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

In this paper, we extend the method proposed by Cochelin and Vergez in [A high order purely frequency based harmonic balance formulation for continuation of periodic solutions, Journal of Sound and Vibration, 324:243–262, 2009] to the case of non polynomial non-linearities. 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 non-linearity 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 d.o.f. vibro-impact system is used to illustrate how an exponential non-linearity 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.