The Harmonic Balance Method and its Applications in Structural Dynamics

In many application areas of structural dynamics it has been shown, that non-linear effects play a central role in the response of technical systems. These nonlinearities are often locally limited and can be described by discrete spring and damper elements with nonlinear characteristic curve and/or elements with hysteresis. Characteristic curves with jumps, as they occur, for example, with the sign function in case of dry friction or piecewise linear functions in case of bilateral contact phenomena can influence the convergence behavior of the solution. Here regularization is often used in practice, which has a positive effect on the numerical solution behavior. In contrast to linear systems with harmonic excitation, the response behavior of nonlinear systems is characterized by the presence of higher harmonic components. For the solution of industrially relevant finite element models, a subdivision is made into linear and non-linear degrees of degrees of freedom. The linear component of the non-linear equations of motion can (optionally) first be condensed via a Craig-Bampton method. A Fourier series approach for the periodic displacement solution leads to a nonlinear algebraic system of equations, which is solved with the help of an arc-length method, whereas the initial solution is found by a continuation approach. The implementation of the HBM method was carried out completely in the commercial FE package PERMAS. This eliminates the time-consuming and error-prone process chains that link commercial FE packages with Matlab or In-house software. The non-linear functions are entered in a user-friendly way by means of a new function class. The back transformation into the time domain and the calculation of secondary results such as the radiated sound power or computation of spring forces is also possible. The overall implementation was carried out from the High Performance Computing point of view. This allows the HBM to be used in an industrial environment. Selected examples from the literature show the efficiency of the method.