Conditions for the existence of isolated backbone curves

Abstract

<jats:p>Isolated backbone curves represent significant dynamic responses of nonlinear systems; however, as they are disconnected from the primary responses, they are challenging to predict and compute. To explore the conditions for the existence of isolated backbone curves, a generalized two-mode system, which is representative of two extensively studied examples, is used. A symmetric two-mass oscillator is initially studied and, as has been previously observed, this exhibits a perfect bifurcation between its backbone curves. As this symmetry is broken, the bifurcation splits to form an isolated backbone curve. Here, it is demonstrated that this perfect bifurcation, indicative of a symmetric structure, may be preserved when the symmetry is broken under certain conditions; these are derived analytically. With the symmetry broken, the second example—a single-mode nonlinear structure with a nonlinear tuned mass damper—is considered. The evolution of the system's backbone curves is investigated in nonlinear parameter space. It is found that this space can be divided into several regions, within which the backbone curves share similar topological features, defining the conditions for the existence of isolated backbone curves. This allows these features to be more easily accounted for, or eliminated, when designing nonlinear systems.</jats:p>

Understanding targeted energy transfer from a symmetry breaking perspective

“…Its backbone curves are shown in figure 9c 1 , which are asymmetric evolutions of those shown in figure 7a 3 . Note that the isolated backbone curves, in this case, are vanished with infinite frequency and amplitude [35]. As the system moves from region (a) to region (c), the critical energy level is less clear, depicted in figure 9c this region, more energy may be dissipated by the NES for a lowẋ 1 (0) than a high value, and a strong oscillation of energy between the primary system and the NES may be seen, see panel (i) forẋ 1 (0) = 0.02.…”

“…To find the backbone curves, the harmonic balance method is used. As the fundamental backbone curves for the two-mass oscillator in figure 1 only exhibit synchronous responses [24,35], where the phase relationships between modal coordinates are either in-phase or anti-phase, 3 the modal responses may be approximated as qi≈ui=Uicos⁡false(ωritfalse), where ui denotes the fundamental component of qi, and where Ui and ωri represent the amplitude and response frequency of ui, respectively. In addition, as this paper considers the realization of TET via fundamental resonant capture, the response frequencies of the two modal coordinates are assumed to be equal, i.e.…”

“…Note that, these isolated backbone curves may vanish with infinite amplitudes when special parameter conditions are satisfied, see ref. [35]. In panel false(b4false), the two single-mode backbone curves, S1 and S2, in panel false(a4false), evolve to mixed-mode backbone curves, S1,2−.…”

“…Its backbone curves are shown in figure 9c1, which are asymmetric evolutions of those shown in figure 7a3. Note that the isolated backbone curves, in this case, are vanished with infinite frequency and amplitude [35]. As the system moves from region ( a ) to region ( c ), the critical energy level is less clear, depicted in figure 9c2.…”

“…To assist in understanding this behaviour through the use of backbone curves, it is useful to consider the system in its modal representation as this domain is normally used in backbone curve analysis [21,[32][33][34]. In addition, in the modal domain, the symmetry of system configurations may be identified via coefficients of the nonlinear terms [35], which assists the analysis of TET from a symmetry breaking perspective. In the modal domain, the corresponding equations of motion for the system in figure 1 can be obtained by introducing linear modal transform, i.e.…”