



" Pk Q > 





\ = 



I* 



SoK ~ 



IV 



for determining Frequencies of Lateral Vibration. 427 

 problem if the scales of the diagrams are such that 

 b _ _ ?i 2 K 



or— by (14)— if 



(18) 



b and k being the actual lengths in figs. 2 and 4 which 

 represent 2} and K , the values of %$ and K at the specified 

 section. 



Equation (18) therefore gives us the value of X— and so, 

 by (14), of the frequency n — for which the ordinates of 

 fig. 1 represent the deflexions Y. Comparing (15) with 

 (lO), we see further that the ordinates m of fig. 3 will 

 represent the corresponding values of M on a scale given by 



M _ n . S3 m 



m~~ /.b 



this last result, however, has little practical importance. 



Having thus shown that the ordinares Y of rig. 1, for some 

 value of n, satisfy the fundamental equation (6), we have 

 now to consider how far the conditions of constraint at the 

 section A may he satisfied : it is evident that we shall not in 

 general satisfy either of them at our fir.st attempt. For 

 purposes of illustration, we shall assume that the specified 

 conditions are that Y and M also vanish at A ; i. e., that A 

 is the other end of the rod, and is simply supported. At the 

 beginning of the instructions given above for constructing 

 figs. 1 and 3, it was stated that Oa and O'a' could be drawn 

 at any convenient angles, their actual slopes being im- 

 material : if we now change the slope of one of these lines, 

 and repeat the construction, we shall obtain different values 

 for Y and M at the end A. Then by synthesis of the two 

 solutions thus obtained — i. e., by writing 



= M 1 + «M 2 ,i 



Y- 



(19) 



and M = 



where Yx and M x correspond to the first, and Y 2 and M* 

 to the second solution, — and by suitably choosing the con- 

 stant a, we can evidently obtain a third solution of the 

 fundamental equations which will satisfy both of the im- 

 posed conditions at the end 0, and one of the imposed 

 conditions (say, the condition M = 0) at the end A. The 



