Stability and Nonlinear Dynamic Analyses of Beam With Piezoelectric Actuator and Sensor Based on Higher-order Multiple Scales Methods

Higher-order multiple scales methods have been developed for a strong nonlinear differential equation modeling the nonlinear transverse dynamic behavior of piezoelectric–elastic–piezoelectric sandwich beams. The proportional and derivative potential feedback controls via sensor and actuator layers are used. Two higher-orders multiple scales methods are developed leading to accurate approximate time-response and frequency and phase-amplitude relationships. As the classical multiple scales method disregards the quadratic nonlinear terms, the developed higher-order multiple scales methods overcame this drawback and lead to more accurate results. Static and dynamic stability criteria are elaborated and tested. Critical bifurcation points and peak amplitudes are characterized in accordance with the excitation amplitude and the feedback parameters.