Dynamic modeling of interactions between fields and matter in MEMS devices

Abstract

In this work, we propose an innovative and general mathematical modeling framework with Lagrangian functions of multi-coupled energy domains for MEMS devices in the material frame. Due to the advantage of material descriptions of continua and the uniformity of Lagrangians for different physical domains, this framework simplifies the numerical analyses of MEMS devices. IntroductionA complete physical model for a MEMS device could lead us into all areas of continuum physics [1][2][3][4][5][6][7][8]. In the present work, we limit ourselves to the mechanical and electromagnetic energy domains. The physical model for a MEMS device is consequently regarded as consisting of two submodels. The first one is a dynamic mechanical submodel describing the mechanics of the elastic deformation structure. The second one is an electromagnetic submodel describing the dynamics of the electromagnetic influence within a MEMS device. These two submodels are mutually coupled.Following the Lagrangian dynamics approach [9-17], we present a new framework for modeling a MEMS device based on the theories of continuum mechanics [18][19][20] and electrodynamics [21][22][23][24][25][26]. In this framework, the deformation field and electromagnetic field are rendered down to unified and consistent representation forms. By the transformations from spatial forms into material forms, fully coupled dynamic equations are rigorously derived. This framework is comprehensive and extensible. It can be extended further to track other energy domains and generate a complete physical model for multi-physical domain MEMS devices. This formalism is not only powerful for theoretical reasoning but also for numerical processing. Because of the uniformity of the approach for different physical domains, a general discretization framework can be set up. Lagrange formalismThe Lagrange formalism determines the model for a dynamic system, the equations of motion, through a variation principle applied to the proposed Lagrangian which stands de facto for the energy functions of the system. This approach exploits the use of so-called generalized coordinates, i.e., any set of variables sufficient in number to locate unambiguously the configuration of the dynamic system. Generalized coordinates are not alternatives to reference coordinate systems, but are descriptions of the kinematic configuration of the dynamic system based on the system degrees of freedom.For the continuum model, which has an infinite number of degrees of freedom, the generalized coordinates constitute fields. By defining the Lagrangian in terms of these fields as a functional of the fields, the Lagrange formalism allows the variational principle to be employed in a variety of physical energy domains.In the Lagrangian formulation, the Lagrangian L of a continuous system in the Euclidean space R 3 is a volume integral over the system domain D:where r z is the gradient operator at z 2 R 3 and the Lagrangian density Lðn; _ n; r z n; z; tÞ depends on the ordered array of fields n 2 R n , its generalized vel...

Design and Analysis of MEMS-Based Actuator

“…4 and Fig. 5 respectively; The dynamic equation is built with Lagrange formalism, the Lagrange formalism determines the model for a dynamic system, the equations of motion, through a variation principle applied to the proposed Lagrange which stands de facto for the energy functions of the system [6]. The kinematics mathematics modeling of the dynamic system is described by using so-called generalized coordinates.…”