192 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 



The rotation and displacement of the center of gravity is then 



+ Xz 



Xh = ix2 - Xi) 



h + f2 



Xv = Xi + (,X2 — X\) 



Ix 



h + h 



X'l — Xi 



(a) 



Rill + i2) 



From this the kinetic energy may be computed 

 T = ^m{xl + xl) + ^mR'd' 



xlil + f2 + it) + xM + (1 + 1) + xl{(, + IzY 



+ 



2{ll + /2)'^ 



2xMl\(2 - tl - 1) - 2xz(z{l\ + (2){xi - ±2) 

 2(^1 + 4)^ 



(b) 



Applying LaGrange's Equation to the above, the equations of motion 

 are obtained: 



1 (^\ _ ^ = p 

 dt \dqr/ dqr 



F\ ^ xM + f3 + 1] + :g2[fif2 - fs - 1] - ^^^3(4 + I2) ^ 

 m {i\ + 12)' 



F2 ^ x^2 - fl - 1] + X2[l\ + fs + 1] + xzhih + ^2) 



/^3 -xihik + 4) + ^y'sOi + ii) + xsCfi + ^2)' 



\ (c) 



w 



(i'l + l2f 



The Equations (3) may be solved for xx , X2 , xz and the results inte" 

 grated, yielding 



Xx = ^- [ClxFx + Cl2i^2 + CxzFzlt' + x,ot + Xio 



2m 



X2 = ^ [C2xFi + C22F2 + C2zFz\f + X20^ + X20 



Xz = ^ [CziF, + C32F2 + CzzFzli' + xz,t + Xz, 

 2m 



(d) 



