A Mathematic Model for Dynamic Transient Response of Electromagnetic Rail on Elastic Foundation

Lei Shi,Xiangzhong Bai

doi:10.5539/mas.v3n10p60

Abstract

The electromagnetic launcher’s rail can be modeled as a beam which is on elastic foundation with cantilevered support by moving load. In this paper Euler beam theory is applied to build the Mathematic model and the complete solution of the equation is derived in detail. At last, a numerical experiment, which analyzes the influences brought by the moving load velocity and the damping force on the transient response of beam, shows that the moving load’s velocity has a quite obvious affect on the response of the beam.

Highlights

Figure.1 shows a schematic of a electromagnetic railgun composed of power source, rail, armature and projectile
When the electric current of armature goes through the rail, it forms a strong magnetic field in the area of their encirclement
The homogeneous equation is a fourth—order partial differential equation, in order to change it into the ordinary differential equation, we solve it by the method of variable separation

Summary

Mathematical Model

Figure. shows a schematic of a electromagnetic railgun composed of power source, rail, armature and projectile. When the electric current of armature goes through the rail, it forms a strong magnetic field in the area of their encirclement. With the reaction by the magnetic field and the electric current, it emerges powerful electromagnetic force, which pushes the armature and projectile to do the accelerating motion along the rail till the projectile be launched out of the rail. Figure .2 is the physical model of the railgun —there is a cantilever beam with one end fixed and the other end free partially subjected to even load sitting on the elastic foundation. Considering the effect of the beam by the damping force and basing on the Euler beam theory, we obtain the governing equation of elastic foundation beam by moving load which is a transient fourth—order differential equation as follows(S.Timoshenko.

Solution of the homogeneous equation

General Solusion

Numerical example

Conclusions