Passive control of one degree of freedom nonlinear quadratic oscillator under combination resonance

Due to study of oscillations of cables (e.g. to investigates vibrations of cable-stayed-bridges and observed resonant cases [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11] and [12]), one consider one degree of freedom system with polynomial nonlinearities (typically quadratic and cubic ones) and parametric nonlinear terms due to coupling with modes of the deck or of the pile of the bridge (see e.g. Boujard [13] or Boujard et al. [14]). In this paper, we consider the following system:equation(1)x¨+ζx˙+ω2x+a2x2=Λ1(t)cos(ω1t+ϕ1)+Λ2(t)cos(ω2t+ϕ2),in the combination resonance case:equation(2)ω1+ω2=ω+σϵ.ω1+ω2=ω+σϵ.Indeed we consider either stationary solicitationequation(3)Λj(t)=Aj,j=1,2,or transient solicitation:equation(4)Λj(t)=Ajexp(-λjt),λj>0,j=1,2.In applications, Eq. (1) may involve many other terms (quadratic and cubic with varying coefficient) but these terms do not interact with the considered resonance: They modify the expression of the response with particular solutions (out of resonance then with small amplitudes) but they can be neglected to analyse the essential phenomenon. We examine passive control of the one degree of freedom system described by Eq. (1) by a purely cubic Nonlinear Energy Sink under solicitation (3). Various analytical developments and numerical simulations are devoted to the study of the responses. Examples of energy pumping are also given under transient external solicitation (4) with the considered resonance. Comparison with tuned mass damper passive control is also given.