First Frequency of Longitudinal Vibration of a Rod with Mass at one End and Partial Placement on Piecewise-Homogeneous Elastic Foundation

Abstract

Statement of the problem. Despite the introduction of engineering practice of numerical methods for the calculation, in a case of engineering problems associated with extended structures, the well-known models «rod on an elastic foundation» and «beam on an elastic foundation» are widely used to describe the behavior of structures. Most of the known solutions were obtained in the conditions of an idealized setting and without differences in the properties of the foundation. This work is devoted to the study of the dynamics of a rod of finite length with a point mass at the end and partially placement on a piecewise-homogeneous elastic foundation, formed by two sections of different stiffness. The aim of this study is to obtain analytical dependences for the first frequency of the longitudinal vibrations of a rod with different boundary conditions at its ends: rigid support and free end. The available research results are obtained for special cases, which makes them difficult to apply in practice. To assess the influence of the inertial pro-perties of the foundation, the case with different linear masses of the rod in the pinched and cantilever sections is considered. Results. Using the Ritz method analytical dependences were obtained for the determination of the first frequency of the longitudinal vibrations of a fix and free rod with a point mass at the end and partially placement on a piecewise-homogeneous elastic foundation formed by two sections of different stiffness. Quantitative evaluation shows good agreement with those of other studies. The results for particular cases are obtained. Conclusions. It is shown that the first natural frequency of longitudinal oscillations is in a certain range and in the nonlinear ratios depending on the lengths of the sections various elastic. The inertial properties of the foundation leads to a decrease in the first frequency.