Sensors and Actuators A 190 (2013) 32–43 Contents lists available at SciVerse ScienceDirect Sensors and Actuators A: Physical jo u rn al hom epage: www.elsevier.com/locate/sna Analytical closed-form solutions for size-dependent static pull-in behavior in electrostatic micro-actuators via Fredholm integral equation Hossein Rokni a,∗ , Rudolf J. Seethaler b , Abbas S. Milani b , Shahrokh Hosseini-Hashemi c , Xian-Fang Li d a Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA b School of Engineering, University of British Columbia (Okanagan), Kelowna, BC V1V 1V7, Canada c School of Mechanical Engineering, Iran University of Science and Technology, Tehran 16848-13114, Iran d School of Civil Engineering, Central South University, Changsha 410075, China a r t i c l e i n f o Article history: Received 7 August 2011 Received in revised form 21 October 2012 Accepted 29 October 2012 Available online xxx Keywords: Micro-actuator Size effect Pull-in voltage Analytical solution a b s t r a c t In this paper, a novel method is proposed for the first time to obtain static pull-in voltages with fring- ing field effects in electrostatically actuated cantilever and clamped-clamped micro-beams where the mid-plane stretching and the residual axial load are taken into account for clamped-clamped boundary conditions. The non-classical Euler–Bernoulli beam model containing one material length scale parame- ter is adopted to effectively capture the size effect. In the solution procedure, the governing fourth-order differential equation of variable coefficients is converted into a Fredholm integral equation. By adopting the first natural mode of the cantilever and clamped-clamped micro-beams as a deflection shape func- tion, the resulting equation is solved for the static pull-in voltages. The accuracy of the present analytical closed-form solution is verified through comparing with the experimentally measured and numerical data conducted in the published works. From the experimental data available in the literature, the value of the material length scale parameter for the (poly)silicon is estimated to be in the order of magnitude of 10 -1 m. Then, the effect of the material length scale parameter on the pull-in voltages of the can- tilever and clamped-clamped micro-beams is investigated. The results indicate that the tensile residual stress can extend the validity range of the classical continuum mechanics to lower beam thicknesses. It is also found that microcantilever beams are more sensitive to the size effect than their corresponding clamped-clamped micro-beams. © 2012 Elsevier B.V. All rights reserved. 1. Introduction Owing to advantages of electrostatic micro-actuators, such as high fast response, favorable scaling property, low energy con- sumption, low cost, low driving power, large deflection capacity, relative ease of fabrication and others, they are widely used in microelectromechanical system (MEMS) [1]. MEMS are usually comprised of a conductive deformable body suspended above a rigid grounded body [2]. An applied direct current potential difference between the two bodies induces the Coulomb force that deflects the deformable body, and consequently changes the system capacitance. When an additional alternating current is applied to excite harmonic motions of the deformable electrode, resonant devices can be obtained. These devices are used in signal filtering, and chemical and mass sensing, see e.g., [3–6]. The applied direct current voltage has an upper limit beyond which the electrostatic force is not balanced by the elastic restoring force ∗ Corresponding author. E-mail address: Rokni@umich.edu (H. Rokni). in the deformable conductor, and the MEMS eventually collapses. This phenomenon, called pull-in instability, has been observed experimentally [7,8]. The critical voltage associated with this insta- bility is called the pull-in voltage. Determination of the pull-in voltage is critical in the design of MEMS devices. In micro-mirrors [9] and micro-resonators [10], the designer avoids this instability to achieve stable motions; while in switching applications [11], the designer exploits this effect to optimize device’s performance. In fact, as a result of the nonlinearity, stable force balance and deflec- tion can only be achieved to one third (in lumped modeling) of the initial gap distance [12]. Accurate determination of pull-in volt- age is very challenging in virtue of the electromechanical coupling effect and the nonlinearity of electrostatic force. Effects such as the fringing field, the mid-plane stretching and the residual axial load further complicate the modeling. Numerous numerical, analytical and experimental investiga- tions have been conducted on the pull-in instability behavior of micro-beams subjected to electrostatic loadings. Among many oth- ers, Osterberg and Senturia [13] reported closed-form empirical relations to estimate the static pull-in voltage of micro-beams. Pamidighantam et al. [14] derived a closed-form expression of 0924-4247/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sna.2012.10.035