DYNAMICS OF PACKAGE CUSHIONINV 427 



On the other hand, for frequency ratios ui/co^ < 0.5 the severity of the 

 shock can be very much less than might be expected from the calculations 

 of Part I. For ver>' small values of coi/a)2 the amplification factor may be seen 

 from (3.2.13) to be equal to 2wi/w2 • For large values of wi/coo (stiff elements) 

 Fig. 3.2.2 shows that the amplification factor is very nearly unit}' and the 

 methods of Part I can be used without additional calculation. 



When damping of the element of the packaged article is considered, the 

 amplification factors are less than without damping. The applicable 

 equations of motion during and after impact are obtained by inserting 

 velocity damping terms in (3.2.4) and (3.2.10): 



niix + Cix -\- kix ^ —miX2 , ^ / ^ — (3.2.14) 



C02 



mix' + ci:t-' + kix' = 0, / ^ -. (3.2.15) 



If we express the damping of the element nii as the fraction of critical 

 damping 



- ^1 



2 vwi^i' 



(as in Section 2.5) equations (3.2.14) and (3.2.15) become 



(3.2.16) 



9 — — TT 



X + 2((3icoi.v + ojiX = —X2 , < t < —, (3.2.17) 



W2 



x' + 2^icoi.v' + col.v' = 0, / ^ -. (3.2.18) 



The amplification factors for equations (3.2.17) and (3.2.18), with boundary 

 conditions (3.2.5) and (3.2.11), respectively, were obtained on the Westing- 

 house Mechanical Transients Analyzer^ for ,81 = 0.005, 0.01, 0.05, 0.10, 0.30, 

 0.50 and 1.00. The curves are shown in Fig. 3.2.2. 



3.3 Application of Half-Sine-Wave Amplification Factors 



As an example of the use of the amplification factor curves of Fig. 3.2.2, 

 let us consider the following problem: 



^ Arrangements lor performing these calculations were made through the courtesy of 

 Mr. A. C. Monteith, Manager of Industry Engineering, and Mr. C. F. Wagner, Manager 

 of Central Station Engineering, Westinghouse Electric and Manufacturing Co. Dr. G. D. 

 McCann, Transmission Engineer, was in immediate charge of the project. For a descrip- 

 tion of the analyzer see "A New Device for the Solution of Transient-Vibration Problems 

 by the Method of Electrical-Mechanical Analogy" by H. E. Criner, G. D. McCann and 

 C. E. Warren, Journal of Applied Mechanics, Vol. 12, No. 3 (1945) pp. A-135 to A-141. 



