Zero-dispersion point in curved micro-mechanical beams

In doubly clamped curved mechanical beams, there are inherent hardening and softening nonlinearities. Thus, their frequency of oscillation is a non-monotonic function of the energy. However, for a sufficiently high energy level there is a zero-dispersion point, where the frequency of oscillation is locally independent of energy even though the beam oscillates deep in its nonlinear regime. This zero-dispersion point is a highly desirable feature in micro-mechanical beams that are used in sensing and time-keeping applications because it effectively eliminates the amplitude-to-frequency noise conversion, and thereby, stabilizes the oscillation frequency. In this paper, we present a detailed analysis of the conservative strongly nonlinear dynamics of curved micro-mechanical beams. Our analysis includes a numerically validated closed-form analytical solution for the strongly nonlinear oscillation of beam, derivation of the condition for the zero-dispersion point in curved beams, and a design scheme for the optimal initial depth of the curved beam that maximizes the frequency of oscillation and the energy level at the zero-dispersion point. We apply our methodology to a physical MEMS device reported in the literature and find the optimal values for this device. Our analysis provides the first step in the development of design tools for exploiting the inherent nonlinearities of curved micro-mechanical beams for frequency stabilization.