The Principle of Total Potential Energy with Stationary Value in Elastic System Dynamics and Its Vibration Analysis

摘要

This paper discusses the inadequacies in Lagrange's equations and in Hamilton's principle and introduces the principle of total potential energy with stationary value in elastic system dynamics and the "Set-in-right-position" rule for formulating matrixes; both were first presented by Zeng Qingyuan 20 years ago. It is shown that however complicated an elastic dynamic system may be, its spacial vibration equations can be methodically and easily formulated by the principle and the rule introduced above. Their peculiar advantages are well embodied in solving the problems in the lateral vibration analysis of train-bridge time-varying system and train-track time-varying system, which are two complex typical dynamic systems whose spacial vibration equations could not be established by using the method of direct equilibrations, Lagrange's equations, or Hamilton principle etc. At home and abroad, the common method evaluating vibration responses of train and bridge (or train and track) is to set up separately two groups of equations for the vehicle vibration and for the bridge (or track) vibration, then to connect them by the interaction force of the wheel and the rail, and finally to solve the two groups of equations by the iterative method. Since there is clearance between the rail and the flange of wheel, and the lateral wheel-rail contact condition can't be listed, the sole solution of lateral vibration equations can't be guaranteed; consequently, no satisfactory calculation results about the lateral vibration responses of train and bridge (or train and track) have been obtained up to now. Zeng Qingyuan and postgraduates regarded the train and the bridge as one whole system and used the aforementioned principle of total potential energy with stationary value and the "Set-in-right-position" rule to establish the spacial vibration equations of train-bridge (or train-track) time-varying system, and obtained, for the first time at home and abroad, some vibration wave figures of the bridge and of the tie, which are very close of those experimental wave figures. At the end of the paper, the concept of the total potential energy in an elastic dynamic system is put forward; based on this concept, the energy criterion distinguishing the stability of motion of a system is also presented. Two examples using our energy criterion to calculate the stability of motion are illustrated, the calculated results of which are in good agreement with the classic solutions gained by the classic theories of the stability of motion, but the process of calculation is simplified to a great extent.