Dynamics of a clamped–clamped microbeam resonator considering fabrication imperfections

Abstract

We present an investigation into the static and dynamic behavior of an electrostatically actuated clamped-clamped polysilicon microbeam resonator accounting for its fabrication imperfections, which are commonly encountered in similar microstructures. These are mainly because of the initial deformation of the beam due to stress gradient and its flexible anchors. First, we show experimental data of the microbeam when driven electrically by varying the amplitude and frequency of the voltage loads. The results reveal several interesting nonlinear phenomena of jumps, hysteresis, and softening behaviors. Theoretical investigation is then conducted to model the microbeam, and hence, interpret the experimental data. We solve the Eigen value problem governing the natural frequencies analytically. We then utilize a Galerkin-based procedure to derive a reduced order model, which is then used to simulate both the static and dynamic responses. To achieve good matching between theory and experiment, we show that the exact profile of the deformed beam needs to be utilized in the reduced order model, as measured from the optical profiler, combined with a shooting technique simulation, which is capable of tracing the resonant frequency branches under very-low damping conditions.

Experimental investigation of snap-through motion of in-plane MEMS shallow arches under electrostatic excitation

“…This implies that this is a super harmonic resonance of order three. The phase portrait in Figure 6d shows two loops at 35 kHz, which also confirms a super harmonic resonance of order two [48].…”

“…The nonlinear equation governing the transverse motion w(x,t) of the arch microbeam shown in figure 1 [38][39][40][41][42][43][47][48][49] is as following…”

In-Plane MEMS Shallow Arch Beam for Mechanical Memory

“…Next, we prove the inherent quadratic nonlinearity of the arch regardless of its actuation method (i.e., whether there is a DC electrostatic force or not), which shows a softening nonlinear response. The nonlinear equation governing the transverse motion w ( x , t ) of the arch beam based on no axial inertia can be written as [ 41 , 42 , 43 , 44 , 45 ]: where x is the spatial position, t is time, and w 0 is the initial curvature of the arch. The arch has a Young’s modulus E , a material density ρ, and is assumed to have a rectangular cross-sectional area A and a moment of inertia I .…”

“…Here, we exploit the softening behavior in an arch-shaped beam for memory application. Unlike electrostatically actuated cantilever microbeams, where the softening nonlinear behavior is solely caused by the DC bias voltage, an arch shaped clamped–clamped microbeam shows softening nonlinearity due to its initial curvature [ 41 , 42 , 43 , 44 , 45 , 46 ]. Hence, the nonlinearity in this case is mainly of mechanical origin, and thus is almost independent of the DC bias (does not put any restriction on the amount of bias needed to trigger the nonlinearity), and can even be actuated using other methods, such as piezoelectric.…”

On the Nonlinear Dynamics of a Doubly Clamped Microbeam Near Primary Resonance

“…Although the model predicted the resonator behavior at low amplitudes, it failed to capture the response at high amplitude of vibration. In [32,33], it was shown that the shooting method is more robust and can predict the stable and unstable periodic solution accurately compared to Runge-Kutta, which depends on robustness and the size of the basin of attraction.…”