360 BELL SYSTEM TECHNICAL JOURNAL 



where C is a constant of integration whose value is determined by the initial 

 conditions that xl = 2gh and .r2 = at the instant of contact. Hence 



C = mogh + J P dx.2. 



Substituting the above value of C in (1.2.11), we have 



>2A-2 + I P dx.2 = m.2g{h + .vo). (1.2.12) 



It may be observed that (1.2.12) is an energy equation in which the 

 terms have the following meanings : 

 hm2X'2 is the instantaneous kinetic energy of m^, 



Jo 



P dx'2 is the energy stored in the spring at any instant. It is also 



''0 



equal to the area under the load-displacement curve up to 

 the displacement X2 , 

 ni'2g{h + .Vo) is the potential energy of the mass at its initial height h + Xi 

 above the instantaneous position .Vo . 

 Hence (1.2.12) expresses the law of conservation of energy. 



Ordinarily // is very much larger than .V2 so that we may write, with good 

 accuracy, 



hm^xl + [ P dx2 = m<2gh. (1.2.13) 



To the same approximation, equation (1.2.3) becomes 



ni.x2 + P = 0. (1.2.14) 



Equation (1.2.14) and its first integral, equation (1.2.13), are convenient 

 forms for calculating events at any time during contact. Their use will be 

 illustrated in Part II. For calculating only maximum displacement and 

 acceleration, the equations become simpler. Let 

 Wi = weight of the packaged article ( = W2g), 

 dm = maximum displacement of the packaged article, 

 Gm — absolute value of maximum acceleration of the packaged article 



in terms of number of times gravity {Gm = \ Xilg |max), 

 Pm = maximum force exerted on packaged article by cushioning. 

 We shall limit our study to the practical regions where P > when 

 x-i > 0. Then it may be seen from (1.2.13) that .T2 is a maximum when 

 ±2 is zero, hence 



P dx.2 = WoJi, (1.2.15) 



/ 



Jo 



