396 BELL SYSTEM TECHNICAL JOURNAL 



Differentiating (2.2.8) twice with respect to /, we find, for the acceleration 

 .^2 = —oiidm sin co2^ = — aj2 'V2gh sin 032t. (2.2.9) 



Hence the absolute magnitude of the maximum acceleration is 



^ _ I ^2 |max _ iO^dm _ Ilkki /"l O 1 n^ 



g g y W-i 



as before. 



W- 9 - 



Fig. 2.2.1 Fig. 2.2.2 



Fig. 2.2.1 — Half-sine-wave pulse acceleration. See equation (2.2.9). 

 Fig. 2.2.2 — Oscillogram of a half-sine-wave pulse obtained with a piezo-crystal 



accelerometer. 



Equation (2.2.9) shows that the acceleration varies sinusoidally with time. 

 It rises from its initial zero value to its maximum in a time 7r/2a)2 , at which 

 time the displacement also reaches its maximum value. The acceleration 

 returns to zero again at time 7r/co2 • At this time the displacement is also 

 zero. This is the end of the range of appHcabiUty of equation (2.2.9); for 

 when / becomes slightly greater than ir/oii , a tension in the spring is required. 

 Since no mechanism, such as a large mass ms (Fig. 0.2.2), has been supplied, 

 to allow a tension in the spring to develop, the system will rebound from the 

 floor at the end of the half period 7r/cu2 . The acceleration is thus a half- 

 sinusoidal pulse of duration 7r/co2 = T^/I and amplitude Gmg as illustrated 

 in Fig. 2.2.1. An oscillogram of such a pulse obtained with a piezo-crystal 

 accelerometer is shown in Fig. 2.2.2. 



2.3 Package Rebound. 



The presence of the mass of an outer container will ajffect the acceleration 

 after the first half cycle of displacement. The outer container is represented 

 by the mass Ws in the general idealized system illustrated in Fig. 0.2.2 and in 

 the simpler system (Fig. 2.3.1) that we shall consider now. 



