DYNAMICS OF PACKAGE CUSHIONING 



439 



3.6 Amplification Factors for the Pulse Acceleration of 

 Cubic Cushioning 



In a rebounding package with undamped Class B cushioning, the pack- 

 aged article (W2) will undergo a pulse acceleration of duration ir/w2 as given 

 by equation (2.8.11). The shape of the pulse is illustrated in Fig. 2.8.2 

 and its functional form is 



X2 = 



4K 



' W2 dm r 2 2 /2K(j}2 1 „ 



—5 2k sn I — K 



TT- |_ \ ■T 



)-] 



f2Koj2t _ 



en I 



\ IT 



.), 



(2.8.14) 



To determine the influence of the shape and duration of this pulse on the 

 amplification factor, we proceed as before by substituting (2.8.14) in the 



r 



////////////////////y 



/// ///77T//////// 



(a) (b) 



Fig. 3.6.1 — Idealized system used in calculating amplification factors for non-linear, 

 undamped cushioning with perfect rebound. 



differential equation governing the relative displacement {x = Xi — x^ 

 between m\ and m-i (see Fig. 3.6.1): 



x-\-oiiX= —Xl. (3.6.1) 



With boundary conditions x(0) = .f(0) = 0, the solution of (3.6.1) may be 

 written as 



X = — I X2 (X) sin coi (X — t) d\ 

 COi Jo 



and the maximum value of x may be expressed by 



1 r''" 

 Xmax = — / XoiX) sin aji(X — tm) d\, 



COi Jo 



where /„, is the time at which the largest value of x occurs. 



(3.6.2) 



(3.6.3) 



