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Global complexity effects due to local damping in a nonlinear system in 1:3 internal resonance


It is well known that nonlinearity may lead to localization effects and coupling of internally resonant modes. However, research focused primarily on conservative systems commonly assumes that the near-resonant forced response closely follows the autonomous dynamics. Our results for even a simple system of two coupled oscillators with a cubic spring clearly contradict this common belief. We demonstrate analytically and numerically global effects of a weak local damping source in a harmonically forced nonlinear system under condition of 1:3 internal resonance: the global motion becomes asynchronous, i.e., mode complexity is introduced with a non-trivial phase difference between the modal oscillations. In particular, we show that a maximum mode complexity with a phase difference of \(90^{\circ }\) is attained in a multi-harmonic sense. This corresponds to a transition from generalized standing to traveling waves in the system’s modal space. We further demonstrate that the localization is crucially affected by the system’s damping. Finally, we propose an extension of the definition of mode complexity and mode localization to nonlinear quasi-periodic motions and illustrate their application to a quasi-periodic regime in the forced response.

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  1. In this article, a linearized system with symmetric structural matrices is considered to feature ‘modal’ (‘non-modal’) damping if its modal deflection shapes are (not) identical to the ones in absence of damping. In the case of modal damping, the damping matrix is diagonal in the modal space. A special but common case of modal damping is proportional, or Rayleigh, damping, where the viscous damping matrix is a linear combination of the mass and the stiffness matrix.

  2. ‘Secondary Hopf bifurcation’ and ‘Neimark-Sacker bifurcation’ are also widely-used terms in this context.


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The work leading to this publication was supported by the German Academic Exchange Service (DAAD) with funds from the German Federal Ministry of Education and Research (BMBF) and the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007–2013) under REA Grant Agreement No. 605728 (P.R.I.M.E.: Postdoctoral Researchers International Mobility Experience). Funding This study was funded by the German Academic Exchange Service within the P.R.I.M.E. program, see acknowledgements.

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Appendix: Derivation of the slow flow equations

Appendix: Derivation of the slow flow equations

In this appendix, we describe the important mathematical steps involved in the derivation of Eqs. (8)–(11). As usual, we expand the solution uv in a perturbation series,

$$\begin{aligned} u\left( \epsilon ,t\right)= & {} u_0\left( \tau ,\eta \right) + \epsilon u_1\left( \tau ,\eta \right) +\mathcal O\left( \epsilon ^2\right) \,,\nonumber \\ v\left( \epsilon ,t\right)= & {} v_0\left( \tau ,\eta \right) + \epsilon v_1\left( \tau ,\eta \right) +\mathcal O\left( \epsilon ^2\right) \,, \end{aligned}$$

with the fast time scale \(\tau \) and the slow time scale \(\eta \) given by,

$$\begin{aligned} \tau = t \,,\quad \eta = \epsilon t\,. \end{aligned}$$

The time derivative operator becomes

$$\begin{aligned} \frac{{\mathrm {d}}}{{\mathrm {d}}t} = \frac{\partial }{\partial \tau } + \epsilon \frac{\partial }{\partial \eta }\,. \end{aligned}$$

Equations (23)–(25) are then substituted into the rescaled version of Eqs. (3)–(4). Balancing the \(\mathcal O\left( 1\right) \) terms yields,

$$\begin{aligned}&\frac{\partial ^2 u_0}{\partial \tau ^2}+ {\omega _1^2}u_0= 0\,, \end{aligned}$$
$$\begin{aligned}&\frac{\partial ^2 v_0}{\partial \tau ^2} + 9{\omega _1^2}v_0 = 0\,. \end{aligned}$$

The corresponding \(\mathcal O\left( \epsilon \right) \) equations read as follows:

$$\begin{aligned} \frac{\partial ^2 u_1}{\partial \tau ^2}+ {\omega _1^2}u_1= & {} -2\frac{\partial ^2 u_0}{\partial \tau \partial \eta } - \frac{\gamma }{2}\left( u_0+v_0\right) ^3 + f_u \,,\end{aligned}$$
$$\begin{aligned} \frac{\partial ^2 v_1}{\partial \tau ^2}+ 9{\omega _1^2}v_1= & {} -2\frac{\partial ^2 v_0}{\partial \tau \partial \eta } - \frac{\gamma }{2}\left( u_0+v_0\right) ^3 + f_v - 2\delta \frac{\partial v_0}{\partial \tau } - 6{\omega _1}\sigma _2v_0\,. \end{aligned}$$

The general solution of Eqs. (26)–(27) is \(u_0 = A(\eta ){\mathrm {e}}^{{\mathrm {i}}{\omega _1}\tau }+\mathrm {c.c.}\) and \(v_0 = B(\eta ){\mathrm {e}}^{3{\mathrm {i}}{\omega _1}\tau }+\mathrm {c.c.}\). Substituting this into Eqs. (28)–(29), and requiring the secular terms to vanish, yields

$$\begin{aligned} 0= & {} -2{\mathrm {i}}{\omega _1}A^\prime - \frac{3\gamma }{2}\left[ A^2\overline{A}+\overline{A}^2B+2B\overline{B}A\right] +F_u{\mathrm {e}}^{{\mathrm {i}}\sigma _1\eta }\,,\end{aligned}$$
$$\begin{aligned} 0= & {} -6{\mathrm {i}}{\omega _1}B^\prime - \frac{\gamma }{2}\left[ A^3+3B^2\overline{B}+6A\overline{A}B\right] -6{\omega _1}\left( \sigma _2+{\mathrm {i}}\delta \right) B \,. \end{aligned}$$

Herein, \({}^\prime \) denotes derivative with respect to \(\eta \). It should be noted that the term \(F_v\) does not occur in these equations. Therefore a weak fundamental harmonic forcing of the out-of-phase mode has only second-order effects. Without loss of generality of the subsequent investigations, we assume \(F_u=F>0\) as a positive real-valued quantity in the following.

We introduce polar coordinates \(A=a_1{\mathrm {e}}^{{\mathrm {i}}\left( \sigma _1\eta +\beta _1\right) }\) and \(B=a_2{\mathrm {e}}^{3{\mathrm {i}}\left( \sigma _1\eta +\beta _2\right) }\) with the real-valued quantities \(a_1\), \(a_2\), \(\beta _1\) and \(\beta _2\). Substitution into Eqs. (30)–(31) and separation of real and imaginary part finally gives rise to the slow flow equations (8)–(11).

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Krack, M., Bergman, L.A. & Vakakis, A.F. Global complexity effects due to local damping in a nonlinear system in 1:3 internal resonance. Arch Appl Mech 86, 1083–1094 (2016).

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  • Modal interaction
  • Mode complexity
  • Damping
  • Dissipation
  • Nonlinear normal modes
  • Multiple scales