Application of Higher Order Hamiltonian Approach to the Nonlinear Vibration of Micro Electro Mechanical Systems

Abstract

This paper implements the higher order Hamiltonian method to analyze an electrostatically actuated nonlinear micro beam-based micro electro mechanical oscillator. First, second and third approximate solutions are obtained, and the frequency responses of the system are compared with energy balance method solution and previously solved Variational Approach (VA) and exact solution. After driving the equation of motion based on the Euler-Bernoulli beam theory, Galerkin method has been used to simplify the nonlinear equation of motion. Higher order Hamiltonian approach has been used to solve the problem and introduce a design strategy. Phase plane diagram of electrostatically actuated micro beam has plotted to show the stability of presented nonlinear system and natural frequencies are calculated to use for resonator design. According to the numerical results, the second approximate is more acceptable and results show that one could obtain a predesign strategy by prediction of effects of mechanical properties and electrical coefficients on the stability and free vibration of common electrostatically actuated micro beam.

Free vibration for a micro resonant pressure sensor with cross-type resonator

“…The solving method of the dynamic equation of the micro-resonators was developed and the calculated accuracy of the dynamic characteristic was discussed in depth. 22–23…”

“…The solving method of the dynamic equation of the micro-resonators was developed and the calculated accuracy of the dynamic characteristic was discussed in depth. [22][23] To sum up, a lot of research work has been done on micro-resonant pressure sensors. Dynamic characteristic is the key to ensure the working performance of resonance sensor, so it has become the most important research work of micro resonance pressure sensor.…”

A modified harmonic balance method for solving forced vibration problems with strong nonlinearity

“…Then Yildirim et al 37 utilized the same method to solve nonlinear oscillators with rational and irrational elastic terms. Sadeghzadeh and Kabiri 38 presented a higher-order Hamiltonian approach to the nonlinear vibration of micro electro-mechanical systems without any damping effect. From the above literature, it is seen that in most cases, variational and Hamiltonian principles are applied for nonlinear free vibration problems.…”

Extension of the Hamiltonian approach with general initial conditions

“…Xu and He (2010) used the Hamiltonian approach to determine the limit cycle motion for strongly nonlinear oscillators. The method was applied for solving some practical problems, such as nonlinear oscillations of a punctual charge in the electric field of a charged ring (Yildirim et al, 2011a), nonlinear vibrations of micro electro mechanical systems (Sadeghzadeh and Kabiri, 2016), nonlinear oscillations in an engineering structure (Akbarzade and Khan, 2012), among others. The Hamiltonian approach has also been applied to improve the accuracy of the solution for nonlinear oscillators, such as higher order approximations (Durmaz et al, 2010;Yildirim et al, 2011c), multiple coupled nonlinear oscillators (Durmaz et al, 2012) and multiple-parameter ones , among others.…”