Nonlinear systems with discrete and continuous elements

Carsten Nordstrøm Jensen

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'Nonlinear dynamical systems with discrete and continuous elements' is a very broad formulation which covers a variety of different physical systems. Two such physical systems form the base of this thesis and the ph.d. project behind it: The dynamics of a train running on a bridge and the dynamical interactions between a pantograph and a catenary system.

Train bridge considerations have two different viewpoints: that of the bridge concerning bridge loads, internal stresses, dynamical amplification factors, etc.; and that of the train concerning train passenger comfort and ride quality, derailment risks, wear, etc. The tendency of bridge design goes towards lighter and more flexible constructions. With the rapid improvements of railway technology and the increasing train speeds, the impact from a train on a bridge and vice versa will increase in the future. Our angle of the investigations differs from other investigations by the inclusion of nonlinearities in the train models. In other areas of railway dynamics, these nonlinearities are crucial.

Pantograph catenary dynamics has greater actuality in Denmark than ever before. The subject of overhead line system designs is very complex and hard to survey. Different concepts have developed in different national railway companies and the different leading railway companies all claim that have the best system.

In Denmark a catenary system design concept was bought when the railways were to be electrified. One of the purposes of this ph.d. project has been to take (some of) the design parameters of the danish overhead line system up to renewed considerations.

Another aim of the project has been to investigate the importance of nonlinearities in overhead line systems. Our investigations differ from other investigations in this field in that we consider the very weak nonlinearities of the cable motions and compare the importance of these with the importance of the bending stiffness of the cables in the overhead line systems. As a contrary to the nonlinearities, the bending stiffness is normally considered to be important, even though it is small. The analysis is divided into a part concerning investigations of the propagation of a pulse on single cable models, and a part concerning the direct investigations of pantograph overhead line system interactions.

In the pulse propagation section we investigate different cable motion formulations. In a simplified cable motion formulation we find that a long waved, low amplitude pulse develops shocks, appearing as discontinuities in the slope of the cable. Investigations of a more complete cable motion description reveals that the longitudinal motion can be satisfactorily described by a slow, local part and a fast, global part, yielding the sum of the nonlinear terms as (K+K_1(t))v_{xx}, where v(x,t) is the vertical motion of the cable.

In the direct pantograph overhead line system investigations, we find that the nonlinearities, as a contrary to the bending stiffness, do become important within the range of realistic train speeds. We find them to have a positive effect on the dynamics. In our opinion the nonlinearities should be included in the cable models.

For our investigations of the dynamics near the {critical} speed, we find that it is {strictly} necessary to include the nonlinearities in the model, to be able to satisfactorily describe the breaking sound barrier like phenomenon, that appears for pantographs accelerating up to and past the critical speed. We find that the breaking sound barrier like phenomenon is much smoother when the nonlinearities of the system are included. Thereby the critical speed is not as critical as henceforth assumed.

We have also investigated the influence of a presag of the contact cable in overhead line systems with simple configurations. The presag is meant to compensate for the variation of the flexibility of the contact cable over a span. However, we think that the presag has been designed from purely statical considerations, and from our dynamical investigations we find that the presag should either not be there, or should be much smaller.

Ph.d thesis 41, 1997

Last modified Feb. 11, 1999
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