Reduced-order models for microelectromechanical rectangular and circular plates incorporating the Casimir force

Abstract

We consider the von Kármán nonlinearity and the Casimir force to develop reduced-order models for prestressed clamped rectangular and circular electrostatically actuated microplates. Reduced-order models are derived by taking flexural vibration mode shapes as basis functions for the transverse displacement. The in-plane displacement vector is decomposed as the sum of displacements for irrotational and isochoric waves in a two-dimensional medium. Each of these two displacement vector fields satisfies an eigenvalue problem analogous to that of transverse vibrations of a linear elastic membrane. Basis functions for the transverse and the in-plane displacements are related by using the nonlinear equation governing the plate in-plane motion. The reduced-order model is derived from the equation yielding the transverse deflection of a point. For static deformations of a plate, the pull-in parameters are found by using the displacement iteration pull-in extraction method. Reduced-order models are also used to study linear vibrations about a predeformed configuration. It is found that 9 basis functions for a rectangular plate give a converged solution, while 3 basis functions give pull-in parameters with an error of at most 4%. For a circular plate, 3 basis functions give a converged solution while the pull-in parameters computed with 2 basis functions have an error of at most 3%. The value of the Casimir force at the onset of pull-in instability is used to compute device size that can be safely fabricated.

Thermal effect on the pull-in instability of functionally graded micro-beams subjected to electrical actuation

“…The relative difference percentage between the experimental results and our theoretical pull-in voltages increases as the beam length L increases and the percentage reaches 4.1% for L = 510 μm in linear analysis and only 2.6% in nonlinear analysis. This indicates that the geometric nonlinearity due to mid-plane stretching should be taken into account, particularly for long microbeams [3,37]. Figure 3 gives the Young's modulus f E , Poisson's ratio Figure 4 shows that the deflection of the micro-beam increases with the growth of the temperature until a certain value.…”

“…The intermolecular force is of the prominent importance in some micro-switches where the micro-beam may collapse onto the fixed ground plane due to the Casimir force only [37]. Once the stiffness of the micro-beam decrease to some extent, the micro-beam may collapse even in the absence of an applied voltage and temperature change.…”

The Influence of Couple Stress Components and Electrostatic Actuation on Free Vibration Characteristics of Thin Micro-Plates

“…es F ) can also be expressed as [8] where the dot super-script refers to partial derivative with respect to dimensionless time. Also, the time scale T, which is determined such that the coefficient of w w becomes unity, is defined as follows Figure 1.…”

“…It is to be mentioned here that utilizing the Galerkin weighted residual method for eliminating the spatial dependence is so usual in the creation of ROMs [3]. In this way, it is proved that employing linear mode-shapes of the system accelerates the convergence of the procedure and significantly reduces the computational costs [2,8,9].…”

Investigation of the nonlinear static and dynamic behaviour of rectangular microplates under electrostatic actuation

“…FEM based software tools are accurate but computationally expensive, especially when it comes to study the nonlinear dynamic behavior. On the other hand Reduced Order Models (ROM) based on the Galerkin approach have got popularity during the last decades because of their accuracy and low computational cost [6][7][8][9][10]. They have the capability to reveal the effect of different design parameters very conveniently.…”