Resonator responses to coupled vibrational modes lead to regions of unstable oscillation

Microelectromechanical (MEMS) resonators have many applications that rely on the resonator’s operation in the linear regime. However, in recent years, scientists have demonstrated that the nonlinear dynamic response of resonators can be used to enhance the performance of oscillators. Operating a resonator in its nonlinear regime allows to couple different vibrational modes and modify the dynamic behavior of the resonator.

To identify the benefits of the nonlinear dynamics that arise in micromechanical systems, Czaplewski et al. have created a bifurcation diagram, a map of the response of a nonlinear MEMS resonator in the presence of resonant mode coupling and different driving conditions to locate the unique and unexpected dynamical regions. “There’s a void in the bifurcation diagram where the device just won’t work,” said co-author David A. Czaplewski.

The researchers found as they increase the driving voltages, the mechanical modes start to interact, and the resonator deviates from the typical Duffing behavior. At sufficiently large driving forces, two dynamical regions are observed, one where stable vibration cannot be sustained and another where only amplitude modulated oscillations are possible.

This system can be modeled by a Hamiltonian with two coupled vibration modes. Because the two modes interact with one another, the researchers applied a linear perturbation to the complex amplitudes of the modes to study the stability of the system. By extracting the eigenvalues of the perturbed Hamiltonian, they found there are two types of instabilities in the system, which correspond to the two regions where the resonator does not exhibit the expected Duffing behavior.

The researchers warn applications using MEMS resonators need to be aware of mode coupling to avoid operating in unstable regions.

Source: “Bifurcation diagram and dynamic response of a MEMS resonator with a 1:3 internal resonance,” by David A. Czaplewski, Scott Strachan, Oriel Shoshani, Steven W. Shaw, and Daniel López, Applied Physics Letters (2019). The article can be accessed at https://doi.org/10.1063/1.5099459 .