A combined FD-HB approximation method for steady-state vibrations in large dynamical systems with localised nonlinearities

The approximation of steady-state vibrations within non-linear dynamical systems is well-established in academics and is becoming increasingly important in industry. However, the complexity and the number of degrees of freedom of application-oriented industrial models demand efficient approximation methods for steady-state solutions. One possible approach to that problem are hybrid approximation schemes, which combine advantages of standard methods from the literature. The common ground of these methods is their description of the steady-state dynamics of a system solely based on the degrees of freedom affected directly by non-linearity—the so-called non-linear degrees of freedom. This contribution proposes a new hybrid method for approximating periodic solutions of systems with localised non-linearities. The motion of the non-linear degrees of freedom is approximated using the Finite Difference method, whilst the motion of the linear degrees of freedom is treated with the Harmonic Balance method. An application to a chain of oscillators showing stick-slip oscillations is used to demonstrate the performance of the proposed hybrid framework. A comparison with both pure Finite Difference and Harmonic Balance method reveals a noticeable increase in efficiency for larger systems, whilst keeping an excellent approximation quality for the strongly non-linear solution parts.