Loaded SJiaft supported in Three Bearings. 655 



whirling speeds higher than the first (see Proc. Roy. Soc. A. 

 vol. xcv. p. 106 (1918)), we have 



JL , JL JL _L 



rv + o 2 2 " o h 2 + (o_ 2 2 ; 



and 



1 1 



Now -^ = -02866 



i2 1 2 n 2 2 cqi 2 c0 2 



1 '^ nnn ml* 



EI'- 



1 mP 



_L--006&2^, 



ft) 2 hi 7 



and -^-2 = -0001482 (~ r ) . 



0)/ft) 2 2 VlLl / 



Hence we obtain the results of § lO^viz. 



■TIT T^T 



1 2 = 32-7^, and &/ = 206^. 

 ml 6 ml 



§ 14. Applying Dunkerley's method to the example of 

 § 12, we have 



1__ JL 1_ 



2 — 2 •" 2 "l~ ' 



I COi co 2 



so that ty i = a 1 + b 2 + c 3 + . = 17*152 from the determinant 



of § 12. 



But ti=§fh •'• N^lSlOr.p.m. 



This value is too low. 



In order to obtain a more correct value of n T , together 

 with an approximate value of fl 2 , we use the extension ot 

 Dunkerley's rule. Then we have approximately 



^ , l + Y , 2 = «l + ^2 + C3+... 



^1^2 — (aj> 2 ) + Wz) + • • ■ + (-Vs) + • • ■ OvW + • • • • 



Now «i+ & 2 + c 3 + . . . = 17-152, 



and (cr L & 2 ) + (a^) + . . . = 54*342. 



Hence ^ = 12-95, ^ 2 = 4*2. 



.-. N 1 =150Qr.p.m., and N 2 = 2700 r.p.m. 



The value of N x is a better approximation to the correct 

 value, but N 2 is rather too low, and a better value would be 

 obtained by taking account of the third and higher whirling- 

 speeds. 



The more precise values are the roots of the determinant^ 

 equation of § 12. 



