Structural dynamics of mechanical systems with local nonlinearities under periodic excitation

Mechanical systems consisting of linear components with many degrees of freedom and local nonlinearities are frequently met in engineering practice. From.a spatial point of view, the local nonlinearities constitute only a small part of the mechanical system. However, their presence can have important consequences for the overall dynamic behaviour. The subject of this paper is the long term behaviour of the above systems, excited by periodic external loads. The number of degrees of freedom of the linear components of the system is reduced by applying a component mode synthesis technique based on free-interface eigenmodes and residual flexibility modes. Periodic solutions are calculated efficiently by solving a two-point boundary value problem using finite differences. How the periodic solution is influenced by a change in a so-called design variable of the system is investigated by applying a path following technique. Floquet multipliers are calculated to determine the local stability of these solutions and to identify local bifurcation points. The steady-state behaviour is also investigated by means of standard numerical time integration. In this case the character of the long term behaviour (peridic, quasi-periodic or chaotic) is identified by caiculation of the Lyapunov exponents. The methods outlined above are applied to a harmonically exCited discretized beam system supported by a one-sided linear spring, which reveals very rich, complex dynamic behaviour.