176 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 



For a one-degree-of-freedom system iji = C21 + C22 f -?rM = 0? whence 



'" = ^h-S](;J + *-«(0 + - 



(5) 



B. Impact Interval 



The change of velocity at point "i" due to an impact at "i" is, by 

 definition of the coefficient of restitution "k'\ 



Ail = — (1 + ki)Xi 



It is assumed here that the action of the stops are true impacts, i.e., 

 the changes in velocity take place while there is negligible motion of the 

 body. The velocity changes then occur as instantaneous rotation about 

 the conjugate axis, leading to the general relation for an impact at 

 point "i": 



VjOn = Vjein-l) + K jiij ie{n-\) (6) 



The first subscript indicates the coordinate, the second subscript 

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

 by the third subscript. The impact transfer coefficient K^ relating a 

 velocity change at point "j" to an impact at point "i": 



Kh = - ^' (1 + A;,) (7) 



Equations (1) through (7) allow any one specific case to be mapped, 

 if the mass distribution and force ratio are kno^^^l. A sample of such 

 mapping of rebound motion for a rectangular two-degree-of-freedom 

 armature appears in Fig. 4. 



v. ANALYSIS OF REBOUND PATTERN — ONE-DEGREE-OF-FREEDOM SYSTEM 



The rebound pattern for the one-degree-of-freedom sj^stem — as de- 

 rived in Appendix II — consists of an infinite series of parabolic arcs of 

 diminishing amplitudes. The structure comes to rest after a finite time 

 interval. The maximum rebound occurs during the fii'st bounce and 

 equals 



^ = - 1 ® 



