Exact solutions for free longitudinal vibrations of non-uniform rods

An exact approach for free longitudinal vibrations of one-step non-uniform rods with classical and non-classical boundary conditions is presented. In this paper, the expression for describing the distribution of mass is arbitrary, and the distribution of longitudinal stiffness is expressed as a functional relation with the mass distribution and vice versa. Using appropriate functional transformation, the governing differential equations for free vibrations of one-step non-uniform rods are reduced to analytically solvable differential equations for several functional relations between stiffness and mass. The fundamental solutions that satisfy the normalization conditions are derived and used to establish the frequency equations for one-step rods with classical and non-classical boundary conditions. Using the fundamental solutions of each step rod and a recurrence formula developed in this paper, a new exact approach for determining the longitudinal natural frequencies and mode shapes of multi-step non-uniform rods is proposed. Numerical examples demonstrate that the calculated longitudinal natural frequencies and mode shapes are in good agreement with the experimental data and those determined by the finite element method, and the proposed procedure is an efficient and exact method.