Application of hybrid differential transformation/finite difference method to nonlinear analysis of micro fixed-fixed beam

Abstract

Analyzing the dynamic response of electrostatic devices is problematic due to the complexity of the interactions between the electrostatic coupling effect, the fringing field effect and the nonlinear electrostatic force. To resolve this problem, this study presents an efficient computational scheme in which the nonlinear governing equation of the electrostatic device is obtained in accordance with Hamilton's principle and is then solved using a hybrid differential transformation/finite difference method. The feasibility of the proposed approach is demonstrated by modeling the dynamic responses of two micro fixedfixed beams with lengths of 250 and 350 lm, respectively. The numerical results show that the pull-in voltage reduces as the beam length increases due to a loss in the structural rigidity. Furthermore, it is shown that the present results for the pull-in voltage deviate by no more than 0.75% from those derived in the literature using a variety of different schemes. Overall, the results presented in this study demonstrate that the proposed hybrid method represents a computationally efficient and precise means of obtaining detailed insights into the nonlinear dynamic behavior of micro fixed-fixed beams and similar micro-electromechanical systems (MEMS)-based devices.

Nonlinear Analysis of Pull-In Voltage of Twin Micro-Cantilever Beams

“…An infinitesimal transformation is applied as an operator on itself and, both functions ξ and η defined in (9), must satisfy the following equality [11]: η xx + (2η xy − ξ xx )ý +(η yy − 2ξ xy )ý2 − ξ yy ý3 + (η y − 2ξ x − 3ξ y ý)ω = ω x + ηω y + ((η x − ξ x )ý− ξ y ý2) ω ý (12) Subsequently, ξ and η can be calculated by decomposed [3][4][5][6] into a system of partial differential equations. The following Lie symmetries which including rotation, translation and scaling was considered to calculate ξ and η. ξ = C 1 + C 2 x + C 3 y η = C 4 + C 5 x + C 6 y (13)…”

“…There are several methods to obtain the governing differential equation. The common methods are using either Hamiltonian [2] or energy [3] methods. Many researchers used several different approaches to linearize and simplify the governing equations given their high non-linearity.…”

Surface effect on dynamic characteristics of the electrostatically nano-beam actuator

“…Considering the first-order fringing field correction of the electrostatic force per unit length of the beam is [25] F elec = ε 0 wV 2 2(g − z) 2 …”

Numerical analysis of entropy generation in mixed convection flow with viscous dissipation effects in vertical channel

“…Kuo [12] applied differential transformation theory to investigate the velocity and temperature distributions associated with a free convection boundary-layer flow over a vertical plate. Chen et al [21] demonstrated that the hybrid differential transformation and finite difference method provide a precise and computationallyefficient means of analyzing the nonlinear dynamic behavior of fixedfixed micro-beams. The same group also used the hybrid method to analyze the nonlinear dynamic response of an electrostaticallyactuated micro circular plate subject to the effects of residual stress and a uniform hydrostatic pressure acting on the upper surface [22].…”