Lateral Vibration of Bars. 583 
Let us assume that we may consider the motion of P to 
be perpendicular to OB , and that the kinetic energy of 
rotation is negligible in comparison to that of translation. 
It may be easily proved (see any book on Hlasticity) that 
the curve is given by the equation 
tu ig ee) 
ia HKAk? 6 ? 
W being the force applied at the end. Then, denoting by 
y, the deflexion of B, 
2 aw(dL—2z te 
1 . . e ° ° e (ie) 
| dy oe pois aa u?(3L—2) i : O 
ee ae fe 
Now the moment about O of the forces causing motion of 
the bar is :-— 
ee oe 
j xyoAdx 
0 
= Sel * 23(3L—a)dx from (2) 
7 a 
nh 
= 7p cAL* Yy- 5 5 = ° ‘ ° - 5 ° ° (3) 
Also the bending moment at Q, 
_ HAR pay a J 
Po da” 
= i Awe Eom 2 (Li eo cs oan 
Equating (3) to (4) we have 
| y, 120 2 E 
This shows us that the bar vibrates in a simple harmonic 
manner and the frequency 
N\= os 2 
20 yy 
Noe (6) 
