194 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 



/I 1 7 \ • (^1^2 — 1) W2 /, , , \ . 



= (1 + ^-l)^! ,,2 , ... = - 77- (1 + ^l)^l 

 (fl +1) (-11 



Ax3 = — (1 — A-Orci — 





= -(l + /cO:^i-^= -^^(1 + A:0ii 

 fi + 1) ^11 



Similarly it can be shown that impacts at points (2) and (3) follow 

 the same pattern. The general impact relations for impact at point "i" 

 are then 



2/iOn = 2/je(n-l) + Kjiyie{n-l) (6) 



The first subscript indicates the coordinate, and the second subscript 

 indicates the beginning (0) or end (e) of the free interval denoted by the 

 third subscript. 



The impact transfer coefficient Kji relating a velocity change at 

 point "f to an impact at point ''i": 



Kji = - ^ (1 + fci) (7) 



Appendix II 



ANALYSIS OF REBOUND PATTERNS — ONE-DEGREE-OF-FREEDOM SYSTEM 



The equation of motion of this system is 



yin = hCt'' + ywnt' + 2/lOn (/) 



where 



C = Cn- ^ (9) 



r 



and is measured from the start of the particular interval of free motion 

 in question. The impact relationship is 



2/lOn = —k\yie{n-l) 



The motion consists of a series of parabolic arcs having periods of 

 2yio/C in general, or 2/C, 2k/ C, 2fcVC, • • •, 2A;'*~VC. The time elapsed 



