ENOC 2008, Saint Petersburg, Russia, June, 30–July, 4 2008 REDUCED-ORDER MODELING OF ELECTROSTATICALLY-ACTUATED MICRO-BEAMS F. Durieu, O. Br¨ uls, V. Rochus, G. S´ erandour, J.-C. Golinval Department of Aerospace and Mechanical Engineering University of Li` ege Belgium Corresponding author: o.bruls@ulg.ac.be Abstract This paper addresses the compact modeling of MEMS devices with nonlinear electromechanical forces. Ide- ally, the reduced-order model should only involve one or two generalized coordinates. We show that a pro- jection method based on the first eigenmode of the lin- earized unactuated structure is a good choice in terms of simplicity and accuracy. To take into account the influence of distributed electrostatic forces, it is neces- sary to approximate the nonlinear force in terms of the reduced coordinate. It is shown that a Pad´ e approxima- tion of order 2 is very attractive in terms of accuracy and numerical efficiency. Key words Model reduction, electromechanical coupling, nonlin- ear dynamics, MEMS. 1 Introduction MEMS are very small devices in which electric as well as mechanical, thermal and fluid phenomena ap- pear and interact. Because of their microscopic scale, strong coupling effects arise between the different physical fields, and some forces, which were negli- gible at macroscopic scale, have to be taken into ac- count. In order to accurately design such micro-electro- mechanical systems, it is important to handle the cou- pling between the electric and the mechanical fields. This work concerns the development of reduced-order models for MEMS devices actuated by nonlinear elec- tromechanical forces. The concept of model order re- duction in the framework of nonlinear systems remains an important challenge in the scientific community. Di- mensionality reduction covers a wide number of appli- cations and constitutes today a sensible subject of re- search. Many authors deal with reduction of MEMS mod- els in the literature. In [Zavracky et al., 1997], the static behavior of a clamped micro-beam is modelled using a single stiffness element to which the electro- static force is applied. This type of model exhibits large errors regarding to the static pull-in value. In [Swart et al., 1998], an automatic tool called ”AutoMM is developed and coupled to the finite element software MEMCAD [Senturia et al., 1992] to generate a reduced model of a three-dimensional MEMS structure. Pa- rameters such as the electrostatic stiffness or the me- chanical stiffness are approximated by polynomials in function of the state variables for different responses corresponding to different excitations of the structure. In [Younis et al., 2003] and [Gabbay, 1998], linear normal modes are used to represent the dynamic be- havior of the structure submitted to non-linear forces. In [Nayfeh et al., 2005], different reduction methods based on the reduction of nodes of the discretized sys- tem and on the reduction of domains are presented. In [Gabbay and Senturia, 1998; Gabbay and Senturia, 2000], a methodology is described to generate mod- els on the basis of shape-functions deduced from a 3D modeling. This methodology to the case of geometric nonlinearities in [Mehner et al., 2000]. In [Hung and Senturia, 1999; Hung et al., 1997; Liang et al., 2002; Lin et al., 2003], the authors study the efficiency of re- duction methods based on the basis of modes obtained from a proper orthogonal decomposition (POD) of sim- ulation data generated by a full 3D model. In [Chen et al., 2004], an Arnoldi type method is presented, which is based on the projection of the system in Krylov sub- spaces. In [Lienemann et al., 2006], the authors em- phasize the necessity of developing compact model- ing of MEMS and specially micro-beams submitted to electrostatic forces. In [Del Tin, 2007], the nonlinear electromechanical forces are represented by equivalent lumped forces applied to a certain number of retained modes in the reduced-order model. The objective of the present work is to obtain a sim- plified model based on a minimal number of degrees- of-freedom. Constructed from an initial finite element model, the reduced model should involve only one or two degrees-of-freedom, and still be able to represent