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



second condition will not also be satisfied, unless by a lucky 

 accident, because we shall nut in general have assumed for 

 the frequency n one of its possible values. 



It is at this point that we have to introduce " trial-and- 

 error" methods. Equation (6) shows that the curvature of 

 figs. 1 and 3 will be increased by an increase in the value of 

 n, and hence we can see that if the result of the synthesis 

 expressed by (19) has been to make Y positive throughout 

 the span OA when a is so chosen that M vanishes at the 

 end A, then a higher value of n must be assumed in our 

 second attempt. The necessary procedure, in the case of a 

 rod constrained in any specified way at two sections only, 

 should now be obvious : — Assuming any value of 11, we can 

 obtain two graphical solutions of the differential equations, 

 each of which satisfies the imposed conditions at the starting- 

 end : by combining these solutions, we obtain a solution satis- 

 fying all but one of the imposed conditions of constraint, and 

 we can calculate the "error/"' or amount by which it fails to 

 satisfy this last condition (<?. g., in the example just con- 

 sidered, the "error" is the value of Y at the end A, since 

 in the correct solution Ya = 0) : repeating the process for a 

 second assumed value of n, we obtain a second resultant 

 error, and by plotting the " error " against n, as in fig. 5, 

 we can construct a curve which enables us very quickly to 

 make a correct guess at the required value of n. 



In practice, hibour may be saved by making the length 

 scale the variable quantity in our successive attempts at 

 •constructing the correct diagrams which correspond to figs. 1 

 and 3 : the lengths h l5 b 2 , . . . . etc. in fig. 2 and k 1? k 2 , .... 

 etc. in fig. 4 are then determined once for all, and only the 

 spacing of the equidistant vertical lines in figs. 1 and o needs 

 to be varied. Equation (18) shows that taking these lines 

 further apart is equivalent to assuming a greater value for 

 \ (or n), and hence we can conveniently proceed bv plotting 

 the resultant error against the assumed value of AB (or of I), 

 instead of against 11 as in fig. 5 : then, having determined 

 by interpolation the value of: I which will result in zero error, 

 we can calculate the corresponding value of II directly by 

 means of (14) and (18). 



It remains to discuss the procedure which is required for 

 dealing with continuous rods or shafts. We have seen that 

 at sections where these are simply supported there must be 



continuity in the values of Y, -~, and M, but not necessarily 



a"NL . dc 



in the value of -=— , since the resultant shear will in general 

 ax • * 



