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Exact longitudinal vibration characteristics of rods with variable cross-sections
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Longitudinal natural vibration frequencies of rods (or bars) with variable cross-sections are obtained from the exact solutions of differential equation of motion based on transformation method. For the rods having cross-section variations as power of the sinusoidal functions of ax+b, the differential equation is reduced to associated Legendre equation by using the appropriate transformations. Frequency equations of rods with certain cross-section area variations are found from the general solution of this equation for different boundary conditions. The present solutions are benchmarked by the solutions available in the literature for the special case of present cross-sectional variations. Moreover, the effects of cross-sectional area variations of rods on natural characteristics are studied with numerical examples.