Computation of quasi-periodic localised vibrations in nonlinear cyclic and symmetric structures using harmonic balance methods

Abstract

In this paper we develop a fully numerical approach to compute quasi-periodic vibrations bifurcating from nonlinear periodic states in cyclic and symmetric structures. The focus is on localised oscillations arising from modulationally unstable travelling waves induced by strong external excitations. The computational strategy is based on the periodic and quasi-periodic harmonic balance methods together with an arc-length continuation scheme. Due to the presence of multiple localised states, a new method to switch from periodic to quasi-periodic states is proposed. The algorithm is applied to two different minimal models for bladed disks vibrating in large amplitudes regimes. In the first case, each sector of the bladed disk is modelled by a single degree of freedom, while in the second application a second degree of freedom is included to account for the disk inertia. In both cases the algorithm has identified and tracked multiple quasi-periodic localised states travelling around the structure in the form of dissipative solitons.

Nonlinear analysis of forced mechanical systems with internal resonance using spectral submanifolds, Part II: Bifurcation and quasi-periodic response

“…In this bifurcation, a unique two-dimensional invariant torus is born out of a periodic orbit for each fixed excitation frequency and amplitude. The quasiperiodic response associated with such an invariant torus has been widely observed in mechanical systems under harmonic excitation, ranging from simple van der Pol oscillator [2] to more complicated systems [3][4][5][6][7][8][9].…”

Effect of gear topology discontinuities on the nonlinear dynamic response of a multi-degree-of-freedom gear train

“…The computational strategy developed hereafter is based on a frequency domain technique, namely the Harmonic Balance Method (HBM). The method has been used by the nonlinear dynamics community for several decades [55] in a wide variety of fields such as bolted structures [23], turbomachinery [49], bladed disks dynamics [12,44], vibro-impact systems [3] and gear dynamics [1,2,62].…”

A Taylor series-based continuation method for solutions of dynamical systems

“…The IDE are a type of equations where the derivatives cannot be expressed explicitly (as for the ODE (19)). It can be written…”

“…In the last few years, progress has been achieved on the continuation of quasi-periodic solutions of ODE, using different methods. In [39] the invariant tori are continued when in [19,43], a two dimensional alternating frequency-time scheme is developed. In [24], the system of equations is recast quadratically and a full quasi-periodic harmonic balance method is applied.…”