The study suggests a method to analyze the vibration of the multi-stepped beam having the different circular cross-sections. The rotatory inertia, the shear deformation and the torque applied at both ends of the beam are considered in the governing equation. The complex displacement and the variable separation are introduced to derive the solution of the equation of each uniform beam element having constant cross-section. Then boundary conditions are applied to solve the total system. This method uses the mathematically exact solutions unlike numerical method such as the finite element method in solving the problem having the simultaneous differential equations of Timoshenko beam theory. the natural frequencies and the corresponding mode shapes are precise, especially the mode shapes are continuous.
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