Non-Linear Periodic and Quasi-Periodic Vibrations in Mechanical Systems - On the use of the Harmonic Balance Methods

Nonlinear vibrations of a mechanical system with non-regular nonlinearities and uncertainties

“…A definition given by Kim and Choi [11] for retaining N h harmonics in a multiple Fourier series can be given in the following form p i=1 |k i | ≤ N h . For the reader comprehension, it can be noted that all harmonics at negative combination frequencies can be replaced by harmonic terms at positive combination frequencies due to the trigonometric relation [8]. So it may be concluded that only terms at positive combination frequencies can be retained in the nonlinear response and nonlinear force expression.…”

“…Furthermore, the need for consideration of nonlinear effects in the description of a dynamical system is well recognized in the field of engineering and numerous studies have been conducted to understand and model the nonlinear phenomena in structural dynamics during the past decades [8]. Even if most of these models deal with deterministic parameters, it is obvious that variations in the geometry or material properties are often present in these systems as it has already been mentioned in the previous paragraph for linear systems.…”

“…In regions of high curvature, it will be necessary to reduce the prediction distance to better estimate the next solution of the system. On the contrary, in the case of an solution that does not change significantly with changes in the control parameter, the prediction distance may be increased to accelerate the calculations while keeping a good estimate of the solution [8].…”

“…Various techniques and numerical procedures exist for these two steps. In this section, we propose to give an overview of the most classical methods for both the predictor and corrector methods [8]. We focus on the resolution of the nonlinear system given by Equation (7).…”

Piecewise constrained optimization harmonic balance method for predicting the limit cycle oscillations of an airfoil with various nonlinear structures

“…Continuation of periodic orbits has now been recognized as the most efficient method for numerically computing the family of periodic orbits and their implementation is well documented within numerous standard methods for step continuation and adaptation (arclength, pseudo-arclength) (Sarrouy and Sinou, 2011). However, if some branches of solutions are disconnected (in the sense that they do not arise from bifurcation points of previous branches), then the continuation algorithm would fail to detect them and new tools would be needed to solve the algebraic system of equations induced by the HBM.…”