Abstract
A new exact approach that combines the general solutions and recurrence formula developed in this paper for determining natural frequencies and mode shapes of multistep flexural beams with varying cross sections is proposed in this paper. The frequency equations of such beams with several concentrated masses, rotational springs, and translational springs at the ends and intermediate points can be easily established from a determinant of order two. The general solutions for free vibrations of five classes of nonuniform beams are found and expressed in terms of Bessel functions, triangular functions, and hyperbolic functions. Numerical examples show that the calculated natural frequencies and mode shapes are very close to the corresponding field measure data, and the proposed procedure is a simple, efficient, and exact method.