909 
It is important to note, however, that in the case of the ball crusher gauge, 
the deformation is permanent® rather than elastice Hence Eqe (I-5) does not 
represent the motion beyond the point of maximum deformation x,- 
In the problem of interest, the initial boundary conditions are: 
x=0, x=Oatt=0 (126) 
It will be the purpose of the following sections to solve Eq. (I-5) for 
various forms of F(t) in the light of these boundary conditions. 
4. Step Function: F(t) = F H(t) F | 
— >t 
Equation (I-5) may be rewritten in the notation of the Cauchy-Heaviside 
operational calculus: 
SG eee = H(t) (1-7) 
where D, is the operator 2 ana H(t) is the unit step function. 
t at 
F/M 
* "WD, + ia) (Dy - 18) Dy 
D, H(t) (T+8) 
Separating Eq- (I-8) into partial fractions: 
1 1 nt F 
aig pete Sere = Zp x(t) 
FE ae? (D, ~ 1) a0" (D, + in) ut 
1 aL (ei -iet,| F s 
ule = ee +e } 7 H(t) (I-9) 
r= (1 = cos wt) H(t) 
To determine the maximum deflection =n and the time of maximum deflection 3 
a) For a discussion of corrections necessary for the slight amount of 
elastic recovery when working at low pressures. 
-94= 
