Self-excited oscillation produced by a phase shift: linear and nonlinear instabilities

Self-excited oscillation is a method to make a resonator vibrate at its natural frequency. This is widely used in vibration sensors such as mass sensors, atomic force microscopes and stiffness sensors. Self-excited oscillation is mainly produced by positive velocity feedback and time-delayed displacement feedback including phase-shifted displacement feedback. In this paper, we consider phase-shifted displacement feedback using a phase shifter. We perform nonlinear analysis to clarify the finite steady-state amplitude of the self-excited oscillation by considering nonlinear damping without a Laplace transform in the frequency domain, and formulate the effect of the phase shifter using an ordinary differential equation. We apply the method of multiple scales to the third-order ordinary differential equations expressing the coupling between the resonator and the phase shifter. This analytically reveals the parameter range of the phase shifter that produces self-excited oscillation in the resonator and the steady-state amplitude depending on the phase shift. We conduct an experiment using a cantilever as the resonator and produce self-excited oscillation via the feedback signal based on a phase shifter in a digital computer. The theoretically predicted characteristics of the self-excited oscillation agree well with the experimental ones.