In this paper we consider the behavior of a two degree-of-freedom mechanical system incorporating static and dynamic friction, assumed to be a decreasing function of the relative sliding velocity. The model consists of two blocks linked by springs, which ride upon a moving belt. The dynamics of the system are described within a four-dimensional phase space. A three-dimensional Poincaré map is discussed together with a simpler one-dimensional map of a scalar variable. Considering the one-dimensional map it is possible to study all the attractors of the system for small belt velocities including the construction of one-dimensional basins of attraction. Thus, albeit in a partial zone of the control-phase space, the global dynamics of the system can be characterized displaying periodic, quasi-periodic and chaotic oscillations.

Continuum and numerical formulations for non-linear dynamics of thin shells are presented in this work. An elastodynamic shell model is developed from the three-dimensional continuum by employing standard assumptions of the ÿrst-order shear-deformation theories. Motion of the shell-director is described by a singularityfree formulation based on the rotation vector. Temporal discretization is performed by an implicit, one-step, second-order accurate, time-integration scheme. In this work, an energy and momentum conserving algorithm, which exactly preserves the fundamental constants of the shell motion and guaranties unconditional algorithmic stability, is used. It may be regarded as a modiÿcation of the standard mid-point rule. Spatial discretization is based on the four-noded isoparametric element. Particular attention is devoted to the consistent linearization of the weak form of the initial boundary value problem discretized in time and space, in order to achieve a quadratic rate of asymptotic convergence typical for the Newton-Raphson based solution procedures. An unconditionally stable time ÿnite element formulation suitable for the long-term dynamic computations of exible shell-like structures, which may be undergoing large displacements, large rotations and large motions is therefore obtained. A set of numerical examples is presented to illustrate the present approach and the performance of the isoparametric four-noded shell ÿnite element in conjunction with the implicit energy and momentum conserving time-integration algorithm. ? 1998 John Wiley & Sons, Ltd.

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The dynamics of a railway bogie are investigated by utilizing methods of nonlinear dynamic as system theory. The railway bogie model is that introduced by Cooperrider and further developed by Kaas-Petersen. "Principal" and "secondary" attractors are defined and periodic and chaotic attractors are described. A "qualitative" distinction of secondary attractors is introduced according to the robustness of the attractors with respect to random perturbations. The results suggest the possible existence of an optimal axle distance between two wheel sets.

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