Dynamics of mechanical systems

9In this dissertation, numerical algorithms are described for theinvestigation of the behaviour of general non-linear mechanicalsystems with a limited number of degrees of freedom. The aim ofthese algorithms is to make feasible a systematic exploration ofthe behaviour. The investigation is narrowed down to systemsdescribed by ordinary differential equations that are autonomousor depend explicitly periodically on time, where the behaviour ina neighbourhood of stationary or periodic solutions is consi-dered. Concerning the differential equations, no further specialproperties are assumed. However, Hamiltonian systems and systemswith a dihedral symmetry are also discussed.In Chapter 2, the theory of dynamical systems is reviewed.Non-linear dynamical systems on manifolds and their limit behav-iour, stability and bifurcations are discussed.In Chapter 3, methods are described for the determination ofstationary and periodic solutions, for the continuation of thesesolutions if a parameter is varied, and for the direct determina-tion and continuation of bifurcation points. For the calculationof stationary solutions, the Newton-Raphson iteration method isused. Periodic solutions are obtained by using shooting methods.A special method for the determination of non-isolated periodicsolutions of Hamiltonian systems has been developed. Solutionsare continued with Fried’s method in a modified form. Bifurca-tion points are distinguished by eigenvalues of the (integrated)linearized equations. Characteristic for the presented methods isthe use of the singular value decomposition, by which overdeter-mined and underdetermined systems can be solved. The methods areillustrated and validated with the help of well-known systemsfrom the literature.In Chapter 4, the developed methods are applied to threesystems with discontinuities. Such systems often occur in mechan-ical engineering and give rise to peculiar difficulties.The first system consists of a wheelset of a railway vehiclewhich moves with a constant forward velocity on a tangent track.The discontinuities arise from the contact between the wheel