On the differential equation for the flexural vibra- 



tions of prismatical rods. 



Prof. Stjepan Timošenko. 



The approximate equation for the flexural Vibration of pris- 

 matikal rods in a principal plane is 



d^y ^F d'^y 



EI ^ -I- 



dx 



Here 



g dt^ 



= 



(1) 



EI denotes the flexural rigidity of the rod 

 F — surface of its cross-section, 



— — the density of material. 

 g 



If we regard the effect of rotatory inertia*), the following equa- 

 tion can be obtained 



EI 



d^y I^ d^y F-id^y 



-T 







(2) 



In the following we point out another correction, which is due 

 to the effect of the shearing force, and obtain a more exact equation 

 for flexural vibrations. 



Let ab cd be a cross-element of a vibrating rod, M and Q the 

 corresponding bending moment and shearing force. We determine 

 the Position of the element by the displacement ;; of their centre of 

 gravity and by the angle of rotation 0- (Fig. 1). The angle between 

 the direction of the central line and the angle 9- is equal to the 

 angle of the shear ß and we can write 





(a) 



Fig. 1. 



*) Cf. Lord Rayleigh, Theory of sound § 186. 



