Tank Munitions and Vector Calculus
In this chapter, we describe the use of vector calculus to model armor piercing tank rounds. Along the way, we review mathematical operations with vectors, as familiarity with the mathematics is prerequisite to understanding the application. Mathematical coverage begins with the basic operations of vector arithmetic and continues with exercises applying the dot product, cross product, and gradient. Vector calculus is specifically applied to the modeling of sabot rounds, a type of tank ammunition described in the following paragraphs. In order to discover the connection between the mathematics and the application, it is important that you do the Exercise Sets. You may have to refer to your multivariable calculus text to do the exercises. For this reason, keeping your textbooks for future references is probably a good idea.
To penetrate the massive armor plating protecting the crew of modern tanks, a projectile must be fast-moving, aerodynamic, and have massive momentum. The sabot round pictured in Figure 2 is such a projectile. Do not let the slim profile mislead. The projectile is made of extremely dense materials and propelled to achieve incredible speeds, giving the penetrator the momentum necessary to puncture protective armor tens of centimeters thick.
The slender penetrator must be guided down the greater diameter tank main gun barrel. The guiding sleeve, or sabot, consists of petals which break apart upon exit from the gun tube, as in Figure 3. Optimal design of the sabot round must ensure the sleeve separates from the penetrator without interfering with the flight dynamics of the penetrator, which will speed to a target. Testing the design by repeated firing of the very expensive rounds is cost ineffective. Computer simulations are an efficient alternative to the destructive and costly trial-and-error methods. Programmed into the simulations are algorithms implementing applications of the vector calculus.
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The Model
In the milliseconds after the explosion which propels the sabot round out the tube of the tank cannon, the round is subject to intense heat and pressure. In order to model the influence of the heat and fluid pressure on the flight dynamics of the projectile, millions of computations are performed in the process of the computer simulation. The effects of heat and pressure on the blocks are simulated using vectors.
Figure 4. Discretized model of sabot round, graphic courtesy of ARL.
The pressure field affecting the sabot round is modeled with a vector-valued function. As an example, the function
represents the pressure field that exists in the gun tube.
(lots of stuff snipped, go there if yer interested)
