Features of the numerical implementation of the rail vibration equation analitical solution on visco-elastic foundation in an electromagnetic accelerator

The accelerator rails vibration under the action of electromagnetic forces is considered. The armature and with it the right boundary of electromagnetic forces application move along the barrel channel from the breech to the muzzle. The rail accelerator is simplistically considered as a Bernoulli–Euler beam of finite length lying on a visco-elastic 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 peculiarities of the formula for the eigenforms for this problem made 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 accuracy 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.