Nonlinear coupling of flexural mode and extensional bulk mode in micromechanical resonators

The effect of coupling between the extensional bulk mode and the flexural mode of a cantilever beam resonator has been experimentally studied by exciting the two modes simultaneously. The modal frequency shift of linear extensional bulk mode shows a quadratic relationship with the square of flexural mode's amplitude displacement, and a frequency shift up to 1492 Hz is observed when the flexural mode is driven by a AC signal V_ac of 3 V and a DC bias of 30 V. The flexural mode shows a Duffing-like behavior with a softening nonlinearity and its frequency shift is influenced not only by its own nonlinear amplitude-frequency effect but also the extensional mode's amplitude as predicted. The nonlinear coupling coefficient is found to be about two orders of magnitude larger than the softening nonlinearity.