A Comparative Analysis of Simply Supported and Clamped-Free Beam Models with Moving Masses on Pasternak Elastic Foundations

Abstract

In this study, we develop a unified, high-fidelity framework to predict the transient dynamics of Euler–Bernoulli beams on a Pasternak elastic foundation subjected to moving masses, explicitly retaining the inertia coupling between the travelling mass patch and the host structure. Spatial discretization is performed with a Legendre–Galerkin projection that enforces boundary conditions exactly and delivers spectral convergence for smooth spatial fields; time integration uses an energy-consistent implicit Newmark–? algorithm that handles the resulting time-varying generalized mass matrix and inertial correction terms without loss of stability. The formulation is applied to two practically important boundary scenarios: simply supported and clamped–free (cantilever) to expose how support constraints alter modal participation, peak deflections, and critical-speed behavior. Parametric studies across foundation moduli (Winkler, Pasternak shear), flexural rigidity, cross-section, mass per unit length and moving-mass width/speed reveal three robust findings: moving mass smooths spatial response but shifts natural frequencies and induces strong mode coupling near critical speeds, Pasternak shear coupling markedly reduces local peaks and elevates critical speeds relative to Winkler support; and boundary condition strongly modulates sensitivity. Cantilevers concentrate response near the free end and are more sensitive to local reinforcement, whereas simply supported spans distribute effects more uniformly. The method is validated against limiting analytic cases and converges rapidly with few modes, offering a practical, reproducible toolset for designers and researchers confronting heavier, faster, and spatially extended moving masses