Active vibration control of an elastic piezoelectric beam subjected to multi-frequency excitation

• Two-frequency excitation increases structure rigidity and softening effects. • Proper gain tuning avoids chaos by steering clear of unstable zones. • Displacement and velocity controlled via piezoelectric sensors and actuators. • 4th-order Runge-Kutta used for nonlinear dynamic control gain analysis first time. This paper focuses on the study of simultaneous secondary resonances of a sandwich beam subjected to two excitation frequencies, with a stacking sequence comprised of piezoelectric/elastomeric/piezoelectric materials. To this end, feedback potential is utilized through piezoelectric sensors and actuators that enable vibration control of the beam. The dynamic model of the sandwich beam is described by a nonlinear differential equation, which is discretized using the Galerkin approximation. The method of first-order and higher-order multiple scales is employed to determine approximate solutions for the generalized displacement at first order and the analytical expressions that link frequency-amplitude and phase-amplitude. The static and dynamic stability criteria of the system are investigated based on the Routh method. The results obtained show that for direct proportional control gain parameters Gp ranging from 20 to 40, the multi-frequency amplitudes improve the behavior of the system, making it more resilient to shocks compared to the mono-frequency responses illustrated in the literature. Under different boundary conditions, the direct proportional control gain parameters G d make the system softer with larger values, while the velocity control gain enhances the system's response with smaller values, effectively shifting the system from low to high frequencies depending on these values. The simply supported boundary condition is more advantageous than the other conditions considered, as it makes the system softer and therefore more resilient. The phase angle increases with significant values of the gain Gp. The Runge-Kutta numerical method is used to illustrate the time responses of the structure for different control gain values.