A computational method for free transverse-vibration analysis of stepped-beams

In this method a stepped beam is substructured into its n spans, each of which, except the ones at extreme ends, is treated as a uniform continuous beam clamped to its adjacent spans. This implies that at each step of the shaft any two adjacent spans have identical displacements, slopes, bending moments and shearing forces. Theoretically speaking, these boundary conditions along with those of the extreme ends (total of 4n), when applied to the vibration equation of corresponding spans, generate a set of 4n equations. Then, this set of simultaneous equations should be solved to generate natural frequencies of the beam as well as the constants of the vibration equation of all spans. An efficient mathematical algorithm is furnished herein to cluster the above-mentioned equations into groups of four and solve them one after the other. The algorithm is programmed in VISUAL C++, and its accuracy justified via a number of examples.