A method for computing the dynamic responses due to the interaction of two non-self-adjoint systems: a linear, one-dimensional (1D) continuum and a linear, multi-degree-of-freedom (MDOF) oscillator travelling over the continuum, is presented. The solution method is applicable to a broad class of 1D continua, whose dynamics may be governed by various linear operators and subjected to different boundary conditions. The problem is reduced to the integration of a system of linear differential equations with time dependent coefficients. These coefficients are found to depend on eigenvalues as well as eigenfunctions and eigenvectors of the continuum and the oscillator. Two examples are included, representing bridge and railway track vibrations, to demonstrate the application of the method and discuss its convergence.