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

Abstract

International audienceIn 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

On the frequency response computation of geometrically nonlinear flat structures using reduced-order finite element models

“…(33) with Q k = 0 and ξ k = 0 ∀k ∈ {1, ..., N w }. In this case, as proposed in [64], a perturbation term −λv k is added to the second line of Eq. (33) at each oscillator k, where λ is an explicit continuation parameter, which enables to add a phase condition.…”

Steady-state, nonlinear analysis of large arrays of electrically actuated micromembranes vibrating in a fluid

“…Truncating the Fourier series of the unknown fields and approximating the Fourier coefficients by finite elements in this case leads to a large coupled nonlinear system. This method is called the multiharmonic or harmonic balance finite element method [13,14,15]. This multiharmonic strategy has already been investigated to study the nonlinear vibration of electrically actuated micromembranes in vacuum [16,8,9], as well as in several other research fields [17,18,19,13,20,21].…”

A generic and efficient Taylor series–based continuation method using a quadratic recast of smooth nonlinear systems

“…'s) can be treated efficiently with this method. The continuation of periodic solutions of dynamical systems is presented in the works of Cochelin et al 18,19 where the ANM is combined with a Fourier series expansion known as the harmonic balance method. Extension to the continuation of quasi-periodic solutions is addressed in the work of Guillot et al 20 The continuation solver Manlab 1.0 has been the first attempt to design a general purpose continuation software working on the ANM principle.…”