The acceleration, a, at entry was then calculated. The velocity and 

 penetration depth at At seconds after soil entry were calculated 

 according to the relations 



V. = V + aAt (17) 



At o 



V V 



x^^ = X + ( ^^ 't ° ) At (18) 



At o / 



where v = entry velocity 

 o 



V. = velocity after At seconds 



At ^ 



X = entry penetration depth (=0) 



X. = penetration depth after At seconds 



At '^ 



A new value of acceleration, a, was then calculated and the 

 iterative process continued. The completion of penetration was taken 

 as the point at which the velocity became equal to zero. 



This procedure would converge to an exact solution as At approached 

 zero. In the calculations presented here, a value of At equal to .01 

 second was used. Other values of At were investigated to determine 

 whether a solution based on this numerical procedure using this value 

 of At would be stable. It was found that varying At between .025 

 second and .001 second altered the predicted ultimate penetration 

 depth by only .3 percent. This was judged to be sufficiently accurate 

 for the problem under consideration. All calculations were performed 

 on a high speed computer. 



The difference between this procedure and that of Smith are as 

 follows : 



1. This procedure considers F , F , and F while Smith's does not. 



2. This procedure uses a different relation for F„p. 



3. The numerical procedure used here involves an integration with 

 respect to time while the Smith technique uses an integration with 

 respect to penetration depth. Aside from the difference in force 

 evaluation techniques, the two approaches would converge for At and 



Ax (penetration depth increment) approaching zero. However, for 

 finite values of At and Ax, the solutions are probably somewhat 

 different. 



It should be noted that the technique used is a "pseudo-static" 

 approach. Static soil bearing capacity equations are used, and all 

 inertial effects are artificially incorporated into added mass and 

 drag terms. This approach was used because existing equations could 

 be used and little additional development was necessary. A better 



16 



