In this paper we present a mixed shooting – harmonic balance method for large linear mechanical systems on which local nonlinearities are imposed. The standard harmonic balance method (HBM), which approximates the periodic solution in frequency domain, is very popular as it is well suited for large systems with many degrees of freedom. However, it suffers from the fact that local nonlinearities cannot be evaluated directly in the frequency domain. The standard HBM performs an inverse Fourier transform, then calculates the nonlinear force in time domain and subsequently the Fourier coefficients of the nonlinear force. The disadvantage of the HBM is that strong nonlinearities are poorly represented by a truncated Fourier series. In contrast, the shooting method operates in time-domain and relies on numerical time-simulation. Set-valued force laws such as dry friction or other strong nonlinearities can be dealt with if an appropriate numerical integrator is available. The shooting method, however, becomes infeasible if the system has many states. The proposed mixed shooting-HBM approach combines the best of both worlds.
This paper presents the improved methodology for the direct calculation of steady-state periodic solutions for electromagnetic devices, as described by nonlinear differential equations, in the time domain. A novel differential operator is developed for periodic functions and the iterative algorithm determining periodic steady-state solutions in a selected set of time instants is identified. Its application to steady-state analysis is verified by an elementary example. The modified algorithm reduces the complexity of steady-state analysis, particularly for electromagnetic devices described by high-dimensional nonlinear differential equations.