Mathematical Modeling And Theoretical Analysis Of Cantilever Beam Flexoelectric Energy Harvesters

With the rapid development of nanotechnology and smart materials,flexoelectricity,which has a coupling effect between the polarization and the strain gradient,attracts increasing attention in the domain of micro electron components.In this paper,the cantilever beam flexoelectric energy harvester is analyzed and discussed systematically.Firstly,based on the electric Gibbs free energy density function of piezoelectric materials by considering flexoelectric effect,the cantilever beam flexoelectric energy harvester with tip mass is studied.Euler-Bernoulli beam model and variational principle method of flexoelectric material are used to get the dynamic electromechanical coupling govern-equation and corresponding electromechanical boundary conditions of this system under the distributed parameter model.Then the govern-equation and corresponding electromechanical boundary conditions of the flexoelectric cantilever beam with tip mass under the open-circuit condition are derived.The natural vibration characteristic equation under the open-circuit condition is obtained further by using the separation of variable method.The effects of flexoelectric coefficients,tip mass and the size of beam on natural frequency and the effective variation of natural frequency between short and open-circuit conditions are discussed.The results indicate that the increase of flexoelectric coefficient could increase the natural frequency of beams,and the increase of the tip mass could decrease the natural frequency of beams.The effective variation of natural frequency increases with decreasing the size of beam,and the value of that will approach a stable value at a critical thickness.Furthermore,this critical thickness is relate to flexoelectric coefficients,and the stable value is relate to the tip mass.Then the natural vibration characteristic equation,modal orthometric conditions and the normalized mode of vibration under the short-circuit condition are obtained by using the separation of variable method.Furthermore,the mode superposition method is used to get the analytical form of mechanical and electrical responses of this system under the extrinsic harmonic mechanical excitation.The relation between flexoelectric coefficients,size of structure,resistance,the frequency of extrinsic excitation and frequency response functions(FRFs)of electromechanical outputs are studied.The results show that the electromechanical coupling performance of flexoelectric energy harvesters will enhance in the small scale.In the meantime,when the beam thickness decreases,the effect of flexoelectricity on output power density will increase firstly and then decreases on the contrary.In addition,when the electrical condition of the system is closer to the short-circuit condition,take the short-circuit resonant frequency as the extrinsic mechanical excitation could get more ideal responses.When the electrical condition of the system is closer to the open-circuit condition,there is an opposite phenomenon.In this paper,the effects of the value of mass and size of tip mass on the output power density performance and the optimal resistance of the system are discussed further.The results indicate that the increase of the value of mass and decrease of the size of tip mass could not only increase the output power and decrease the resonant frequency of energy harvester,but also adjust the optimal resistance.The optimal resistance of the energy harvester system will increase with the increase of the value of mass and the decrease of size of the tip mass.In the meantime,when the value of mass of tip mass is not so large in some way,the size of tip mass can be neglected to simplify analysis.These results are conducive to understand the size effect of flexoelectricity of nano electron components,and could be a helpful theoretical guidance to the optimal design of flexoelectric energy harvesters in a way.