Anomalous amplitude-frequency dependence in a micromechanical resonator under synchronization

It is well known that the oscillation frequency relates approximately quadratically with amplitude in a Duffing nonlinear oscillator while the frequency is independent of amplitude in a linear oscillator. In this article, the dynamics of a micromechanical oscillator during synchronization is studied and anomalous amplitude-frequency (a-f) dependence in a micromechanical resonator is observed. We theoretically and experimentally observed that in a linear resonator the amplitude is tuned quadratically with frequency while tuned linearly in a hardening as well as a softening nonlinear resonator, when the self-sustained resonator is synchronized to an external weak perturbation. Further investigation shows that the tunable range of the oscillation amplitude of a certain oscillator directly relies on the synchronization bandwidth, perturbation amplitude and frequency difference. The slope of the dependence can be tuned by phase delay in the feedback loop, while the feedback force dominantly determines the properties of the dependence from nonlinear relation to linear relation. This anomalous a-f effect provides a convenient technique for precise amplitude control.