Analytical solution for nonlinear free vibrations of viscoelastic microcantilevers covered with a piezoelectric layer

Abstract

Nonlinear vibrations of viscoelastic microcantilevers with a piezoelectric actuator layer on the top surface are investigated. In this work, the microcantilever follows a classical linear viscoelastic model, i.e., Kelvin–Voigt. In addition, it is assumed that the microcantilever complies with Euler–Bernoulli beam theory. The Hamilton principle is used to obtain the equations of motion for the microcantilever oscillations. Then, the Galerkin approximation is utilized for separation of time and displacement variables, thus the time function is obtained as a second order nonlinear ordinary differential equation with quadratic and cubic nonlinear terms. Nonlinearities appear in stiffness, inertia and damping terms. Using the method of multiple scales, the analytical relations for nonlinear natural frequency and amplitude of the vibration are derived. Using the obtained analytical relations, the effects of geometric factors and material properties on the free nonlinear behavior of this beam are investigated. The results are also verified by numerical analysis of the equations.