733 
=/OTA 
(i) Precision. The standard deviation of a single ball-crusher gauge 
on a single shot was from 2 to 3 percent, calculated from the difference of 
readings of the two gauges in the same pair for a number of shots. A de- 
tailed analysis of many ball-crusher results from different sources by 
Brown [8] is in agreement with this estimate. 
(c) Theory. -~ The ballecrusher gauge was designed and constructed by 
the Naval Ordnance Laboratories and used by them to evaluate underwater ex— 
plosion effectiveness empirically in terms of the deformation of the copper 
sphere. Static calibrations of deformation in terms of applied pressure on 
the gauge were also available, later, Hartmann developed the theory for the 
ball-crusher gauge to enable the calculation of the absolute value for the 
peak pressure. This involved the solution of the differential equation, 
EP leo en Piet 
m 
for the gauge. Here Mis the mass of the moving parts (piston, water behind 
it, and a portion of the conper), x-the piston displacement, x the piston 
acceleration, k the force constant of the copper sphere, A the piston area, 
Pm the peak pressure and j(= 1/8) the decay constant of the shock wave, and 
t the time measured from the incidence of the shock wave on the gauge. The 
solution is: 
esr A (4 sinwt — cosat + ener ) 
Fo moe ae Nee 
where 
@ =\ 
A partially analytical treatment of this equation has been carried through 
by E. B, Wilson, Jr. [22], who found that the equation 
P aL /2 
ot = 0.500 + 0.806(H\ 
2 k 
a 
expresses the relation with sufficient accuracy over the range (tie) 1/2 
from 0 to 1,0. Ps is the apparent peak pressure calculated from the force 
constant of the sphere and its deformation according to Pa = kx/A. Sub- 
stitution yields 
oh 04500 + 04806{ ¢ 
For 3/8-in. diameter copper spheres with a force constant of 349 x 105, the 
eae becomes Pm/x = 7.5 + 6.724Mwhere x is in 1o73 in. and P is 1b/in? 
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