412 BELL SYSTEM TECHNICAL JOURNAL 



In performing the numerical integrations of equations (2.9.2) and (2.9.3), 

 it is found that the integrand becomes infinite when X2 = dm since at this 

 point the velocity is zero. In order to avoid this difficulty, it was assumed^ 

 that, for a small distance in the neighborhood of dm , the acceleration is 

 constant with magnitude Gmg as obtained from equation (1.9.4). The 

 procedure is described in further detail in Section 2.12. 



Figure 2.9.2 gives the acceleration-time curve for perfect rebound. If 

 rebound does not occur, the acceleration is a periodic vibration, each suc- 

 cessive half period having the shape shown, with alternating sign. 



2,10 Acceleration-Time Relation for Abrupt Bottoming 



By abrupt bottoming, we mean bilinear cushioning (Class D) as treated 

 in Section 1.12. The load-displacement relation is (see equation (1.4.4) 

 and Fig. 1.4.4) 



P = koXi ^ X2^ ds 1 



P = kbX2 — (kb — ko)db X2 > ds] 



Considering, again, the system illustrated in Fig. 1,2.1, the equation of 

 motion of W2 , before bottoming, is 



OJo 



where 



_ . / kp 



The time (/«) at which .^2 reaches ds is found from (2.10.3); 



(2.10.4) 



1 . _i (^ods 1 . _,ds 



t» = — sm /-— = — sm J , (2.10.5) 



coo \/2gn Wo "0 



where 



/' 



A . /2W2h 



eta — 



^ See Timoshenko, "Vibration problems in Engineering," D. Van Nostrand Co.. New 

 York, Second Edition (1937) page 123. 



