Dynamics of structures with wideband autoparametric vibration absorbers: theory

This article analyses the dynamics of a resonantly excited single–degree–of–freedom linear system coupled to an array of nonlinear autoparametric vibration absorbers (pendulums). Each pendulum is also coupled to the neighbouring pendulums by linear elastic springs. The case of a 1:1:…:2 internal resonance between pendulums and the primary oscillator is studied for stationary (harmonic) and non–stationary (slow frequency sweep) excitations. The method of averaging is used to obtain amplitude equations that determine the first–order approximation to the nonlinear response of the system. The amplitude equations are analysed for their equilibrium as well as non–stationary solutions as a function of the parameters associated with the absorber pendulums. For stationary excitation, most steady–state solutions correspond to modes in which only one pendulum and the primary system execute coupled motions. Conditions for the existence of manifolds of equilibria are revealed when the averaged equations are expressed in modal coordinates. In the non–stationary case with linear frequency sweep through the primary resonance region, delays through pitchforks, smooth but rapid transitions through jumps, and transitions from one stable coupled–mode branch to another are studied using numerical simulations of the amplitude equations. The array of autoparametric pendulums is shown to effectively attenuate the large–amplitude resonant response of structures over a wide band of excitation frequencies.