Longitudinal vibrations of elastic rods of stepwise variable cross-sectioncolliding with a rigid obstacle

A frequency method of solving the problem of the longitudinal vibrations of elastic rods of stepwise-variable cross-section is proposed, taking into a ccount or ignoring the energy dissipation when they collide with a rigid ob stacle. A Laplace transformation is applied to the equation of longitudinal vibrations of the rod when there are non-zero initial conditions. For the inhomogeneous differential equation obtained, the boundary-value problem of finding the Laplace-transformed longitudinal boundary forces as functions of the boundary displacements is solved. The equations of equilibrium of th e junction points, which are a system of equations for the unknown junction displacements, are then set up. Since the corresponding coefficients are o btained by exact integration, there is no constraint on the length of the r od sections. An inverse transformation is carried out by using extremal poi nts of the amplitude-phase frequency characteristics [1] or by direct integ ration. A rod of constant cross-section of finite length is considered as a test example. The result is compared with the well-known wave solution [2] . The proposed approach is described here for the first time and imposes pr actically no constraints on the class of problems that can be considered, w hereas the existing approach leads to unsurmountable difficulties when ther e are several sections of the rod. (C) 2001 Elsevier Science Ltd. All right s reserved.