Vibration of a Beam on a Pasternak Foundation Under the Action of Moving Loads

Abstract

The article is devoted to solving an essential applied problem in deformable solid mechanics. When using refined models for an elastic foundation, obtaining more accurate numerical results for practical applications is possible, albeit with increased mathematical complexity. The study aims to develop a mathematical vibration model for an Euler–Bernoulli isotropic elastic beam with infinite length on the two-parameter Pasternak foundation. The stress-strain state under the action of moving uniform loads was studied. The initial differential equation was solved using the direct and inverse Fourier transforms. The proposed transformations eliminated the need for contour integration of the corresponding function. As a result, an analytical solution to the differential equation of beam vibrations was obtained. Numerical results of evaluating normal displacements of the beam points were also obtained and analyzed. The integrand function of the improper integral was graphically analyzed, and its change at different loads was shown. The accuracy of the boundary conditions was verified. The integration interval convergence also checked the convergence of the corresponding series. Diagrams of beam deflections at different velocities and time steps were shown. The oscillations of the beam point were calculated depending on the velocity of the moving force. Overall, the developed mathematical model makes it possible to reliably determine the strain state of beams on an elastic foundation under the action of moving loads. The developed model can be applied to solve specific problems related to solid mechanics.