Dynamic Response of a Non-Uniform Rayleigh Beam Under Accelerating Distributed Masses on a Bi-Parametric Elastic Foundation

This study investigates the problem of a non-prismatic Rayleigh beam subjected to moving loads with arbitrarily prescribed velocities. The research addresses a complex structural dynamics problem that has significant implications for various engineering applications, including railway systems, bridge structures, and industrial machinery where moving loads interact with flexible structural elements. The non-uniform Rayleigh beam model incorporates both translational and rotational inertia effects, providing a more accurate representation of real-world structural behaviour compared to classical Euler-Bernoulli beam theory. The beam's non-uniformity is characterised by spatially varying material properties and geometric parameters, which significantly influence the dynamic response patterns. The bi-parametric elastic foundation is modelled using Winkler and Pasternak foundation parameters, accounting for both normal and shear interactions between the beam and its supporting medium. The accelerating distributed masses represent a realistic loading scenario encountered in many practical applications, where the velocity and acceleration of moving loads vary continuously along the beam span. This loading condition introduces complex inertial effects and coupling between the beam's natural vibration modes and the motion characteristics of the distributed masses. Gerlakin's weighted residual method is employed to treat this vibrating system problem, as it was in the preceding section. This technique is first used to transform the fourth-order partial differential equation with singular and variable coefficients governing the motion of this vibrating beam. The resulting system of equations, called Gerlakin's equations, is further simplified using the asymptotic method of Struble to obtain a second-order ordinary differential equation, which is then solved using the Duhamel integration method.