The stationary response of piezoelectric cantilever beam model excited by colored noise

Jie Liu,Gen Ge

doi:10.21595/jve.2021.22052

Abstract

This article provides a method to predict the steady-state response of a piezoelectric cantilever under the excitation of Gaussian colored noise which is a better approximation to the ambient random excitation than the white noise. In this paper, a cantilever energy harvester model, which is fully-paved with piezoelectric ceramic and excited by Gaussian colored noise, is built by the Lagrange’s equations and is nondimensionalized. Then the dynamic equation and piezoelectric equation are transformed into a pair of stochastic differential equations about transient equivalent amplitude and transient phase. After that, the stochastic averaging method is applied to simplify the stochastic differential equations into an Itô type equation about the equivalent amplitude, the drift coefficient and the diffusion coefficient are obtained. This method provides another approach which is different from the stochastic linearization method and moment method. On this basis, some crucial outputs such as the steady-state probability density function (PDF) of the equivalent amplitude, the beam’s displacement, the transient output voltage, the joint probability density function of displacement and velocity as well as the mean square value of output voltage are all obtained. With different noise intensity values and delay factors, the Monte Carlo numerical simulation has verified the correctness of the theoretical prediction.

Highlights

Piezoelectric cantilever model which can transfer the ambient energy into electrical energy has attracted many researchers’ attention in recent decades
At the beginning of this century, Sndano et al [1] and Erturk et al [2] designed a type of mono-stable-state piezoelectric cantilever harvester and theoretically predicted the vibrational voltage output
Based on the minimum mean-square error criterion, a kind of equivalent linearization method was proposed by Wen-an Jiang et al [25] to investigate the output voltage of a piezoelectric harvester excited by white noise

Summary

Introduction

Piezoelectric cantilever model which can transfer the ambient energy into electrical energy has attracted many researchers’ attention in recent decades. Based on the minimum mean-square error criterion, a kind of equivalent linearization method was proposed by Wen-an Jiang et al [25] to investigate the output voltage of a piezoelectric harvester excited by white noise. These scholars have made great contribution to the model research about the white noise excitation. The detailed response characters of colored noise excited piezoelectric cantilever model, such as stationary probability density function (PDF) of displacement, PDF of voltage, joint PDF of displacement and velocity and mean square values of voltage, haven’t been studied yet. Parameters in Eq (15) and Eq (16) are as follows: ω denotes the fundamental frequency, α is the cubic nonlinearity coefficient, μ is the damping ratio, β and θ are electromechanical coupling coefficient in piezoelectric equation and that in dynamic equation, η(t) denotes the colored noise

Stochastic averaging

Stationary responses

Numerical simulation

Conclusions