A nonclassical vibration absorber for pendulation reduction

A passive vibration absorber for reducing the motion of a planar pendulum is developed. The system is excited by the horizontal motion of the support. The design transforms the original one-degree-of-freedom pendulum into a double pendulum by adding a small secondary pendulating sacrificial mass between the main system and the base excitation point and two pretensioned springs that generate negative restoring moments (i.e., of opposite sign to that of the gravity-induced restoring moments). Optimal conditions for enhancing the transfer of energy from the main (lower) to the secondary (upper) pendulum are sought. The damping is assumed to be of a linear viscous-type. Due to the action of the springs, the transfer function between the pendulation angle of the main system and the disturbance, in the undamped linearized case, can be reduced to zero for any excitation frequency. This is accomplished by requiring that the two spring stiffnesses satisfy an algebraic tuning relation. Due to the inherent inertial coupling, the two normal coordinates are coupled through off-diagonal terms in the damping matrix. Hence, the vibration absorber acts to block the transfer of disturbance energy to the main system while enhancing the transfer of energy due to initial conditions from the main pendulum to the secondary pendulum. The absorber design is based on a frequency-domain approach borrowed from linear theory Therefore, to corroborate the effectiveness of the absorber in the nonlinear operating regime (for higher excitation levels), representative responses to initial conditions and frequency-response curves are computed by applying a path-following algorithm to the full nonlinear governing equations. The overall effect of the design is somehow a "linearization" of the system behavior with increased damping properties. In fact, the proposed absorber reduces the response of the system by more than 30 decibels near resonance, exhibits good attenuation characteristics in a broad range of frequencies away from resonance, and remarkably improves the initial-condition response.