In this article, OPSEM (Orthonormal Polynomial Series Expansion Method) is developed as a new computational approach for the evaluation of thin beams of variable thickness transverse vibration. Capability of the OPSEM in assessing the free vibration frequencies and mode shapes of an Euler–Bernoulli beam with varying thickness is discussed. Multispan continuous beams with various classical boundary conditions are included. Contribution of BOPs (Basic Orthonormal Polynomials) in capturing the beam vibrations is also illustrated in numerical examples to give a quantitative measure of convergence rate. Furthermore, OPSEM is adopted for the forced vibration of a thin beam caused by a moving mass. Dynamics of beams supported by flexible elastic base like free to free beam on elastic foundation are also regarded. Verifications are made via eigenfunction expansion method and GMLSM (Generalized Moving Least Square Method). The very close observed agreement between the results of the two recently mentioned methods and that of OPSEM can be regarded as a guarantee of validity for the newly introduced technique. In comparison with eigenfunction expansion method, the simplicity and handiness of OPSEM in coping with different boundary conditions of the beam can be considered as its benefit for engineering practitioners.