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



and its restoring effect is equivalent to an intensity of 

 lateral loading id which is given hy the equation 



•+£-« A 



the direction of z# being opposite to that of the deflexion 

 (and so tending to restore the straight configuration), and 

 its intensity, as given by (2), being measured in pounds 

 per foot run. 



In the vibrating bar, this " effective lateral loading " 

 produces the required acceleration of the bar towards the 

 central position : in the " whirling " shaft it produces 

 the normal acceleration towards the axis of rotation. In 

 either case, the magnitude of the acceleration is given 

 (in foot-second units) by the equation 



a = 47r 2 n 2 y, (3) 



the instantaneous deflexion y being a function of x only in 

 the case of the whirling shaft, and represented in the case 

 of the vibrating bar by the expression 



j/ = Ysin27rni, (4) 



where Y is a function of x only. 



We have therefore, in both problems, the equation 



pa 47r 2 n 2 / o 

 w g=g y > (5 > 



and by eliminating w and M from (1), (2), and (5) we 

 obtain the differential equation 



***'*>_ av»w 



9 

 which reduces to 





both when y = Y simply, and when y is given by (4). 



Equation (6) governs Y as a function of #, and thus 

 defines the curve of detlexion. The arbitrary constants in its 

 complete solution are determined in any particular instance 

 by the special conditions which define the constraints : thus, 

 at an end which is " simply supported " (as by a swivelling 

 bearing, in the problem of the whirling shaft) we have 



y=.0; M = 0: (7) 



