Nonlinear Dyn DOI 10.1007/s11071-014-1275-7 ORIGINAL PAPER Effects of the van der Waals force, squeeze-film damping, and contact bounce on the dynamics of electrostatic microcantilevers before and after pull-in Mansour Abtahi · Gholamreza Vossoughi · Ali Meghdari Received: 21 April 2013 / Accepted: 27 January 2014 © Springer Science+Business Media Dordrecht 2014 Abstract The operational range of microcantilever beams under electrostatic force can be extended beyond pull-in in the presence of an intermediate dielectric layer. In this paper, a systematic method for deriving dynamic equation of microcantilevers under electro- static force is presented. This model covers the behav- ior of the microcantilevers before and after the pull-in including the effects of van der Waals force, squeeze- film damping, and contact bounce. First, a polynomial approximate shape function with a time-dependent variable for each configuration is defined. Using Hamil- ton’s principle, dynamic equations of microcantilever in all configurations have been derived. Comparison between modeling results and previous experimental data that have been used for validation of the model shows a good agreement. Keywords Microcantilever beams · Electrostatic force · Dynamic modeling · Floating · Pinned and flat configurations M. Abtahi · G. Vossoughi (B ) · A. Meghdari Center of Excellence in Design, Robotics & Automation (CEDRA), Department of Mechanical Engineering, Sharif University of Technology, Azadi Ave, 1458889694 Tehran, Iran e-mail: vossough@sharif.ir 1 Introduction The electrostatic actuation has appeared as one of the most common means of actuations for microsystems due to its relative simplicity in implementation [1]. Electrostatically actuated beams or plates are used in a wide range of applications such as microswitches, capacitive pressure sensors, accelerometers, micromo- tors, resonant sensors, and many others [1, 2]. Model- ing and simulation of electrostatically actuated devices play an important role in the design phase for predicting device characteristics. The behavior of electrostatic MEMS parallel plate actuators before pull-in is studied extensively in the literature [3–7]. Many MEMS devices operate beyond the pull-in, e.g., capacitive switches [8, 9], zipper var- actors [10, 11], and tunable coplanar waveguide (CPW) resonators [12, 13]. Little research has been performed to predict the behavior of the electrostatically actuated microbeams beyond pull-in. Gilbert et al. [14, 15] used CoSolve-EM, a coupled solver for 3D quasi-static electro-mechanical prob- lems, to simulate MEMS actuators beyond pull-in. To find contact length and stiction force after pull-in, some approaches use static analysis to approximate beam deflection by iterative methods [16–19]. Gorthi et al. [20] depicted three possible static configurations of the beam over the total operational voltage range: floating, pinned, and flat configurations (defined and shown in the next section). Using numerical methods, static gov- erning equation has been solved numerically in three 123