     +       2     π   2       ⁢     m   p1         )     ⁢     (         2     π   2       ⁢     m   p2       +     m   r       )       +                     (       m   w1     +       2     π   2       ⁢     m   p1         )     ⁢     (         2     π   2       ⁢     m   p1       +     m   w2     +       2     π   2       ⁢     m   p2         )       )     2             =     4   25                   (     5   ′     )               
&lt;Fifth Embodiment ( FIGS. 16 and 17  )&gt;
 
   In any of the foregoing embodiments, the vibration-generator  40  comprises two piezoelectric elements  31  and  32 . But, the vibration-generator can comprise three or more piezoelectric elements. In the fifth embodiment, three piezoelectric elements are used to compose a vibration-generator. 
   To the vibration-generator as shown in  FIG. 4 , a third piezoelectric element is added, which is disposed in series between the piezoelectric elements  31  and  32 . Further, respective weights are disposed between the piezoelectric element  31  and the third piezoelectric element, and between the piezoelectric element  32  and the third piezoelectric element. 
   In another case, as shown in  FIG. 16 , a piezoelectric sheet  70  is rolled up, and a rod (not shown) is fixed thereto so that a vibration-generator is composed. The embodiment shown in  FIG. 16  is constructed in the same manner as in the embodiment shown in  FIG. 14 . That is, electrodes  71 ,  72  and  73  with certain intervals are applied on the piezoelectric sheet  70 , and then, the sheet  70  is rolled up to provide an integral vibration-generator. The fields of the sheet on which the electrodes are not applied can not expand or contract, and thus can serve as weight-sections. In this regard, electrodes (not shown) can be applied also to the fields of the sheet which are to be functioned as weights, like in the embodiment shown in  FIG. 15 . 
   Also in this vibration-generator comprising three piezoelectric elements, the relationship of f 1 =f 2 /2=f 3 /3 can be obtained for the first to the third resonance frequencies of the vibration-generator, where f 3  is the third resonance frequency. This is realized by determining the masses and the spring constants of the respective members of the systems in accordance with the modeling based on the Force-Voltage Method as described above. 
   The same is also true in a vibration-generator comprising four or more piezoelectric elements.  FIG. 18  shows an embodiment, wherein “n” piezoelectric elements are used. In  FIG. 18 , one end of the first piezoelectric element  131  is fixed to the first weight  121 , and one end of the second piezoelectric element  132  is fixed to the rod  25 . Between the first and the second piezoelectric elements, there is provided “n−2” piezoelectric elements. Thus, in total, “n” piezoelectric elements are used. Between each of the “n” piezoelectric elements, the respective weight is provided, only the weights  122  and  123  being illustrated in  FIG. 18 . Also in such the embodiment, the relationship of f 1 =f 2 /2= . . . =fn/n can be obtained in accordance with the modeling based on the Force-Voltage Method, where fn is “n”th resonance frequency. 
   In the fifth embodiment, the first to the third resonance frequencies can be utilized. Thus, more ideal vibration waveform can be obtained for the rod, as compared with any of the vibration-generator in which the first and the second resonance frequencies are utilized. As shown in  FIG. 17 , the inclination of the rising portion of the vibration waveform becomes more gentle in the embodiment utilizing the first to the third resonance frequencies, as compared with the embodiment utilizing the first and the second resonance frequencies. Thus, the vibration waveform is more close to an ideal sawtooth-like waveform shown in  FIG. 2 . 
   Although the present invention has been fully described in connection with the preferred embodiments thereof with reference to the accompanying drawings, it is to be noted that various changes and modifications are apparent to those skilled in the art. Such changes and modifications are to be understood as included within the scope of the present invention as defined by the appended claims unless they depart therefrom.