Abstract:
A vibration-based power generator has a variable stiffness oscillator connected to a base. The oscillator comprises an inertial mass moving relative to the base in response to vibrations. The oscillator has a neutral position corresponding to a position of the oscillator when no vibrations are transmitted to the base. The oscillator has a first position where the mass is at a first distance and a second position where the inertial mass is at a second distance from a position of the mass when the oscillator is in neutral position. The second distance is greater than the first distance. A stiffness of the oscillator at the second position is greater than a stiffness of the oscillator at the first position. A transducer generating electric power in response to movement of the inertial mass is associated with the oscillator. A method of optimizing a vibration-based power generator is also presented.

Description:
CROSS-REFERENCE 
     This application claims priority from U.S. Provisional Application No. 61/071,957, filed May 28, 2008, the entirety of which is incorporated by reference. 
    
    
     FIELD OF THE INVENTION 
     The present invention is related to an apparatus for energy harvesting, more specifically for harvesting energy from waste or environmental vibrations. 
     BACKGROUND OF THE INVENTION 
     Batteries are the power source of choice for many sensor systems. Maintenance, replacement, and disposal of batteries are expensive, time consuming, and environmentally hazardous tasks for large sensor networks. In fact, these tasks are practically impossible to perform for embedded sensors, such as those embedded inside structures (bridges, roads, buildings) or airframes. Using micro-power generators (MPGs) to recharge batteries or alternative energy storage devices is an effective solution to this problem. In addition, MPGs can supply energy at higher levels than batteries which allows localized computing and enables new applications such as autonomous wireless sensor networks. 
     A MPG is a vibration-based apparatus consisting of an oscillator embodied by an inertial mass  4  attached to a spring  2 , and of a mechanical damper  3  ( FIG. 1 ). The spring  2  is attached to a housing  6 , which is itself rigidly attached to a surface of a host body that experiences ground vibrations (typically a bridge, a car, or a bicycle). As a result, the housing  6  moves with the host body, and the inertial mass  4  moves with respect to the housing  6 . A transducer  1  drains some of the apparent kinetic energy from the relative displacement of the inertial mass  4  and converts it to electric energy. The electric energy is routed through electric connectors  11  to a power conditioning circuit  22 , and then to a storage device or an electric load  5 . 
     MPGs are designed differently depending on their transduction mechanism. Common transduction mechanisms used in vibration-based MPG include electromagnetic, electrostatic, and piezoelectric mechanisms. For simplicity, similar elements of the MPGs described below with respect to  FIGS. 2 to 4  have been labelled with the same reference numerals and will only be described once. 
     A typical electromagnetic MPG, such as shown in  FIGS. 2   a  and  2   b , consists of an inertial mass  4  and a coil  10  supported by a cantilever beam  8  attached to a wall or base  9  of the housing  6  (shown in  FIG. 1 ) and moving in a magnetic field. The coil  10  and magnetic field combination constitutes the transduction mechanism; it ensures the drain of electric energy. The coil  10  is made of a large number of turns of a small gage copper wire or alternatively another conducting non-magnetic material. A system of four permanent magnets  13  maintains the magnetic field in the air gap. The magnets  13  are attached to the inside of a yoke  12  made of steel (or another ferromagnetic material) to increase the flux density in the air gap. Different arrangements of magnets  13  and support structures made of a magnetic material can also be used to maintain a high flux density magnetic field in the air gap (portion in phantom). The relative motion of the inertial mass  4  with respect to the housing  6  causes the beam  8  and the coil  10  it carries to oscillate in the air gap. As a result, an electric current flows through the coil  10  which dampens the relative motion of the beam  8 -mass  4  system (or oscillator). Alternatively, the yoke  12  and magnets  13  can act as an inertial mass  4  carried by the beam  8  and move with respect to a fixed coil  10 . In either case, the relative motion of the beam  8 -mass  4  system is damped (energy is extracted) by the current flowing through the coil  10 . 
     In a typical electrostatic MPG, such as shown in  FIGS. 3   a  and  3   b , the transducer consists of a variable capacitor supported by the beam  8 . A voltage source (not shown) maintains a potential difference between the capacitor plates  14 . The relative motion of the inertial mass  4  with respect to the housing  6  causes the beam  8  and the capacitor plate  14  it carries, to oscillate in the air gap. As a result, the capacitance of the capacitor changes, electric current flows through the circuit connected to the capacitor, and a capacitive force dampens the relative motion of the beam  8 -mass  4  system. Alternatively, the capacitor plates  14  can be carried on flexible structures that move relative to the housing  6  which allow them to move with respect to each other. In either case, the relative motion of the beam  8 -mass  4  system is damped (energy is extracted) by a capacitive force opposing the motion. 
     In a typical piezoelectric MPG, such as shown in  FIGS. 4   a  and  4   b , the transducer consists of a piezoelectric patch  15  attached to the beam  8 . The relative motion of the inertial mass  4  with respect to the housing  6  creates stresses in the beam  8  and the piezoelectric patch  15 . The patch  15  transforms this stress to a potential difference between the top and bottom sides of the patch  15 . Alternatively, piezoelectric patches  15  can be attached on either side of the cantilever beam  8 . In either case, the relative motion of the beam  8 -mass  4  system is damped (energy is extracted) by a piezoelectric force opposing the motion. 
     All these embodiments use the motion of a linear oscillator (the beam  8 -mass  4  system) generated by environmental vibrations to create electric energy. Regardless of the transduction mechanisms, the collected energy can be maximized by minimizing dissipation in the mechanical oscillator and parasitic losses in electric circuits, or by maximizing the inertial mass of the MPG to increase the input kinetic energy. 
     The usability of vibration-based MPGs is severely limited by the random nature of environmental vibrations. Vibration-based MPGs are tuned to harvest energy within a narrow frequency band in the neighborhood of a natural frequency of the oscillator (MPG bandwidth). Outside this band, the output power is too low to be conditioned and utilized. This limitation is exacerbated by the fact that MPGs are also designed to minimize energy dissipation, further narrowing the MPG bandwidth. On the other hand, vibrations in most environments are random and wideband. As a result, vibration-based MPGs can only harvest energy for a relatively limited fraction of time, which imposes excessive constraints on their usability. 
     Therefore, there is a need for a MPG which would increase the amount of collected energy by increasing the bandwidth of vibration frequencies that can be harvested. 
     SUMMARY OF THE INVENTION 
     It is an object of the present invention to ameliorate at least some of the inconveniences present in the prior art. 
     This invention provides a new concept of wideband vibration-based MPGs, henceforth dubbed MPG. The new MPG architecture expands the bandwidth of vibration-based energy harvesters by employing variable-stiffness oscillators as kinetic energy harvesting elements. These oscillators are designed to passively change their effective stiffness up and/or down with the amplitude of ground/environmental vibration, thereby maintaining the oscillator in resonance and the MPG&#39;s harvested energy near maximum for a wider frequency band than was possible for an oscillator with a fixed/constant stiffness. 
     It is an object of the present invention to provide a wideband vibration-based MPG. 
     It is another object of the present invention to provide a method of optimizing such an MPG. 
     In one aspect, the invention provides a vibration-based power generator comprising a base and a variable stiffness oscillator connected to the base. The oscillator comprises an inertial mass. The inertial mass moves relative to the base in response to vibrations transmitted to the base. The oscillator has a neutral position corresponding to a position of the oscillator relative to the base when no vibrations are transmitted to the base. The oscillator has a first position where the inertial mass is at a first distance in a first direction from a position of the inertial mass when the oscillator is at the neutral position. The oscillator has a second position where the inertial mass is at a second distance in the first direction from the position of the inertial mass when the oscillator is at the neutral position, the second distance being greater than the first distance. A stiffness of the oscillator at the second position is greater than a stiffness of the oscillator at the first position. An electric energy transducer is associated with the oscillator. The electric energy transducer generates electric power in response to movement of the inertial mass relative to the base. 
     In a further aspect, the electric energy transducer has at least a portion connected to the oscillator. 
     In an additional aspect, the electric energy transducer is an electromagnetic transducer comprising at least one magnet and a coil, one of the at least one magnet and the coil being connected to the base, and an other one of the at least one magnet and the coil being connected to the oscillator. 
     In another aspect, the inertial mass is formed by the at least one magnet. 
     In a further aspect, the electric energy transducer is an electrostatic transducer comprising a capacitor having first and second capacitor plates, the first capacitor plate is connected to the base, and the second capacitor plate is connected to the oscillator and generally faces the first capacitor plate. 
     In an additional aspect, the electric energy transducer is a piezoelectric transducer comprising a piezoelectric patch connected to the oscillator. 
     In another aspect, the oscillator further comprises at least one spring connected to the inertial mass, the spring having one end connected to the base. 
     In a further aspect, the oscillator further comprises at least one stopper, the inertial mass contacting the at least one stopper when the oscillator is in the second position. 
     In an additional aspect, the inertial mass forms a portion of the electric energy transducer. 
     In another aspect, the at least one stopper is a spring. 
     In a further aspect, the base houses the oscillator. The oscillator further comprises a plate connected to the inertial mass and the at least one stopper. The plate is connected to the base via at least a pair of springs. The plate contacts the at least one stopper when the oscillator is in the second position. 
     In an additional aspect, the oscillator further comprises a beam connected to the inertial mass. The beam has one end connected to the base. 
     In another aspect, the beam is at least two beams. 
     In a further aspect, the beam has a cross-section which varies along a length of the beam. 
     In an additional aspect, the oscillator further comprises a first stopper. The beam contacts the first stopper when the oscillator is in the second position. 
     In another aspect, the at least one stopper is supported by a movable carriage. 
     In a further aspect, the oscillator further comprises a first stopper. The beam further comprises a ledge extending from the inertial mass. The ledge contacts the first stopper when the oscillator is in the second position. 
     In an additional aspect, the oscillator further comprises a first stopper. The first stopper contacts the inertial mass when the oscillator is in the second position. 
     In another aspect, the oscillator has a third position where the inertial mass is at a third distance in the first direction from the position of the inertial mass when the oscillator is at the neutral position. The third distance is greater than the second distance. A stiffness of the oscillator at the third position is greater than a stiffness of the oscillator at the second position. The oscillator further comprises a second stopper. The beam contacts the second stopper when the oscillator is in the third position. 
     In a further aspect, the beam comprises at least two beams. 
     In an additional aspect, the oscillator has a third position where the inertial mass is at a third distance in a second direction from the position of the inertial mass when the oscillator is at the neutral position. The oscillator has a fourth position where the inertial mass is at a fourth distance in the second direction from the position of the inertial mass when the oscillator is at the neutral position. The fourth distance is greater than the third distance. A stiffness of the oscillator at the fourth position is greater than a stiffness of the oscillator at the third position. The oscillator further comprises a second stopper. The beam contacts the second stopper when the oscillator is in the fourth position. 
     In another aspect, the beam is a first beam. The oscillator further comprises a second beam. The oscillator has a third position where the inertial mass is at a third distance in the first direction from a position of the inertial mass when the oscillator is at the neutral position. The oscillator has a fourth position where the inertial mass is at a fourth distance in the first direction from the position of the inertial mass when the oscillator is at the neutral position. The fourth distance is greater than the second distance. A stiffness of the oscillator at the fourth position is greater than a stiffness of the oscillator at the third position. The oscillator has a fifth position where the inertial mass is at a fifth distance in the second direction from the position of the inertial mass when the oscillator is at the neutral position. The oscillator has a sixth position where the inertial mass is at a sixth distance in the second direction from the position of the inertial mass when the oscillator is at the neutral position. The sixth distance is greater than the fifth distance. A stiffness of the oscillator at the sixth position is greater than a stiffness of the oscillator at the fifth position. The oscillator further comprises a third stopper and a fourth stopper. The second beam contacts the third stopper when the oscillator is in the fourth position. The second beam contacts the fourth stopper when the oscillator is in the sixth position. 
     It is also an object of the present invention to provide a method of optimizing a vibration-based power generator. 
     In another aspect the invention provides a method of optimizing a vibration-based power generator, the vibration-based power generator having a variable stiffness oscillator. The method comprises obtaining a probability density function of vibrations of an environment in which the power generator is to operate; obtaining a frequency-response function of the power generator; obtaining a figure of merit for the probability density function by convoluting the probability density function of the environment with the frequency-response function of the power generator; and adjusting the variable stiffness oscillator so as to optimize the figure of merit. 
     In a further aspect, adjusting the variable stiffness oscillator so as to optimize the figure of merit includes obtaining a figure of merit for different configurations of the variable stiffness oscillator, constituting a family of figures of merit, and selecting an absolute optimal from the family of figures of merit. 
     In an additional aspect, adjusting the variable stiffness oscillator so as to optimize the figure of merit includes obtaining a figure of merit for different configurations of the variable stiffness oscillator, constituting a family of figures of merit, and selecting a suboptimal from the family of figures of merit, the suboptimal having a wider range of frequencies than an absolute optimal of the family of figures of merit. 
     In another aspect, the variable stiffness oscillator comprises a cantilever beam connected to an inertial mass and having one end connected to a base, and a stopper for contacting one of the inertial mass and the cantilever beam at a point of impact while the cantilever beam is moving relative to the base. Adjusting the variable stiffness oscillator based on the figure of merit includes adjusting a position of the stopper along the cantilever beam. 
     In a further aspect, the variable stiffness oscillator is further adjusted by selecting a distance of the stopper to the cantilever beam that minimizes a velocity of the cantilever beam at the point of impact. 
     For purposes of this application, the term “beam” includes, but is not limited to, beam, plate, and tether. 
     Embodiments of the present invention each have at least one of the above-mentioned aspects, but do not necessarily have all of them. It should be understood that some aspects of the present invention that have resulted from attaining the above-mentioned objects may not satisfy these objects and/or may satisfy other objects not specifically recited herein. 
     Additional and/or alternative features, aspects, and advantages of embodiments of the present invention will become apparent from the following description, the accompanying drawings, and the appended claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       For a better understanding of the present invention, as well as other aspects and further features thereof, reference is made to the following description which is to be used in conjunction with the accompanying drawings, where: 
         FIG. 1  is a schematic of a generic prior art vibration-based micro-power generator (MPG); 
         FIG. 2   a  is a top view of a prior art electromagnetic MPG; 
         FIG. 2   b  is a side view of the electromagnetic MPG of  FIG. 2   a  with a portion in phantom showing a position of the MPG in operation; 
         FIG. 3   a  is a top view of a prior art electrostatic MPG; 
         FIG. 3   b  is a side view of the electrostatic MPG of  FIG. 3   a;    
         FIG. 4   a  is a top view of a prior art piezoelectric MPG; 
         FIG. 4   b  is a side view of the piezoelectric MPG of  FIG. 4   a;    
         FIG. 5   a  is a top view of a first version of an electromagnetic MPG according to a first embodiment of the invention; 
         FIG. 5   b  is a side view of the electromagnetic MPG of  FIG. 5   a  with a portion in phantom showing a position of the MPG when in contact with a stopper; 
         FIG. 5   c  is a perspective view of the electromagnetic MPG of  FIG. 5   a  showing the stopper mounted on a carriage; 
         FIG. 6   a  is a top view a second version of an electromagnetic MPG according to the first embodiment of the invention; 
         FIG. 6   b  is a side view of the electromagnetic MPG of  FIG. 6   a;    
         FIG. 7   a  is a top view of an electrostatic MPG according to the first embodiment of the invention; 
         FIG. 7   b  is a side view of the electrostatic MPG of  FIG. 7   a  with a portion in phantom showing a position of the MPG when in contact with a stopper; 
         FIG. 8   a  is a top view of a piezoelectric MPG according to the first embodiment of the invention; 
         FIG. 8   b  is a side view of the piezoelectric MPG of  FIG. 8   a;    
         FIG. 8   c  is a side view of a piezoelectric MPG according to an alternative design of the first embodiment of the invention; 
         FIG. 9  is a plot illustrating the force versus displacement of a MPG according to the first embodiment of the invention; 
         FIG. 10   a  is a top view of an electromagnetic MPG according to a second embodiment of the invention; 
         FIG. 10   b  is a side view of the electromagnetic MPG of  FIG. 10   a  with a portion in phantom showing a position of the MPG when in contact with a stopper and with a partial cross-section showing the configuration at rest; 
         FIG. 11  is a plot illustrating the force versus displacement of a MPG according to the second embodiment of the invention; 
         FIG. 12   a  is a top view of an electromagnetic MPG according to a third embodiment of the invention; 
         FIG. 12   b  is a side view of the electromagnetic MPG of  FIG. 12   a  with a portion in phantom showing a position of the MPG when in contact with a stopper, and with a portion removed for clarity; 
         FIG. 13  is a plot illustrating the force versus displacement of a MPG according to the third embodiment of the invention; 
         FIG. 14  is a plot of experimentally measured voltage across a resistive load of an electromagnetic MPG realized according to the prior art architecture (dashed line) and according to the invention, in a frequency up-sweep (dashed-dotted line) and a frequency down-sweep (solid line); 
         FIG. 15   a  is a top view of a first version of an electromagnetic MPG according to a fourth embodiment of the invention; 
         FIG. 15   b  is a side view of the electromagnetic MPG of  FIG. 15   a  with a portion in phantom showing a position of the MPG when in contact with a stopper, and with a portion removed for clarity; 
         FIG. 16   a  is a top view of a second version of an electromagnetic MPG according to the fourth embodiment of the invention; 
         FIG. 16   b  is a side view of the electromagnetic MPG of  FIG. 16   a;    
         FIG. 17   a  is a top view of a piezoelectric MPG according to the fourth embodiment of the invention; 
         FIG. 17   b  is a side view of the piezoelectric MPG of  FIG. 17   a;    
         FIG. 18  is a plot illustrating the force versus displacement of a MPG according to the fourth embodiment of the invention; 
         FIG. 19   a  is a top view of an electromagnetic MPG according to a fifth embodiment of the invention; 
         FIG. 19   b  is a side view of the electromagnetic MPG of  FIG. 19   a  with a portion removed for clarity; 
         FIG. 20  is plot illustrating the force versus displacement of a MPG according to the fifth embodiment of the invention; 
         FIG. 21   a  is a top view of an electromagnetic MPG according to a sixth embodiment of the invention; 
         FIG. 21   b  is a side view of the electromagnetic MPG of  FIG. 21   a;    
         FIG. 22   a  is a top view of a piezoelectric MPG according to the sixth embodiment of the invention; 
         FIG. 22   b  is a side view of the piezoelectric MPG of  FIG. 22   a;    
         FIG. 23   a  is a top view of a first version of an electrostatic MPG according to the sixth embodiment of the invention; 
         FIG. 23   b  is a side view of the electrostatic MPG of  FIG. 23   a;    
         FIG. 24   a  is a top view of a second version an electrostatic MPG according to the sixth embodiment of the invention; 
         FIG. 24   b  is a side view of the electrostatic MPG of  FIG. 24   a;    
         FIG. 25   a  is a top view of a third version an electrostatic MPG according to the sixth embodiment of the invention; 
         FIG. 25   b  is a side view of the electrostatic MPG of  FIG. 25   a;    
         FIG. 26   a  is a top view of an electromagnetic MPG using the micro-electro-mechanical system (MEMS) technology according to the sixth embodiment of the invention 
         FIG. 26   b  is a side view of the electromagnetic MPG of  FIG. 26   a  taken along the line  26   b - 26   b  of  FIG. 26   a;    
         FIG. 27  is plot illustrating the force versus displacement of a MPG according to the sixth embodiment of the invention; 
         FIG. 28   a  is a top view of an electrostatic MPG according to a seventh embodiment of the invention; 
         FIG. 28   b  is a side view of the electrostatic MPG of  FIG. 28   a;    
         FIG. 29   a  is a top view of an electromagnetic MPG using the MEMS technology according to the seventh embodiment of the invention; 
         FIG. 29   b  is a side view of the electromagnetic MPG of  FIG. 29   a  taken along the line  29   b - 29   b  of  FIG. 29   a ; and 
         FIG. 30  is a plot illustrating the force versus displacement of a MPG according to the seventh embodiment of the invention. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     With reference to  FIGS. 5   a  to  30 , various embodiments of MPGs (or harvesters) will be described. For simplicity, elements of these embodiments which are similar to each other have been labelled with the same reference numerals and will be described only once, unless otherwise necessary. 
     Turning now to  FIGS. 5   a  to  8   b  various embodiments of electromagnetic, electrostatic, and piezoelectric MPGs designed according to a first embodiment of the present invention will be described. 
     Referring to  FIGS. 5   a  and  5   b , an electromagnetic MPG according to the first embodiment consists of an inertial mass  104 , a cantilever beam  108 , a coil  110 , and magnets  113  attached to a yoke  112 , in a similar arrangement to the one described above in the prior art ( FIGS. 2   a  and  2   b ). Alternatively, the inertial mass  104  could be made of a yoke  112  and magnets  113  assembly while the coil  110  could be stationary (such a version is shown in  FIG. 10   b ). The energy produced by the coil  110  and magnet  113  arrangement is routed through electric connectors  111  to a power conditioning circuit  122  and then to a storage device or an electric load  105 . 
     The electromagnetic MPG of the first embodiment of the invention, is further equipped with a subsystem consisting of a rigid stopper  116  in the vicinity of the moving cantilever beam  108 . The stopper  116  is a bolt or a screw, but could be any element (flexible or rigid) that would interfere or stop the motion of the inertial mass  104  when the stopper  116  comes into contact with the beam  108 . 
     The subsystem is used for extending the frequency domain where energy is harvested. The stopper  116  is positioned so that it interferes with the regular envelope of the beam  108  (or spring  102  depending on the design, see  FIGS. 6   a,    6   b,    18   a  and  18   b ) motions resulting from ground vibrations. The stopper  116  engages the beam  108  when the amplitude of the oscillations is above a certain threshold. As a result of the contact of the moving beam  108  with the stopper  116 , the effective length of the beam  108  is reduced from L to a shorter length L−l o . This in turn changes the stiffness of the beam  108 -mass  104  system.  FIG. 9  shows the force versus beam displacement of a MPG equipped with a stopper  116  placed inside the envelope of motion of the beam  108 . The stopper  116  creates a two-stage stiffness relationship, where displacements in a first stage face a softer resistance (embodied by a low stiffness k 1 ) and displacements in a second stage face a harder resistance (embodied by a higher stiffness k 2 ). This bi-linear stiffness relationship transforms the harvester from a linear oscillator (prior art) into a bi-linear oscillator. 
     As shown in  FIG. 5   c , a carriage  111  moves along a pair of tracks  117  to place the stopper  116  at a fixed horizontal offset l o  from a base  109  (or support) of the cantilever beam  108 . A screw mechanism in the carriage  111  rigidly supports the stopper  116  at a fixed height h o  with respect to the rest or neutral position of the cantilever beam  108 . The height h o  and fixed horizontal offset l o  can be set to other values as per the design requirements. As it will be explained in more details below, a value of h o  and l o  influences a stiffness of the MPG, and a design methodology can be carried out to select an optimized pair (h o , l o ). This first embodiment is well suited for meso-sized MPGs (i.e. gross dimensions of few to several centimeters in each direction). 
     An alternative version of an electromagnetic MPG according to the first embodiment of the invention, shown in  FIGS. 6   a  and  6   b , uses two springs  102  in place of the cantilever beam  108 . The springs  102  support a magnet  113  which acts as the inertial mass  104  of the MPG. In addition, the MPG is equipped with two more symmetrically arranged springs  118  that contact the magnet  113  when the strike exceeds a fixed amplitude h o . Each spring  118  works as a variable resistance stopper  116 . This alternative to the first embodiment is well suited for low frequency applications (below 50 Hz). 
     Referring now to  FIGS. 7   a  and  7   b , an electrostatic MPG according to the first embodiment of the invention will be described. The electrostatic MPG uses a variable capacitor  114  as a transduction mechanism. The electrostatic MPG is equipped with the same subsystem consisting of the rigid stopper  116  placed in the vicinity of the moving beam  108  described above. The electrostatic MPG according to the first embodiment of the invention is well suited for micro-sized MPGs (i.e. gross dimensions of less than a centimeter in each direction). 
     Referring now to  FIGS. 8   a  and  8   b , a piezoelectric MPG according to the first embodiment of the invention uses a piezoelectric patch  115  as the transduction mechanism. The piezoelectric MPG is equipped with the subsystem consisting of the rigid stopper  116  placed in the vicinity of the moving beam  108  as described above. The piezoelectric MPG according to the first embodiment of the invention is well suited for micro-sized MPGs (gross dimensions of less than a centimeter in each direction). 
     Alternatively, as shown in  FIG. 8   c , it is contemplated that in the case of the inertial mass  104  not being at a tip of the beam  108 , the rigid stopper  116  could be placed within the envelope of motion of the cantilever beam ledge  123 . That position has a similar effect to the stopper  116  being positioned directly within the envelope of motion of the beam  108 , and in turns creates a bi-linear oscillator. 
     Typical frequency-response curves of the MPG according to the first embodiment (and regardless of the transduction mechanism) are shown in  FIG. 14 . More specifically, the variation of the root mean square (RMS) voltage across a resistive load  105  for an electromagnetic MPG according to the first embodiment and for a typical electromagnetic MPG of the prior art (shown in  FIG. 1 ) is plotted against the frequency of environmental vibrations. 
     Starting at point A and sweeping up the frequency of the environmental vibrations, the RMS voltage across the load  105  increases monotonically and identically in the MPG according to the first embodiment and in the prior art MPG until point B. At point B, the slope of the frequency-response curve of the MPG according to the first embodiment drops abruptly as the cantilever beam  108  engages the stopper  116 , while the frequency-response curve of the prior art MPG continues to increase smoothly. From point B to point D (up-sweep), the RMS of the load voltage of the first embodiment increases slowly as the speed at which the beam  108  engages the stopper  116  increases. At point D the RMS load voltage of the first embodiment MPG drops to match the level of the prior art MPG. The up-sweep bandwidth of the first embodiment MPG is equal to the difference between the locations of points B and D along the frequency spectrum and is larger than the bandwidth of the prior art MPG (the difference between the locations of points B and C). The responses of the first embodiment MPG and the prior art MPG are identical from this point up to point E. 
     Starting now at point F and sweeping down the frequency range, the RMS voltage of the resistive load increases monotonically and identically in the first embodiment MPG and in the prior art MPG from point F to point C. At point C, the slope of the frequency-response curve of the first embodiment MPG drops abruptly as the cantilever beam  108  engages the stopper  116 , while the frequency-response curve of the prior art MPG continues to increase smoothly. An abrupt slope change is seen at point B in the frequency-response curve of the first embodiment MPG where it becomes once again identical to that of the prior art MPG from this point and onward for the rest of the down-sweep. The down-sweep bandwidth of the first embodiment MPG is equal to the difference between the locations of points B and C along the frequency spectrum and is, therefore, identical to the bandwidth of the prior art MPG. 
     The increase in the size of the up-sweep bandwidth over the down-sweep bandwidth depends on the stiffness ratio of the first to the second stages of the bi-linear spring in the first embodiment. As it will be seen below, a piecewise-linear oscillator can be made by having more than one stopper  116 , resulting in an oscillator having more than two stiffness. In that case, the up-sweep bandwidth depends on the stiffness ratios of the first to second and first to third stages (or more) of the spring. In either case, as the stiffness ratio increases beyond unity the up-sweep bandwidth expands until it saturates at a maximum up-sweep bandwidth. The up-sweep bandwidth of the piecewise-linear MPG saturates to the maximum up-sweep bandwidth faster (the multiple stoppers of a piecewise-linear oscillator will need to interfere less with the envelope of beam motions) than the first embodiment. 
     The MPG according to the first embodiment delivers more power than the prior art MPG whenever the frequency of environmental vibrations varies from a value inside the down-sweep bandwidth in any pattern that includes values outside the down-sweep bandwidth. As shown in M. S. M. Soliman, E. M. Abdel-Rahman, E. F. El-Saadany, and R. R. Mansour, “A Wideband Vibration-Based Energy Harvester”, Journal of Micromechanics and Microengineering, 2008, Vol. 18, paper #115021., the entirety of which is incorporated by reference, the MPG is robust to external disturbances in the up-sweep bandwidth. As a result, it can maintain these advantages under adverse conditions. However, the output power of this wideband MPG is less than the output power of the prior art MPG for environmental vibrations inside the down-sweep bandwidth ( FIG. 14 ). Therefore, a design procedure is used to determine the stopper  116  height ho and offset distance lo that will allow the wideband MPG to collect a maximum energy for a given probability density function of the environmental vibrations frequency. 
     The design methodology is as follow: (i) minimizing mechanical and electrical energy losses via structural design, circuit, and material selection, (ii) minimizing the MPG non-contact damping ratio by increasing the inertial mass  104  and stiffness of the MPG linearly (while maintaining their ratio k/m fixed) until size effects cause the rate at which energy losses increase to accelerate to a rate higher than that linear rate, (iii) tailoring the output power and bandwidth to fit the probability density function of environmental vibrations. To do so, a figure of merit is devised to quantify the quality of this fit. In “Optimization of Energy Collection in Vibration-Based Micro-Power Generators” by M. S. M. Soliman, E. M. Abdel-Rahman, E. F. El-Saadany, and R. R. Mansour, enclosed in an appendix herein, an example of using this figure of merit to configure a wideband MPG constructed according to the first embodiment of this invention to fit a Gaussian probability density function with a standard deviation of 2.5 Hz is provided. 
     More specifically, to determine the figure of merit, one must first obtain the probability density function of the environment vibrations, before engaging the following iterative procedure. The stopper is fixed at a first position determined by its distance to the base  109   l   1  and a nominal height above the beam h 1  and the frequency response of the MPG having the stopper  116  in that position is obtained, The frequency response is convoluted with the probability density function to obtain a figure of merit (representing the probability of the MPG collecting environmental vibration energy) with respect to the frequencies of the environment vibrations. Once that figure of merit is determined, the stopper  116  is moved to another position. The probability density function and the figure of merit are re-calculated. Once the range of distances to the base  109  has been swept, a position that maximizes the figure of merit is selected. The optimization criteria can be twofold. It could be the absolute maximum of the convolution product, which in turn is the maximum energy collected, or a suboptimum of the convolution product, which collects less energy but allows a wider range of frequencies to be collected. The suboptimum is preferred when there is some uncertainty about the probability density function of environment vibrations. By this procedure the offset distance lo has been adjusted to a value slightly larger than the minimum threshold necessary for an up-sweep bandwidth that is larger than the bandwidth of the probability density function of environmental vibrations. 
     Once the offset distance lo from the support is determined, the stopper  116  height h 1  is adjusted to minimize the velocity of the cantilever beam  108  at the point of impact, in order to minimize energy losses. To do so the stopper height ho is set as high as possible to minimize the impact velocity, while maintaining the up-sweep bandwidth larger or equal to the bandwidth of interest in the probability density function. 
     Referring now to  FIGS. 10   a  and  10   b , a MPG designed according to a second embodiment of the invention will be described. In this second embodiment, two (or more) carriages  111  are used to place two (or more) stoppers  116  at different locations along the beam  108  l 1  and l 2  at two (or more) increasing heights h 1  and h 2  inside the envelope of motions of the beam  108 , so that the beam  108  engages the two (or more) stoppers  116  progressively in three (or more) stages. The first stage is when the oscillation intensity is below a threshold corresponding to where the beam  108  does not contact the first stopper  116 . The first stopper  116  is the stopper  116  closest to the base  109 . The oscillator has then a first stiffness k 1 . When the vibrations reach a sufficient intensity, the beam  108  encounters the first stopper  116 . The encounter of the beam  108  with the first stopper  116  constitutes the second stage. Upon contact with the first stopper  116 , the beam  108  deflects. The beam  108  has its effective length reduced to L−l 1 , which in turn forces the oscillator into a second stiffness k 2  greater than k 1 . If the vibrations intensity increases further, the beam  108  encounters a second stopper  116 , and the oscillator enters in the third stage. The second stopper  116  is at a height h 2  greater than the height h 1  and at a location l 2  greater than the location l 1  of the first stopper  116  such that the contact between the beam  108  and the stoppers  116  is effectively happening in three stages. Upon contact with the second stopper  116 , the beam  108  deflects even more, and the beam  108  has its effective length reduced to L-l 1 -l 2 , which in turn forces the oscillator into a third stiffness k 3  greater than k 2 . The presence of the two stoppers  116  produces a tri-linear stiffness relationship (shown in  FIG. 11 ). The harvesting element becomes a tri-linear oscillator, as opposed to a bi-linear oscillator for the first embodiment having a single stopper. 
     It is possible to create a piecewise-linear oscillator by having additional stoppers  116  located at increasing heights and lengths along the envelope of motion of the beam  108  so as to obtain an oscillator with a variable stiffness. Piecewise-linear oscillators (oscillators with two or more linear stiffness stages) exhibit resonance in a broad bandwidth in the neighbourhood of a natural frequency. The response of the piecewise-linear oscillator and linear oscillator (no stopper  116 , prior art) are identical as long as the beam  108  does not engage one of the stoppers  116 . When the beam  108  engages a stopper  116 , the effective stiffness of the piecewise-linear oscillator increases in proportion to the interval the beam  108  spends engaged with the stopper  116  per cycle. The higher effective stiffness caps the beam  108 -mass  104  amplitude at a lower level and increases the effective natural frequency of the oscillator causing the resonance of the piecewise-linear oscillator to persist over a wider band of the frequency spectrum than a linear oscillator. As a result, the bandwidth of a MPG equipped with a piecewise-linear oscillator expands to a larger band. 
     It is to be noted that a similar optimization methodology as seen in the first embodiment would apply to a MPG using a piecewise-linear oscillator, having a two or more stopper  116 . Piecewise-linear oscillators are useful where it is not possible to reach a wide enough up-sweep bandwidth using a single stopper  116 . This is due to design requirements placing restrictions on the localisation of the stopper  116 . As an example, it is possible that the height ho of the stopper  116  could not be decreased enough to bring the stopper  116  closer to the beam  108 . 
     Referring now to  FIGS. 12   a  and  12   b , a MPG designed according to a third embodiment of the invention will be described. In this third embodiment (bi-linear oscillator), two carriages  111  are used to place two stoppers  116  at the same location along the beam  108  on either side of the beam  108 . This arrangement produces a symmetric bi-linear stiffness relationship (shown in  FIG. 13 ). 
     Referring now to  FIGS. 15   a  to  17   b , MPGs designed according to a fourth embodiment of the invention will be described. In this fourth embodiment, a single stopper  116  is placed directly within the envelope of motion of the inertial mass  104 . 
     A first version of an electromagnetic MPG according to the fourth embodiment of the invention is presented in  FIGS. 15   a  and  15   b . The MPG consists of the same elements as the electromagnetic MPG according to the first embodiment of the invention presented in  FIGS. 5   a  and  5   b , except that the stopper  116  is placed so as to interfere directly with the inertial mass  104  during its motion, as opposed to interfering with the cantilever beam  108 . More specifically, the stopper  116  is placed at a fixed height ho and horizontal offset lo within the envelope of motion of the inertial mass  104 . When the level of vibration is sufficient, the inertial mass  104  impacts the stopper  116  and thus terminates its motion in the direction of the stopper  116 . As shown in  FIG. 18 , the impact induces a very large almost infinite stiffness. The beam  108 -mass  104  oscillator according to this embodiment is a bi-linear impact oscillator, which is a limiting case for the bi-linear oscillators. 
     A second version of an electromagnetic MPG according to the fourth embodiment of the invention is presented in  FIGS. 16   a  and  16   b . In this alternative version, a single spring  102  is used instead of the cantilever beam  108 . The spring  102  supports the magnet  113  which acts as the inertial mass  104  of the MPG. It is contemplated that any spring  119  with a hardening nonlinearity could be used to construct this MPG. The stopper  116  is placed on the upper part of the base  109  so as to contact the magnet  13  when the strike exceeds a threshold amplitude h o . 
     A piezoelectric MPG according to the fourth embodiment of the invention is shown in  FIGS. 17   a , and  17   b . Similarly to the electromagnetic MPG according to the fourth embodiment, the rigid stopper  116  is placed directly within the envelope of motion of the inertial mass  104  of a piezoelectric MPG. 
     Referring now to  FIGS. 19   a  and  19   b , a MPG designed according to a fifth embodiment of the invention will be described. In this fifth embodiment, the inertial mass  104  is supported by a set of two (or more) symmetrically (or asymmetrically) arranged beams  119 . The beams  119  are characterized by the fact that they have a cross-section which varies along a length of the beam  119 . The beams  119  could alternatively be plates, or other tether-like structures. The immovable supports result in the tethers behaving as hardening-type springs with smoothly increasing stiffness as the displacement increases (as shown in  FIG. 20 ). The harvesting element acquires a hardening-type nonlinearity and becomes a hardening-type oscillator. 
     Referring now to  FIGS. 21   a  and  21   b , a MPG designed according to a sixth embodiment of the invention will be described. In this sixth embodiment, two (or more) symmetrically (or asymmetrically) arranged rigid stoppers  116  are placed above (or below) two (or more) beams  119  (or tethers) supporting the inertial mass  104  of an electromagnetic MPG. The beams  119  and stoppers  116  are used to create a two-stage (or more) nonlinear spring with two (or more) increasingly stiffer stages (such as shown in  FIG. 27 ). The hardening-type nonlinearity of the harvesting element grows thereby enhancing the hardening-type behavior of the oscillator and further expanding the up-sweep bandwidth of the MPG. 
     In a second version of the sixth embodiment of the invention, shown in  FIGS. 22   a  and  22   b , the MPG comprises a piezoelectric transduction mechanism. A single rigid stopper  116  is placed above (or below) one of the two beams  108  (or tethers) supporting the inertial mass  104 . This configuration creates a two-stage nonlinear spring. As a result, the hardening-type nonlinearity of the harvesting element grows thereby enhancing the hardening-type behavior of the oscillator and further expanding the up-sweep bandwidth of the MPG. 
     A third version of the sixth embodiment of the invention is shown in  FIGS. 23   a  and  23   b . In this third version, two rigid stoppers  116  are asymmetrically placed above (or below) two beams  108  (or tethers) supporting the inertial mass  104  of an electrostatic MPG. Each stopper  116  is placed within the envelope of motion of each beam  108  to create a three-stage nonlinear spring with three increasingly stiffer stages. As a result, the hardening-type nonlinearity of the harvesting element grows thereby enhancing the hardening-type behavior of the oscillator and further expanding the up-sweep bandwidth of the MPG. 
     A fourth version of the sixth embodiment of the invention is shown in  FIGS. 24   a  and  24   b . In this fourth version, two ridges  116  (or more) are symmetrically (or asymmetrically) placed above (or below) two of four beams  108  (or tethers) supporting the inertial mass  104  of an electrostatic MPG. The ridges  116  interfere within the envelope of motion of the pair of beams  108  and create a two (or more) stage nonlinear spring with two (or more) increasingly stiffer stages. As a result, the hardening-type nonlinearity of the harvesting element grows thereby enhancing the hardening-type behavior of the oscillator and further expanding the up-sweep bandwidth of the MPG. 
     A fifth version of the sixth embodiment of the invention is shown in  FIGS. 25   a  and  25   b . In this fifth version, four rigid stoppers  116  are symmetrically placed above and below two beams  108  (or tethers) supporting the inertial mass  104  of an electrostatic MPG. The rigid stoppers  116  interfere within the envelope of motion of the beams  108  to create a two-stage nonlinear spring. As a result, the hardening-type nonlinearity of the harvesting element grows thereby enhancing the hardening-type behavior of the oscillator and further expanding the up-sweep bandwidth of the MPG. Alternatively, it is contemplated that the inertial mass  104  could be placed on a single beam  108  connected at both ends to the base  109  instead of the two beams  108 . 
     A sixth version of the sixth embodiment of the invention is shown in  FIGS. 26   a  and  26   b . In this sixth version, four flexible end stoppers  118  are placed two at either end of the stroke of an electromagnetic MPG fabricated using micro-electro-mechanical system (MEMS) technology. The inertial mass  104  is supported by four beams  112  (or tethers) connected to a plate  120 , and can engage the flexible end-stoppers  118 . The end-stoppers  118  add another stage to the hardening nonlinear spring of the freely moving inertial mass  104  ( FIG. 10 ) resulting from the four restrained tethers  112 . The MPG under this arrangement has a symmetric two-stage nonlinear spring like that shown in  FIG. 27 . As a result, the hardening-type nonlinearity of the harvesting element grows thereby enhancing the hardening-type behavior of the oscillator and further expanding the up-sweep bandwidth of the MPG. This arrangement of the sixth embodiment is well suited for MPGs fabricated using micro-electro-mechanical systems fabrication technology. 
     Referring now to  FIGS. 28   a  and  28   b , a MPG designed according to a seventh embodiment of the invention will be described. In this seventh embodiment, one rigid stopper  116  is placed above (or below) the inertial mass  104  of an electrostatic MPG supported by two (or more) beams  108  (or tethers). The rigid stopper  116  adds an infinite stiffness wall on one side of the smooth nonlinear spring created by the tethers as shown in  FIG. 30 . The hardening-type oscillator becomes an impact oscillator which expands the up-sweep bandwidth of the MPG to its maximum value. 
     A second version of the seventh embodiment of the invention is shown in  FIGS. 29   a  and  29   b . In this second version, two rigid end-stoppers  116  are placed at either end of the stroke of an electromagnetic MPG fabricated the MEMS technology. The stoppers  116  add an infinite stiffness wall on either side of the smooth nonlinear spring created by the tethers. In this configuration, the hardening-type oscillator becomes an impact oscillator which expands the up-sweep bandwidth of the MPG to its maximum value. 
     It is contemplated that various combinations of the above and other bi-linear, piecewise-linear, and nonlinear springs of the hardening-type could be assembled that could be used to support the inertial mass  104  of the harvesting element in an electromagnetic, electrostatic or piezoelectric MPG. 
     Modifications and improvement to the above described embodiments of the present invention may become apparent to those skilled in the art. The foregoing description is intended to be exemplary rather than limiting. Furthermore, the dimensions of features of various components that may appear on the drawings are not meant to be limiting, and the size of the components therein can vary from the size that may be portrayed in the figures herein. The scope of the present invention is therefore intended to be limited solely by the scope of the appended claims.