Free vibrations of beam-mass-spring systems: analytical analysis with numerical confirmation

Free vibrations of a beam-mass-spring system with different boundary conditions are analyzed both analytically and numerically. In the analytical analysis, the system is divided into three subsystems and the effects of the spring and the point mass are considered as internal boundary conditions between any two neighboring subsystems. The partial differential equations governing the motion of the subsystems and internal boundary conditions are then solved using the method of separation of variables. In the numerical analysis, the whole system is considered as a single system and the effects of the spring and point mass are introduced using the Dirac delta function. The Galerkin method is then employed to discretize the equation of motion and the resulting set of ordinary differential equations are solved via eigenvalue analysis. Analytical and numerical results are shown to be in very good agreement.