In this paper, equations of motion appropriate to long trains are derived which include the effects of undulating terrain. The formulation of a nonlinear dynamic programming problem to be applied to “programmed braking” is presented and the class of problems within the capability of available computing equipment is discussed. An analysis of the effect of undulating terrain on the operational response of long traveling at constant speed is presented in detail and numerical results for particular train configurations are given. In particular, the critical velocity for longitudinal resonance as a function of train length and terrain characteristic is given, and from the specialized results for the uniform train it is possible to discern the influence of changes in car mass, coupling stiffness, and so on.

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