Dynamic Characteristics of Micro-Beams Considering the Effect of Flexible Supports

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.

Micromechanical Characterization of Polysilicon Films through On-Chip Tests

“…Furthermore, small imperfections, like, e.g., non-zero values of u and θ at no actuation (i.e., for ), could lead to a drift or bias in the solution obtained with the particle filter. It has been also assumed that the beam anchors behave ideally, even if several studies have shown that stress absorption and deformation at the anchors [ 52 , 53 ] can induce an additional compliance and cause discrepancies between the model and measured responses. The effects of these additional sources of uncertainty at the micron or even sub-micron scale will be considered in future investigations.…”

Mass sensor using mode localization in two weakly coupled MEMS cantilevers with different lengths: Design and experimental model validation

“…As the surface micromachining process does not allow ideal clamping conditions to be achieved, a torsional stiffnessk e is introduced in the model at the supposed fixed ends. Normally, a complete modelization of the real boundary condition should include a translational spring [29,30] but we assume that the small deflection at the fixed end can be neglected. On the long cantilever, we also add a discrete point mass located at a distancex m from the fixed end in order to consider the mass perturbation.…”

Parametrically Excited Electrostatic MEMS Cantilever Beam With Flexible Support

“…Further research of the dynamic effect of nonideal boundary conditions for various types of beams has been reported in Refs. [21][22][23].…”

“…With a lower restoring force at any given deflection, the beam can deflect to a larger amplitude under the same magnitude of electrostatic force. Unlike previous studies on nonideal supports [19][20][21][22][23], which justified dynamic behaviors by modeling the base with springs attached, we purposely added a flexible support to increase vibration amplitudes.…”