[Paper Review] Dynamics of Electrostatic MEMS Actuators

Yisong Yang, Ruifeng Zhang+1 more|arXiv (Cornell University)|Jan 26, 2012

Advanced MEMS and NEMS Technologies12 references22 citations

TL;DR

This paper provides a rigorous mathematical analysis of the dynamics of undamped electrostatic MEMS actuators with one degree of freedom, using energy conservation and Hamiltonian methods to prove that below a critical pull-in voltage, the system exhibits symmetric one-stagnation-point periodic motion, while above it, finite-time touch-down (collapse) occurs. The study establishes explicit expressions for pull-in voltage, pull-in coordinate, and stagnation level in the linear elastic force case, and shows these quantities converge monotonically to the linear case in the zero nonlinear-force limit for cubic nonlinear elasticity.

ABSTRACT

Electrostatic actuators are simple but important switching devices for MEMS applications. Due to the difficulties associated with the electrostatic nonlinearity, precise mathematical description is often hard to obtain for the dynamics of these actuators. Here we present two sharp theorems concerning the dynamics of an undamped electrostatic actuator with one-degree of freedom, subject to linear and nonlinear elastic forces, respectively. We prove that both situations are characterized by the onset of one-stagnation-point periodic response below a well-defined pull-in voltage and a finite-time touch-down or collapse of the actuator above this pull-in voltage. In the linear-force situation, the stagnation level, pull-in voltage, and pull-in coordinate of the movable electrode may all be determined explicitly, following the recent work of Leus and Elata based on numerics. Furthermore, in the nonlinear-force situation, the stagnation level, pull-in voltage, and pull-in coordinate may be described completely in terms of the electrostatic and mechanical parameters of the model so that they approach those in the linear-force situation monotonically in the zero nonlinear-force limit.

Motivation & Objective

• To provide a precise mathematical description of the dynamic behavior of undamped electrostatic MEMS actuators, which are typically studied only via numerical simulations due to strong electrostatic nonlinearity.
• To address the lack of analytical studies on the dynamical wave-equation nature of MEMS actuators, as most prior work focuses on steady-state or parabolic models.
• To establish sharp theorems on the existence of periodic motion below the pull-in voltage and finite-time touch-down above it, under both linear and nonlinear elastic force models.
• To extend the analysis to cubic nonlinear elastic forces and demonstrate the method's applicability to more general nonlinearities.
• To identify universal conditions under which dynamic touch-down occurs due to sufficiently high applied voltage, even in damped systems.

Proposed method

• The authors use energy conservation as a first integral to derive the equation of motion for an undamped one-degree-of-freedom electrostatic actuator, modeling it as a Hamiltonian system.
• For the linear elastic force case, they solve the energy equation explicitly to determine the stagnation level, pull-in voltage, and pull-in coordinate in closed form.
• They prove that periodic motion occurs when the applied voltage is below the critical pull-in voltage, characterized by a single stagnation point.
• For the nonlinear (cubic) elastic force case, they use a similar energy-based approach and show that the key dynamic quantities converge monotonically to the linear case as the nonlinearity vanishes.
• They generalize the method to a normalized Hamiltonian with a general elastic potential function, identifying the necessary convexity condition for the method to remain valid.
• They establish a universal touch-down condition under damping by analyzing the asymptotic behavior of the solution under large applied voltage, using differential inequality techniques.

Experimental results

Research questions

• RQ1 What is the precise dynamic behavior of an undamped electrostatic MEMS actuator with linear elastic force, and does it exhibit periodic motion below a critical voltage?
• RQ2 Can the pull-in voltage, pull-in coordinate, and stagnation level be computed explicitly in the linear elastic force case?
• RQ3 How does the system's behavior change when the elastic force is nonlinear (e.g., cubic), and do the dynamic quantities converge to the linear case as nonlinearity diminishes?
• RQ4 Under what conditions does the system exhibit finite-time touch-down (collapse) for voltages above the pull-in threshold?
• RQ5 Is the dynamic touch-down phenomenon universal for sufficiently high applied voltages, even in damped systems?

Key findings

• In the linear elastic force case, the pull-in voltage, pull-in coordinate, and stagnation level are explicitly determined and match the numerical results of Leus and Elata.
• The system exhibits symmetric one-stagnation-point periodic motion when the applied voltage is below the pull-in voltage.
• Above the pull-in voltage, the solution collapses in finite time, with the movable electrode reaching the fixed electrode (touch-down) in finite time.
• For the cubic nonlinear elastic force case, the stagnation level, pull-in voltage, and pull-in coordinate are completely described in terms of the electrostatic and mechanical parameters of the model.
• As the nonlinear force parameter tends to zero, these quantities converge monotonically to their linear-force counterparts.
• Under general conditions, including damping, dynamic touch-down occurs universally when the applied voltage exceeds a threshold, regardless of initial conditions, due to the dominance of the singular electrostatic force.