652 Prof. H. H. Jeffcott on the Whirling Sjieeds of a 



The positions s and magnitudes /x or the section changes 

 are as follows : — 



s -0703 -118 -163 -348 '94 



fi 1-602 16- -039 2-445 -129 



s' -147 -878 



li 5-97 -0983 



For an approximate solution the loads may be considered 

 as concentrated at the positions t, and to be of the magnitudes 

 in, given by 



t -254 -436 -566 '695 '84 



m 426-3 719-6 709'6 709*6 392'8 



*' '2SS "5 -727 



mf ... ,359-4 1232 388'5 



m represents the weights in kilogrammes. The couples 

 exerted by the loads are regarded as negligible. The first 

 portion of the shaft is 160 mm. in diameter, so that 

 1 = 3220 cm. 4 ; E = 2-0xl0 6 kg. per sq. cm.; # = 981 cm. 

 per sec. per sec. The lengths between bearing centres are 

 1 = 2705 mm. and V — 211b mm. 



We proceed to determine the values of the constants for 

 each load in turn, following the tabular form (Table II., 

 pp. 650, 651). 



Note that the coefficients of X and X' in the expressions 

 for ui, u 2 , w 3 , m 4 , u 5 , vi, v 2 , v d , are approximately symmetrical 



about the principal diagonal (after multiplying the v co- 



efficients by -^ ), in accordance with Bayleiglr's reciprocal 



theorem, which expresses that the deflexion at a point A due 

 to a load at B is equal to the deflexion at B due to the 

 same load placed at A. 



\ \ e now put 2L X = — : , A 2 = -, etc., since m 



represents the weight. Write ^=— gjr, and we then have 



•fu x = 1-784m 1 + 3-4w 2 + 2'945» 3 + 2'15w 4 + '57tt 5 + -376^ 



+ l-232r 2 + -2635r s , 



^r W3 = l-995ft 1 + 4-6 ( / 2 4 4-28// 3 + 3-24w 4 + -821w 5 -f -3845^ 



+ l-27r 2 + -2745r 3 , 



^ti i =l-749M 1 + 4-285M 2 + 4-305M 1 + 3-28i^ + -806M B + '327rj 



+ l-084r 2 + -2345r 3 , 



