424 Mr. R. V. Southwell on a Graphical Method 



method of determination which is now to be described pro- 

 ceeds by constructing a curve of deflexion which satisfies 

 equation (6), and by modifying this (on " trial-and-errof "" 

 principles) until it satisfies nil the constraint conditions 

 simultaneously : to each modification corresponds a change 

 in the frequency H tor which (6 is satisfied, and thus we 

 arrive finally at the required solution. 



Our graphical construction employs the principle of the 

 "funicular polygon." We construct two diagrams (figs. 1 

 and 3), the first representing the deflexions Y, and the 

 second representing the corresponding bending-moments M, 

 defined by equation (10). Either end of the rod may be 

 taken as a starting-point, and constraint conditions of the 

 tvpe (11) will determine the values at this end of some pair 



dY dM. 



of the Quantities Y, -r— , M and - . By assuming values 

 1 ax ax 



for the remaining two quantities, we are enabled to proceed 

 with the construction of the curves, making use for this 

 purpose of equation (10) above, and of equation (6) re- 

 written in the equivalent form 



d' 2 W 



Y + K^==0, (12) 



ax/ 



where the symbol K is substituted for gj^VL 2 p, a positive 

 quantity of dimensions [L][T] 2 /.[M], which varies in some 

 specified manner with x. 



In the engineering problem at any rate, it will often 

 be desirable to vary the length of our diagrams, and we 

 must therefore give careful consideration to the question 

 of scale. Let us for the moment replace x by the non- 

 dimensional variable *, defined by the equation 



x = lz, 



in which I denotes the distance between any specified pair 

 of sections (for example, two adjoining sections of con- 

 straint). Then equation (6) may be rewritten in the form 



£(«?)- ?=°> • • • • (13 > 



whence it is evident that the natural frequencies will be 

 obtainable from expressions of the type 



4tt 2 I1 2 1 :B {u) 



g K l 



—p^ $ and K denoting the values of p, S3 and K at some 



