Abstract

Normally, the boundaries are assumed to allow small deflections and moments for MEMS beams with flexible supports. The non-ideal boundary conditions have a significant effect on the qualitative dynamical behavior. In this paper, by employing the principle of energy equivalence, rigorous theoretical solutions of the tangential and rotational equivalent stiffness are derived based on the Boussinesq's and Cerruti's displacement equations. The non-dimensional differential partial equation of the motion, as well as coupled boundary conditions, are solved analytically using the method of multiple time scales. The closed-form solution provides a direct insight into the relationship between the boundary conditions and vibration characteristics of the dynamic system, in which resonance frequencies increase with the nonlinear mechanical spring effect but decrease with the effect of flexible supports. The obtained results of frequencies and mode shapes are compared with the cases of ideal boundary conditions, and the differences between them are contrasted on frequency response curves. The influences of the support material property on the equivalent stiffness and resonance frequency shift are also discussed. It is demonstrated that the proposed model with the flexible supports boundary conditions has significant effect on the rigorous quantitative dynamical analysis of the MEMS beams. Moreover, the proposed analytical solutions are in good agreement with those obtained from finite element analyses.

Highlights

  • Micro-beams [1,2,3] are widely used as the key components of diverse sensing and actuation systems [4,5,6]

  • It is clear that the proposed analytical solutions are in good agreement with the finite element results

  • We have quantitatively studied the effect of the flexible supports boundary conditions on the dynamic characteristics of MEMS beams

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Summary

Introduction

Micro-beams [1,2,3] are widely used as the key components of diverse sensing and actuation systems [4,5,6]. Alkharabsheh and Younis [27] demonstrated that non-ideal boundary conditions can have significant effect on the qualitative static or dynamic behavior of MEMS beams, which includes lowering the natural frequencies from the expected range of operation and causing unpredictable dynamic pull-in. In this regard, support boundary characterization is important in the applications such as flexible optical waveguides [28] and AFM cantilever probes [29]. The closed-form solution derived by the method of multiple timescales provides direct insight into the relationship between the boundary conditions and vibration characteristics of the system

The Tangential Equivalent Stiffness
The Rotational Equivalent Stiffness
The Comparison and Validation
Dynamical Model and Analysis
The Resonance Frequency and Mode Shape Analysis
Frequency-Response Analysis
Results and Discussion
Conclusions
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