On the motion of a pendulum attached with tuned absorber near resonances

In this paper, the motion of two degrees of freedom of a dynamical system consists of a simple pendulum attached with tuned absorber subject to harmonic excitation is investigated. The governing equations of motion are obtained using Lagrange's equations in terms of the generalized coordinates. The multiple scales technique (MST) is used to gain the solutions of the equations of motion up to the third order of approximation. Two resonance cases namely; main (primary) external resonance and the internal one have been investigated to obtain the modulation equations. The amplitude and phase variables are obtained to investigate the possible steady state solutions and stability conditions. Time histories of the dynamical motions are discussed and presented graphically at any instant. In addition, resonance curves are graphically presented. The steady state solutions are obtained and their stabilities are checked.