Analysis and control of primary resonance in an oscillatory cantilever beam excited transversely at its free end

—In this paper, an oscillatory cantilever beam that is excited transversely at its free end is studied. This type of excitation causes the appearance of external and parametric forces that enhance the unwanted nonlinear vibrations of the beam especially at resonance cases. This dynamical behavior is modeled in a nonlinear differential equation to be solved analytically and numerically in an approximate sense. In addition, positive position feedback (PPF) control algorithm is applied through piezoelectric actuators implemented all over the beam’s surface in order to reduce such vibrations. Furthermore, the simultaneous primary and internal resonance case is investigated to see how the control law can overcome it. The beam’s overall characteristics are pictured with the aid of the multiple scales method in order to judge the effectiveness of the controller. Accordingly, the steady-state behavior’s stability is tested applying Lyapunov’s first (indirect) method along with Routh-Hurwitz criterion. The bifurcation analysis of the cantilever beam is shown before and after control with comparing the beam’s behavior pre and post control. Numerical verifications have been conducted in order to certify the applied control algorithm via time responses and phase portraits.