Optimal design of a bistable switch

Abstract

Determining optimally designed structures is important for diverse fields of science and engineering. Here we describe a procedure for calculating the optimal design of a switch and apply the method to a bistable microelectromechanical system relay switch. The approach focuses on characterizing the unstable transition state connecting the two stable equilibria to control the force displacements. Small modifications in component shape lead to a substantial improvement in device operation. Fabrication of the optimized devices confirms the predictions.

Fig. 1.

Schematic for the operation of the relay switch. Cantilevered starting zippers (Top) initiate the zipping by closing the gap and then allowing the beam to zip closed when the middle electrode is charged (Middle), causing the switch to close. Turning on the top electrode (Bottom) causes the top starting zippers to engage, and the switch opens.

Fig. 2.

(A) Force displacement characteristics for the switch, assuming the force is applied in the center of the switch. The ratio of the maximum force to the minimum force is ≈2. (B) Tension τ²/(2π)² in the beam as a function of the displacement in the center. Through the middle part of the cycle the tension is constant with τ ² = 16π ².

Fig. 3.

(Upper) Evolution of the moment of inertia of the beam during the optimization. The initial ψ(x) is uniform; with increasing τ₂/τ₀ the beam becomes modulated. Note that the beam thickness is exaggerated relative to its length by ≈10³. (Lower) Evolution of the R = (τ₂/τ₀)²/((τ₂/τ₀)² – 2) as a function of ψ_min the minimum moment of inertia of the beam.

Fig. 4.

Optimized thickness profile of the beam for ψ_min = 0.5, 0.25, and 0.15, with R of 1.5, 1.29, and 1.02, respectively. Note that the beam thickness is exaggerated relative to its length by ≈10³.

Fig. 5.

(Top) Force displacement characteristics for the structure with ψ_min = 0.5. R ≈ 1.43. (Middle) Force displacement characteristics for the structure with ψ_min = 0.25. R ≈ 1.18. (Bottom) Force displacement characteristics for the structure with ψ_min = 0.125. R ≈ 1.05.

Fig. 6.

Experiments for the optimized beam with ψ_min = 0.5, 0.25, and 0.125, respectively. The fabricated beam has length 15 mm, minimum thickness 20μm, and d = 250 μm. The optimization analysis designed these structures to have R ≈ 1.43, 1.18, and 1.05, respectively, compared with the measured values R = 1.64, 1.37, and 1.28.