180 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 



In addition it is useful to set down energy equations in order to 

 simplify evaluation of greatest rebound for the various groups of re- 

 bound patterns. The kinetic energy function T is evaluated in Appendix 

 I. A potential energy term V — the work done against Fi and F2 from the 

 equihbrium position — is introduced. If To is the total energy of the 

 system prior to the "zero" impact, then 



T + F .2 , Mi2 .2 2ilfi2 . . 



—^ — = ?/i + p^ y^ - ~-f^ y^y^ 



-2(7(2/1+ /2/2) 

 The energy loss due to n front impacts is 



+ V 



-A 



(T + 



\ To 



= (1 - A-f) (1 - M^,)yU 

 For a complete front series n — > 00 , and 



-A 



n^) 



= (1 - Mu)yl 



(18) 



(19) 



(20) 



If a complete front series follows the "zero" impact, yieo = 1 and 



-A 



T -\- V 



= (1 - Mn) 



(21) 



After completion of this "initial" front series, the system maintains 

 only one degree of freedom (rotation about the front) until a heel 

 impact occurs. By setting |/i = ?/i = ?/2 = in (21) we obtain the heel 

 impact approach velocity yo = P12 . 



Apparently energy loss due to n front impacts is a function of 71/ 12 , 

 ki , and the approach velocity of the first impact. 



C. Heel Series 



An analysis similar to the above can be made for partial and com- 

 plete heel series following the "zero" impact. This is demonstrated in 

 Appendix III, yielding, for ki = /c2* 



^Pi2(l + kf [APn 

 B 



yuoo = 



?/u« 



1 + k 



- ky 



'2APu 

 B 



k{l 



k{\ - k) 

 1 + k 



■k) 



- Muk 



(1 + k) 



- Mr2(l + k) 



(22) 



The more general form ki 9^ k2 can be obtained as indicated in Appendix III. 



