CROSS REFERENCE TO RELATED APPLICATIONS 
   The present application is related in general subject matter to the following applications, each of which is being filed concurrently with the present application, and each of which is incorporated by reference into the present application: 
   U.S. application Ser. No. 11/551,525; 
   U.S. application Ser. No. 11/551,388; 
   U.S. application Ser. No. 11/584,304; 
   U.S. application Ser. No. 11/584,305. 
   FIELD 
   The present disclosure relates to piezoelectric devices, and more particularly to a nonlinear piezoelectric generator that generates electrical power from a mechanical input. 
   BACKGROUND 
   Piezoelectric devices are presently being employed in greater numbers of applications and in a wide ranging area of technologies. Piezoelectric devices make use of one or more piezoelectric ceramic wafers that are adapted to bow or deform in response to an electric current applied to the wafer. Such piezoelectric wafers also produce an electrical output when they are flexed or deformed from an initial, non-flexed configuration. Thus, piezoelectric wafers are especially useful in applications involving actuators and vibration energy harvesting apparatuses. The following U.S. patents and applications involve various implementations of piezoelectric materials, and are each hereby incorporated by reference into the present application: U.S. Pat. No. 6,858,970 and U.S. Ser. No. 10/909,784, filed Jul. 30, 2004. 
   Another device which has only recently achieved practicality is a Stirling engine. Stirling engines have existed in various forms for many years, however, it has been recent breakthroughs in the design of engine chamber seals that has made these devices practical. A Stirling engine utilizes temperature gradients to convert thermal energy into mechanical energy. Typically, the Stirling engine includes one or more pistons that are driven in a reciprocating fashion by converting thermal energy into mechanical energy. Recently, Stirling engines have shown promise as a low cost, high efficiency solar powered generator for U.S. power grid and spacecraft electric power generation systems. The ability of the Stirling engine to meet or exceed the performance of concentrated photovoltaics has been recently recognized by engineers and researchers interested in exploring alternative power generation systems for use in spacecraft. 
   One drawback with a typical Stirling engine is that the mechanical energy is typically converted to electrical energy using a very large AC electromagnetic generator. A large electromagnetic generator, however, can be a serious drawback for spacecraft applications, where weight is an important consideration. 
   Thus, it would be highly desirable to provide some means for generating electric power from a mechanical input device, for example, from one or more pistons of a Stirling engine. It would further be highly desirable if such a device formed a small, lightweight, and highly efficient apparatus for converting mechanical energy to electrical power. Such a device would significantly enhance the utility of other components, such as Stirling engines. Such a device could enable a Stirling engine to be used in various power generating applications which, at the present time, are not feasible because of the size and weight of typical electromagnetic generators presently employed for use with Stirling engines in power generating applications. 
   SUMMARY 
   The present disclosure relates to a system and method for forming a non-linear, piezoelectric mechanical-to-electrical converter or power generator. 
   In one implementation, a non-linear, piezoelectric power generator is provided that includes a flexible element and an electrically responsive member that are arranged mechanically in series. 
   In one embodiment the flexible element is formed by a flexible beam. The flexible beam and electrically responsive member are further coupled such that a first end of the flexible beam is braced against a fixed structure, while a first end of the electrically responsive member is similarly braced or secured against a second structure. In one implementation, the first and second structures are fixed structures. The second ends of the flexible beam and the electrically responsive member are operably coupled together. The flexible beam and the electrically responsive member are further dimensioned such that the flexible beam assumes a first bowed configuration that exerts a compressive force on the electrically responsive member when no external force is being applied to the flexible beam. The reciprocating mechanical output component is positioned adjacent the flexible beam at an intermediate point along the length of the flexible beam. When the mechanical device presses against the bowed flexible beam, it tends to flatten the beam, which increases the compressive stress on the electrically responsive member, thus causing the electrically responsive member to generate an electrical output signal. When the mechanical force is removed from the flexible beam, or the beam is further flexed into another bowed configuration as the flexible beam is moved past an “over center” orientation, the compressive force on the electrically responsive member is concurrently reduced. This action also produces an electric current as the electrically responsive member is allowed to decompress, but one having a plurality opposite to that of the previously generated electric current. Thus, a reciprocating motion of the mechanical device causes an alternating flattening-bowing-flattening action on the flexible beam, which in turn causes an alternating compressing and decompressing of the electrically responsive member, and a resulting alternating polarity electric current generated by the member. Forming the flexible element in a beam-like configuration and placing the flexible element in a buckled configuration enables a force-to-displacement advantage to be achieved that would be difficult to achieve with other forms or configurations of biasing elements. 
   In one specific implementation, a mechanical output device comprises a piston of a Stirling engine. In this exemplary embodiment, the piston is disposed at a midpoint of the length of the flexible beam. The flexible beam and the electronically responsive member act to convert the mechanical reciprocating motion of the piston to electric power. 
   In one particular embodiment, the apparatus and method of the present disclosure enables the flexible beam to effectively act as a strain amplifier. The flexible beam generates a compressive pressure on the electrically responsive member that is approximately four times greater than what would be achieved by a pressure being exerted directly along the length of the electrically responsive member. 
   In one particular embodiment the electrically responsive member is formed by a multilayer, piezoceramic stack. In another embodiment the electrically responsive member is formed by a magnetostrictive material. 
   Further areas of applicability will become apparent from the description provided herein. It should be understood that the description and specific examples are intended for purposes of illustration only and are not intended to limit the scope of the present disclosure. 


   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The drawings described herein are for illustration purposes only and are not intended to limit the scope of the present disclosure in any way. 
       FIG. 1  is a simplified block diagram of a non-linear piezoelectric mechanical-to-electrical generator in accordance with one particular embodiment of the present disclosure, and showing a flexible beam and a piezoceramic stack of the apparatus in their initial orientations prior to an external mechanical force being applied to the flexible beam, with a Stirling engine being the device that is generating the mechanical input; 
       FIG. 2  is a simplified block diagram of the apparatus of  FIG. 1 , but with a piston of the Stirling engine extended to flatten the flexible beam, thus causing an increased compressive force to be exerted on the piezoceramic stack; 
       FIG. 3  is a simplified block diagram of the apparatus of  FIG. 2 , but with the piston of the Stirling engine fully extended to force the flexible beam past an over center position to a new bowed orientation; 
       FIG. 4  is a perspective view of the flexible beam; 
       FIG. 5  is a simplified block diagram of one embodiment of an alternative implementation of the apparatus and method in which a piezoceramic stack is used to form an electrical-to-mechanical transducer to drive a load spring; 
       FIG. 6  is a block diagram of the apparatus of  FIG. 5  but with a flexible beam of the apparatus being bowed by the force of the expanded piezoceramic beam, to thus drive an input member associated with a load spring in a linearly manner; 
       FIG. 7  is a simplified illustration showing how the apparatus of  FIG. 5  could be configured to form a crankshaft-like driving arrangement to impart rotational movement to a rotationally supported member, to thus drive a motor. 
   


   DETAILED DESCRIPTION 
   The following description is merely exemplary in nature and is not intended to limit the present disclosure, application, or uses. 
   Referring to  FIG. 1 , there is shown a non-linear, mechanical-to-electrical generator apparatus  10  in accordance with one exemplary embodiment of the present disclosure. In this example, a Stirling engine  12  having a reciprocating output piston  14  is illustrated as the device that provides a mechanical input signal to the apparatus  10 . However, it will be appreciated that any device that generates a reciprocating mechanical signal can be used with the apparatus  10 . It is anticipated that the apparatus  10  will find particular utility in connection with a Stirling engine, as the apparatus  10  is able to readily convert the mechanical reciprocating motion of an output piston of such an engine to electric power. Stirling engines are discussed in the following patents, the disclosure of each of which is hereby incorporated by reference into the present disclosure: U.S. Pat. No. 6,871,495; U.S. Pat. No. 6,735,946 and U.S. Pat. No. 6,871,495. 
   The apparatus  10  includes a flexible beam  16  having a first end  18  and a second end  20 . An electrically responsive member  22  in one form comprises a piezoceramic stack (i.e., a unitary stack of piezoelectric wafers), and has a first end  24  and a second end  26 . Alternatively, the electrically responsive member  22  may be formed by a magnetostrictive material. The use of a piezoceramic stack or magnetostrictive material for the electrically responsive member  22  are both viewed as being equally applicable for use with the apparatus  10 . Accordingly, while the following description will reference the electrically responsive member as “piezoceramic stack  22 ”, it will be appreciated that a magnetostrictive material could readily be substituted in place of the piezoceramic material. 
   The flexible beam  16  may be formed from spring steel having a planar shape, as shown in  FIG. 4 , or as a planar leaf spring, or from any other material that is suitably flexible. The first end  18  of the flexible beam  16  is fixedly secured to a suitable support structure or frame member  28 , while the first end  24  of the piezoceramic stack  22  is similarly fixedly secured to a support structure or frame member  30 . The second ends  20  and  26  of the flexible beam  16  and the piezoceramic stack  22  may be secured directly to one another, or to an intermediate coupling assembly  32 . Coupling assembly  32  includes a plurality of wheels  34  that are adapted to ride within a guide track or rail  36  to thus facilitate the smooth application and removal of a compressive force to/from the piezoceramic stack  22 . 
   The flexible beam  16  and the piezoceramic stack  22  are also arranged such that their opposing free ends (i.e., ends  18 ,  20 ,  24  and  26 ) are all generally aligned along a common longitudinal axis extending through the piezoceramic stack  22 . The Stirling engine  12  is preferably supported so that its piston  14  extends generally normal to the longitudinal axis extending through the piezoceramic stack  22 . It will be appreciated that the stroke length of the piston  14  will be a factor that needs to be considered in determining the precise dimensions, and particularly the length, of the flexible beam  16 . 
   With brief reference to  FIG. 4 , the flexible beam  16  has an overall (i.e., unbowed or “unbuckled”) length “L”, a thickness, and a width that may each vary widely to suit a specific application. In one exemplary form the length (“L”) of the beam  16  is about 2.0 inches (50.4 mm), the width is about 0.5 inch (12.7 mm), and the thickness is about 0.030 inch (0.762 mm). The piezoceramic stack  22  may have a length that varies in accordance with the particular application with which the apparatus  10  is being interfaced with, and will be in part dependent on the length and stiffness of the flexible beam  16 . In one exemplary form the uncompressed length of the piezoceramic stack may be about 0.5 inch (12.7 mm). The piezoceramic stack  22  may also take a variety of cross sectional shapes, for example rectangular, round, oval, square, or any other shape that might best suit the need of a particular application. In the present example, the piezoceramic stack  22  has a circular cross sectional shape having a diameter from about 0.375 inch to about 0.5 inch (9.525 mm-12.7 mm). 
   Referring now to  FIGS. 1-3 , the operation of the apparatus  10  will be described. Referring initially to  FIG. 1 , the flexible beam  16  assumes one stable position. In this orientation the flexible beam is exerting a first, or minimum, degree of compressive force on the piezoceramic stack  22 , but a force that is not sufficient to tangibly compress the piezoceramic stack  22 . As the output piston  14  of the Stirling engine  12  initially begins to extend, it exerts a mechanical input force on the flexible beam  16 . This flattens the beam  16 , as indicated in  FIG. 2 . As the flexible beam  16  is flattened, it exerts a significantly increased compressive force on the piezoceramic stack  22 , which causes the piezoceramic stack  22  to generate an electric current pulse output from electrical contacts (not shown) connected to the various layers thereof. This electrical pulse has a first polarity, for example a positive polarity. The orientation of the flexible beam  16  shown in  FIG. 2  also represents an “over center” position. By “over center”, it is meant that the flexible beam  16  will rapidly flex past this point as it is being moved toward the orientation shown in either  FIG. 1  or  FIG. 3 , but will be unstable in the orientation of  FIG. 2 . In the position of  FIG. 2 , the flexible beam  16  is exerting a maximum degree of compressive force on the piezoceramic stack  22 . 
   With reference specifically to  FIGS. 2 and 3 , as the piston  14  of the Stirling engine  12  extends to its maximum stroke length, it moves the midpoint of the flexible beam  16  past the over center position shown in  FIG. 2 . The flexible beam  16  then begins to release the compressive force on the piezoceramic stack  22 . The compressive force continues to decrease as the flexible beam  16  moves into the orientation shown in  FIG. 3 . Once in the orientation shown in  FIG. 3 , the flexible beam  16  is again exerting the same minimum force as it did in the orientation shown in  FIG. 1 . The orientation shown in  FIG. 3  forms a second stable position for the flexible beam  16 . As the flexible beam  16  moves from the orientation shown in  FIG. 2  to that shown in  FIG. 3 , the decompression of the piezoceramic stack  22  causes the stack  22  to generate another electrical output pulse. This electrical pulse will, however, be of the opposite polarity to the pulse that was created by the compressive movement described in connection with the movement of the flexible beam  16  from the orientation shown in  FIG. 1  to that shown in  FIG. 2 . Thus, each complete extension or complete retraction of the piston  14  generates two electrical pulses from the piezoceramic stack  22 . One complete cycle of the piston  14  (i.e., one extension stroke and one retraction stroke) thus generates four electrical pulses from the piezoceramic stack  22 . 
   A significant advantage of the bowed configuration of the flexible beam  16  is that the flexible beam effectively operates as a “strain amplifier”. By this it is meant that a relatively small mechanical motion (i.e., short mechanical stroke) applied at the midpoint of the flexible beam  16  will cause the beam  16  to generate a significantly large compressive pressure on the piezoceramic stack  22 . For example, the compressive pressure generated on the second end  26  of the piezoceramic stack  22  may be up to or greater than 100 times the compressive pressure that would otherwise be generated by a linear linkage applying a force to the second end  26  of the piezoceramic stack  22 . Obviously, the degree of amplification achieved will depend on the stiffness of the flexible beam  16 , the length of the beam  16  and other design criteria. The stiffness and length of the flexible beam  16  can be tailored to meet the needs of a particular application. 
   The change in length of the piezoceramic stack  22 , as a result of a compressive pressure from the flexible beam  16 , is represented by dimension  38  in  FIG. 2 . Mathematically, this displacement can be expressed by the following formula: 
   
     
       
         
           
             Δ 
             stack 
             Piezoceramic 
           
           = 
           
             
               
                 π 
                 2 
               
               ⁢ 
               
                 D 
                 2 
               
             
             
               4 
               ⁢ 
               L 
             
           
         
       
     
   
   where “D” represents the distance separating a line bisecting the free ends of the flexible beam  16 , and a line tangent to the midpoint of the beam  16  ( FIG. 1 ); and 
   The “critical” force required to move the flexible beam  16  between its two stable positions described above may also vary to suit the needs of a specific application. The critical force is also sometimes referred to as the “critical buckling load”. In the exemplary embodiment being discussed, the critical force “Pcr” can be expressed by the formula: 
   
     
       
         
           Pcr 
           = 
           
             
               
                 π 
                 2 
               
               ⁢ 
               EI 
             
             
               L 
               2 
             
           
         
       
     
   
   where “E” is the elastic modulus of the material of the flexible beam  16 ; 
   where “I” is the area moment of inertia of the flexible beam  16 ; and 
   where “L” is the length of the flexible beam  16 . 
   The apparatus  10  can also be used in connection with a Stirling engine to form a “refrigerator”, by intermittently applying and removing an electric current to the piezoceramic stack  22  that causes intermittent bowing and unbowing of the stack  22 . The apparatus  10  is also expected to find utility in other applications where an electrical power output signal is desired in response to linear movement of a mechanical member. 
   Referring to  FIG. 5 , a transducer in the form of an actuator apparatus  100  is shown in accordance with an alternative implementation of the teachings of the present disclosure. Components in common with those described in connection with apparatus  10  will be denoted by reference numerals increased by a factor of 100 over those used to describe the embodiment shown in  FIGS. 1-4 . 
   The apparatus  100  is substantially similar to the apparatus  10 , and includes an electrically responsive member  122  which is installed under compression by a bowed flexible beam  116 , which itself is also installed under compression to assume a slightly bowed or buckled shape. Again, the electrically responsive member  122  may be formed by a piezoceramic stack or by magnetostrictive material. For convenience, component  122  will be referred to as the “piezoceramic stack”. The principal difference between apparatus  10  and apparatus  100  is that with apparatus  100 , an electrical signal (e.g., a voltage) is alternately applied to and removed from the piezoceramic stack  122 , which causes a corresponding alternating expansion (i.e., lengthening) and contraction (i.e., shortening lengthwise) of the stack  122 . However, the flexible beam  116  in this embodiment does not flatten or move over center, as with the apparatus  10 . 
   When the piezoceramic stack  122  lengthens, it urges coupling assembly  132  to move to the left, as indicated in  FIG. 6 . As this occurs, the flexible beam  116  is further bowed into the shape shown in  FIG. 6 . A linear member  114  in contact with the flexible beam  116  at an intermediate point along the length of flexible beam  116  is driven linearly as the flexible beam  116  bows. The member  114  moves in an up and down oscillating motion in accordance with the expansion and retraction of the piezoceramic stack  122 . The member  114  may form a portion of a load spring  112  or any other device able to receive an oscillating mechanical signal. 
   An advantageous feature of the apparatus  100  is that the piezoceramic stack  122  provides a maximum available force at the beginning of its lengthening stroke, where the apparatus provides maximum stroke multiplication. At the end of the piezo stack  122  motion where available force is minimum, the motion amplification is minimum resulting in the ability of the apparatus to transfer more energy to a spring-like load than would be possible if the motion multiplication has a linear relationship. 
   The following equation shows the relationship between piezoceramic stack  122  motion and beam  116  center motion: 
   
     
       
         
           
             
               Δ 
               ⁢ 
               
                   
               
               ⁢ 
               Piezoelectric 
               ⁢ 
               
                   
               
               ⁢ 
               stack 
             
             D 
           
           = 
           
             
               
                 π 
                 2 
               
               ⁢ 
               D 
             
             
               2 
               ⁢ 
               L 
             
           
         
       
     
   
   where “D” is the distance separating a line bisecting the free ends of the beam  116  ( FIG. 5 ) and a line tangent to the midpoint of the beam  116 ; and 
   where “L” is the unbent (i.e., flator unbowed) length of the beam  116 . 
   Referring to  FIG. 7 , a motor apparatus  200  is illustrated in accordance with another embodiment of the present disclosure. The motor apparatus  200  is identical in construction to the apparatus  100  with the exception that an output member  214  is coupled pivotally at one end to a flexible beam  216 . At its opposite end, the member  214  is pivotally coupled via a linkage assembly  214   a  to a rotationally mounted crank  214   b . Apparatus  200  can thus be used to drive a crankshaft like output arrangement from the alternating bowing motion of the piezoceramic stack  222 , to thus create rotational motion from the alternating compressing and decompressing of the piezoceramic stack  222 . 
   While various embodiments and/or implementations have been described for the subject matter of the present disclosure, it will be appreciated that these are merely exemplary, and that other forms of transducers could be implemented from the teachings presented herein.
A non-linear, piezoelectric mechanical-to-electrical generator especially well adapted for use with a Stirling engine, to thus form an electrical power generation system. In one form the generator includes a flexible beam that is configured in a bowed orientation to exert a compressive stress on a piezoceramic stack. A mechanical, linearly reciprocating member is positioned against the flexible beam at a midpoint of the beam. The mechanical member applies a force to the flexible beam that initially tends to flatten the flexible beam, which increases the compressive stress applied to the piezoceramic stack, thus compressing the stack and causing it to generate an electrical output signal. When the mechanical member removes the force, the flexible beam reverts to its initial, bowed configuration. This allows the piezoceramic stack to decompress, and it generates another electrical signal. This alternating compressing and decompressing of the piezoceramic stack causes a series of electrical signals to be generated from the stack. The apparatus forms a lightweight, compact means for converting a linear, reciprocating mechanical output signal into electrical power.