RELAY ARMATURE REBOUND ANALYSIS 181 



The energy relationships for heel series are 



-(^0 



= (1 — /02 ; - — -^f^ y^eo KJ'O) 



For a complete series n — > oo , and 



JT +V\ Mio(l - M12) .2 ,„,x 



If a complete heel series follows the "zero" impact, ?/2eo = ^12(1 + k\), 

 and 



-A (^^^) = ^^12(1 - i^^i2)(l + k,f {2r^) 



Finally, for the special case where a complete heel series follows an 

 initial complete front series y^eo = Pn , and 



-A (J^^YT^) = MAMn - 1) (26) 



It is to be noted that the energy loss due to a partial heel series is a 

 function of il/12 , Pn , ki , and the approach velocity of the first impact, 

 but that the equation for a complete heel series does not contain ki . 

 Finally, a complete initial heel series is a function of only ilf 12 and ki . 



D. Complete Mapping of Problem 



Equations (1) through (26) make it possible to completely map the 

 two-degree-of -freedom rebound problem. The relative maximum ampli- 

 tude 2CYi and the rebound pattern will be determined. 



Examination of the necessary equations, show that 2CFi is in all 

 cases a function of four parameters: fci , /c2 , M12 and P12/. Of these, k2 

 enters only if a partial heel series occurs prior to the time of maximum 

 rebound. If it is assumed that for this limited group of cases k^ = ki = k, 

 the number of parameters is reduced to three: k, Mn , Pnj. 



In Figs. 5 to 10, 2CFi is plotted against Pnj for the most useful range 

 of 1/8 < M12 < 1/2.5, 0.3 < fc < 0.6 and < P12/ < 10. 



As P12/ is increased from zero to infinity (corresponding to an increase 

 in the heel tension F^, the rebound pattern goes through some or all of 

 five regions. The criterion for location in any one region is based upon 

 the parameter 



1 -4- — f 

 ^ _ ^ mJ _ k (1 + k) f 



^~ l + Pnf'k—k- ^^^^ 



