This study presents an approximate solution to Euler-Bernoulli beams on biparametric elastic foundation via Legendre polynomials and Newmark-beta method subjected to a moving distributed mass, with full retention of inertia coupling between the travelling mass patch and the host structure. The continuum formulation yields a non-autonomous fourth-order PDE whose spatial operators are projected onto a compact Legendre–Galerkin basis that enforces boundary conditions and delivers spectral convergence for smooth mode shapes. The projection produces a reduced-order, time-dependent system, in which varies with the moving mass distribution and contains both distributed forcing and inertial coupling terms. Time integration employs an energy-consistent implicit Newmark–β scheme augmented with consistent inertial corrections for the non-stationary mass matrix, ensuring unconditional stability and accurate capture of transient, near-resonant, and steady-state dynamics. Comprehensive parametric studies such as foundation moduli, flexural rigidity, cross-section, mass, damping, load intensity and speed  demonstrated that: distributed moving mass smooths spatial response but can shift natural frequencies and induce strong mode coupling at critical speeds, Pasternak shear coupling notably reduces local peaks and raises critical velocities relative to Winkler support; and damping and increased stiffness mitigate dynamic amplification while moving-mass inertia produces speed-dependent amplification or attenuation depending on modal alignment. The method is validated against limiting analytical solutions and converges rapidly with few Legendre modes, offering a robust, reproducible toolkit for design and control of structures subject to high-speed moving masses.