424 BELL SYSTEM TECHNICAL JOURNAL 



undamped cushioning and perfect rebound. Figure 3.2.1 illustrates the 

 idealized system, and it may be noted that the mass mz is omitted, as is 

 required for perfect rebound (Section 2.3), At first we shall consider that 

 the mass Wi is undamped and later we shall consider the effect of damping 

 in this element. 



The mass mi is taken to be small in comparison with m^, so that the motion 

 of the latter is the same as we found it to be in Section 2.2 where mi was not 

 considered. Hence the acceleration of W2 is a half-sine wave pulse: 



x-i = — C02 s/lghsinoiit, (0 ^ / ^ 7r/oo2). (3.2.1) 



The equation of motion of mi is 



mixi + ^i(xi — x^ = 0. (3.2.2) 



Let X be the relative displacement of mi with respect to m^ , i.e., 



X = a;i — .'v:2 . (3.2.3) 



X is proportional to the force in the spring (^i^;) and to the acceleration 

 of mi and hence is proportional to the deflection, strain and stress in the 

 element which the system mi , ki represents. 

 Substituting (3.2.3) in (3.2.2), we find: 



mix + kix = —miX2 . (3.2.4) 



This equation holds for the duration x/co2 of the pulse ^2 . The initial 

 conditions for x are 



M^=o = W^=o = (3.2.5) 



so that the solution of (3.2.4) is 



X 



= V^. [- sin C.I / - sin C02 ^1 , U^t^^. (3.2.6) 



It may be seen that x is composed of a forced displacement at the accelera- 

 tion frequency co2 , on which is superposed a free vibration at the natural 

 frequency, coi , of the element. The maximum value of the relative displace- 

 ment is 



■\/lah. 2nir 



•^max — ' i C Sin 





J^-lY'^^+l' V ^' ^co2/- (3.2.7) 



\W2 / W2 



in which w is a positive integer chosen so as to make the sine term as large 

 as possible while the argument remains less than tt. 



(3.2.7) gives the maximum dynamic response of the element mi during 

 the interval of impact. To find the amplification factor we must compare 



