Coulomb-actuated microbeams revisited: experimental and numerical modal decomposition of the saddle-node bifurcation

Abstract

Electrostatic micromechanical actuators have numerous applications in science and technology. In many applications, they are operated in a narrow frequency range close to resonance and at a drive voltage of low variation. Recently, new applications, such as microelectromechanical systems (MEMS) microspeakers (µSpeakers), have emerged that require operation over a wide frequency and dynamic range. Simulating the dynamic performance under such circumstances is still highly cumbersome. State-of-the-art finite element analysis struggles with pull-in instability and does not deliver the necessary information about unstable equilibrium states accordingly. Convincing lumped-parameter models amenable to direct physical interpretation are missing. This inhibits the indispensable in-depth analysis of the dynamic stability of such systems. In this paper, we take a major step towards mending the situation. By combining the finite element method (FEM) with an arc-length solver, we obtain the full bifurcation diagram for electrostatic actuators based on prismatic Euler-Bernoulli beams. A subsequent modal analysis then shows that within very narrow error margins, it is exclusively the lowest Euler-Bernoulli eigenmode that dominates the beam physics over the entire relevant drive voltage range. An experiment directly recording the deflection profile of a MEMS microbeam is performed and confirms the numerical findings with astonishing precision. This enables modeling the system using a single spatial degree of freedom.

Coulomb actuated microbeams: A Chebyshev-Edgeworth approach to highly efficient lumped parameter models

“…Finally, the application of our zero-mode LPM Eq. (47) to the simulation results and experimental findings of Melnikov et al 16 , reveal a very good agreement.…”

“…Within the realm of Euler-Bernoulli theory, these boundaries are independent of the shape of the beam cross section. While this fact certainly is known from numerical studies 16 , it is derived here based on an analytical model, probably for the first time.…”

“…The zero-mode results are compared to the results of Gilbert et al 31 and to the 3D ANSYS simulation by Melnikov et al 16 in Fig. 3.…”

“…The picture substantially changes however with the observation of Melnikov et al 16 that the lowest Euler-Bernoulli eigenmode ψ 0 (ξ) is by far dominating the physics of Coulomb actuated prismatic clamped-clamped micro-beams in practical applications. This observation implies that the use of higher modes in a LPM for a prismatic Euler-Bernoulli beam is hardly justified, unless higher kinetic energies are involved.…”

“…In this paper we systematically derive a single degree of freedom lumped-parameter model (LPM), describing the physics of prismatic clamped-clamped Coulomb actuated microbeams with high precision as compared to FEM simulations and in line with experimental findings. In a previous publication, Melnikov et al 16 demonstrated that the stable and unstable states of prismatic Coulomb actuated Euler-Bernoulli micro-beams, clamped at both ends, can be successfully simulated combining FEM with arc-length solvers. The resulting model predictions were experimentally scrutinised by combining direct optical observations with a modal analysis regarding Euler-Bernoulli eigenmodes.…”