RELAY ARMATURE REBOUND ANALYSIS 



197 



Expressions for an initial soiics may be obtained by setting ijieo = 1 

 y-ieo = y-2eo = 0, and, finally, for an initial cc^mplete series m — > oo and 

 hence A;"" -^ 0, and Equations (1) and (m) become 



1/2C00 = 



?/2eco = 



2/vi 



^(1 + hY L A 



Bk^ P ' 



—r- — Jrn 



1 



1 - fci 



'2Bh 



Pi2(l + A-O 



(17) 



B. Complete Heel Series 



For heel series, Equations (1) and (m) may be used by interchanging 

 the initial velocities, accelerations, and impact transfer coefficients for 

 those relating to heel motion : 



yion — yieo = 



'2 A k-2 - k2 Mioil + itz) 1 - kf 



yion — yioi = 



B 1 - k2 



2Ulj-_kr') 

 B{1 - ko. 



1 - ko 



1/2 eO 



yieOy2eO 



+ 



' 2A{kl - 2k^^' + A-D _ 2Mi2A:2(l - k^ - kr' + A-^~^) ' 

 52(1 - kl) BPu(l - kl) 



(n) 



(o) 



.2 

 'i/2f0 



An initial heel series occurs when the heel strikes first after the "zero" 

 impact. The first heel impact then occurs Ti = 2Pn/B(l + A;i) after 

 the zero impact and the initial conditions are 



?/2el = Pl2(l + A-i) 



9 4 P,, 

 2/1.1 = -A-i +ATr = ^p" (1 + A-i) - A-: 



yui = -k,T, + \ATl = ?^Ml + A;i) 



B 



(1 + A^i) - A-i 



Substitution of the above into {n) yields 



yion = — A:i + 



1 + A-, 

 1 - h 



2 _ 



2AFi 



(1 - ^2)" -ilfl2(l + A:2)(l - /v2") 



(P) 



The corresponding expression for 7/ion is quite involved. For the special 

 case of A: = A;i = k2 



