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Nonlinear vibrations and stability of an axially moving beam with an intermediate spring support: two-dimensional analysis

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Abstract

The nonlinear coupled longitudinal-transverse vibrations and stability of an axially moving beam, subjected to a distributed harmonic external force, which is supported by an intermediate spring, are investigated. A case of three-to-one internal resonance as well as that of non-resonance is considered. The equations of motion are obtained via Hamilton’s principle and discretized into a set of coupled nonlinear ordinary differential equations using Galerkin’s method. The resulting equations are solved via two different techniques: the pseudo-arclength continuation method and direct time integration. The frequency-response curves of the system and the bifurcation diagrams of Poincaré maps are analyzed.

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Correspondence to Mergen H. Ghayesh.

Appendix: Why should the longitudinal displacement be considered in the analysis?

Appendix: Why should the longitudinal displacement be considered in the analysis?

In order to emphasize the effect of the longitudinal motion on the system dynamics, the dynamics of the system considering both longitudinal and transverse motions is presented in sub-figures (a, b) of Figs. 18, 19, 20, 21 for different cases studied in the paper. The transverse dynamics (obtained by setting the longitudinal displacement equal to zero) is presented in sub-figures (c, d) of each figure. As seen in Figs. 1821, comparing the results shown in sub-figures (a, b) with those plotted in sub-figures (c, d), we find that the longitudinal displacement plays an important role in the nonlinear dynamics of the system not only quantitatively, but also qualitatively. More specifically:

  1. (i)

    As seen in Figs. 18(a)–18(d), the system in which only the transverse displacement is considered, shows complex motions such as quasiperiodic and chaotic in the vicinity of f 1=0.15 and 0.25; this is diminished in the system in which the longitudinal displacement is taken into account (Fig. 18(a), 18(b)), and only there is a simple quasiperiodic motion at f 1=0.25.

    Fig. 18
    figure 18

    (a, b) Bifurcation diagrams of Fig. 11, obtained by considering the coupled longitudinal-transverse vibrations of the system. (c, d) The counterparts of (a, b), obtained by considering only the transverse vibrations of the system

  2. (ii)

    Figures 19(a)–19(d) show that taking into account the longitudinal displacement changes the sequence of different attractors in the bifurcation diagram. For instance, in the vicinity of f 1=0.5, the motion is periodic for the system in which both longitudinal and transverse vibrations are considered (Fig. 19(a)), whereas Fig. 19(c), corresponding to only the transverse vibrations, displays chaotic oscillations.

    Fig. 19
    figure 19

    (a, b) Bifurcation diagrams of Fig. 13, obtained by considering the coupled longitudinal-transverse vibrations of the system. (c, d) The counterparts of (a, b), obtained by considering only the transverse vibrations of the system

  3. (iii)

    Figure 20 shows that, for this case, due to introduced longitudinal displacement, the number of chaotic attractors increase.

    Fig. 20
    figure 20

    (a, b) Bifurcation diagrams of Fig. 16, obtained by considering the coupled longitudinal-transverse vibrations of the system. (c, d) The counterparts of (a, b), obtained by considering only the transverse vibrations of the system

  4. (iv)

    As seen in Fig. 21, if the longitudinal displacement is not considered in the analysis, quasiperiodic and chaotic motions occur in the vicinity of v=0.4 (Figs. 21(c), 21(d)), which disappear when the longitudinal displacement is considered (Figs. 21(a), 21(b)).

    Fig. 21
    figure 21

    (a, b) Bifurcation diagrams of Fig. 17, obtained by considering the coupled longitudinal-transverse vibrations of the system. (c, d) The counterparts of (a, b), obtained by considering only the transverse vibrations of the system

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Ghayesh, M.H., Amabili, M. & Païdoussis, M.P. Nonlinear vibrations and stability of an axially moving beam with an intermediate spring support: two-dimensional analysis. Nonlinear Dyn 70, 335–354 (2012). https://doi.org/10.1007/s11071-012-0458-3

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