Parametric Thermally Induced Vibration of an Electrostatically Deflected FGM Micro-Beam

Abstract

The present paper deals with the study of nonlinear vibration of a functionally graded cantilever micro-beam imposed on a bias DC voltage and superimposed on a sinusoidal heat source. The governing equation of motion is derived extremizing the Lagrange’s equation and Hamilton’s principal under the assumption of Euler–Bernoulli beam theory. The thermo-elastic equation is obtained utilizing the first law of thermodynamics under the assumption of the classical Fourier heat conduction model. Due to the displacement dependency of the electrostatic force and time variability of the heat source, the governing differential equations of the system are nonlinear implicitly parametrically electro-thermo-elastic coupled equations. To evaluate the dynamic response of the micro-beam, the coupled equations are discretized applying a Galerkin-based reduced order model and then integrated numerically by the Runge–Kutta method. By solving the equations, the stable and unstable regions at different bias DC voltages are identified. By picking some special points from these regions and depicting the time history and phase portrait diagrams, their behaviors are investigated in detail. In addition to the classical dynamic pull-in, in which a homoclinic orbit separates stable periodic orbits from the unbounded solutions, a new kind of dynamic pull-in is presented, which separates unstable solutions, due to parametric resonance response, from unbounded rapidly growing solutions owing to the existence of saddle and singular fixed points in the system.