Vibration Assessment of the Beams via the Characteristic Orthogonal Polynomials

One of the major issues facing structural engineers is assessing the effects of dynamic loads on structural systems, including beams. The importance of this matter arises in moving vehicles such as cars and trains on bridge structures that are usually simulated by beam structures. Hence, in several studies, in order to explore the dynamic response of the beam structures under the excitation of dynamic loads, various analytical and numerical methods have been utilized. In this study, to examine the easier and faster procedures aimed at finding the dynamic response of Euler-Bernoulli, Timoshenko and Higher-Order beams, a simple semi-analytical method based on the characteristic orthogonal polynomials and trigonometric functions compatible with boundary conditions is presented. To this end, discrete equations of motion are derived for the three mentioned theories due to a moving mass according to the Hamilton’s principle. Then, the governing equations are transformed into ordinary differential equations in the time domain, and by applying an approximate method, displacement field of the beam is achieved. In order to consider the efficiency, convergence rate and accuracy of this method, two numerical examples are provided to compare the results of this paper with those presented by other researchers. In this regard, in the former one, free vibration frequencies of the beam with various theories for different boundary conditions were obtained, and it is shown that, the results for all three theories, give a good convergence rate and a high accuracy. Furthermore, in the latter example, the dynamic response of the beams subjected to a moving mass for different values of the base beam slenderness was achieved and compared with other studies. Analysis of the maximum dynamic response of the beam and the time history diagrams illustrated the obtained results are in a close agreement with those issued from the numerical method; despite using the lower number of the shape functions.