448 BELL SYSTEM TECHNICAL JOURNAL 



The initial conditions of the system are 



M(=o = 0, (4.2.4) 



\fL = - '-•=> 



The boundary conditions are 



Nx=o = 0, (4.2.6) 



k(\ — \ = -mo —r . (4.2.7 



Equation (4.2.7) expresses the requirement that the force on the upper end 

 of the cushioning must balance the inertia force of the packaged article. 

 For continuous cushioning kf should be replaced by EA, where A is the 

 cross-sectional area of the cushioning. 



A solution of (4.2.1) satisfying conditions (4.2.4) and (4.2.6) is 



w = X] ^In sin — ^ sin co„/, (4.2.8) 



n=i a 



where co„ is the w"" root of a transcendental equation to be obtained from 



(4.2.7) and yln is a constant to be determined by (4.2.5). Substituting 



(4.2.8) in (4.2.7) and equating coefficients of like terms of the series, we 

 obtain the transcendental equation 



^^an'^^ = ^. (4.2.9) 



a a mz 



Substituting (4.2.8) in (4.2.5) we obtain, by the usual methods of expansion 

 into trigonometric series, 



2v 

 An = ~ ^ T— A ■ (4.2.10) 



in [ — + I sm • 1 



\ a a / 



Hence the complete solution of the problem is 



2v sm sm co„ t 



= -E —7 r . (4.2.11) 



co„ I + ^ sm 1 



\ a a / 



Our chief interest is in the acceleration of W2 . Making use of (4.2.7) 

 and (4.2.9) we find, from (4.2.11), that this acceleration is 





= z'coo 2^ Bn sin co„/, (4.2.12) 



