A Finite Difference Approach to Modeling High Velocity/Variable Loads using the Timoshenko Beam Model

Abstract

Electromagnetic launchers (railguns) are set to replace traditional large caliber ship mounted cannons in the near future. The success of the railgun depends heavily upon a comprehensive understanding of beam behavior during periods of heavy dynamic loading. It is hypothesized that the combination of velocity transition effects, electromagnetic loading, and other non-linear or design specific effects contribute to areas of high stresses/strains over the length of the rail/beam during launch.

This paper outlines the use of the Timoshenko beam model, a model which builds upon the traditional Bernoulli-Euler beam theory with the addition of shear deformation and rotary inertia effects, a necessity for high wave velocities. Real-world experimental setups are simplified and approximated by a series of linear springs and dampers for model prediction and validation.

The Timoshenko beam model is solved using finite difference (FD) methods for the approximation of spatial derivatives and MATLAB ordinary differential equation (ODE) solvers. The model shows good convergence and precision over a large range of system parameters including load velocities, foundation stiffness values, and beam dimensions. Comparison to experimental strain data has validated model accuracy to an acceptable level. Accuracy is further enhanced with the inclusion of damping and non-linear or piecewise effects used to mimic experimental observations. The MATLAB software package presents a valid preliminary analysis tool for railgun beam and foundation design while offering advantages in ease of use, computation time, and system requirements when compared to traditional FEA tools.