Transient and quasi-steady-state vertical vibrations of a multi-span beam steel bridge located on a single-track railway line are considered, induced by a superfast passenger train, moving at speed 120-360 km/h. Matrix dynamic equations of motion of a simplified model of the system are formulated partly in the implicit form. A recurrent-iterative algorithm for solving these equations is presented. Excessive vibrations of the system in the resonant zones are reduced effectively with passive dynamic absorbers, tuned to the first mode of a single bridge span. The dynamic analysis has been performed for a series of types of bridges with span lengths of 10 to 30 m, and with parameters closed to multi-span beam railway bridges erected in the second half of the $20^{th}$ century.