178 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 



this treatment, to reduce Fi , the greatest excursion at the front. In 

 order to simphfy mapping, this maximum excursion will V)e expressed 

 as 2CYi , the ratio of ]'i to Y^ as given by Efiuation (8) for the case of 

 A; = 1. Thus 2CYi is the ratio of the greatest excursion of the two- 

 degree-of-freedom system under consideration to the greatest excursion 

 of the corresponding perfectly elastic one-degree-of freedom system. 

 We first introduce two basic constants which are functions of the mass 

 distribution relative to the stop locations: 



M.J -^ (11) 



This constant represents a mechanical coupling coefficient. As 

 Mij = M ji , the two-degree-of -freedom system under consideration here 

 has only one such non-trivial constant ilf 12 . 



The second constant represents a force transformation factor from the 

 j" coordinate to the "z" coordinate: 



ii •}} 



Pij = ^ (12) 



In the analysis of the two-degree-of -freedom system only P12 is important. 



If there is to be any heel motion, the "zero" impact at the front must 

 impart a positive velocity to the heel. By Equations (6), (7), and (12), 

 this recjuires that P12 be negative, which in turn implies that (ik > 1. 

 For the limiting case of Hiii = 1, P\i = il/12 = and no coupling exists 

 between the heel and the front. Physically this means that the two 

 stops are the centers of percussion of each other and the system will 

 act as a simple hinge. 



With the above foundation, it is possible to analyze the patterns of 

 motion and maximum rebound amplitudes. 



A. Motion Immediately Following "Zero" Impact 



After the "zero" impact at the front, both front and heel will lift off 

 in accordance wdth impact Equation (6) and continue to move in ac- 

 cordance with the free interval Equations (4). Whether the next impact 

 occurs at the front or the heel depends on their respective periods, 

 ti and ^2 : 



(13) 



