On Applicability of Mode Truncation Method for Damped Micro Beam

Abstract

In this work the dynamics behaviors of microbeam under the influence of various physical parameters such as damping, nonlinearity of beam equation, alternating current (AC) and direct current (DC) voltage is carried out. The small viscous damping and bending stiffness are taken into consideration. Furthermore, it is supposed that current load is composed of direct current (DC) voltage and a small harmonic alternating current (AC) voltage. Mathematically the oscillations of simply supported microbeam are formulated as nonlinear partial differential equations (PDEs) with Dirichlet type homogeneous boundary conditions. The governing PDEs are discretized via Fourier expansion method. The formal approximations of the solutions of partial differential equation for nanobeam are obtained by means of two time-scale perturbation method. It turned out that certain resonances occur only when the natural frequency of microbeam is zero or type of frequency of the harmonic (AC) voltage, that is Ω ≈ 0. The stability of the microbeam system is investigated in terms of amplitude response of the system for non-resonant as well as resonant cases. In non-resonant case, the amplitude response is obtained under the influence of damping, nonlinear parameter of bending stiffness and DC and AC voltage parameters. In zero-resonant case, that is, when the ‘exciting frequency is around the zero natural frequency of micro beam, the deflection of the system is examined under the influence of detuning parameter together with the aforementioned physical parameters. The amplitude is found out to be damped as time increases. Whereas the stability of system for resonant case is examined for two-mode qualitatively with the variation of damping, nonlinear parameter of bending stiffness and DC voltage parameters