Performance Analysis Of Nonlinear Piezoelectric Energy Harvester With Bimorph Laminated Beam Under Parametric Excitation And Direct Excitation

With the development of micro-electromechanical systems in recent years,vibration energy harvesting systems have received extensive attention from researchers.Since most environmental vibrations occur randomly in a wide frequency range,compared with linear energy harvesting system,introducing nonlinearity into the design of the energy harvester can improve the efficiency of the harvesting system and broaden its working frequency band.In virtue of the low-frequency and multi-directional features of environmental vibration,this paper studies the nonlinear characteristics of the vibration energy harvester of the cantilever piezoelectric composite beam under the combined action of parametric excitation and direct excitation.The main research contents are as follows:(1)Based on the principle of thermodynamics and Legendre transformation,four forms of constitutive equations and corresponding thermodynamic functions of piezoelectric materials are derived in detail.Based on the Hamilton variational principle of three-dimensional piezoelectric elastic solids,it is demonstrated that the variational principle is equivalent to the differential equation of motion of piezoelectric elastic solids,the divergence equation satisfied by electric displacement and the boundary conditions of force and charge.The general form of Hamilton variational principle for the electromechanical coupling problem of piezoelectric composite beams is obtained.Using single modal method,the infinite degree of freedom mechanical system is approximated as the two degree of freedom system of beam deflection and output voltage.The optimal impedance under resonance and anti-resonance conditions is derived in detail,and it has been proved that under given conditions(2ζ_m/K~2＜＜1),the optimal power generated in the case of resonance and anti-resonance has the same magnitude.(2)The nonlinear dynamics of three-layer cantilever piezoelectric bimorph composite beam under the simultaneous action of parametric excitation and external excitation are studied.The beam is assumed to be Euler-Bernoulli beam model with axial inextensibility.Based on the generalized Hamilton principle,the nonlinear distributed parameter model of the cantilever piezoelectric composite beam energy collector is established,which includes geometric nonlinearity and inertial nonlinearity.Using Galerkin method and harmonic balance method,the analytical expressions of the displacement,output voltage and output power of the cantilever piezoelectric composite beam are presented when the first mode is dominant.The effects of damping,load resistance,electromechanical coupling coefficient and excitation amplitude on the frequency response curves,and the difference between the nonlinear lumped parameter model and the distributed parameter model in energy collection performance is compared.The results show that only in the case of parametric excitation,the energy harvesting system has an initial excitation threshold,below this threshold,no energy can be harvested.The damping coefficient and load resistance will affect the initial threshold.The energy conversion efficiency of energy harvesting system can be improved effectively under the combined action of parametric excitation and external excitation.(3)Based on Hamilton’s principle,considering geometric nonlinearity and inextensibility conditions of the beam,a differential equation of motion of a five-layer cantilever piezoelectric bimorph composite beam energy harvester under direct and parametric excitation is established.The analytical solutions of the displacement,output voltage and output power of the energy harvester are obtained by Galerkin method and harmonic balance method.The stability condition of the solution of the nonlinear equation is presented by introducing the time-varying perturbation.The structure-performance relationship of the piezoelectric energy harvester is comprehensively analyzed,and the influences of passive layer configuration,thickness ratio between passive layer and active layer and material characteristics on the performance of the piezoelectric energy harvesting system are studied.In the case of the constant thickness of five-layer piezoelectric bimorph composite beam,the research results show that the energy harvesting efficiency can be improved effectively by choosing reasonable thickness ratio of passive layer to active layer,elastic modulus of passive layer,thickness ratio of passive layer and load resistance.(4)Based on Hamilton principle and the inextensible condition of beam,the distributed parameter differential motion equation of nonlinear piezoelectric energy harvester is established for five layer piezoelectric bimorph cantilever beam with tip mass under direct excitation and parametric excitation.The analytical expression of the approximate solution of the equation is obtained by Galerkin method and harmonic balance method.Under the conditions of parametric excitation,direct excitation and combined excitation,the effects of beam’s tip mass,load resistance and base thickness ratio on the performance of nonlinear piezoelectric energy harvester are studied.The results show that the initial threshold and resonance frequency of the parametric excitation system decrease with the increase of the beam end mass.By reasonably selecting the beam’s tip mass,the efficiency of the vibration energy acquisition device can be effectively improved and its resonance frequency can be reduced.(5)The dynamic equation of nonlinear piezoelectric energy harvesting system is solved by the harmonic balance method of fundamental harmonic and super-harmonic components,and the influence of super-harmonic effect on energy harvesting system performance is explored.The results of numerical simulation analysis show that because the nonlinear vibration energy harvesting system has relatively low linear stiffness or low linear natural frequency,even slight excitation may cause obvious super-harmonic effects.If the super-harmonic effect is not considered,a large deviation may occur between the predicted system response and the actual system response.The research results indicate that effective nonlinear vibration energy harvesting system design requires a clear understanding the characteristics of the design parameters and input excitation so as to use super-harmonic effects to improve the efficiency of vibration energy harvesting system.