A new formulation is given to the problem of vibration of beams or rods carrying a concentrated mass in which the Dirac δ-function is used in the differential equation to describe the effect of a concentrated mass. The homogeneous differential equation which describes the free vibration is solved first by separation of variables and the resulting ordinary differential equation is then solved by Laplace transform to yield the eigenfunctions of the problem. Orthogonality relation, derived by treating the beam as a variable-density beam, is used to solve the general initial-value problem as well as the forced-vibration problem.