Pull-in solutions to MEMS model of parallel plate capacitor - Turganov, Alkham (Nazarbayev University School of Sciences and Humanities, 2025)
In the MEMS model of a parallel plate capacitor, pull-in instability occurs when the voltage exceeds the threshold value. This phenomenon represents a performance limit in most MEMS devices, whereas microscale switches and accelerometers operate by this. Although single-degree-of-freedom (SDOF) spring–mass models are commonly employed to forecast static and dynamic pull-in, precise time-domain solutions cannot be represented by elementary functions. Current analyses typically either linearize the governing nonlinear ordinary differential equation or rely on numerical simulations and semi-analytic solutions. This thesis presents an implicit analytical pull-in solu tion for the pull-in time of the undamped, constant-voltage single-degree-of-freedom model, expressed clearly in relation to complete elliptic integrals. We proceed by developing two complementary Puiseux-series expansions for the transient trajectory. The first expansion is derived using the Lagrange inversion theorem in relation to the touchdown point, while the second is aligned with the zero-initial-condition deriva tives through a truncated series and a linear system. Both series illustrate the final approach to collapse through the use of fractional exponents, shedding light on the singularity structure of the solution. The use of high-precision solvers for numerical integration demonstrates a strong alignment between the implicit formula and the exact trajectory, with L2 errors remaining below 1e − 9 throughout the entire pa rameter range of K > 1 8. The Puiseux series approximation achieves an accuracy of less than one percent for K ≥ 05, and when used together, they offer a practical and clear approximation throughout the entire transient period. This thesis combines elliptic-integral solutions with convergent series in the neighborhood of the pull-in, providing a complete analytical toolkit for rapidly predicting the pull-in dynamics in standard MEMS capacitors. This advancement allows for significant reductions in computational costs and improves our understanding of nonlinear collapse, which is crucial for effective device design and optimization.
Keywords: Type of access: Embargo; MEMS; Elliptic Integrals; Nonlinear ODE