Computational methods for continuation and bifurcation of nonlinear dissipative dynamical systems | Christian Kaas-Petersen
| | Abstract | Oscillations may be seen in railway bogies when they run with high speed along a straight, perfect, and horizontal track. These oscillations take place across the running direction and cause wear on wheels and rails and therefore large maintenance costs. Oscillations of the bogie introduce through springs and dampers oscillations of the car body; these oscillations are unwanted since the passenger comfort is reduced. The oscillations can be periodic, bi-periodic (i.e. the oscillations consist of two coupled periodic oscillations), or chaotic (i.e. the oscillations depend critically on how they are triggered). We have used Cooperrider's models of railway bogies and railway vehicles. The models are described by ordinary differential equations. The forces between the wheels and the rails are assumed to be nonlinear, to be dissipative, and to depend of the running speed. Furthermore are the forces between the wheel-flanges and the rails assumed to be nonlinear. Spring- and damper-forces are assumed to be linear. Since non forces depend explicitly on time, the system of ordinary differential equations is said to be autonomous. The equations and the results we have obtained have been presented in (22,32,33), and will be presented in (26,27). We have developed computational methods to examine the equations mentioned. The methods are applicable to all autonomous ordinary differential equations. Theses methods have also been used to examine ordinary differential equations periodic or bi-periodic in time. All theses systems are denoted dynamical systems, and we restrict ourselves to dissipative dynamical systems. Besides determining the solution, we also determine the stability of the solution. We trace the solution in dependence of a parameter in the system - socalled path of solutions - with a path following method. The solutions can be stable on a part of the path and unstable on the other part of the path. When the stability chances, a new path of solutions branches off from the path we are following. Such branch points are called bifurcation points and we perform a bifurcation analysis in order to determine the direction of the bifurcating path. A dynamical system can therefore have more than one path of solutions - necessary condition for this is that the system is nonlinear. The computational methods have been developed and used on a computer. A computer is necessary when the dynamical system (as is the case here) consists of many coupled equations, i s strongly nonlinear, and have to be solved over long time intervals. The systems we have considered and the results we have obtained have been presented in (20,21,23,24). The methods are implemented in a software package coded in fortran (25). | | Type | Ph.D. thesis [Academic thesis] | | Year | 1985 | | Publisher | Laboratory of Applied Mathematical Physics, The Technical University of Denmark | | Series | Lic.techn.-thesis | | Note | Thanks to supervisors associate professor lic.tech. Hans True, ht@imm.dtu.dk, and associate professor lic.techn. Ove Skovgaard | | Electronic version(s) | [pdf] | | BibTeX data | [bibtex] | | IMM Group(s) | Mathematical Physics |
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