Stability and perturbation analysis of a one-degree-of-freedom doubly clamped microresonator with delayed velocity feedback control

Abstract

In this paper, a study on a doubly clamped microresonator actuated by two symmetrical electrodes is carried out to investigate its dynamic properties with delayed velocity feedback control. A stability chart of the linearized system depicting delay time versus feedback gain is drawn first, which is actually nonperiodic. Moreover, stability switches do exist in this system. Then, the method of multiple scales is used to determine the existence, stability and dynamic properties of small amplitude vibration in the neighborhood of different equilibrium positions. It is shown that the stability condition via perturbation analysis overestimates the system stable region. The delayed stability condition via linearized analysis is more suitable for stability estimation. The following analytical and numerical results are presented to investigate frequency responses and frequency/damping trimming properties with various system and control parameters. Moreover, explicit formulas for optimum direct current (DC) voltage and equivalent natural frequency, corresponding to an approximate linear-like state, are deduced, respectively. Two typical design sketches depicting the initial gap width versus DC voltage are drawn with different beam lengths and thicknesses. Finally, a case study is carried out to verify the correctness of our analytical results about linear-like state prediction and frequency/damping trimming.

Linear and nonlinear dynamics of micro and nano-resonators: Review of recent advances

“…Perturbation methods are applied to obtain the modulation equations that govern the amplitude and phase response of the resonator. Examples of these studies can be found in microbeams [107,166,217,[261][262][263][264][265][266][267][268][269][270][271][272][273][274][275][276][277][278][279], torsional mirrors [280,281], self-excited resonators [282][283][284][285], PZT based MEMS models [286][287][288], MEMS filters [171], MEMS arch resonators [169,289], carbon Nanotubes [290], and coupled oscillators [291]. The influence of secondary-resonance excitations on MEMS has been also studied.…”

Nonlinear analysis of electrostatic micro-electro-mechanical systems resonators subject to delayed proportional–derivative controller

“…They found that positive feedback control gain can strengthen system stability, whereas negative feedback can lead to an unstable response gain. Moreover, Han et al (2018) extended this study to large delay times and showed the appearance of a stability switches.…”

“…Thus, some researchers have been interested to analyze the effect of parameters of delayed feedback controllers on the system performances. The authors Shao et al (2013) and Han et al (2018) analyzed the effects of delayed velocity feedback control on an electrostatic microactuator actuated by one electrode and by two symmetrical electrodes. They found that positive feedback control gain can strengthen system stability, whereas negative feedback can lead to an unstable response gain.…”

Multistability of the Vibrating System of a Micro Resonator

“…It is of great significance to study the multistability and necessary conditions for inducing it either for avoiding this phenomenon or making use of it. Thus, multistability of vibrating systems of micro resonators has been studied experimentally and numerically during these decades [16]. Via experiments, Mohammadreza investigated the dynamic response of an electrostatic micro-actuator in the vicinity of the primary resonance and the parametric one [17].…”