518 Mr. S. Lees on the Whirling of an 



From (i.) and (iv.) we get for the total kinetic energy T 

 o£ the system, 



2T = M \~* + z 2 co 2 + r\ cb + co) 2 + r 2 cj> 2 9 2 sin 2 <£ 



+ 2~r((j) + &j) sin (/) — 2zrco(<j) + co) cos <f> \ 



+ l 2 \c]> 2 ^co 2 -\-2cbco-^cod 2 + d 2 -co 2 6 2 \ . (v.) 



§ 4. We have next to write down an expression for the 

 potential energy of the system in the displaced position 

 considered. Assuming that the shaft is ot circular solid 

 section of area A, moment of inertia of cross-section AP, it 

 is known * that the shape of the bent shaft is given by 



f=(3r-L(9).r 2 /L 2 + (L(9-2^)^/lA . . (vi.) 



and the potential energy, accordingly, by f 



V=2EAk\Zz 2 -ZzL0 + JJd 2 IV. . . (vii.) 



In these expressions, f is the deflexion of the shaft at a 

 distance x from the constrained end, L is the length of the 

 shaft, while E denotes the Young's Modulus of the shaft. 

 We have neglected the effect of gravity on the potential 

 energy. For our purposes, this will be small ; whilst if the 

 shaft is vertical, the expression (vii.) is absolutely correct. 



§ 5. We now utilize Lagrange's equations of the general 

 form : — 



where yjr is any coordinate. Taking for yjr the coordinates 

 respectively z, 0, and <p, we get, putting 2EAZ; 2 /L 3 = B J , 



'i + r(0 + &>) 2 cos</> + rcf)s'mcf)-cjo 2 z + 3B (2~- L(9)/M =0, 



... (viii.) 



9- Mr*$ 2 sin 2 </>/I 2 -f (<£o> + co 2 ) 6 + B( - 3: L + 2L 2 0) /I 2 = 0, 



. . . (ix.) 

 r 2 (^+2(9^(9sin 2 <^ + </)l9 2 sin 2 + ^ 2 6' 2 sin0cos</)) 



+ r(.-'sinc/>-2i(wcos(£ — rar£in<f>) + L(4>-2co00)/M=O. 



. . (x.) 



§ 6. Before going further, it will be necessary to discuss- 

 the possible states of steady motion. To get these, we have- 

 to equate to zero all differential coefficients of the variables- 



*Ray]eigh, 'Sound/ §183. 



t Neglecting torsional oscillations. 



