考虑材料非线性时压电发电悬臂梁的主共振响应分析

Primary resonance of cantilevered piezoelectric energy harvesters considering nonlinearities of piezoelectric material

针对单晶压电悬臂梁发电系统进行了动力学建模及主共振响应分析。首先在考虑压电材料非线性的情况下,利用广义Hamilton原理、Rayleigh-Ritz法、Euler-Bernoulli梁理论等建立了单晶压电悬臂梁发电结构的机电耦合模型;其次利用多尺度法对系统进行了主共振二次近似响应分析,得到了谐振频率附近解的特性与系统参数的关系,揭示了压电材料非线性、外激励参数及负载电阻对系统响应的影响规律;最后通过数值计算验证了解析解的正确性。结论表明,压电材料的非线性特性会导致近似解的共振峰向左偏移,呈现软特性的非线性特征;当激励频率变化时,系统响应存在多解、跳跃等现象,多解区有2个稳定焦点和1个不稳定鞍点;主共振解的真正实现取决于系统的稳定性条件及初始条件的选取;系统存在最佳匹配负载电阻范围,对应系统的发电功率较大。

Dynamic modeling and primary resonance response analysis of cantilevered piezoelectric energy harvesters were presented.Firstly,the electromechanical coupling model of a cantilevered piezoelectric harvester was established considering nonlinearities of piezoelectric material using Hamilton principle,Rayleigh-Ritz method,Euler-Bernoulli beam theory and constant electrical field across the piezoelectric element.Then,the second order approximate solution to the primary resonance response of the system was investigated by using the multi-scale method.The influences of piezoelectric material nonlinearity and external excitation on the system response,and those of load resistance on the output power of the system were analyzed.By exploring the nonlinear characteristics of the response of the system near the resonant frequencies,the nature of the multi-solution and jump phenomena in the resonance region were revealed.It was shown that the realization of primary resonance depends on the stability conditions and initial conditions of the system;in addition,the generation power of the system reaches the maximum by properly selecting resistance.Finally,numerical integral results verified the correctness of the analysis.