116 Mr. Morrow on Lateral Vibration of Bars of 



to find a more accurate expression for M, inserting this in 

 equation (2) and again solving. 



The process can be repeated to any extent, but in common 

 with other methods it neglects the rotatory inertia of the rod. 



In the following work the bars will be considered to be of 

 uniform sectional area, density, and flexural rigidity except 

 when otherwise stated. 



3. Clamped-Free Bar. 



Taking first the case of a rod " clamped " or " built-in " at 

 one end and free at the other, let the origin, 0, be at the free 

 end. 



If p = density of material of bar, 



o> = cross- sectional area, 



then the Moment of the Inertia forces about X (distant x 

 from 0) 



pa>y z {a:-z)dz, 



-c 



and therefore the differential equation of the centre line of 

 the beam is, by (1) and (2), 



-d-miS^-^ dz ' • • • • w 



pco being taken from under the integral sign on account of 

 the uniformity of the density and sectional area throughout 

 the length. 



The object is now to find the solution of this equation in 

 ascending powers of x, and hence to obtain the period of 

 vibration. 



Assume a value of y 2 which satisfies the end conditions. 



Thus let 



y = A + B A ' + (> 2 + D^ 3 4-E^. 



The end conditions at the free end, # = 0, are 



dhj _ 

 dx 2 



o, 



•'• 



G = 



o 7 







d?y = 

 dx z 



o, 



• • 



D = 



0, 







y = 



and at the' fixed end, x - 





y 



A = 

 = 0, 



: yv 



dy = 



dx 



o, 



B = 





3/> 





E = 



. yi 





