Abstract
The equations of motion of longitudinal vibrations of rods according to the theory of Bishop are rederived using Hamilton's principle, in the process of which the associated boundary conditions are also obtained. The frequency equations and mode shapes for ten different combinations of boundary conditions are given. The influence of the secondary effects an the fundamental frequency of rods is illustrated and it is shown that the effect of lateral inertia is almost three times as important as the effect of shear stiffness. Studies of free and forced vibrations of rods by the eigenfunction expansion process show that the higher-order theory predicts a maximum response which is less than the one predicted by the elementary theory of rods. Comparison with the available experimental data show that the Bishop theory represents a better model for predicting the response of a free-free rod subjected to a suddenly applied force.
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