文摘
A frequency method of solving the problem of the longitudinal vibrations of elastic rods of stepwise-variable cross-section is proposed, taking into account or ignoring the energy dissipation when they collide with a rigid obstacle. 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 the junction points, which are a system of equations for the unknown junction displacements, are then set up. Since the corresponding coefficients are obtained by exact integration, there is no constraint on the length of the rod sections. An inverse transformation is carried out by using extremal points of the amplitude-phase frequency characteristics [1] or by direct integration. 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 practically no constraints on the class of problems that can be considered, whereas the existing approach leads to unsurmountable difficulties when there are several sections of the rod.
