The paper studied the transverse vibrations of a continuous support with rigid, elastic or mixed supports, which was excited by a class of moving internal load. The initial imperfection of the support, which is the result of technological imperfection of the structure or the action of static influences on the structure in an unexcited state, was also taken into consideration. Following the idea of the force method, the influence of intermediate supports is replaced by an equivalent reactive load, which is defined by the Dirac function by a set of dynamic reactions $R_i(t)$, $(i=1,2,\dots,k_1)$ in rigid intermediate supports and dynamic reactions $\tilde R_j(t)$, $(j=1,2,\dots,k_2)$ in elastic intermediate supports. Starting from Hamilton's principle, the task is reduced to a boundary value problem defined by the partial differential equation (3) and the conditions in the intermediate supports (5). The problem was reduced to a matrix integral equation of the Volterra type using the Fourier method: \[ ıt^t_0\utilde{R}_j(t)\utilde{G}(t,au)dau+\utilde{C}\utilde{R}^T(t)=A(t) \] from which it is possible to directly determine the dynamic reactions in the intermediate supports, which are defined by the $k$-dimensional vector $\utilde{R}(t)$, where $k=k_1+k_2$ is the number of intermediate supports on the support.