  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.
