A Perturbation Method Based Solution for Beam Vibration Excited by Moving Mass

The study of the dynamic effect of moving mass on structures such as vehicles moving on bridges or a crane trolley moving on girder beam has become an important research area in the recent past. The problems were originally modelled as moving force problems, but later on moving mass inertia effects have been also included and recently models are further modified considering vehicles as moving oscillators. Available literatures indicate that response characteristics are mostly obtained using numerical integration. This, however, does not give qualitative insight into the dependence of critical response behaviours on the variable parameters. The present work attempts to obtain an analytical expression for the response using a novel perturbation-based method, where moving mass-to-beam mass ratio is taken as the perturbation parameter. For a typical simply supported beam, expressions are derived in polynomial series form, up to second-order term. Analytical response, although truncated after the second-order polynomial term, is found to be in good agreement with numerically integrated response. Sensitivity studies are performed with respect to two parameters: mass ratios and velocity ratios. In addition, magnitude of deflection maxima and corresponding vehicle location are investigated for different velocity ratios and it is shown that low velocity ratio around 0.2 produces minimum deflection maxima than the velocities in the higher range. This is a significant observation as in case of a suspected degradation in a bridge, vehicular velocity can be monitored near a ratio of 0.2 to have minimum deflection and stresses on the bridge structure.