118 Mr. Morrow on Lateral Vibration of Bars of 

 ... _^ = * 12 ^ ft) ^ ^663084 Z 8 — -912718 Vx + -341435 ZV 



- -093915 ZV + -002480 a? 8 



- -000367 y + -000002 ~). 



This is a more accurate expression for the vibration-curve 

 than is (6), but it can be seen that the difference is not 

 great. 



-^ = •0809227^;, 

 and , T 3-5153 MJ 



which is correct to 0'02 per cent. o£ the value given by 

 exact methods. 



4. Bar ^Supported" at Both Ends. 



This is the case in which the directions at the ends are 

 free, but the ends themselves are constrained to remain at 

 rest by " supporting " forces of the required magnitude. 



For a uniform bar these forces must each equal one half 

 of the total force due to the inertia of the bar, that is, taking 

 O at one end, 



^i Jo 



Equation (2) then becomes 



; Elg=ip^ . (7) 



Taking y = A + B* + Gx 2 -f Dx d + E^ 4 + F# 5 , 



.-. A = C = 



*=-o, 



d*y 

 V da*'' 



= 0, 



I 



x= 2 



^=0 

 dx U ' 



•' 



I 



*=2 



y=y» 



: 



x= I 



y=Q, 



.' 



x = I 



dx* U > 



•' 



= BZ + DZ 3 + EZ 4 + FZ 5 

 .-. 0=6DZ + 12E/ 2 + 20FZ 3 ; 



