Second-order approximation of primary resonance of a disk-type piezoelectric stator for traveling wave vibration

Considering the quadratic nonlinear constitutive relations of piezoelectric materials, a traveling wave dynamic model for a lead zirconate titanate stator of a traveling wave ultrasonic motor is established using Hamilton's principle and the Rayleigh-Ritz method. Applying the method of multiple scales, the second-order approximation of the primary resonance for traveling wave vibration of the stator is investigated. The second harmonic component is found in the primary response of the stator, which arises from the quadratic stiffness in the condition of weak excitation. In the region of the resonance, the two coupled modals are split and the lower-order peak bends to the left, hence a jump and delay exist in the response. In this way numerical results are given to verify the feasibility of the analytical approach. The results provide a theoretical foundation for further nonlinear dynamic analysis and design of the traveling wave ultrasonic motor.