Abstarct: This thesis concerns the study of vibrations of a viscous cantilever beam with a viscous end condition under moving load and its dynamic response. The equations of motion are obtained by extended Hamilton's principle and the Eigen value problem develop by applying the boundary conditions and the separation theorem. The equation of lateral displacement or mode shape of beam is effectiveness of numerical solution of Eigen value problem. Modal analysis and procedure of solving equations is differing from existing common problems due to presence of distributed viscously damping. The self-adjointness and orthogonality of eigen function is one of the considerable cases; The orthogonality of eigen function is obtained without parametric symbolization. The dynamic response of harmonic moving load with constant velocity in analytical and either numerical solution is in center focus of consideration. In both analyses, especially on non-zero initial condition, implicit solution will become complicated, appropriate solution and comparison is available with take advantage of subroutine. Discussion about tow instance problems and step by step numerical solution of them according to obtained formulas, give a reasonable approach to understanding the tabled topics; by varying the boundary and initial conditions, beam characteristics and loading type, we would can inspect and calculate similar cases even different matters.