Abstract
In this paper, the function for describing the distribution of mass of a non-uniform rod is arbitrary, and the distribution of longitudinal stiffness is expressed as a functional relation with the mass distribution and vice versa. The governing differential equations for free longitudinal vibration of rods with variable cross-section are reduced Bessel's equations or other analytically solvable differential equations by selecting suitable expressions, such as power functions and exponential function for the functional relations. Simple formulas to predict the longitudinal vibration frequencies and mode shapes of one-step rod with continuously varying cross-section are presented. The obtained exact solutions are used to obtain the frequency equation of a multi-step non-uniform rod. This approach which combines the transfer matrix method and closed-form solutions of one step non-uniform rods leads to a single frequency equation for a multi-step non-uniform rod with any number of steps.
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