The free vibration of an elastically restrained symmetric non-uniform Timoshenko beam resting on a non-uniform elastic foundation and subjected to an axial load is studied. The two coupled governing characteristic differential equations are reduced into two separate fourth order ordinary differential equations with variable coefficients in the angle of rotation due to bending and the flexural displacement. The frequency equation is expressed in terms of the four normalized fundamental solutions of the associated differential equation. A simple and efficient algorithm is developed to find the approximate fundamental solutions of the governing characteristic differential equation. The relation between problems with elastically restrained boundary conditions and those with tip-mass boundary conditions is explored. Finally, several limiting cases are examined and examples are given to illustrate the validity and accuracy of the analysis. It is noted that the proposed analysis can also be applied to stepped beam problems.
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