Abstract
In this contribution, recently published new semi-analytical solution for the moving mass problem [1] is extended to account for the transient terms that adapt the initial part of the complete solution in a way to match the initial conditions. It is assumed that a mass and a vertical force with harmonic component move by constant velocity along a horizontal infinite beam posted on a two-parameter visco-elastic foundation. The new semi-analytical solution is presented as a sum of truly steady-state terms, harmonic terms induced by the moving mass and transient terms adapting the initial conditions. Closed-form formula is given for the first two types of vibrations. It is concluded that transient terms have in most cases almost negligible effect on the full solution and that the initial conditions can significantly affect the amplitudes of the induced harmonic vibrations, but the induced frequencies are kept without any changes.
Highlights
Vibration analyses of beam structures under moving loads undoubtedly contributed to the design of modern railway lines
Deep understanding of dynamic phenomena related to train-track-soil interactions, and questions regarding the moving load and moving mass problems still attract the scientific community
Analytical and semianalytical solutions have the unquestionable advantages of closed form solutions and quickly obtainable highprecision results solely in places of interest without the necessity to test numerical parameters ensuring the result convergence
Summary
Vibration analyses of beam structures under moving loads undoubtedly contributed to the design of modern railway lines. New modelling approaches and solution methods are always welcomed to underline the necessary understanding In this context, analytical and semianalytical solutions have the unquestionable advantages of closed form solutions and quickly obtainable highprecision results solely in places of interest without the necessity to test numerical parameters ensuring the result convergence. Considering the fact that someone may want to calibrate a numerical model, or control the unstable region, it is pertinent to identify the instability velocity interval, and determine the exact vibration pattern that will lead to instability. With this in mind the new semi-analytical solution was derived in closed form.
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