A high-order, purely frequency based harmonic balance formulation for continuation of periodic solutions: The case of non-polynomial nonlinearities

By Bruno Cochelin, Christophe Vergez. 2008

280 citations recorded by the source corpus.

Abstract In this paper, we extend the method proposed by Cochelin and Vergez [A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions, Journal of Sound and Vibration, 324 (2009) 243–262] to the case of non-polynomial nonlinearities. This extension allows for the computation of branches of periodic solutions of a broader class of nonlinear dynamical systems. The principle remains to transform the original ODE system into an extended polynomial quadratic system for an easy application of the harmonic balance method (HBM). The transformation of non-polynomial terms is based on the differentiation of state variables with respect to the time variable, shifting the nonlinear non-polynomial nonlinearity to a time-independent initial condition equation, not concerned with the HBM. The continuation of the resulting algebraic system is here performed by the asymptotic numerical method (high order Taylor series representation of the solution branch) using a further differentiation of the non-polynomial algebraic equation with respect to the path parameter. A one dof vibro-impact system is used to illustrate how an exponential nonlinearity is handled, showing that the method works at very high order, 1000 in that case. Various kinds of nonlinear functions are also treated, and finally the nonlinear free pendulum is addressed, showing that very accurate periodic solutions can be computed with the proposed method.

Subfields: Computational Mechanics, Control and Systems Engineering, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/5073c3c82f7b77270e9dfbd87abe5734fed9300a

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Citations in the corpus

Numerical computation of nonlinear normal modes in mechanical engineering (2016, methodology): “[51] a high-order frequency-based HB method which recasts system (3) into quadratic form d( ̇̂ z(t)) = c+ l1(ẑ) + l2(ẑ, ẑ) (21) where d and l1 are linear operators and l2 is a bilinear operator.”
The harmonic balance method for bifurcation analysis of large-scale nonlinear mechanical systems (2015, methodology): “with arc-length continuation [21] or the so-called asymptotic numerical method [22, 23] in the manlab package.”
A comparative study of the harmonic balance method and the orthogonal collocation method on stiff nonlinear systems (2014, methodology): “64 The present study compares the high-order purely frequency HBM pre65 sented in [1] and extended in [2], with the piecewise polynomial collocation 66 method.”
Model Order Reduction in Fluid Dynamics: Challenges and Perspectives (2014, methodology): “This problem is solved either using the alternating frequency/time domain method, as was done in [57, 90], or by the asymptotic numerical method, as was done in [34].”
The Harmonic Balance Method for Advanced Analysis and Design of Nonlinear Mechanical Systems (2014, unknown): “, [11, 12]) and to the development of a continuation package manlab[13, 14].”
Bifurcation tracking by Harmonic Balance Method for performance tuning of nonlinear dynamical systems (2017, methodology): “the arclength continuation based on tangent prediction steps and orthogonal corrections [19, 20] or the so-called asymptotic numerical method [21].”