19G THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 



At the heel, from (g), the heel velocity preceding the n*'^ impact is 



2/2e(„-:) = ?/20(n-l) + Bt' (k) 



The velocity change during the (n — 1) interval is then equal to 

 BTn-i . From Equations (6), (7) and (12), the change in velocity during 

 the n*'' impact is — Pi2(l + ki)ki'~^yiea • 



The total change of ?/2 between impacts is then 



2/20n - 2/20(n-l) = BTn-1 " Pl2(l + ki)kl~^yieO 



Similarly in preceding intervals: 



yiO(n-l) — y20(n-2) = BTn-2 " -Pl2(l + ki)ki~^ yuo 



^202 — ?/2oi = BTi - Pi2(l + ki)kiyeo 



^201 — 2/2eO = - Pl2(l + kl)yieO 



By addition of the above 



y20n — y2e0 



^ Bi^Tm- Pud + h) Z k". 



n-1 



E 



m=0 



yuo 



= -^ yuo 2Z ki — Pnil + ki)yuo 22 k" 



■A. m=l 



n-1 



E 



TO=0 



The summations may be evaluated, yielding 

 ^2B ki - ki 



y20n 



y2e0 = 



- Pl2(l + ki) 



1 - ki' 



yuo 



(1) 



A I - ki "' ' ^' 1 - ki 



To evaluate the displacements at the heel, Equation (g) yields 



y20n — 1/20 (n-1) = ^20(n-l) 7^n-l + ^BT n-\ 



Adding these expressions for intervals to n; the total change in 1/2 is 



n— 1 n— 1 



2/20n — 2/201 = ^ y20mT„, + ^^ 23 ^T^ 

 m=l m=l 



2h{i - kr') . . 



~ yieOy2eO 



A{1 - h) 



r 2Biki - 2fcr+^ + ki 

 2Pi2fci(i - /cr - kr'' + /ci"-')"" 



(m) 



^(1 - hy 



yuo 



