82 Mr. E. R. Darnley on the Transverse Vibrations 



In this connexion it has found important practical appli- 

 cations. 



Dunkerley* and Chree f do not apply the strict Kirchhoff 

 theory to beams or shafts of more than two bays or sections, 

 but tiiey deal with three bays and with many cases of loaded 

 shafts by means of an approximate method suggested by 

 Rayleigh, in which the form of the vibrating shaft is assumed 

 to be defined by various types of algebraic formulse. 



§ 2. For an account of the methol of forming the equations 

 of motion reference may be made to the, books by Rayleigh 

 and Love. 



Let m be the mass of the beam per unit length, 

 E its modulus of elasticity, 



I the moment of inertia of the cross-section about a 

 diameter. 



Take the origin at one end of the beam, measure x along 

 the beam, and let y be the lateral displacement, supposed 

 small. 



Then, neglecting the effect of the rotation of elements 

 about an axis perpendicular to the beam, the equation of 

 motion is 



-_ . cPy d 2 y x 



which must be satisfied at all points along the beam, and the 

 following are the conditions at the ends and at supported 

 intermediate points. 



At a simply supported end, the ordinate and the bending 



moment are zero, i. e. y = 0, and —4=0. 



air 



At an end fixed in direction, the ordinate and inclination 

 are zero, i. e. ?/ = 0, and -—- =0. 



At a free end, the bending moment and the shear are 



d 2 y „ , d 6 y ~ 

 «ero, ,.«. ^=0, and ^=0. 



At a simply supported intermediate point the ordinate is 

 zero, and both the inclination and the bending-moment are 



continuous, i. e. y = 0, and ~- and ~~ are continuous. 



* Dunkerley, " Whirling- and Vibrations of Shafts," Phil. Trans A 

 vol. clxxxv. pt. 1 (1894). 



t Chree, " Whirling and Transverse Vibrations of Kotating Shafts '* 

 Phil. Mag. May 1894. 



