双稳态压电悬臂梁发电系统的动力学特性研究
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摘要
微电子设备低功耗的特点使利用环境振动进行自供电成为可能。压电式结构是常用的发电结构形式之一，目前研究的压电发电系统多是线性的，具有单一的固有频率，只有当激励频率和系统固有频率接近时，才会具有较高的发电效率。但实际环境振动经常是非周期、低频和宽频激励，这使系统很难工作在共振状态。而双稳态压电系统在周期或随机激励下，可在低宽频激励下产生大幅运动，是最具可能实现利用振动进行发电的结构之一，因此成为近两年国际研究的一个热点。
本文利用数值计算、解析分析和实验的方法研究了双稳态压电发电系统的响应特性及影响其产生大幅运动的因素，对双稳态压电发电系统的设计及应用具有重要意义。主要研究工作和成果如下：
1、全面总结了压电发电的发展、主要研究方法、实现宽频响应的方式，着重介绍了双稳态压电发电系统的研究现状，简单介绍了本文所用非线性振动分析方法及用于研究混沌运动的Melnikov方法。
2、建立了两永久磁铁之间的磁力模型和简谐激励下双稳态压电悬臂梁发电系统的一阶模型。通过数值计算分析了简谐激励下双稳态运动的频率响应范围，指出系统在较低的激励频率和幅值下能发生大幅运动。利用增量谐波平衡法(IHB)对双稳态压电发电系统的发电性能进行了研究，发现双稳态压电发电系统的电压幅值解存在跳跃现象；当非线性系数增加时，幅频曲线向右偏移，大幅运动的响应频带变宽，最大幅值变大；存在匹配电阻使得系统的发电功率最大。与线性系统相对比，在低频激励下，双稳态压电发电系统具有更高的发电效率。
3、分别用一阶和二阶Melnikov函数计算了双稳态压电发电系统的混沌阈值，分析了混沌阈值表达式中激励频率和阻抗对混沌阈值的影响，得到对混沌阈值影响最大的激励频率和电阻值。结果表明，二阶Melnikov函数的计算精度比一阶Melnikov函数高44%，最后给出了不同激励和电阻参数下系统产生大幅运动的吸引域。
4、用数值方法研究了随机激励下噪声强度、负载阻抗、初始值对系统响应的影响。发现噪声强度的增加会减少双稳态系统两个稳定平衡点之间的跳跃时间；与简谐激励下双稳态压电发电系统的响应相似，随机激励下系统也存在匹配阻抗使得输出功率最大；通过给出大幅运动的吸引域说明初始值会影响系统的运动状态，从而影响输出功率，同时也证明负载阻抗值会影响系统的运动。
5、设计了双稳态压电悬臂梁发电装置，通过实验分析了在简谐激励和随机激励下系统的响应特性。在简谐激励下，分析了激励频率、激励幅值、磁铁距离、负载阻抗和初始值对系统响应的影响，证明了双稳态压电发电系统适于低频激励；激励幅值越大，系统具有越高的能量逃离势阱产生大幅运动，并且系统的输出电压也越高，响应频带越宽；通过对比双稳态系统和线性系统的幅频曲线，得到在低频下双稳态系统的幅值比线性系统高；存在匹配负载阻抗使得系统的输出功率最大；初始值可以将系统从小幅运动吸引到大幅运动状态上。在随机激励下，分析了激励强度和负载阻抗对系统响应的影响，实验结果与理论计算相吻合。
通过上述工作，较为全面地获得了双稳态压电发电系统的动力学和发电特性，深入把握了主要因素对系统产生大幅周期运动和混沌运动的影响规律。由于从物理参数建立了磁力模型和机电耦合动力学模型，本文结论可具体指导双稳态压电发电结构的设计和优化。
Low-power requirements of microelectromechanical system (MEMS) enableenergy extraction from ambient vibrations. One of the common power generationstructures is the piezoelectric power generator. There are many researches on linearpiezoelectric power generation systems which have high output efficiency atresonance excitation. But the ambient vibration is generally non-periodic, random andbroandband, it is hard for linear system to work at resonance state. While bistablepiezeoelectric power generators can produce large amplitude motion under periodicand random excitation, which are most possible to realize power generation usingambient vibration. So it becomes a research focus in recent two years.
The response characteristics and influencing factors of large amplitude motion ofbistable piezoelectric power generators are studied by numerical calculation,analytical analysis and experiment, it has important significance for designing andapplication of the bistable piezoelectric power generation system. The main researchwork and results in the thesis are as follow:
1. The main research methods of piezoelectric power generation system, themethod of realizing resonance on a wide frequency and the research status of bistablepiezoelectric power generation system are summarized. Besides, the general analyticmethods of nonlinear system and Melnikov method are introduced.
2. A magnetic force model and a lumped parameter model of piezoelectriccantilever power generation system are established. The response characteristics ofthe system are studied numerically. The results show that the frequency response has abroadband and the beam undergoes large-amplitude motion at low frequency and lowamplitude of the excitation. By the Incremental Harmonic Balance method(IHB), theelectrical performance of the bistable piezoelectric power generation system isanalyzed. It is shown that the jump phenomenon is present on theamplitude-frequency response curve, and with magnetization increasing, the curveoffsets to the right, the frequency band is broadened, and the value of maximumamplitude is increased. In addition, the generation power reaches the maximum byproper selections of resistance. Comparison with the linear piezoelectric powergeneration system shows the bistable piezoelectric power generation system hashigher power under a lower broadband frequency.
3. The chaos threshold of the bistable piezoelectric power generation system iscalculated by the first and second order Melnikov function respectively. The effects ofexciting frequency and impedance item in chaos threshold value expression areanalyzed, and the excitation frequency and resistance values are found which have themost influence on chaos threshold value. The precision of the second order Melnikovfunction is44%more than the first order. The attraction of the large amplitude motionunder different excitation and resistance parameters is given.
4. The effect of noise intensity, load impedance and initial value on the systemresponse under random excitation are studied numerically. The time interval of thejump between two equilibrium points is reduced as the noise intensity increasing.There is a matching impedance making the generation power maximum under randomexcitation being same as harmonic excitation. The initial value and impedance willinfluence motion of the system, being proved by drawing the attraction domain oflarge amplitude motion.
5. A bistable piezoelectric cantilever power generation device is designed andmanufactured. The response of the bistable system under harmonic excitation andrandom excitation are measured respectively. Under harmonic excitation, thefollowing conclusions are proved by measuring the responses under different excitingfrequency, excitation amplitude, load impedance, magnet distance and initial value.firstly, the bistable piezoelectric cantilever power generation system is suitable forworking at low frequency; Secondly, the system has high energy escaped frompotential well when the excitation amplitude increases, the output voltage is higher, aswell as the frequency range is wider; Thirdly, under a low excitation frequency, thebistable system has higher amplitudes than that of a linear system; Finally, thegeneration power reaches the maximum by proper selection of load resistance and thesystem will be attracted to a large amplitude montion by changing the initial value.Under random excitation, the effects of excitation intensity and load impedance on theresponse of the bistable system are measured, and the experiment results are identicalwith the theoretical calculations.
By aforementioned work, the dynamic and generating characteristics of thebistable piezoelectric power generation system are overall obtained, and the influencelaw of main factors on large amplitude periodic motion and chaos are grasped.Because the model of piezoelectric cantilever power generation system is establishedbased on actual physical parameters, the conclusions of this paper can provide concrete guidance for designing and optimization of the bistable piezoelectric powergeneration structure.
本文利用数值计算、解析分析和实验的方法研究了双稳态压电发电系统的响应特性及影响其产生大幅运动的因素，对双稳态压电发电系统的设计及应用具有重要意义。主要研究工作和成果如下：
1、全面总结了压电发电的发展、主要研究方法、实现宽频响应的方式，着重介绍了双稳态压电发电系统的研究现状，简单介绍了本文所用非线性振动分析方法及用于研究混沌运动的Melnikov方法。
2、建立了两永久磁铁之间的磁力模型和简谐激励下双稳态压电悬臂梁发电系统的一阶模型。通过数值计算分析了简谐激励下双稳态运动的频率响应范围，指出系统在较低的激励频率和幅值下能发生大幅运动。利用增量谐波平衡法(IHB)对双稳态压电发电系统的发电性能进行了研究，发现双稳态压电发电系统的电压幅值解存在跳跃现象；当非线性系数增加时，幅频曲线向右偏移，大幅运动的响应频带变宽，最大幅值变大；存在匹配电阻使得系统的发电功率最大。与线性系统相对比，在低频激励下，双稳态压电发电系统具有更高的发电效率。
3、分别用一阶和二阶Melnikov函数计算了双稳态压电发电系统的混沌阈值，分析了混沌阈值表达式中激励频率和阻抗对混沌阈值的影响，得到对混沌阈值影响最大的激励频率和电阻值。结果表明，二阶Melnikov函数的计算精度比一阶Melnikov函数高44%，最后给出了不同激励和电阻参数下系统产生大幅运动的吸引域。
4、用数值方法研究了随机激励下噪声强度、负载阻抗、初始值对系统响应的影响。发现噪声强度的增加会减少双稳态系统两个稳定平衡点之间的跳跃时间；与简谐激励下双稳态压电发电系统的响应相似，随机激励下系统也存在匹配阻抗使得输出功率最大；通过给出大幅运动的吸引域说明初始值会影响系统的运动状态，从而影响输出功率，同时也证明负载阻抗值会影响系统的运动。
5、设计了双稳态压电悬臂梁发电装置，通过实验分析了在简谐激励和随机激励下系统的响应特性。在简谐激励下，分析了激励频率、激励幅值、磁铁距离、负载阻抗和初始值对系统响应的影响，证明了双稳态压电发电系统适于低频激励；激励幅值越大，系统具有越高的能量逃离势阱产生大幅运动，并且系统的输出电压也越高，响应频带越宽；通过对比双稳态系统和线性系统的幅频曲线，得到在低频下双稳态系统的幅值比线性系统高；存在匹配负载阻抗使得系统的输出功率最大；初始值可以将系统从小幅运动吸引到大幅运动状态上。在随机激励下，分析了激励强度和负载阻抗对系统响应的影响，实验结果与理论计算相吻合。
通过上述工作，较为全面地获得了双稳态压电发电系统的动力学和发电特性，深入把握了主要因素对系统产生大幅周期运动和混沌运动的影响规律。由于从物理参数建立了磁力模型和机电耦合动力学模型，本文结论可具体指导双稳态压电发电结构的设计和优化。
Low-power requirements of microelectromechanical system (MEMS) enableenergy extraction from ambient vibrations. One of the common power generationstructures is the piezoelectric power generator. There are many researches on linearpiezoelectric power generation systems which have high output efficiency atresonance excitation. But the ambient vibration is generally non-periodic, random andbroandband, it is hard for linear system to work at resonance state. While bistablepiezeoelectric power generators can produce large amplitude motion under periodicand random excitation, which are most possible to realize power generation usingambient vibration. So it becomes a research focus in recent two years.
The response characteristics and influencing factors of large amplitude motion ofbistable piezoelectric power generators are studied by numerical calculation,analytical analysis and experiment, it has important significance for designing andapplication of the bistable piezoelectric power generation system. The main researchwork and results in the thesis are as follow:
1. The main research methods of piezoelectric power generation system, themethod of realizing resonance on a wide frequency and the research status of bistablepiezoelectric power generation system are summarized. Besides, the general analyticmethods of nonlinear system and Melnikov method are introduced.
2. A magnetic force model and a lumped parameter model of piezoelectriccantilever power generation system are established. The response characteristics ofthe system are studied numerically. The results show that the frequency response has abroadband and the beam undergoes large-amplitude motion at low frequency and lowamplitude of the excitation. By the Incremental Harmonic Balance method(IHB), theelectrical performance of the bistable piezoelectric power generation system isanalyzed. It is shown that the jump phenomenon is present on theamplitude-frequency response curve, and with magnetization increasing, the curveoffsets to the right, the frequency band is broadened, and the value of maximumamplitude is increased. In addition, the generation power reaches the maximum byproper selections of resistance. Comparison with the linear piezoelectric powergeneration system shows the bistable piezoelectric power generation system hashigher power under a lower broadband frequency.
3. The chaos threshold of the bistable piezoelectric power generation system iscalculated by the first and second order Melnikov function respectively. The effects ofexciting frequency and impedance item in chaos threshold value expression areanalyzed, and the excitation frequency and resistance values are found which have themost influence on chaos threshold value. The precision of the second order Melnikovfunction is44%more than the first order. The attraction of the large amplitude motionunder different excitation and resistance parameters is given.
4. The effect of noise intensity, load impedance and initial value on the systemresponse under random excitation are studied numerically. The time interval of thejump between two equilibrium points is reduced as the noise intensity increasing.There is a matching impedance making the generation power maximum under randomexcitation being same as harmonic excitation. The initial value and impedance willinfluence motion of the system, being proved by drawing the attraction domain oflarge amplitude motion.
5. A bistable piezoelectric cantilever power generation device is designed andmanufactured. The response of the bistable system under harmonic excitation andrandom excitation are measured respectively. Under harmonic excitation, thefollowing conclusions are proved by measuring the responses under different excitingfrequency, excitation amplitude, load impedance, magnet distance and initial value.firstly, the bistable piezoelectric cantilever power generation system is suitable forworking at low frequency; Secondly, the system has high energy escaped frompotential well when the excitation amplitude increases, the output voltage is higher, aswell as the frequency range is wider; Thirdly, under a low excitation frequency, thebistable system has higher amplitudes than that of a linear system; Finally, thegeneration power reaches the maximum by proper selection of load resistance and thesystem will be attracted to a large amplitude montion by changing the initial value.Under random excitation, the effects of excitation intensity and load impedance on theresponse of the bistable system are measured, and the experiment results are identicalwith the theoretical calculations.
By aforementioned work, the dynamic and generating characteristics of thebistable piezoelectric power generation system are overall obtained, and the influencelaw of main factors on large amplitude periodic motion and chaos are grasped.Because the model of piezoelectric cantilever power generation system is establishedbased on actual physical parameters, the conclusions of this paper can provide concrete guidance for designing and optimization of the bistable piezoelectric powergeneration structure.
引文
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