Establishing the Exact Relation Between Conservative Backbone Curves and Frequency Responses via Energy Balance 191 Fig. 1 illustration of the energy-based prediction of frequency responses from conservative backbone curves. Plot (a) shows the phase condition between the l-th harmonic of Qf (t), indicated with the notation [Qf ]l(t), and the forcing such that Ml:1(s) has the form in Eq. (4). This latter function is shown in plot (b) for three different conservative motions, also highlighted in plot (c) with corresponding colors. Plot (c) shows the conservative backbone curve (blue line) and the frequency response (red line, the solid part depicts asymptotically stable motions, while the dashed part unstable ones) in the classic amplitude Aand frequency Ωplane where wl is the amplitude of the l-th harmonic in the Fourier series of the periodic function ˙Qf(t) = ˙q 0 (t)F, under the assumption that the l-th harmonic of the Fourier series of Qf(t) = q 0 (t)F has the opposite phase of forcing, as shown in Fig. 1(a). First, we note that only primary resonances m=l =1, so that T =τ, and subharmonic resonances, m=1, l > 1, are possible, whenever the forcing direction is not orthogonal to the conservative motion. The term subharmonic refers to the fact that the forcing frequency is higher than that of the response. Other types of resonances occurring when m> 1, i.e., superharmonic, ultrasubharmonic, or ultrasuperharmonic ones, are always destroyed by the damping for small monoharmonic forcing and weak positive definite damping. When m=1, the forcing contribution wl must overcome the resistanceRof the damping in order to generate a response; otherwise the Melnikov function shapes as the yellow line in Fig. 1(b). We also note that, for subharmonic resonances l > 1, the motion amplitude hence needs to have a contribution on thel-th harmonic and this may only occur for sufficiently large amplitudes or nonlinear terms, since the linearized resonant response only features a single harmonic. Therefore, subharmonic resonances typically appear as isolated response curves from the main branch of the frequency response. If wl >R, then the energy balance in (3) has two zeros s+, s−in a forcing period where M1:l (s+) =M1:l (s−) =0, M 1 :l (s+) >0 and M 1 :l (s−) <0, as shown with the green line in Fig. 1(b). This means that two forced-damped frequency responses bifurcate from the conservative limit q0(t). If the damping is acting positively, these two solutions have opposite stability type as discussed in [4]. For hardening-type behaviors ω (h0) > 0, the solution occurring from the phase shift s+ is unstable, while the other one froms− is asymptotically stable, cf. Fig. 1(c). The converse is true for locally softening conservative backbone curves at which ω (h0) < 0. This analytical conclusion matches with numerical and experimental studies, thus explaining for multi-degree-of-freedom systems the hysteretic behavior occurring when the frequency response is in the vicinity of conservative backbone curves [1, 2, 9, 10]. The case wl =Rindicates that a single, forced-damped solution persists fromq0(t), having a phase shift equal to T/4, which is depicted in Fig. 1(b). Specifically, the displacement response projected onto the forcing direction, i.e., the product q 0 (t)F, is in phase lag quadrature with respect to the forcing signal. When this occurs, then the conservative periodic solution q0(t) is very close to either maximal or minimal responses of the frequency response curve, as highlighted in Fig. 1(c). In other words, this analytical evidence extends the phase lag quadrature criterion to more complex damping shapes with respect to previous studies [9]. We also remark that the phase lag needs to be evaluated looking at the same displacement location where the forcing excites the system and with reference to the same harmonic of the forcing, namely the l-th harmonic. This is particularly important for systems that exhibit non-synchronous NNM motions. 3 Conclusion We have shown some among the conclusions that can be derived from applying exact energy arguments for evaluating the survival of conservative backbone curves as forced-damped motions. These are justified from a mathematically rigorous Melnikov analysis, which can be also exploited for investigating existence and stability of motions arising from other nonconservative forces (e.g., parametric forcing) with respect to those considered in this paper.
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