The bifurcation of the periodic response of a micro-machined gyroscope with cubic supporting stiffness and fractional electrostatic forces is investigated. The pull-in phenomenon is analyzed to show that the system can have a stable periodic response when the detecting voltage is kept within a certain range. The method of averaging and the residue theorem are employed to give the averaging equations for the case of primary resonance and 1:1 internal resonance. Transition sets on the driving/detecting voltage plane that divide the parameter plane into 12 persistent regions and the corresponding bifurcation diagrams are obtained via the singularity theory. The results show that multiple solutions of the resonance curves appear with a large driving voltage and a small detecting voltage, which may lead to an uncertain output of the gyroscope. The effects of driving and detecting voltages on mechanical sensitivity and nonlinearity are analyzed for three persistent regions considering the operation requirements of the micro-machined gyroscope. The results indicate that in the region with a small driving voltage, the mechanical sensitivity is much smaller. In the other two regions, the variations in the mechanical sensitivity and nonlinearity are analogous. It is possible that the system has a maximum mechanical sensitivity and minimum nonlinearity for an appropriate range of detecting voltages.

The dynamic response and its stability of a blade rotor with coupled rubbing in the labyrinth seal and tip seal are investigated. The dynamic equations are established based on the Hertz contact rubbing force at the labyrinth seal and the tip rubbing force considering both the contact deformation of the tip seal and the bending deformation of the blade. Numerical simulations show that the coupled rubbing response includes periodic motions, almost periodic motions, and chaotic motions. Compared with the single rubbing fault, coupled rubbing increases the range of rotation velocity of contact. A new continuation shooting method is used in the solution and stability analysis of the periodic response to give the bifurcation diagrams. The paths of the system for entering and exiting chaos are analyzed.

The nonlinear dynamic response of a MEMS resonator with a triangular tuning comb is studied. The motion equation with dis-smooth tuning electrostatic force is derived according to Newton’s second law. The analytical solution of the periodic response is obtained using the harmonic balance method and section integral method. The singularity theory is then applied to investigate the bifurcation of the periodic response of the untuned system. The transition sets on the DC-AC voltage plane dividing the planes into several persistent regions are obtained. The bifurcation diagrams’ topological structures and jump phenomena corresponding to different parameter regions are analyzed. We explore the effects of tuning voltage on the response. This demonstrates that the amplitude–frequency curves present more hardening characteristics with increased tuning voltage. Many twists, bifurcation points, and unstable solutions appear, leading to complicated jump phenomena. Two bifurcation points exist on the response curves: the smooth and dis-smooth bifurcation points, with the latter occurring on the switching plane of non-uniform fingers.

The inclination effect caused by fabrication errors on the periodic response of a symmetric MEMS gyroscope is investigated. The dynamic equation is established considering the inclination effect on support stiffness and electrostatic forces. The periodic response is obtained by the averaging method. The two-variable singularity theory is employed to study the bifurcation characteristics and give transition sets on the DC-AC voltage plane, which divide the plane into four persistent regions. The amplitude-frequency curves demonstrate that only the two persistent regions with low voltages are feasible for the gyroscope. Both over-etching and under-etching reduce the feasible region. The effect of parameters on the performance is present. The mechanical sensitivity and nonlinearity increase with the voltages. With the increase in the inclination angle, the mechanical sensitivity and nonlinearity decrease first and then increase. The full temperature stability of the mechanical sensitivity is also considered. The variation in mechanical sensitivity with temperature is small at a large voltage and negative inclination angle. Under-etching, which leads to small nonlinearity and good temperature stability, is more beneficial to the performance of the gyroscope than over-etching.