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This work focuses on the dynamic response(s) of Rayleigh beams under moving concentrated masses with time dependent boundary conditions. This dynamic problem with time dependent boundary conditions, are more difficult to solve unlike the non classical boundary conditions which is common in literature and straight forward to solve. The novelty in this work is a systemic approach that can be categorized into two: 1. The technique of solution that are not direct but required several transformations to reduce the problem to second order solvable equations without the loss of any parameter. 2. The use of Strubble's techniques coupled with the programming of the closed form solution using QBasic programming which took between thirty two to one hundred & twenty (32-120) pages to write the codes for each and every of the solutions to arrive at the numerical solutions and to plot them into graphs and finally compare the results with the existing literature for agreement or disagreement. Irecommend this book to prospective master's and Doctoral students of vibration analysis in applied Mathematics, engineering. Prof. A.W.Gbolagade.