Nonlinear Dynamics of Spring Softening and Hardening in Folded-MEMS Comb Drive Resonators

• Published in: Journal of microelectromechanical systems 2011-08, Vol.20 (4), p.943-958
• Main Authors: Elshurafa, A. M., Khirallah, K., Tawfik, H. H., Emira, A., Abdel Aziz, Ahmed K. S., Sedky, S. M.
• Format: Article
• Language: English
• Subjects: Applied sciences; Beams (structural); Capacitance; Economic models; Electronics; Electrostatic actuation; Electrostatics; Equations; Exact sciences and technology; Exact solutions; folded suspension; Instruments, apparatus, components and techniques common to several branches of physics and astronomy; Integrated circuits; Integrated circuits by function (including memories and processors); Mathematical analysis; Mathematical model; Mathematical models; Matlab; Measurements common to several branches of physics and astronomy; Mechanical instruments, equipment and techniques; Metrology, measurements and laboratory procedures; microelectromechanical systems (MEMS) resonators; Micromechanical devices and systems; nonlinear dynamics; perturbation; Physics; Resonant frequency; Resonators; Semiconductor electronics. Microelectronics. Optoelectronics. Solid state devices; Softening; spring hardening and softening; Springs; Studies; Velocity, acceleration and rotation
• ISSN: 1057-7157; 1941-0158
• DOI: 10.1109/JMEMS.2011.2148162

This paper studies analytically and numerically the spring softening and hardening phenomena that occur in electrostatically actuated microelectromechanical systems comb drive resonators utilizing folded suspension beams. An analytical expression for the electrostatic force generated between the combs of the rotor and the stator is derived and takes into account both the transverse and longitudinal capacitances present. After formulating the problem, the resulting stiff differential equations are solved analytically using the method of multiple scales, and a closed-form solution is obtained. Furthermore, the nonlinear boundary value problem that describes the dynamics of inextensional spring beams is solved using straightforward perturbation to obtain the linear and nonlinear spring constants of the beam. The analytical solution is verified numerically using a Matlab/Simulink environment, and the results from both analyses exhibit excellent agreement. Stability analysis based on phase plane trajectory is also presented and fully explains previously reported empirical results that lacked sufficient theoretical description. Finally, the proposed solutions are, once again, verified with previously published measurement results. The closed-form solutions provided are easy to apply and enable predicting the actual behavior of resonators and gyroscopes with similar structures.