Path continuation for the concept of non-linear normal modes using a normal flow algorithm - Jerschl, Martin; Süß, Dominik; Willner, Kai; Jerschl, Martin (Proceedings of the International Conference on Noise and Vibration Engineering ISMA, 2014)
Non-linear normal modes (NNMs) can be considered as a kind of parallel theory to the description of linear systems with linear normal modes (LNMs) but for non-linear ones. In this paper a NNM is not defined as harmonic or synchronous oscillation but as periodic one. Therefore one has to search for periodic solutions of the governing equations of the system to get an NNM. This periodicity condition leads to a mathematical formulation as a two-point-boundary-value problem. Small systems with a low number of degrees of freedom and non-linear couplings (cubic springs) are investigated. The approach is also adopted to a finite-element model with a cubic non-linearity. With increasing the energy in the system the progressive non-linearity leads to a hardening effect. One typical dynamical property of non-linear systems is the frequency-energy dependency of their oscillations. A good graphic illustration is to plot such dependency in a so called frequency-energy plot (FEP) because here this dependency of energy and eigenfrequency is given explicitly. Thereby the points on a branch in the FEP represent a family of NNM oscillations with qualitatively equal motion properties, quasi a family of periodic orbits. A NNM branch can be calculated by a numerical continuation method with starting at low energy level in a quasi linear regime and increasing the energy and reducing the period of the oscillation iteratively. In non-linear systems internal resonances and other phenomena can occur. Several tongues can bifurcate from a NNM branch. Therefore normal continuation methods fail at such bifurcation points. To overcome this problem a predictor-corrector-method is used. As correction a probability one homotopy algorithm is implemented, in particular the normal flow algorithm. The predicted solution point gets iterated by a Newton-Raphson approach. The gradual iteration steps to the solution of the periodicity condition are along a path which is normal to the so-called Davidenko flow. The Davidenko flow can be seen as family of solution curves around the periodic solution caused by small perturbations to the non-linear system equations and varying these perturbations. These correction steps are continued until convergence is achieved. Different corrector approaches are compared to the normal flow algorithm and the results are discussed.
UnivIS-Import:2015-10-26:Pub.2014.tech.FT.FT-TM.pathco