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BELL SYSTEM TECHNICAL JOURNAL 



C02 = 



(4.3.1) 



The maximum acceleration of m^ is thus i'co2 . If this magnitude of ac- 

 celeration were reached very slowly, so as not to excite transient longitudinal 

 waves in the bar, the maximum force between the bar and m^ would be the 

 product of the acceleration and the mass of the bar: 



F = voi.pAf. (4.3.2) 



Hence the strain at the end of the bar attached to ^2 would be 



VOi-2 pC 



fo 



E 



(4.3.3) 



Our problem is to find the ratio of the maximum transient strain to eo 



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^1 = TTa 



Fig. 4.3.2 — Amplification factors for an element of the packaged article having dis- 

 tributed mass and elasticity. The package has linear undamped cushioning and perfect 

 rebound. See equations (4.3.15), and (4.3.14). 



Let u be the displacement of a transverse plane section of the bar distant x 

 from the end attached to mo . Then, the equation of motion of the bar is 



d''u 



d'u 



(4.3.4) 



where a is the velocity of propagation of longitudinal waves in the bar: 



(4.3.5) 



E 

 P 



Taking the instant of first contact of the cushioning with the floor to be 

 / = 0, we know, from Part II, that the system will leave the floor when / = 



