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In this paper, we extend the method proposed by Cochelin and Vergez [A highorder purely frequency-based harmonic balance formulation for continuation ofperiodic solutions, Journal of Sound and Vibration, 324 (2009) 243-262] to thecase of non-polynomial nonlinearities. This extension allows for thecomputation of branches of periodic solutions of a broader class of nonlineardynamical systems. The principle remains to transform the original ODE systeminto an extended polynomial quadratic system for an easy application of theharmonic balance method (HBM). The transformation of non-polynomial terms isbased on the differentiation of state variables with respect to the timevariable, shifting the nonlinear non-polynomial nonlinearity to atime-independent initial condition equation, not concerned with the HBM. Thecontinuation of the resulting algebraic system is here performed by theasymptotic numerical method (high order Taylor series representation of thesolution branch) using a further differentiation of the non-polynomialalgebraic equation with respect to the path parameter. A one dof vibro-impactsystem is used to illustrate how an exponential nonlinearity is handled,showing that the method works at very high order, 1000 in that case. Variouskinds of nonlinear functions are also treated, and finally the nonlinear freependulum is addressed, showing that very accurate periodic solutions can becomputed with the proposed method.

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