Exact solutions for longitudinal vibration of multi-step bars with varying cross-section

Using appropriate transformations, the equation of motion for free longitudinal vibration of a nonuniform one-step bar is reduced to an analytically solvable equation by selecting suitable expressions, such as power functions and exponential functions, for the area variation. Exact analytical solutions to determine the longitudinal natural frequencies and mode shapes for a one step nonuniform bar are derived and used to obtain the frequency equation of multi-step bars. The new exact approach is presented which combines the transfer matrix method and closed form solutions of one step bars. A numerical example demonstrates that the calculated natural frequencies and mode shapes of a television transmission tower are in good agreement with the corresponding experimental data, and the selected expressions are suitable for describing the area variation of typical high-rise structures.