Crossover/Veering in V-Shaped MEMS Resonators

We investigate experimentally and theoretically mode crossing and mode veering between the first two symmetric and anti-symmetric modes of an electrothermally and electrostatically actuated in-plane V-shaped micro-beam. The micro-beam is sandwiched between four electrodes to electrostatically activate both modes of vibrations and to offer various actuation options. When tuning the electrothermal voltage, the natural frequencies of the two modes approach each other until they cross. Under an asymmetrical electrostatic actuation, it is shown experimentally that the system undergoes a transition between modes crossing to veering. In addition, an analytical study is presented based on a Galerkin-based reduced-order model of a nonlinear Euler-Bernoulli shallow arch beam equation. The analytical results are compared to experimental data showing excellent agreement. The effects of the shape profile, Cosine and Arc, and micro-beam length are theoretically investigated on the transition between the crossing and veering behavior for the first four vibration modes.