A purely frequency based Floquet-Hill formulation for the efficient stability computation of periodic solutions of ordinary differential systems

Abstract

Since the founding theory established by G. Floquet more than a hundred years ago, computing the stability of periodic solutions has given rise to various numerical methods, mostly depending on the way the periodic solutions are themselves determined, either in the time domain or in the frequency domain. In this paper, we address the stability analysis of branches of periodic solutions that are computed by combining a pure Harmonic Balance Method (HBM) with an Asymptotic Numerical Method (ANM). HBM is a frequency domain method for determining periodic solutions under the form of Fourier series and ANM is continuation technique that relies on high order Taylor series expansion of the solutions branches with respect to a path parameter. It is well established now that this HBM-ANM combination is efficient and reliable, provided that the system of ODE is first of all recasted with quadratic nonlinearities, allowing an easy manipulation of both the Taylor and the Fourier series. In this context, Hill's method, a frequency domain version of Floquet theory, is revisited so as to become a by-product of the HBM applied to a quadratic system, allowing the stability analysis to be implemented in an elegant way and with good computing performances. The different types of stability changes of periodic solutions are all explored and illustrated through several academic examples, including systems that are autonomous or not, conservative or not, free or forced.

Comparison of Reduction Methods for Finite Element Geometrically Nonlinear Beam Structures

“…Their specific locations were selected in order to underline the different possible predictions given by all tested methods. The backbone curves were obtained numerically with a continuation method using an asymptotic-numerical method combined with harmonic balance, which was implemented in the Manlab software [64][65][66]. The reference solution was obtained by using Equation (4) with ten modes.…”

“…A reference solution was obtained via numerical continuation on all degrees of freedom by using a code with parallel implementation of the harmonic balance method and pseudo arc-length [68]. On the other hand, as the reduced dynamics were composed of a single master mode, the backbones were obtained numerically by continuation using a method combining harmonic balance and an asymptotic-numerical method implemented in Manlab [64][65][66].…”

“…Consequently, the solution is perturbed with a non-zero solution in the v direction, allowing the continuation method to retrieve the coupled branch. On the other hand, the FRFs of the reduced dynamics are computed with Manlab [64][65][66], which reports stability and detects pitchfork bifurcations.…”

Non-intrusive reduced order modelling for the dynamics of geometrically nonlinear flat structures using three-dimensional finite elements

“…This geometric nonlinearity is then at the root of complex behaviours, that also need dedicated computational strategies in order to derive quantitative predictions. On the phenomenological point of view, structural nonlinearities give rise to numerous nonlinear phenomena that have been analysed in a number of studies: frequency dependence on amplitude [14,20,25], hardening/softening behaviour [45,46], hysteresis and jump phenomena [24,36], mode coupling through internal resonances [9,24,39], bifurcations and loss of stability [10,44], chaotic and turbulent vibrations [3,5]. On the computational point of view, nonlinear couplings break the invariance property of the linear eigenmodes.…”

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

“…• sorting the eigenvalues consists in keeping the 2 eigenvalues associated with the most symmetric eigenvectors [19,31]. This approach appears to be more robust, although there is no formal mathematical proof regarding its convergence.…”