666 BELL SYSTEM TECHNICAL JOURNAL 



This is the transform equation. To find x we use the inverse Laplace 

 transform and the solution of (7) is 



_ 00)^ ["/ / _ sin coA _ 9 /(^ ~ ^) _ sin co(/ — b)\ 



{t-h) 



^(^/_^_sin^^ 



(7) 



The expression u{t — h) simply means that the term to which it is attached 

 is zero for all values oi t < b. 



Let us now consider what this solution consists of. 



There are apparently three terms which take effect at successive intervals. 



The initial whip can be considered to consist of three different displace- 

 ments starting at successive times 0, b and 2b. With the displacement of 

 the base there is a corresponding displacement of the mass m. After the 

 time b the second term or displacement takes hold and an associated dis- 

 placement of mass m except that the initial conditions are the end conditions 

 of the first displacement. After the time 2b the third displacement enters 

 and the final result is the displacement-time pulse or whip. To make the 

 problem somewhat simpler we introduce the following modifications: 



1°. Because the motion is a simple harmonic of known frequency after 

 the whip has passed we will only consider the maximum ampHtude. 



2°. Only the displacement-time function of the mass m during the pulse 

 interval will be examined. 



3°. The dimensional magnitudes of the motion of mass m will be expressed 

 as ratios of those of the pulse. 



If fl is the maximum ampUtude of the whip, and To = 2b its time interval 

 (usually expressed in miUiseconds), then we define 



X 



—= 8 Amplitude ratio of pulse displacement and response of mass m dur- 

 ing pulse interval only. 



To 2b 2b icb at j /• r i 



X ~ T ~ 9~~7 ~ ^ Natural frequency of mass m expressed as a 



ratio of the pulse length, 

 t 

 '^ ~ ji Elapsed time expressed as a ratio of the pulse length. 



^ = — - Ratio of maximum amplitude to pulse displacement after pulse 



interval. 

 Substituting these values in equation (7) and rearranging we obtain 



« = 2r - ^H^^ _ , f(2, - 1) _ Sin M2r - 1)\ ^^^^ _ 



(8) 



+ ( 2(r - t) - Si-t 2xv(r - 1) \ ^(^ _ j^ 



\ Tip / 



