Effect of thermoelastic damping on nonlinear vibrations of a microbeam under electrostatic actuation using nonlinear normal modes

A Galerkin-based nonlinear normal mode approach based on the concept of invariant manifolds is adopted to analyze the effect of thermoelastic damping on the nonlinear vibrations of a microbeam with mid-plane stretching and electrostatic actuation. The derivation of the mathematical model starts with two coupled equations: a general Euler-Bernoulli equation for microbeam vibrations, which includes thermal moment and thermal force terms due to the thermoelastic damping effect; and a thermal equation describing the generation of the thermal moment due to bending vibrations. Given the available information about the material properties and geometry of conventional microbeams, and a physical reasoning based on it about the time constant of the thermal equation, the two governing equations are integrated into a single nonlinear partial differential equation. Galerkin discretization is used to transform the governing equation into a system of coupled nonlinear ordinary differential equations that are uncoupled at linear order. The modal analysis is then followed by finding an invariant manifold leading to a nonlinear ordinary differential equation that governs the vibration behavior of the system. The results of the case study showed that the amount of thermoelastic damping is dependent on the excitation voltage, and the damping increases with increasing voltage. Furthermore, in the presence of thermoelastic damping, by applying the step voltage and starting the vibrations, the frequency increases over time until it reaches a certain limit. With a higher excitation voltage, the frequency increases at a higher rate, but to a lower final limit. In an undamped system, the frequency of oscillations is constant over time, but a higher excitation voltage causes lower frequency vibrations. The results of pull-in analysis showed that thermoelastic damping has no significant effect on static or dynamic pull-in voltage.