Abstract <p>The vibration of accelerator rails under the action of electromagnetic forces is considered. The armature and with it the right boundary of the application of electromagnetic forces move along the barrel’s bore from the breech to the muzzle. The rail accelerator is simplified and considered as a Bernoulli–Euler beam of finite length lying on a viscoelastic foundation with cantilever support from the breech of the accelerator. The vibration of the rail is described by a partial differential equation of the fourth order in coordinate and the second order in time. Using the method of superposition of modes, an analytical solution of the equation is obtained taking into account the change in the velocity of the armature along the length of the channel. Taking into account the features of the formula for the eigenforms for this problem makes it possible to simplify calculations and take into account the necessary number of harmonics along the entire trajectory of the armature, which is difficult with the traditional application of the method, and thereby increase the precison of predicting the amplitude and nature of rail vibrations during acceleration. The proposed approach makes it possible to test and debug numerical methods for solving complex problems, as well as to design the rail accelerator channel more correctly.</p>