446 BELL SYSTEM TECHNICAL JOURNAL 



Amplification factors for small values of 001/002 may be calculated very 

 accurately if it is observed that the initial slope of the amplification factor 

 curve for a pulse acceleration is proportional to the area under the accelera- 

 tion-time curve. For example, noting that the initial slope of the amplifica- 

 tion factor curve for the half-sine wave pulse is 2, we assign the value 2 to 

 the area under the half -sine wave. On the same scale, the area under a 

 square wave pulse is tt and under a triangular pulse is 7r/2. Accordingly, 

 the initial slopes of the amplification factor curves for the latter two pulses 

 are ir and 7r/2 respectively. 



As an additional aid in finding amplification factors for unusual cases. 

 Fig. 3.8.2 is given to show the effect of asymmetry of an acceleration pulse. 

 The pulse is triangular in shape but the time (t/>) taken to reach the peak 

 value of acceleration may have any value from zero to the total duration 

 (T2) of the pulse. 



PART IV 



DISTRIBUTED MASS AND ELASTICITY 



4.1 Introduction 



It is important to be aware of the conditions under which the assumption 

 of lumped parameters is permissible. In Parts I and II the cushioning 

 medium was assumed to be massless, so that wave propagation (or surges) 

 through it was ignored. Such surges will contribute to the acceleration 

 imposed on the packaged article and we should be able to predict both the 

 magnitudes and frequencies of the additional disturbances. If this is done, 

 the information in Part III may be used to obtain at least a rough estimate 

 of the resulting effects. In Part III itself the effects of accelerations were 

 determined by studying the response of a system having only one degree of 

 freedom; that is, an element of the packaged article was assumed to be a 

 single mass supported by a massless spring. Every real element, of course, 

 has an infinite number of degrees of freedom, so that it is important to 

 discover the contribution, of the higher modes of vibration of an element, 

 to the overall response. 



Both of these problems (distributed parameters of mass and elasticity in 

 the cushioning medium and in an element of the packaged article) are 

 studied in this part. One example of each type is considered, and the choice 

 of the example in each case was influenced by considerations of expediency, 

 namely that the mathematical derivations be relatively simple and lead to 

 solutions for which not too lengthy computations are necessary to yield 

 results that can be applied practically. At the same time, the examples 

 chosen are believed to give some insight into several of the physical phe- 



