A Perturbation Method Based Solution for Beam Vibration Excited by Moving Mass
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ORIGINAL CONTRIBUTION
A Perturbation Method Based Solution for Beam Vibration Excited by Moving Mass Animesh Chatterjee1 • Tanuja Vaidya2
Received: 18 May 2019 / Accepted: 15 October 2020 Ó The Institution of Engineers (India) 2020
Abstract 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-tobeam 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 & Tanuja Vaidya [email protected] 1
Department of Mechanical Engineering, Visvesvaraya National Institute of Technology, Nagpur, India
2
Department of Mechanical Engineering, Anjuman College of Engineering and Technology, Nagpur 440010, India
bridge, vehicular velocity can be monitored near a ratio of 0.2 to have minimum deflection and stresses on the bridge structure. Keywords Moving mass Vibration of beam Perturbation method Vehicle-bridge dynamics
Introduction The subject of studying dynamic response of beam like structures, such as a highway bridge under the action of a moving vehicular mass, has become an important research area in the recent past. Here the force is not stationary in space but is moving along with the vehicle position. This type of problems, commonly termed as moving force problem (MFP), was initially modelled as moving load problem [1], where only the weight of the vehicle was considered acting at continuously progressing points on the beam. Later on, researchers improved the model by including moving mass inertia effects and further extended it by considering the vehicle as a moving oscillator. The equation of motion of moving mass problem becomes more complicated as the inertia effects introduce three pseudo-excitation terms, due to
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