398 BELL SYSTEM TECHNICAL JOURNAL 



2.4 Motion After Rebound 



If rebound occurs, the equations of motion for the two masses, W2 and Wg , 

 are 



W2i-2 + ^2(^2 — xz) = m^g, (2.4.1) 



nizXz - kii^Xi — X3) = nisg. (2.4.2) 



Multiplying (2.4.1) by W3 and (2.4.2) by m^ and subtracting, we find 



my + k2y = 0, (2.4.3) 



where 



y = X2 — X3 , (2.4.4) 



W2W3 

 ni2 -\- ms 



m = ■ . (2.4.5) 



Fig. 2.4.1— Oscillogram illustrating the half-sine pulse followed In- the higher frequency, 

 lower amplitude vibration of the packaged article in a rebounding package. 



Equation (2.4.3) is the equation governing the vibration of the two-mass 

 system as a simple oscillator. The circular frequency of the vibration is 



Vm 



(2.4.6) 

 m 



and it may be noticed that this frequency is always greater than uo (equation 

 (2.2.6)). This fact is important in estimating the effect of vibrations on 

 elements of the packaged item (Section 3.5). 



CO is also the frequency of vibration of the packaged article during the 

 interval of free fall. This vibration (usually of small amplitude) is initiated 

 by the sudden release of the dead load displacement of the packaged article. 



As an intermediate step in obtaining the acceleration after rebound we 

 shall find the magnitude of the relative displacement (y) of the two masses. 

 To do this it is necessary to solve equation (2.4.3) with the appropriate 

 boundary conditions. Calling tr the time at which nis leaves the floor, we 



