for determining Frequencies of Lateral Vibration. 431 



the "error diagram" (PL VII. fig. 5) it is necessary to have 

 all errors referred to some standard value for this slope. 



The diagrams illustrate an obvious simplification which 

 can be made when, as in the present instance, a shaft is 

 symmetrical about its centre. It is only necessary to 

 continue the construction as far as the first point of sub- 

 division past the centre, since the slope of the lines df and 

 d'f gives a measure of the error involved : obviously, if the 

 complete diagrams had zero error they would be symmetrical 

 about their centres, and so df and d'f would be horizontal. 

 It is convenient to take as a definite measure of the " error " 

 the total rise of these lines over the horizontal distance OA, 

 so that when the diagrams obtained in the first and second 

 attempts have been combined in such a way that d'f is 

 horizontal, the total rise of df over this length may be taken 

 as the " error/' and plotted against the assumed length 

 of OA. 



In the first application of the construction (length OA 

 taken as 5*71 inches) the line df was found, after correcting 

 the bending-moment diagram, to fall by 0*044 inch over the 

 total length. A fall indicates that the figure assumed for 

 I (or n) was too small, and hence at the second application 

 a length of 6 inches was assumed : this gave an error of the 

 opposite sign and of amount 0*28 inch. Plotting the 

 corresponding points on the error diagram (fig. 5) and 

 joining them by a straight line, a length of 5*75 inches was 

 obtained as the estimated length corresponding to zero error. 

 This was tried, and an error of the same sign as in the first 

 attempt was obtained, the amount of the error being now 

 reduced to 0*004 inch. Plotting this as a third point in the 

 error diagram, and estimating again, the value of I for zero 

 error was given as 5*755 inches, and the corresponding value 

 of X was obtained from equation (18) as 173. The correct 

 answer, as shown above, is 175*4, approximately, and hence 

 the method has resulted in this instance in an error of 2'4 in 

 175— slightly under 1*4 per cent. 



From (14) it follows that the whirling speed n is given 

 correctly to well within 1 per cent., and it is clear that prac- 

 tically as accurate a solution could have been obtained by 

 dispensing with the third application of the construction, 

 and accepting the estimated value of 5*75 inches for /. 



Trinitv College, Cambridge, 

 June 4th, 1920. 



