The energy conservation principle has been applied to derive the formulation of the frequencies of free transverse vibration of beams for a given mode shape. For uniform beams with a tip mass and with either a clamped/free or a clamped/sliding or a hinged/sliding constraint, under various loads, such a centrifugal force, axial acceleration and concentrated force, the mode shape functions for the free uniform beams have been employed to develop empirical formulae, which are capable of predicting their frequencies. The predictions show a good agreement with those given by finite element analysis.