Exact solutions to the free vibration of stepped beams under axial load

Axial loads on shafts have been shown to cause substantial changes in the vibration response characteristics of beams. In particular, the lateral vibration natural frequencies tend to increase with increasing tension in accordance with the system acting somewhere in between a free beam and a tensioned string. Conversely under a compressive axial load the fundamental frequency reduces towards zero at the buckling load. Previous research has developed exact solutions for a uniform beam under axial load and stepped beams under no load. The present research extends this work to provide exact analytical solutions for stepped beams under axial loads for a variety of boundary conditions. Solutions are shown to conform with those from uniform and unloaded conditions and are conveniently tabulated for general use. In addition, a nonlinear finite element analysis (FEA) is performed to confirm analytical predictions for more general conditions. The results are useful for obtaining efficient solutions and confirming FEA results of more complicated systems