EXACT SOLUTIONS FOR FREE LONGITUDINAL VIBRATIONS OF NON-UNIFORM RODS

✍ Scribed by Q.S. LI

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 194 KB
• Volume: 234
• Category: Article
• ISSN: 0022-460X
• DOI: 10.1006/jsvi.1999.2856

✦ Synopsis

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 sti!ness is expressed as a functional relation with the mass distribution and vice versa. Using appropriate functional transformation, the governing di!erential equations for free vibrations of one-step non-uniform rods are reduced to analytically solvable di!erential equations for several functional relations between sti!ness 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 "nite element method, and the proposed procedure is an e$cient and exact method.