Adjustable piezoelectric energy harvesters

Abstract

An example device includes a first piezoelectric member; and a linear actuator coupled to a stopper member, where the linear actuator is configured to control a resonant frequency of the device by adjusting a gap between the first piezoelectric member and elastic stopper member. An example device includes a first piezoelectric member coupled to a slider, and a second piezoelectric member coupled to the slider, where the first piezoelectric member and second piezoelectric member are separated by a gap.

Description

Docket Number: 103362 005WO1ADJUSTABLE PIEZOELECTRIC ENERGY HARVESTERSCROSS REFERENCE TO RELATED APPLICATIONS

[0001] This application claims the benefit of U.S. provisional patent application No.63 / 654,423, filed on August May 31, 2024, and titled “ADJUSTABLE PIEZOELECTRIC ENERGYHARVESTERS,” the disclosure of which is expressly incorporated herein by reference in itsentirety. STATEMENT REGARDING FEDERALLY FUNDED RESEARCH

[0002] This invention was made with government support under 2140523 awarded bythe National Science Foundation. The government has certain rights in the invention.BACKGROUND

[0003] Mechanical vibrations are mechanical motions that occur around an oscillationpoint. For example, a car hood can oscillate up and down in response to the motions causedby a spinning engine beneath the hood, or a computer case can vibrate due to themovement of fans and air inside. Vibrations represent energy that is effectively lost to theenvironment over time as a byproduct of various mechanical processes.SUMMARY

[0004] In some aspects, implementations of the present disclosure include a deviceincluding: a first piezoelectric member; a linear actuator coupled to a stopper member,wherein the linear actuator is configured to control a resonant frequency of the device byadjusting a gap between the first piezoelectric member and elastic stopper member.

[0005] In some aspects, implementations of the present disclosure include a device,wherein the stopper member includes a second piezoelectric member.Docket Number: 103362 005WO1

[0006] In some aspects, implementations of the present disclosure include a device,wherein the stopper member further includes an extrusion coupled to the secondpiezoelectric member.

[0007] In some aspects, implementations of the present disclosure include a device,wherein the extrusion includes an elastic material.

[0008] In some aspects, implementations of the present disclosure include a device,wherein the extrusion includes a spring.

[0009] In some aspects, implementations of the present disclosure include a device,wherein the stopper member includes an elastic material.

[0010] In some aspects, implementations of the present disclosure include a device,wherein the first piezoelectric member includes a first end and a second end, the first endand second end defining a long axis of the first piezoelectric member, and wherein the firstend of the first piezoelectric member is coupled to a base.

[0011] In some aspects, implementations of the present disclosure include a device,wherein the second piezoelectric member includes a first end and a second end, the firstend and second end defining a long axis of the second piezoelectric member, and whereinthe first end of the second piezoelectric member is coupled to the linear actuator.

[0012] In some aspects, implementations of the present disclosure include a device,wherein a mass is fixed to the first piezoelectric member.

[0013] In some aspects, implementations of the present disclosure include a device,wherein a mass is fixed to the second piezoelectric member.

[0014] In some aspects, implementations of the present disclosure include a deviceincluding: a first piezoelectric member coupled to a slider; a second piezoelectric membercoupled to the slider; wherein the first piezoelectric member and second piezoelectricmember are separated by a gap.

[0015] In some aspects, implementations of the present disclosure include a device,wherein the first piezoelectric member and second piezoelectric member are parallel.

[0016] In some aspects, implementations of the present disclosure include a devicewherein the slider is a linear slider.

[0017] In some aspects, implementations of the present disclosure include a device,wherein the slider is configured to adjust the gap between the first piezoelectric memberDocket Number: 103362 005WO1and second piezoelectric member to control a resonant frequency of the first piezoelectricmember or second piezoelectric member.

[0018] In some aspects, implementations of the present disclosure include a device,wherein the first piezoelectric member has a first end and a second end defining a long axisof the piezoelectric member, and wherein the first end is coupled to the slider.

[0019] In some aspects, implementations of the present disclosure include a device,wherein the second piezoelectric member has a first end and a second end defining a longaxis of the piezoelectric member, and wherein the first end of the second piezoelectricmember is coupled to the slider.

[0020] In some aspects, implementations of the present disclosure include an energyharvesting system including: an energy harvesting device, and a controller operably coupledto the energy harvesting device, the controller including a processor and a memory, thememory having computer executable instructions stored thereon that, when executed bythe processor, cause the controller to: determine a source frequency of a source ofvibrational energy; determine, based on the source frequency, an optimized resonantfrequency for the device; adjust, by the linear actuator, the gap so that the resonantfrequency of the energy harvesting device is the optimized resonant frequency.

[0021] In some aspects, implementations of the present disclosure include a system,wherein the optimized resonant frequency is a frequency that maximizes an amount ofelectrical energy harvested by the energy harvesting device.

[0022] In some aspects, implementations of the present disclosure include a system,wherein the optimized resonant frequency is a frequency that matches an excitationfrequency of the source of vibrational energy.

[0023] In some aspects, implementations of the present disclosure include a system,wherein determining a source frequency includes receiving, from a sensor, a frequencyspectra of the source of vibrational energy.

[0024] It should be understood that the above described subject matter may also beimplemented as a computer controlled apparatus, a computer process, a computingsystem, or an article of manufacture, such as a computer readable storage medium.

[0025] Other systems, methods, features and / or advantages will be or may becomeapparent to one with skill in the art upon examination of the following drawings anddetailed description. It is intended that all such additional systems, methods, featuresDocket Number: 103362 005WO1and / or advantages be included within this description and be protected by theaccompanying claims.BRIEF DESCRIPTION OF THE DRAWINGS

[0026] The components in the drawings are not necessarily to scale relative to eachother. Like reference numerals designate corresponding parts throughout the several views.

[0027] FIG. 1A illustrates an example piezoelectric energy harvester with a cantileverbeam coupled to a base and an elastic stopper coupled to a linear actuator, according toimplementations of the present disclosure.

[0028] FIG. 1B illustrates an example piezoelectric energy harvester with two cantileverbeams coupled to a base, according to implementations of the present disclosure.

[0029] FIG. 1C illustrates an example piezoelectric energy harvester with two cantileverbeams, where one piezoelectric beam is coupled to a base and the other is coupled to aslider, according to implementations of the present disclosure.

[0030] FIG. 1D illustrates an example piezoelectric energy harvester with two cantileverbeams, where one piezoelectric beam is coupled to a base and the other is coupled to alinear actuator, according to implementations of the present disclosure.

[0031] FIG. 2 illustrates an example system including a piezoelectric energy harvesterwith adjustable resonant frequency, according to implementations of the presentdisclosure.

[0032] FIG. 3 illustrates an example computer implemented method of operatingpiezoelectric energy harvesters, according to implementations of the present disclosure.

[0033] FIG. 4 illustrates an example computing device.

[0034] FIG. 5 illustrates an example mechanical model of a first mode of a piezoelectriccantilever beam with actuated elastic stopper, according to implementations of the presentdisclosure.

[0035] FIG. 6 illustrates schematics of an experimental setup, according to a study of anexample implementation of the present disclosure.

[0036] FIG. 7A illustrates an experimental setup including an example implementation ofthe present disclosure.Docket Number: 103362 005WO1

[0037] FIG. 7B illustrates a piezoelectric beam, elastic stopper, actuator and shakeraccording to an example implementation of the present disclosure.

[0038] FIG. 8 illustrates a stiffness measurement of a piezoelectric tip, according to anexample implementation of the present disclosure.

[0039] FIG. 9 illustrates experimental displacement transmissibility of piezoelectric beam,according to a study of an example implementation of the present disclosure.

[0040] FIG. 10 illustrates displacement transmissibility of piezoelectric cantilever beam,according to a study of an example implementation of the present disclosure.

[0041] FIG. 11A illustrates optimal response and linear response for an open state, 0 gap,and closed state, according to a study of an example implementation of the presentdisclosure.

[0042] FIG. 11B illustrates optimal gap size, according to a study of an exampleimplementation of the present disclosure.

[0043] FIG. 12 illustrates transition ratio of gap size to tip deflection, according to a studyof an example implementation of the present disclosure.

[0044] FIG. 13 illustrates an example voltage reading, according to a study of an exampleimplementation of the present disclosure.

[0045] FIG. 14 illustrates an experimental response envelope and experimentaldisplacement transmissibility, according to a study of an example implementation of thepresent disclosure.

[0046] FIG. 15 illustrates an example piecewise linear response and linear systemresponse, according to a study of an example implementation of the present disclosure.

[0047] FIG. 16 illustrates time histories for the linear displacement and piecewise linearsystem displacement, according to a study of an example implementation of the presentdisclosure.

[0048] FIG. 17 illustrates peak amplitude time history, according to a study of an exampleimplementation of the present disclosure.

[0049] FIG. 18 illustrates a mechanical model of a first mode of a piezoelectric cantileverbeam with an actuated second piezoelectric cantilever beam, according to a study of anexample implementation of the present disclosure.

[0050] FIG. 19A illustrates a bench test setup of an example implementation of thepresent disclosure.Docket Number: 103362 005WO1

[0051] FIG. 19B illustrates a dual piezoelectric beam system according to an exampleimplementation of the present disclosure.

[0052] FIG. 20 illustrates a stiffness measurement of a piezoelectric beam, according to astudy of an example implementation of the present disclosure.

[0053] FIG. 21 illustrates experimental time domain signals including an upper envelopeand lower envelope, according to a study of an example implementation of the presentdisclosure.

[0054] FIG. 22A illustrates experimental and computational displacement transmissibilityof a lower piezoelectric cantilever beam, according to a study of an exampleimplementation of the present disclosure.

[0055] FIG. 22B illustrates experimental and computational displacement transmissibilityof an upper piezoelectric cantilever beam, according to a study of an exampleimplementation of the present disclosure.

[0056] FIG. 23A illustrates an optimal response envelope and linear response for a lowerpiezoelectric beam, according to a study of an example implementation of the presentdisclosure.

[0057] FIG. 23B illustrates an optimal response envelope and linear response for an upperpiezoelectric beam, according to a study of an example implementation of the presentdisclosure.

[0058] FIG. 24A illustrates an optimal response envelope and linear response for acombined lower and upper piezoelectric beam, according to a study of an exampleimplementation of the present disclosure.

[0059] FIG. 24B illustrates an optimal gap size for an example implementation of thepresent disclosure that was studied.

[0060] FIG. 25 illustrates a comparison between a response envelope, observed results,and linear responses, according to a study of an example implementation of the presentdisclosure.Docket Number: 103362 005WO1DETAILED DESCRIPTION

[0061] Unless defined otherwise, all technical and scientific terms used herein have thesame meaning as commonly understood by one of ordinary skill in the art. Methods andmaterials similar or equivalent to those described herein can be used in the practice ortesting of the present disclosure. As used in the specification, and in the appended claims,the singular forms “a,” “an,” “the” include plural referents unless the context clearlydictates otherwise. The term “comprising” and variations thereof as used herein is usedsynonymously with the term “including” and variations thereof and are open, non limitingterms. The terms “optional” or “optionally” used herein mean that the subsequentlydescribed feature, event or circumstance may or may not occur, and that the descriptionincludes instances where said feature, event or circumstance occurs and instances where itdoes not. Ranges may be expressed herein as from "about" one particular value, and / or to"about" another particular value. When such a range is expressed, an aspect includes fromthe one particular value and / or to the other particular value. Similarly, when values areexpressed as approximations, by use of the antecedent "about," it will be understood thatthe particular value forms another aspect. It will be further understood that the endpointsof each of the ranges are significant both in relation to the other endpoint, andindependently of the other endpoint. While implementations will be described forpiezoelectric energy harvesting, it will become evident to those skilled in the art that theimplementations are not limited thereto, but are applicable for other types of energyharvesting.

[0062] Implementations of the present disclosure include systems, devices and methodsthat can enhance the energy generated in systems that use piezoelectric materials bymanipulating the gap between the end of one or more cantilevered piezoelectric beams. Insome implementations of the present disclosure, a piezoelectric beam interacts with amechanical stopper that is adjusted by a linear actuator to modify the resonant frequencyto match the frequency of excitation and more efficiently produce energy. The system canalso connect to one or more piezoelectric beams that are adjusted by linear actuator toadjust the gaps between the beams. The use of multiple beams can result in additionalenergy produced by each beam that will be tuned to operate at resonance. These multiplebeams can all produce more than if they were uncoupled because the multiple beams inDocket Number: 103362 005WO1implementations of the present disclosure can be configured to operate at resonance byadjusting the gaps between adjacent beams. Piezoelectric beams, also referred to herein as“piezoelectric members” are structures made of piezoelectric materials that can convertmechanical stress into electrical energy by the piezoelectric effect. Example types ofpiezoelectric materials include types of crystals and ceramics.

[0063] As non limiting examples, implementations of the present disclosure can be usedin self powered sensors (eliminating the need for battery replacement or laying conduit toinaccessible areas, i.e., on bridges, locations on industrial machinery, aircraft). Alternativelyor additionally, implementations of the present disclosure can also be used for other lowpower applications that are not easily connected to the grid and remove the need forreplacing batteries.

[0064] FIG. 1A illustrates an example device 100 according to implementations of thepresent disclosure. The device includes a first piezoelectric member 102. Optionally, asshown in FIG. 1A, the first piezoelectric member 102 can be coupled to a base 104, and thecoupling between the base 104 and the first piezoelectric member 102 can optionally be acantilever coupling (e.g., one end of the piezoelectric member is coupled to the base, whileanother end is free) as shown in FIG. 1A. In the cantilever arrangement shown in FIG. 1A,the first piezoelectric member can define a first end 112a and second end 112b, so that thefirst end 112a is coupled to the base 104, while the second end 112b is free.

[0065] The base 104 can be any surface or group of surfaces. The base 104 can be avibration source (e.g., anything that vibrates), so that vibrations of the base 104 can vibratethe device 100 coupled to the base 104.

[0066] The device 100 can further include an actuator 106. Optionally, the actuator 106 isa linear actuator 106. The actuator can be coupled to the base 104. The actuator can becoupled to a stopper 108, which can optionally be an elastic stopper (i.e., a stopper formedfrom an elastic member). The arrangement of actuator 106, base 104, piezoelectric member102, and stopper 108 can be configured in any way so that there is a gap 110 between theelastic stopper 108 and piezoelectric member 102 when the system is at rest (e.g., notvibrating). It should be understood that the arrangement of piezoelectric member ofactuator 106, base 104, piezoelectric member 102, and stopper 108 shown in FIG. 1A areintended only as a non limiting example.Docket Number: 103362 005WO1

[0067] The actuator 106 can be configured to adjust the gap 110 by moving the stopper108 relative to the piezoelectric member 102. When the device 100 is excited, thepiezoelectric member 102 can move relative to the base 104 so that it contacts the stopper108 (e.g., the end 112b vibrates relative to the base). Because the motion of thepiezoelectric member 102 is related to the resonance frequency, the gap 110 can beadjusted to control the resonance frequency of the motion of the piezoelectric member 102to maximize the electrical energy generated by the piezoelectric member 102.

[0068] As shown in FIG. 1B, implementations of the present disclosure include devices120 where the stopper 108 shown in FIG. 1A can be implemented as a second piezoelectricmember 122. The second piezoelectric member 122 can optionally include an extrusion 126,so that the gap 110 is defined as the distance between the second piezoelectric member122 and the first piezoelectric member, or between the first piezoelectric member 102 andthe extrusion 126. The extrusion 126 can be formed from different materials. In someimplementations, the extrusion 126 is stiff, whereas in other implementations, the extrusion126 can be formed from an elastic material and / or spring. In some implementations, anextrusion (not shown) can be formed on the first piezoelectric member 102 as an alternativeto, or in addition to, the extrusion 126 shown in FIG. 1B. The extrusion 126 can also beconfigured using different amounts of weight. As described in greater detail with referenceto FIG. 1D, and example 2, herein, in some implementations, the extrusion 126 canoptionally be formed from bolts / nuts coupled to the piezoelectric members 102, 122 thatserve the purpose of the extrusion by extending from the piezoelectric members 102, 122 todefine the gap 110.

[0069] The second piezoelectric member 122 can also be configured to generateelectrical energy by its motion relative to the base 104, and the gap 110 can be controlledusing the actuator 106 to maximize the electrical energy generated by either the firstpiezoelectric member 102, second piezoelectric member 122, and / or the overall energygenerated by the device 120.

[0070] As shown in FIG. 1B, the piezoelectric members 102, 122, can be coupled to thelinear actuator 106 and base 104 so that either or both piezoelectric members 102, 122form cantilever beams (beams attached at only one end). The second piezoelectric member122 shown in FIG. 1B includes a first end 124a and a second end 124b, where the first end124a is coupled to the actuator 106, and the second end 124b is free. As shown in bothDocket Number: 103362 005WO1FIGS. 1A and 1B, the first end 112a, 124a and second ends 112b, 124b define long axes ofthe first piezoelectric member 102 and second piezoelectric member 122.

[0071] Optionally, the implementations of the present disclosure shown in FIGS. 1A 1Ccan include masses (not shown) attached at any point on the piezoelectric members 102,122. It should be understood that the resonant frequencies of the piezoelectric members102, 122 can be tuned by changing the length, mass, shape, and other mechanicalproperties of the piezoelectric members 102, 122, for example to change the efficienciesand / or frequencies of implementations of the present disclosure.

[0072] As shown in FIG. 1C, implementations of the present disclosure can beimplemented as devices 150 using a slider 152. The device 150 includes a first piezoelectricmember 102 and second piezoelectric member 122 coupled to the slider 152. The slider 152can optionally be a linear slider (e.g., a linear actuator or other coupling that can adjust theposition and / or orientation of the first piezoelectric member 102 and second piezoelectricmember 122. Optionally, the first and second piezoelectric members 102, 122 areconfigured to be parallel to one another. Alternatively or additionally, the first piezoelectricmember 102 and / or second piezoelectric member 122 can be configured as cantileverbeams coupled to the slider 152 at one or both first ends 112a, 124a.

[0073] A gap 110 is defined by the spacing of the piezoelectric members 102, 122.Optionally, the first and / or second piezoelectric members 102, 122 can include an extrusion(not shown) as described with reference to FIG. 1B. As described with reference to FIGS.1A 1B, the gap 110 can be adjusted to change the resonant frequency of the device 150and / or piezoelectric members 102, 122 and thereby maximize energy production bymatching resonant frequencies with a vibration source (e.g., the excitation frequency).

[0074] With reference to FIG. 1D, yet another example implementation is shownaccording to implementations of the present disclosure. The structure includes a dualcantilever beam that can operate according to the principles described with reference toFIG. 1C, and Example 2 herein. In the implementation of FIG. 1D, each of the piezoelectricmembers 102, 122 can include a tip mass 160a and 160b, respectively, that serve thefunction of the extrusions described with reference to FIG. 1B. The tip masses 160a, 160bcan optionally be formed from bolts or nuts to adjust the size of the gap 110 between thepiezoelectric members 102, 122.Docket Number: 103362 005WO1

[0075] With reference to FIG. 2, implementations of the present disclosure includesystems 200 configured to optimize the harvesting of electrical energy, for example usingthe devices 100, 120, 150 shown in FIGS. 1A 1C. The system 200 can include an energyharvesting device 210 with an adjustable gap 220 (e.g., the gap 110 described withreference to FIGS. 1A 1C), and a controller 250 configured to control the adjustable gap tomaximize energy production. The controller 250 can include any or all of the components ofthe example computing device 400 shown in FIG. 4.

[0076] In some implementations, the system 200 can include a sensor 270. The sensor270 can be operably coupled to the controller 250, and configured to measure one or moreexcitation frequencies acting on the device 210. The controller 250 can optionally use themeasurements of the one or more excitation frequencies to determine an optimizedresonant frequency of the device and control the adjustable gap 220 to match the optimizedresonant frequency. Optionally, the sensor 270 can be configured to measure and / or outputa spectrum of energies (e.g., the power level at different frequencies or ranges offrequencies).

[0077] With reference to FIG. 3, the controller 250 shown in FIG. 2 can be configured toimplement computer implemented methods to control the device 210 of FIG. 2 to maximizeenergy production or other desired outputs of the device 210. An example method 300includes determining, at step 302, an optimized resonant frequency of the device 210, andadjusting, at step 304 the adjustable gap 220 to maximize the energy generated by thedevice 210 or any other output of the device. Optionally, the optimized resonant frequencycan match an excitation frequency of a source of vibrational energy. Optionally, multipleexcitation frequencies can be present, and the method 300 can include selecting anoptimized resonant frequency is between two or more of the excitation frequencies.

[0078] Additional examples of control strategies and outputs for the systems of thepresent disclosure are shown and described with reference to the Appendices A and B,attached hereto.

[0079] As used herein, the terms "about" or "approximately" when referring to ameasurable value such as an amount, a percentage, and the like, is meant to encompassvariations of ±20%, ±10%, ±5%, or ±1% from the measurable value.

[0080] It should be appreciated that the logical operations described herein with respectto the various figures may be implemented (1) as a sequence of computer implemented actsDocket Number: 103362 005WO1or program modules (i.e., software) running on a computing device (e.g., the computingdevice described in Fig. 4), (2) as interconnected machine logic circuits or circuit modules(i.e., hardware) within the computing device and / or (3) a combination of software andhardware of the computing device. Thus, the logical operations discussed herein are notlimited to any specific combination of hardware and software. The implementation is amatter of choice dependent on the performance and other requirements of the computingdevice. Accordingly, the logical operations described herein are referred to variously asoperations, structural devices, acts, or modules. These operations, structural devices, actsand modules may be implemented in software, in firmware, in special purpose digital logic,and any combination thereof. It should also be appreciated that more or fewer operationsmay be performed than shown in the figures and described herein. These operations mayalso be performed in a different order than those described herein.

[0081] Referring to Fig. 4, an example computing device 400 upon which the methodsdescribed herein may be implemented is illustrated. It should be understood that theexample computing device 400 is only one example of a suitable computing environmentupon which the methods described herein may be implemented. Optionally, the computingdevice 400 can be a well known computing system including, but not limited to, personalcomputers, servers, handheld or laptop devices, multiprocessor systems, microprocessorbased systems, network personal computers (PCs), minicomputers, mainframe computers,embedded systems, and / or distributed computing environments including a plurality of anyof the above systems or devices. Distributed computing environments enable remotecomputing devices, which are connected to a communication network or other datatransmission medium, to perform various tasks. In the distributed computing environment,the program modules, applications, and other data may be stored on local and / or remotecomputer storage media.

[0082] In its most basic configuration, computing device 400 typically includes at leastone processing unit 406 and system memory 404. Depending on the exact configuration andtype of computing device, system memory 404 may be volatile (such as random accessmemory (RAM)), non volatile (such as read only memory (ROM), flash memory, etc.), orsome combination of the two. This most basic configuration is illustrated in Fig. 4 by box402. The processing unit 406 may be a standard programmable processor that performsarithmetic and logic operations necessary for operation of the computing device 400. TheDocket Number: 103362 005WO1computing device 400 may also include a bus or other communication mechanism forcommunicating information among various components of the computing device 400.

[0083] Computing device 400 may have additional features / functionality. For example,computing device 400 may include additional storage such as removable storage 408 andnon removable storage 410 including, but not limited to, magnetic or optical disks or tapes.Computing device 400 may also contain network connection(s) 416 that allow the device tocommunicate with other devices. Computing device 400 may also have input device(s) 414such as a keyboard, mouse, touch screen, etc. Output device(s) 412 such as a display,speakers, printer, etc. may also be included. The additional devices may be connected to thebus in order to facilitate communication of data among the components of the computingdevice 400. All these devices are well known in the art and need not be discussed at lengthhere.

[0084] The processing unit 406 may be configured to execute program code encoded intangible, computer readable media. Tangible, computer readable media refers to any mediathat is capable of providing data that causes the computing device 400 (i.e., a machine) tooperate in a particular fashion. Various computer readable media may be utilized to provideinstructions to the processing unit 406 for execution. Example tangible, computer readablemedia may include, but is not limited to, volatile media, non volatile media, removablemedia and non removable media implemented in any method or technology for storage ofinformation such as computer readable instructions, data structures, program modules orother data. System memory 404, removable storage 408, and non removable storage 410are all examples of tangible, computer storage media. Example tangible, computer readablerecording media include, but are not limited to, an integrated circuit (e.g., fieldprogrammable gate array or application specific IC), a hard disk, an optical disk, a magnetooptical disk, a floppy disk, a magnetic tape, a holographic storage medium, a solid statedevice, RAM, ROM, electrically erasable program read only memory (EEPROM), flashmemory or other memory technology, CD ROM, digital versatile disks (DVD) or other opticalstorage, magnetic cassettes, magnetic tape, magnetic disk storage or other magneticstorage devices.

[0085] In an example implementation, the processing unit 406 may execute programcode stored in the system memory 404. For example, the bus may carry data to the systemmemory 404, from which the processing unit 406 receives and executes instructions. TheDocket Number: 103362 005WO1data received by the system memory 404 may optionally be stored on the removablestorage 408 or the non removable storage 410 before or after execution by the processingunit 406.

[0086] It should be understood that the various techniques described herein may beimplemented in connection with hardware or software or, where appropriate, with acombination thereof. Thus, the methods and apparatuses of the presently disclosed subjectmatter, or certain aspects or portions thereof, may take the form of program code (i.e.,instructions) embodied in tangible media, such as floppy diskettes, CD ROMs, hard drives, orany other machine readable storage medium wherein, when the program code is loadedinto and executed by a machine, such as a computing device, the machine becomes anapparatus for practicing the presently disclosed subject matter. In the case of program codeexecution on programmable computers, the computing device generally includes aprocessor, a storage medium readable by the processor (including volatile and non volatilememory and / or storage elements), at least one input device, and at least one output device.One or more programs may implement or utilize the processes described in connection withthe presently disclosed subject matter, e.g., through the use of an application programminginterface (API), reusable controls, or the like. Such programs may be implemented in a highlevel procedural or object oriented programming language to communicate with acomputer system. However, the program(s) can be implemented in assembly or machinelanguage, if desired. In any case, the language may be a compiled or interpreted languageand it may be combined with hardware implementations.

[0087] Examples

[0088] The following examples are put forth so as to provide those of ordinary skill in theart with a complete disclosure and description of how the compounds, compositions,articles, devices and / or methods claimed herein are made and evaluated, and are intendedto be purely exemplary and are not intended to limit the disclosure. Efforts have been madeto ensure accuracy with respect to numbers (e.g., amounts, temperature, etc.), but someerrors and deviations should be accounted for. Unless indicated otherwise, parts are partsby weight, temperature is in C or is at ambient temperature, and pressure is at or nearatmospheric.

[0089] Example 1:Docket Number: 103362 005WO1

[0090] An example implementation of a piezoelectric energy harvester according to FIG.1A was designed, modeled, and tested. The example implementation is a piecewise linear(PWL) piezoelectric energy harvester. The study shows the performance of the PWL energyharvester using a piezoelectric cantilever beam as the energy generator, which can besuitable for smaller scale harvesting applications. An optimized design was created using theproperties of a piezoelectric cantilever beam, and a simplified controlling algorithm isimplemented. This design can actively adjust the resonance frequency to maximize powergeneration over a larger frequency range to make self powered sensors more viable. Theresonance frequency can be optimized by adjusting the gap size between the Piezoelectriccantilever beam and an elastic stopper using a combination of linear actuators, circuits andmicroprocessors. The design shows an increased performance in maintaining an optimizedvibrating amplitude in the pre computed frequency range. An experimental setup wastested for the given frequencies to compare to the computational prediction and validatethe design's effectiveness.

[0091] The study herein includes implementations of a piezoelectric sensor and energyharvester with a controllable gap. The system reads the voltage generated from thepiezoelectric material and adjusts the gap to stay in the effective harvester range. A controlalgorithm based on the BAA method can be used to control the gap between thepiezoelectric cantilever beam and the elastic stopper. This energy harvester is tested at apredetermined frequency range, and the system dynamics are captured. These results arethen compared to the experimental data from conventional linear designs of the samefrequency range. Furthermore, an experimental investigation of the effect on the position ofthe elastic stopper is presented. A comparison of three different positions of the elasticstopper is included and analyzed. Advantages of the example implementation over existingpiezoelectric systems include an optimized energy approach whereby the sensinginformation transmitted is the same as what is required for the control algorithm and ademonstrated control algorithm that uses minimum inputs (i.e., just the voltage from thepiezoelectric material and the stopper location) to control the stopper position.

[0092] Computational Methodologies

[0093] The example piezoelectric energy harvester with the actuated elastic stopper isshown in FIG. 1A. The left end of the piezoelectric cantilever beam is clamped at the fixedDocket Number: 103362 005WO1base, and the right end of the piezoelectric beam, referred to at the tip, is free to vibrate. Inthis system, an external vertical excitation is applied as the base to create a verticaldisplacement at the tip of the piezoelectric cantilever beam. The vibration created bythe relative displacement generates AC power in the piezoelectric material thatcan be collected at the base of the beams. The cantilever beam can be a continuous systemthat can be modelled with an infinite number of degrees of freedom (DOFs). However,practically a subset of modes are typically used to characterize the motion of the systemover a specified frequency range. In this work, the interest is in capturing the most energy,which occurs at the highest tip deflections. Therefore a single mode representation usingthe fundamental mode of the cantilevered beam will capture the desired dynamics at thetip

[0021] . The elastic stopper is set to make contact with the piezoelectric cantilever beam atits tip, introducing the piecewise nonlinearity. Note that the clamping position of thepiezoelectric cantilever beam can be adjusted to the fixed base. The more contact area thepiezoelectric beam and the fixed base are, the smaller the resonant frequency will be. Themechanical model of this single DOF piecewise linear system can be represented as thesystem shown in FIG. 5. This model

[0019] includes a contact mass (in addition to a contactstiffness and damping) to predict the system dynamics more accurately.

[0094] In this model, a piezoelectric cantilever beam with mass , stiffness anddamping coefficient is positioned at the base. Note that the clamping position has animpact on the effective mass and stiffness. The mass and stiffness will decrease if thecontact surface between the beam and the fixed base increases. This property can be usedto adjust the resonance frequency of the open state system, which will be used in thePhysical Investigation. An elastic stopper with mass , stiffness and damping coefficientis attached to the linear actuator. The damping coefficients for both components areassumed to be linear and viscous. The external excitation , tip deflection and gapsize refer to the same positions as in FIG. 1A.

[0095] The equation of motion for the model is shown in Eq. (1). The variableis the relative motion of the piezoelectric cantilever beam tip to the base, andis the combined mass of the beam and contact mass.Docket Number: 103362 005WO1

[0096] The subscript denotes the closed state system when , and the subscriptdenotes the open state system when .

[0097] Dividing the closed system equation of motion by and the open systemequation of motion by , Eq. (1) can be rewritten as:¨2 ¨ , whenwhen .

[0098] The new parameters in Eq. (2) are:

[0099] Time related dimensionless variables can also be introduced:and

[0100] To apply the BAA method to the model and match its periodic excitation, thesystem is assumed to be subjected to a harmonic excitation sin where isthe excitation frequency, and the is the excitation amplitude. Eq. (2) then be written as:

[0101] For specific , and. The conversion between these time relateddimensionless variables for the open and closed systems is:

[0102] The last step is to introduce the displacement related dimensionless variables:Docket Number: 103362 005WO1

[0103] After scaling and by the base excitation amplitude , the final equation ofmotion listed is given in Eq. (8):¨ 21 sin2 , 8¨ 2sin ,when

[0104] System Dynamics: To efficiently solve for the steady state dynamics of thepiecewise linear system, the BAA method is applied to Eq. (8). Each cycle of the PWLresponse comprises an open state and a closed state. Furthermore, each state occurs onlyonce in each vibration cycle. The overall period can be expressed as , wheredenotes the Time interval in the open state and denotes the time interval in the closestate. The system can therefore be written as the sum of the steady state and transientresponses from linear vibration theory with some minor alterations to account for thenonlinearity::and. Variables and are scalar coefficients of thetransient response; and are the phase angle of the transient response; is the phaseshift between the excitation and steady state response due to the piecewise nonlinearity.Finally, a nonlinear solver is applied to minimize the residual of the following compatibilityconditions:Docket Number: 103362 005WO1

[0106] The equations in Eq. (10) represent the transition between open and closed states:the first four equations are for displacement compatibility, and the last two are for velocitycompatibility. Also, an extra unknown value, , is introduced in the equations to match theboundaries between states. The Matlab function

[0022] 'Isqnonlin' is used to solve the set ofequations. The solution of these unknown values is used to construct the entire cycle for thegiven gap size. The detailed overview of the BAA method has previously been published

[0014] . The optimal gap size from the computational tool can be compared to theexperimental result and thus used as the base of the simplified control algorithm.

[0107] Experimental Methodology: The study assessed the performance of thepiezoelectric cantilever beam system with the adjustable gap size. The energy used andgenerated in the system can be monitored to understand the benefits of this new system,but an external power supply is used for powering the system in this benchtop experiment.The schematic of the experimental setup is shown in FIG. 6.

[0108] The experimental setup is shown in FIGS. 7A and 7B. The PWL energy harvester iscomprised of a piezoelectric energy harvesting module (PPA 1011, MIDE), a linear actuator(L12 I, Actuonix) and an elastic stopper. The piezoelectric energy harvesting module and thelinear actuator are mounted on the base and connected to an electrodynamic shaker (ET126B, Labworks). The shaker generates a base excitation in the vertical direction, whichenables the piezoelectric energy harvesting module to vibrate at its tip. The elastic stopper,which serves as the piecewise linear component, is mounted on the top of the linearactuator. The gap size is then altered by adjusting the length of the linear actuator. Thelinear actuator is powered and controlled by an Arduino with 5 V and a pause widthmodulation Signal. The linear actuator can provide a maximum force of 80 N. An AC voltageis generated from the vibration of the beam inducing a stress in the piezoelectric materialand collected at the base of the piezoelectric beam. The AC voltage is centered at 0 V and isperiodic. A voltage divider is connected to modify the AC voltage and enable the Arduino toread and analyze the system dynamics. The Voltage divider divides the AC voltageDocket Number: 103362 005WO1proportionally and then adds an offset before sending the voltage to the Arduino. Theresulting voltage falls in the detectable range of the Arduino board ( 0 V 5 V ) and can beused to analyze the dynamics of the beam. An Arduino UNO R4 WiFi is used as the processorand sensor. The Arduino has a built in analogue to digital converter that can transformanalogue signals into values. The analogue input is collected and analyzed to find thefrequency and peak voltage for each AC voltage cycle. The 5 V pin and PWM digital outputpin on the Arduino board provide power and a controlling algorithm to the linear actuatorto change the gap size based on the piezoelectric cantilever beam's frequency andestimated tip displacement.

[0109] To monitor the performance of the system, two laser displacement sensors (IL100, Keyence) are implemented above the piezoelectric to detect its base excitationand tip motion . The system parameters were identified at the beginning of theinvestigation to build an accurate model of the PWL system. Both static and dynamic testswere conducted to identify the mass, damping, and stiffness of the beam and the stopper tobuild the computational model.

[0110] The stiffness and were identified using a static analysis

[0020] . A spring balancewas used to apply a static force at the tip, as shown in FIG. 8. The deformation at the tip wasmeasured using a laser displacement sensor. The equivalent stiffness of the first mode ofthe cantilevered beam was calculated by dividing the force by the displacement. A similarprocess was applied for the elastic stopper to identify the stiffness .

[0111] The cantilevered beam can be approximated with a one mode approximationusing the tip motion as a coordinate. Under this assumption, the damping ratio andresonant frequency can be identified separately for the beam and the elastic stopperfrequency responses. The single DOF system of each component was subjected to asinusoidal base excitation that slowly swept across the desired frequency. The two laserdisplacement sensors are positioned at the base and tip for both components. Thedisplacement transmissibility was then calculated using the power spectral densities of tipmotion and base excitation

[0023] . The experimental displacement transmissibilities for thepiezoelectric cantilever beam and the elastic stopper are shown in FIG. 9. Note that themeasured response of the system does have significant noise due in part to its large sizethat is required due to the large actuator used in the current setup. The Matlab function'movmean' was used to smooth the displacement transmissibility and the averagedDocket Number: 103362 005WO1transmissibility is shown in FIG. 10. The damping ratio was then estimated using the halfpower bandwidth method given in Eq. (11).

[0112] In this equation,and are the corresponding half power points, and peak isthe resonance frequency. The piezoelectric beam's half power points and resonancefrequency are noted in FIG. 10. The mass and damping coefficient was then calculated usingthe following equations:

[0113] where and , peak are known. The experimental displacement transmissibility isplotted against the computational displacement transmissibility using the identifiedparameters in FIG. 10. The same process was also used to model the elastic stopper. Theparameters for both components are given in Table. 1.

[0114] TABLE 1: Estimated system parameters

[0115] Control Algorithm: The optimal gap size for different frequencies was computedusing the BAA method and the identified parameters. The control algorithm for the actuatoris discussed in this section and uses the precomputed optimal gap size that is shown in FIG.11A and FIG. 11B. Compared to the electromagnetic PWL energy harvester previouslystudied

[0019] , the piezoelectric energy harvester has a significantly smaller resonance peakwith a wider frequency range near resonance due to the increased damping in this system.Unfortunately, this lowers the effectiveness of the controlled PWL energy harvester;however, as seen in FIG. 11B, a significant improvement over the linear system can begained if the excitation frequency drifts significantly. There is also a limit to the effectivenessof the system due to the nature of the linear actuator used in this work. The actuator movesin a discrete stepwise manner and is not continuous; therefore, the corresponding gapDocket Number: 103362 005WO1control has to be implemented in a stepwise manner, which lowers the effectiveness of theapproach.

[0116] To simplify the system, the voltage generated from the energy harvester is used toindicate the estimated tip displacement. According to the piezoelectric energy harvestingmodule, its tip deflection and the voltage have an approximately fixed proportional relationunder the same fixed base and tip mass. The displacement sensors in previous studies [19,20] are replaced by a direct voltage reading at the power output of the piezoelectric beamto estimate the expected displacement.

[0117] A linear control algorithm is then introduced to the piezoelectric energy harvesterbased on FIG. 12. The figure is the ratio of optimal gap size over optimal displacementtransmissibility. An approximately linear relation is observed between the excitationfrequency of open state system ( 1 ) to the excitation frequency of 0 gap system (1.288 ). This linearity is used for the control algorithm.

[0118] This linear relation is denoted as and can be solved by the two boundaryconditions at the open state resonant frequency and 0 gap resonant frequency.Furthermore, can be represented in Eq. (13):

[0119] The and are from Eq. (7) and are both normalized by . Equation. (13) can befurther denormalized using and rewritten as:14

[0120] where g is gap size and is tip deflection. As mentioned above, tip deflection andvoltage reading from the piezoelectric also have an approximately linear relation, noted as / , which leads to the final equations:

[0121] In this equation, v is the AC voltage reading from the piezoelectric beam during tipvibration, and is the adjusted linear relation subject to tip deflection and voltagereading ratio. This equation is used in the linear control algorithm to calculate the gap. Thealgorithm then adjusts the gap size linearly between open state resonant frequency and 0gap resonant frequency. The steps of the control process are given as follows:Docket Number: 103362 005WO11. The linear relation and open state resonant frequency are calculated basedon the BAA method.2. The AC voltage and corresponding time stamp is measured and stored.3. The estimated frequency est and peak voltage est are identified from the storeddata. 4. Find the optimized gap size est / est . Eq.155. Send the optimal gap size to the linear actuator to adjust the actual gap size.6. Steps 2 4 can be repeated during the operation.

[0122] The signal estimation in step 2 can be done after each cycle. An example voltagereading stored in the Arduino is shown in FIG. 13. The time period of one full cycle can becalculated directly using the time stamps of the two most recent consecutive voltage peaks.The frequency can be estimated using 1 / . The most recent peak voltage is denotedto be the est used in the control algorithm. The algorithm is robust and only takes ACoutput voltage from the piezoelectric beam as the input to adjust the gap size.

[0123] Physical Investigation: This section first compares the displacementtransmissibility of the experimental PWL system with the computational model and thencompares the PWL system with a corresponding linear system. The energy harvester isexcited under a fixed voltage signal input to the shaker. The displacement transmissibilitywas generated using the base excitation and tip deflection measurements over a fixedperiod. The signal was slowly swept through a frequency range that covered the resonanceof the open system and the resonance at the 0 gap position. The resulting experimentaldisplacement transmissibility is compared against the ideal results from the computationalmodel in FIG. 14. The experimental result tends to lie below the ideal computational resultdue mainly to the step behavior in the linear actuator. The displacement transmissibility ofthe PWL energy harvester is able to keep track of the response envelope over the excitationfrequency of 1 to 1.25 , which matches the corresponding frequency from the open state to0 gap state in the computational model. Note that the experimental result exceeds theresponse envelope at certain frequencies. This is probably due to the system's noise. Thesudden decrease of the system happens when the linear actuator changes its stroke length.Note that the microcontroller controls the linear actuator entirely through the controlalgorithm discussed in this work.

[0124] Next, the PWL energy harvester response is compared with the linear systemresponse, and the results are summarized in FIG. 15. The resonance peak is adjusted byDocket Number: 103362 005WO1changing the clamp position of the piezoelectric beam. As discussed previously, theresonant frequency of the open state system increases as the clamped position moves fromthe fixed end of the piezoelectric beam to the free end. The clamping position does notaffect the resonant peak value of the piezoelectric cantilever beam (PPA1011). The averagedisplacement transmissibility is compared between the two systems through the effectivefrequency ranges.

[0125] The PWL response has an effective excitation frequency from 0.9 to 1.288. Theaverage value of the linear response centered at 1.12 is 9.5585, and the average of the PWLresponse is 13.0843. The PWL system shows a 36.89% increase in displacementtransmissibility compared to the linear system. Note that if a lower damping piezoelectricbeam can be used, the effectiveness of the PWL system over the linear system will becomeeven more apparent since the resonance peak will become much sharper and higher. Todemonstrate the effectiveness of the piezoelectric energy harvester, a comparison betweenthe PWL system and the linear system is shown in FIG. 16.

[0126] Both systems switched between two excitation frequencies, 1.12 and 1.25, andstayed for 20 seconds. Note that Matlab function 'movmean'

[0022] is used in the result tosmooth the displacement transmissibility and frequency. It is clear that the system with acontrolled gap size performs better overall at both excitation frequencies than the linearsystem. Meanwhile, it can be observed that the PWL system maintained its expecteddisplacement transmissibility at each frequency switch, which further proves its stability. Acomparison between the tip deflection is included in FIG. 17 on one axis. It is also clear thatthe deflection in the PWL system performs better than the deflection in the linear system. Acomparison of frequency is also included according to the other axis. The operatingfrequencies measured during the experiment are similar for both systems, which isexpected.

[0127] The study shows the performance of a piecewise linear (PWL) piezoelectric energyharvester with a controllable gap size. The resonance frequency of the free end of thepiezoelectric beam is modified by adjusting the gap size between the piezoelectric cantileverbeam and an elastic stopper. A simple control algorithm has been implemented anddemonstrated. The algorithm enables the active control of the piezoelectric energyharvester within the precomputed frequency range using only the voltage response ofpiezoelectric material as input. The study further shows the PWL energy harvester'sDocket Number: 103362 005WO1improved performance over a traditional linear energy harvester. Although thisimprovement in the performance matches the computational prediction, the increase in theperformance and the frequency range can optionally be improved greatly if a lower dampedpiezoelectric beam were used. The elastic stopper can also be modified to enhance theperformance further. Additionally, although the linear actuator has a low voltagerequirement that the Arduino can satisfy, it has a much higher supplied force than thesystem requires.

[0128] Example 2:

[0129] An additional study was performed to test the performance of implementationsincluding dual beam piezoelectric energy harvesters, including the implementation shown inFIG. 1D. The study included both computational and experimental validation of the newpiezoelectric energy harvester. The example design replaces the mechanical stopper withanother piezoelectric cantilever beam and the vibration of both beams contributes to thepower production. A modified bilinear amplitude approximation (BAA) [25A 29A] method isused to capture the system dynamics of the dual piezoelectric VEH design. An experimentalsystem is built and tested at different excitations conditions and the system dynamics aremonitored. These results are then compared to traditional linear designs, as well as anumerical simulation of the proposed system.

[0130] Computational Methods: Dual Piezoelectric Cantilever Beam Model. The dualpiezoelectric energy harvester used in this work is shown in FIG. 1D. The left ends of thepiezoelectric cantilever beams are clamped at the base and on the linear actuator. The rightends of the piezoelectric beams, referred to as the tips, are free to vibrate. In this system, anexternal vertical excitation is applied at the base to create vertical displacementsand at the tip of the piezoelectric cantilever beams. The relative displacementsand create stress in the piezoelectric material that oscillates due to thevibration. The changing stress in the piezoelectric material produces an alternating current(AC) that can be collected at the base of the beam. Theoretically, the cantilever beam can bemodeled with an infinite number of degrees of freedom (DOFs) using a continuous analyticalmodel of the beam. Subsets of modes are typically used to characterize the motion of thesystem over a specified frequency range. The example implementation can be configured tooptimize the power from the piezoelectric material, which can be achieved by creating largerelative motions in the blade tips. This large relative motion of the tip is achieved mostDocket Number: 103362 005WO1effectively by exciting the first bending mode of the cantilevered beam. Therefore, in thisexample implementation, the motion of the beam can be captured by projecting it onto thefirst bending mode [30A]. The two piezoelectric beams are designed to make single pointcontact at their tips by fixing bolts at the tips, which introduces a PWL nonlinearity. Theeffective stiffness and masses of the beams are adjusted by changing the clamping positionand adding tip masses. The mechanical model of this PWL system can be represented as thesystem shown in FIG. 18.

[0131] In this model, one piezoelectric cantilever beam with stiffness and dampingcoefficient is positioned on the linear actuator. Another piezoelectric cantilever beamwith stiffness and damping coefficient is positioned on the fixed base. The damping forboth components is assumed to be linear and viscous. Note that bolts are used as tipmasses for the piezoelectric cantilever beams to make single point contact during collision.After adjusting the clamping positions and the tip masses (bolts), the effective mass for thepiezoelectric cantilever beam 1 isand the effective mass for the piezoelectric cantileverbeam 2 is . The external excitation , tip deflections and and gap sizerefer to the same positions as in FIG. 1D. Note that the gap size refers to the distancebetween the bottom of the tip mass for piezoelectric cantilever beam 2 and the tip of thepiezoelectric cantilever beam 1 in their undeformed states and varies in time only due tomotion induced by the linear actuator.

[0132] A modified BAA method is used to find the dynamics of this system. The BAAmethod assumes the system dynamic is periodic and one vibration cycle can be divided intotwo linear states. The first state is the open state where the two masses vibrateindependent of each other. This corresponds to two decoupled single DOF systems. Thesecond state is the closed state where the two masses are in contact. In this closed state,the two masses are considered to be one solid piece. This makes the system one DOF duringcontact. The states switch from closed to open when mass one and mass two detach andswitch back when mass one and mass two attach. The equation of motion is developed foreach of the two states.

[0133] The equation of motion for the system is shown in Eq. (1). The variablesare the relative motions of the piezoelectric cantilever beam tips to baseexcitation. is the combined mass of the beams.Docket Number: 103362 005WO1

[0134] The subscript denotes the closed state system when , and thesubscript denotes the open state system when . Their relation can becharacterized as:when ,2when .

[0135] The closed state system equations can then be normalized by dividing by . Thetwo open state system equations can be normalized by dividing byand , respectively.Equation. (1) can then be rewritten as:

[0136] The new parameters in Eq. (3) are:and

[0137] To apply the BAA method, the system is assumed to be periodic and subjected to aharmonic excitation sin where is the excitation frequency, and is theexcitation amplitude. Then the displacement related dimensionless variables can be definedas:Docket Number: 103362 005WO1

[0138] After scalingand by the base excitation amplitude , pre computedresponses can be made independent of the base excitation. The equation can finally bewritten as:

[0139] As mentioned in the previous section, the modified BAA method is used to solvethe equations of motion given in Eq. (6). The method assumes that the response of thesystem is periodic and that one vibration cycle has only one interval of the closed state andopen state. The overall time for one vibration period is 2 / . The time perioddivides into two time intervals where is the time interval in the closed stateand is the time interval in the open state. Note that since the gap is controlled bythe linear actuator and not the vibration, 0. The displacement of the system in theclosed state and the open state can be represented by a combination of the transientresponse and linear steady state response, respectively:Docket Number: 103362 005WO1,coefficients of the transient response; the phase angle of the transient response arerepresented by , . ; and is the phase shift between the excitation and steadystate response due to the intermittent contact. A nonlinear solver is used to minimize theresidual for the compatibility conditions below:

[0141] Equation (8) represents the boundary conditions between the open and closedstates. The first four equations are for the displacement compatibility at each transition. Thefifth to seventh equation are the velocity compatibility at each transition. The collisionoccurs at the transition from the open state to the closed state and is assumed to be perfectinelastic. The separation occurs at the transition from the closed state to the open state.The velocity of the combined mass and the separated masses are assumed to be the sameat the transitions. The last equation is the force compatibility at the transition from theclosed state to the open state. The forces at this transition must be balanced. An extraunknown is introduced in the boundary condition equations that relates to the amount oftime the system remains in the closed state out of the full period . The MATLAB function'lsqnonlin' [31A] is used to solve Eq. (7) by minimizing the residuals in Eq. (8). The number ofunknowns ( , , , , , and) match the number of compatibilityequations given in Eq. (8). Note that the function 'Isqnonlin' finds the set of unknowns withthe lowest residual. If the residual is higher than a threshold value, this means either thecurrent gap size is beyond the maximum gap size the compatibility conditions are valid foror the assumptions of the methodology do not hold (i.e., the response is chaotic ornonstationary). A detailed overview of the general BAA method has previously beenpublished [25A], but this method does not account for the motion in the stopper mass whenDocket Number: 103362 005WO1not engaged with the primary mass. This assumption was fine when the stopper responsewas at a much higher frequency, but is not valid for the system investigated in this work.

[0142] Experimental Setup. The study assessed the performance of the dual piezoelectricenergy harvester with adjustable gap size. The power generated by the system was analyzedfor various configurations to demonstrate the capability of the proposed controllable PWLVEH. An external power supply is used for the bench top experiment to power the linearactuator. The overall bench top setup is shown in FIGS. 19A and 19B.

[0143] The PWL VEH is composed of two piezoelectric energy harvesting modules (PPA1011, MIDE) and a linear actuator (L12I, Actuonix). One piezoelectric energy harvestingmodule is mounted on the linear actuator. The linear actuator and a second piezoelectricenergy harvesting module is mounted on the base which is connected to the electrodynamicshaker (ET 126B, Labworks). A vertical periodic excitation is produced by the shaker,creating vibrations in the two piezoelectric cantilever beams. The gap size between the twopiezoelectric cantilever beams is altered by adjusting the linear actuator. To adjust theeffective masses of the piezoelectric cantilever beams, pairs of bolts and nuts are mountedas tip masses. The vibrations of the beam induce stresses in the piezoelectric materials thatgenerate AC voltages collected at the base of the piezoelectric beams. Note that the ACvoltage is approximately proportional to the tip deflection [32A], thus can be used toestimate the relative motion of the tip deflections. Laser displacement sensors (IL 100,Keyence) are used to track the base excitation. The relative motions of the piezoelectricbeams are collected using a data acquisition system (DT9816, Digilent).

[0144] In order to setup the computational model for comparison with the bench testsetup, key parameters of the physical system needed to be identified. Both piezoelectricenergy harvesting modules were tested independently to identify their individual effectivemass, damping and stiffness.

[0145] A spring balanced method [33A] was used to identify the stiffness and . Aforce is applied to the tip of the upper piezoelectric beam as shown in FIG. 20. Thedeformation of the tips were measured using the laser displacement sensor. The equivalentstiffness was calculated by dividing the force by the displacement. The similar processwas then applied to the other piezoelectric beam to calculate .

[0146] Since the dynamics of the piezoelectric beam at the tip can be estimated by a onemode approximation, the damping ratio and resonant frequency of each beam can beDocket Number: 103362 005WO1computed from a slow frequency sweep of each beam. Each piezoelectric beam is excited bythe same periodic sinusoidal base excitation. The base excitation is designed to be an upchirp wave, which means that the excitation frequency changes slowly from low to highwithin a preset frequency range. The relative motion of the piezoelectric beam is identifiedfrom the AC voltage collected during the frequency sweep and the base excitation from theshaker is identified by the laser displacement sensor. The time domain relative motion ofthe piezoelectric beam is shown in FIG. 21.

[0147] Due to the slow sweep of the base excitation, the frequency response of thepiezoelectric beam can be approximated by computing the envelopes of the time domainAC voltage reading. The slow sweep excitation starts from a frequency of 60 Hz at 0 secondsand increases to a frequency of 120 Hz at 60 seconds. The frequency increase is distributedequally in time, which means that the time can be converted to a corresponding frequency.A similar process was carried out for the other piezoelectric beam as well. The upperenvelope and lower envelope of the time domain signals were computed using the MATLABfunction 'envelope' [31A]. By subtracting the upper envelope of the piezoelectric beams bytheir lower envelope and dividing the results by the amplitudes of the base excitation, thefrequency response of the piezoelectric beams were computed, and plotted in FIGS. 22A22B.

[0148] TABLE 1A: ESTIMATED SYSTEM PARAMETERS

[0149] The half power bandwidth method was used to compute the damping ratioof the two piezoelectric beams. This method, given in Eq. (9), uses data pointsextracted from FIGS. 22A 22B.Docket Number: 103362 005WO1

[0150] The frequencies , , and are the half power points and peak and peak arethe resonant frequencies for piezoelectric beam 1 and 2. The effective mass and dampingcoefficient can then be calculated using the following equations:

[0151] , peak and peak are known. The experimental displacementtransmissibility is plotted against the computational displacement transmissibility using theidentified parameters in Fig. FIGS. 22A 22B for comparison. The parameters for both beamsare given in Table. 1A.

[0152] Results. In this section, results from the bench top system being tested arecompared to the simulation for the same conditions. The system parameters were firstconverted into the variables used in the computational tool and are shown in Table. 2A.

[0153] TABLE 2A: COMPUTATIONAL VARIABLE VALUES

[0154] After incorporating these values into the computational method, the responseenvelope over the frequency range bounded by the linear responses of the twopiezoelectric beams was constructed and is shown in FIGS. 23A 23B.Docket Number: 103362 005WO1

[0155] The combined envelope with the optimized gap is shown in FIGS. 24A 24B. Notethat in the simulation tool, the maximum acceptable residual for the compatibilityconditions is set to be 10 to find the gap size. It is notable that the least residual for thesystem will be higher than the threshold of 10 after the frequency reaches 101.2 Hz . Thisindicates that the system dynamics might be different from those predicted by BAA afterthe frequency reaches 101.2 Hz . The envelope and gap size after 101.2 Hz with the largeresidual is denoted by the dashed lines.

[0156] For the computational validation, the experimental system is tested underdifferent fixed gap sizes while the base excitation has a constant amplitude and slow upsweep frequency for each of the gap sizes. A total of seven frequency points were chosen tocompare to the simulation results. The peak frequencies and total amplitudes werecollected and plotted at the selected frequency points in FIG. 25.

[0157] TABLE 3A: COMPUTATIONAL VARIABLE VALUES

[0158] The details of the measured points are included in Table. 3A. This is a two DOFsystem, and there are two resonant frequencies for each positive gap size. A zero gap size isconsidered to be optimal for a range of frequencies, which indicates that the linear actuatordoes not have to move the beam over a range of frequencies to reach optimal displacementtransmissibility. Also note that even though the 104.9 Hz frequency point has a small errorcompared to the computational result. In this case an elastic collision is observed during theexperiment, which means that it does not meet the assumptions of BAA and matches thehigh residuals predicted by BAA at this frequency.Docket Number: 103362 005WO1

[0159] Finally, the average displacement transmissibility is compared between thecombined PWL system and the combined two linear state systems. The PWL system has anapproximately 65% increase compared to the combined two linear responses.

[0160] The study showed the performance of a dual piecewise linear piezoelectric energyharvester with a controllable gap size was presented. A modified BAA methodology for atwo degree of freedom system was discussed for cases where the second (stopper)element's motion, when not in contact, could not be ignored. An experimental setup of thisdual piezoelectric harvester was created to validate the computational methodology. A newmeasurement method was discussed to lower noise effects in the frequency responseanalysis. The identified system parameters from the measurements were used to generatethe computational model to compare to the experimental result.

[0161] The study shows that the performance of the piecewise linear piezoelectric energyharvester can be further improved by using a second piezoelectric beam as the mechanicalstopper. In addition to the increase in the overall performance, the operating frequencyrange is also increased significantly. The piecewise linear dual piezoelectric energy harvesteris able to harvest energy between the resonant frequencies of the two piezoelectric beams,which is significantly more than the piezoelectric energy harvester with a mechanicalstopper that had been previously explored. The BAA method was not capable of catchingthe dynamics of the system in all frequencies due to different collision types. A morecomprehensive method is required to fully analyze the system dynamics when they are notentirely stationary.

[0162] Although the subject matter has been described in language specific to structuralfeatures and / or methodological acts, it is to be understood that the subject matter definedin the appended claims is not necessarily limited to the specific features or acts describedabove. Rather, the specific features and acts described above are disclosed as exampleforms of implementing the claims.

[0163] REFERENCES[1] P. J. Paul, W. K. Richards, R. S. D. Tutu and Jerome, V. M. "Project power shoe:Piezoelectric wireless power transfer A mobile charging technique.": pp. 334 339. 2015.IEEE Global Humanitarian Technology Conference (GHTC), Seattle, WA, USA. DOI10.1109 / GHTC.2015.7343993.[2] Zhang H., Zhou Z. Zhu P. Du W., Qin W. "Piezomagnetoelastic energy harvesting fromDocket Number: 103362 005WO1bridge vibrations using bi stable characteristics." Energy Vol. 263 (2023): pp. 125859. DOI10.1016 / j.energy.2022.125859.[3] Zhang Y, Deng L, Cai SC. "Piezoelectric based energy harvesting in bridge systems."Journal of Intelligent Material Systems and Structures Vol. 25 (2014): pp. 1414 1428. DOI10.1177 / 1045389X13507354.[4] Mousavi, Mohammad, Ziaei Rad, Saeed and Karimi, Amir Hossein. "Piezoelectric basedenergy harvesting from bridge vibrations subjected to moving successive vehicles byfunctionally graded cantilever beams Theoretical and experimental investigations."Mechanical Systems and Signal Processing Vol. 188 (2023): p. 110015. DOI10.1016 / j.ymssp.2022.110015.[5] De Marqui Junior, Carlos, Erturk, Alper and Inman, Daniel J. "An electromechanical finiteelement model for piezoelectric energy harvester plates." Journal of Sound and VibrationVol. 327 No. 1 (2009): pp. 9 25. DOI https: / / doi.org / 10.1016 / j.jsv.2009.05.015.[6] Poulin, G., Sarraute, E. and Costa, F. "Generation of electrical energy for portabledevices: Comparative study of an electromagnetic and a piezoelectric system." Sensors andActuators A: Physical Vol. 116 No. 3 (2004): pp. 461 471. DOI 10.1016 / j.sna.2004.05.013.[7] Fang, Shitong, Miao, Gang, Chen, Keyu, Xing, Juntong, Zhou, Shengxi, Yang, Zhichun andLiao, WeiHsin. "Broadband energy harvester for low frequency rotations utilizing centrifugalsoftening piezoelectric beam array." Energy Vol. 241 (2022): p. 122833. DOIhttps: / / doi.org / 10.1016 / j.energy.2021.122833.[8] Salem, Marwa S., Ahmed, Shimaa, Shaker, Ahmed, Alshammari, Mohammad T., Al Dhlan,Kawther A., Alanazi, Adwan, Saeed, Ahmed and Abouelatta, Mohamed. "BandwidthBroadening of Piezoelectric Energy Harvesters Using Arrays of a Proposed PiezoelectricCantilever Structure." Micromachines Vol. 12 No. 8 (2021). DOI 10.3390 / mi12080973.[9] Goldschmidtboeing, Frank and Woias, Peter. "Characterization of different beam shapesfor piezoelectric energy harvesting." Journal of Micromechanics and Microengineering Vol.18 No. 10 (2008): p. 104013. DOI 10.1088 / 09601317 / 18 / 10 / 104013.

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[0012] Chen, Tingting, Wang, Kai, Cheng, Li, Pan, Hongbin, Cui, Haichao and Zhou, Jiaxi."Theoretical and Experimental Research on a Quasi Zero Stiffness Enabled NonlinearPiezoelectric Energy Harvester." Communications in Nonlinear Science and NumericalSimulation (2024): p. 107863DOI https: / / doi.org / 10.1016 / j.cnsns.2024.107863.

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[0015] Tien, M. and D'Souza, K. "A Generalized Bilinear Amplitude and FrequencyApproximation for Piecewise Linear Nonlinear Systems with Gaps or Prestress." NonlinearDynamics Vol. 88 No. 4 (2017): pp. 2403 2416. DOI https: / / doi.org / 10.1007 / s11071 0173385 5.

[0016] M. Tien, T. Hu and D'Souza, K. "Generalized Bilinear Amplitude Approximation and X Xrfor Modeling Cyclically Symmetric Structures With Cracks." Journal of Vibration andAcoustics Vol. 140 No. 4 (2018). DOI https: / / doi.org / 10.1115 / 1.4039296.

[0017] M. Tien, T. Hu and D'Souza, K. "Statistical Analysis of the Nonlinear Response of BladedDisks with Mistuning and Cracks." AIAA Journal Vol. 57 No. 11 (2019). DOIhttps: / / doi.org / 10.2514 / 1.J058190.

[0018] M. Tien, M. Lu and D'Souza, K. "Efficient Analysis of Piecewise Linear Nonlinear SystemsModeled Using General State Space Representations." Journal of Computational andNonlinear Dynamics Vol. 17 No. 8 (2022). DOI https: / / doi.org / 10.1115 / 1.4054152.

[0019] Veney, Jacob and DSouza, Kiran. "Frequency tunable electromagnetic vibration energyharvester using piecewise linear nonlinearity." Proceedings of the Royal Society A:Mathematical, Physical and Engineering Sciences Vol. 479 (2023). DOI10.1098 / rspa.2023.0207.Docket Number: 103362 005WO1

[0020] Tien, Meng Hsuan, Lee, Keng Yen and Huang, ShihChun. "Analyzing the backbone curveof piecewiselinear non smooth systems using a generalized bilinear frequencyapproximation method." Mechanical Systems and Signal Processing Vol. 204 (2023): p.110765. DOI 10.1016 / j.ymssp.2023.110765.

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Claims

Docket Number: 103362 005WO1WHAT IS CLAIMED:

1. A device comprising:a first piezoelectric member; anda linear actuator coupled to a stopper member, wherein the linear actuator isconfigured to control a resonant frequency of the device by adjusting a gapbetween the first piezoelectric member and elastic stopper member.

2. The device of claim 1, wherein the stopper member comprises a second piezoelectricmember.

3. The device of claim 2, wherein the stopper member further comprises an extrusioncoupled to the second piezoelectric member.

4. The device of claim 3, wherein the extrusion comprises an elastic material.

5. The device of claim 3, wherein the extrusion comprises a spring.

6. The device of any one of claims 1 5, wherein the stopper member comprises anelastic material.

7. The device of any one of claims 1 6, wherein the first piezoelectric membercomprises a first end and a second end, the first end and second end defining a longaxis of the first piezoelectric member, and wherein the first end of the firstpiezoelectric member is coupled to a base.

8. The device of any one of claims 2 7, wherein the second piezoelectric membercomprises a first end and a second end, the first end and second end defining a longaxis of the second piezoelectric member, and wherein the first end of the secondpiezoelectric member is coupled to the linear actuator.Docket Number: 103362 005WO19. The device of any one of claims 1 8, wherein a mass is fixed to the first piezoelectricmember.

10. The device of any one of claims 2 9, wherein a mass is fixed to the secondpiezoelectric member.

11. A device comprising:a first piezoelectric member coupled to a slider; anda second piezoelectric member coupled to the slider; wherein the firstpiezoelectric member and second piezoelectric member are separated by a gap.

12. The device of claim 11, wherein the first piezoelectric member and secondpiezoelectric member are parallel.

13. The device of claim 11 or claim 12, wherein the slider is a linear slider.

14. The device of any one of claims 11 13, wherein the slider is configured to adjust thegap between the first piezoelectric member and second piezoelectric member tocontrol a resonant frequency of the first piezoelectric member or secondpiezoelectric member.

15. The device of any one of claims 11 14, wherein the first piezoelectric member has afirst end and a second end defining a long axis of the piezoelectric member, andwherein the first end is coupled to the slider.

16. The device of any one of claims 11 15, wherein the second piezoelectric member hasa first end and a second end defining a long axis of the piezoelectric member, andwherein the first end of the second piezoelectric member is coupled to the slider.Docket Number: 103362 005WO117. An energy harvesting system comprising:the device of any one of claims 1 16; anda controller operably coupled to the device, the controller comprising aprocessor and a memory, the memory having computer executable instructionsstored thereon that, when executed by the processor, cause the controller to:determine a source frequency of a source of vibrational energy;determine, based on the source frequency, an optimized resonantfrequency for the device;adjust, by the linear actuator, the gap so that the resonant frequencyof the device is the optimized resonant frequency.

18. The system of claim 17, wherein the optimized resonant frequency is a frequencythat maximizes an amount of electrical energy harvested by the device.

19. The system of claim 17, wherein the optimized resonant frequency is a frequencythat matches an excitation frequency of the source of vibrational energy.

20. The system of any one of claims 17 19, wherein determining a source frequencycomprises receiving, from a sensor, a frequency spectra of the source of vibrationalenergy.

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