On the Transverse Vibrations of Prismatic Bars. 745 



The position of the element during vibration will be deter- 

 mined by the displacement of its centre of gravity and by 

 the angular rotation <fi in the (a?, y) plane : the axis Ooc may 

 be taken as coinciding with the initial position of the axis 

 of the bar. 



The angle at which the tangent to the curve into which 

 the axis o£ the bar is bent (the curve of deflexion) is inclined 

 to the axis Ox will differ from the angle </> by the angle 

 of shear 7. Hence, for very small deflexions, we may write 



%=*+•* ^ 



For determining M and Q we have the familiar expressions 



M=-El|£ Q = XCn7 = \Cn(!|-A . (4) 



where C denotes the modulus of rigidity, for the material of 

 the bar, and X is a constant which depends upon the shape 

 of the cross-section. 



The equations of motion will now be : — 



for the rotation — 



if we substitute from equations (4) ; 

 for translation in the direction of Oy— 



to**!*?**' 



P -^^(^¥)-0 (6) 



g at 2 \dx* da?/ v ; 



Eliminating </> from (5) and (6), we obtain the required 

 equation in the form 



Introducing the notation 



m l _ , i , 2 



we may write equation (7) in the form 



or 



or 



