A Generalized Equation of Motion for Dynamic Analysis of a Uniform Beam Subjected to Multiple Moving Loads

PurposeThis paper presented a generalized equation of motions, where the effects of damping forces of the entire vibrating system and inertial force of each moving load were considered, and the forced vibration problem regarding the uniform beams subjected to multiple moving loads can be easily solved.MethodsTo the above end, each mode of vibration was considered as one degree of freedom (DOF) of the entire vibrating system. Then, the mode-superposition method (MSM) and Rayleigh damping theory were incorporated, and the equation of motion: mn ' xn 'eta<spacing diaeresis>n ' x1+cn ' xn 'eta(center dot)n ' x1+kn ' xn 'eta n ' x1=fn ' x1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\left[m\right]}_{{n}<^>{\prime}\times {n}<^>{\prime}}{\left\{\ddot{\eta }\right\}}_{{n}<^>{\prime}\times 1}+{\left[c\right]}_{{n}<^>{\prime}\times {n}<^>{\prime}}{\left\{\dot{\eta }\right\}}_{{n}<^>{\prime}\times 1}+{\left[k\right]}_{{n}<^>{\prime}\times {n}<^>{\prime}}{\left\{\eta \right\}}_{{n}<^>{\prime}\times 1}={\left\{f\right\}}_{{n}<^>{\prime}\times 1}$$\end{document}, was obtained. Finally, solving the matrix equation for {eta}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ \eta \}$$\end{document} by using the Newmark's direct integration method, one may obtain the vertical deflections of the beam at any position x and time t.ResultsThe numerical examples revealed that the presented method is easy to tackle the dynamic problem regarding the uniform beams under any number of moving loads with effects of damping forces and inertial forces considered (or neglected), and the obtained results are in good agreement with those obtained from the FEM.ConclusionAlthough the forgoing "generalized" equation of motions for the "analytical" method is similar to the "conventional" one for the "numerical" FEM, the CPU time required by the former was much less than that required by the latter, because the total mode numbers n '\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n}<^>{\prime}$$\end{document} (in general n '\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n}<^>{\prime}$$\end{document} <= 15) considered by the presented method is much smaller than the total DOFs (in general n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n}$$\end{document} >= 80) considered by the FEM.