Abstract
The regular and chaotic vibrations of a nonlinear structure subjected to self-, parametric, and external excitations acting simultaneously are analysed in this study. Moreover, a time delay input is added to the model to control the system response. The frequency-locking phenomenon and transition to quasi-periodic oscillations via Hopf bifurcation of the second kind (Neimark–Sacker bifurcation) are determined analytically by the multiple time scales method up to the second-order perturbation. Approximate solutions of the quasi-periodic motion are determined by a second application of the multiple time scales method for the slow flow, and then, slow–slow motion is obtained. The similarities and differences between the van der Pol and Rayleigh models are demonstrated for regular, periodic, and quasi-periodic oscillations, as well as for chaotic oscillations. The control of the structural response, and modifications of the resonance curves and bifurcation points by the time delay signal are presented for selected cases.
Highlights
Interactions among different vibration types can occur in various engineering systems
In both self-excitation models, for the appropriate selection of parameters, the equilibrium position of the system becomes unstable and the solution tends towards a limit cycle (LC), which is stable in such a case
We investigate a 1-DOF Duffing-type oscillator, which includes all possible excitation types, self-excitation, parametric terms, external forcing, and a time delay signal, which can be treated as a control input
Summary
Interactions among different vibration types can occur in various engineering systems. In the above-mentioned literature, self-excited oscillations were studied based on a phenomenological model represented by van der Pol or equivalent Rayleigh nonlinear damping. In both self-excitation models, for the appropriate selection of parameters, the equilibrium position of the system becomes unstable and the solution tends towards a limit cycle (LC), which is stable in such a case. We compare two different selfexcitation types given by van der Pol or Rayleigh terms and demonstrate their differences The solution of such a general model is determined analytically by the multiple time scales method, taking into account the periodic motion (slow flow) and quasi-periodic oscillations (slow–slow flow).
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