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Dimension Reduction and Domains of Attraction of Nonlinear Dynamical Systems

Hochlenert D.

Machine Dynamics Research

2011

Vol. 35, No. 4

32-48

The appropriate modeling of technical systems usually results in dynamical systems having many or even an infinite number of degrees of freedom. Moreover, nonlinearities play an important role in many applications, so that the arising systems of nonlinear differential equations are difficult to analyze. However, it is well known that the asymptotic behavior of some high dimensional systems can be described by corresponding systems of much smaller dimension. The present paper deals with the dimension reduction of nonlinear systems close to a bifurcation point. Using the ideas of normal form theory, the asymptotic dynamics of the system is extracted by a nonlinear coordinate transformation. The solutions of the reducedordsr system are analyzed analytically with respect to their stability and their domains of attraction. Furthermore, the inverse of the near-identity transformations is used to construct adapted Lyapunov functions for the original system to estimate the attractors of the solutions as well. The procedure is applied to the Duffing equation and the equations of motion of a railway wheelset and compared with numerical solutions.

Bifurcation Behavior and Attractors in Vehicle Dynamics

Hochlenert D., Von Wagner U., Hornig S.

Machine Dynamics Problems

2009

Vol. 33, No. 2

57-73

Nonlinear self-excited systems in vehicle dynamics are discussed using the examples of squealing automotive disk brakes and the stability behavior of a railway wheelset. Both systems show self-excited vibrations for specific operation states. The self-excited vibrations are due to friction forces between pad and disk in the case of the automotive disk and due to contact forces in the case of the railway wheelset respectively. The analysis of the nonlinear equations of motion shows that the trivial solution looses stability either through a sub- or through a supercritical Hopf bifurcation depending on the system's parameters. In the case of a subcritical Hopf bifurcation two stable solutions coexist and the initial conditions determine which solution emerges. The properties of the nonlinear systems such as critical velocities, limit cycle amplitudes and attractors of coexisting solutions are calculated using center manifold reduction and normal form theory.