1890.] vibrations of a revolving cylinder or bell. 103 



represented by the arrows at A, E in Fig. 2. At C, G the particles 

 are moving outwards, and this will retard their angular velocity. 

 The particles at B, F are moving with greater total angular 

 velocity than the rest; this will increase their "centrifugal force" 

 and give them a relative acceleration outwards. Those at D, H 

 are moving with the least total angular velocity, and the diminution 

 in centrifugal force will give them a relative acceleration inwards. 

 Hence the rotatory motion of the mass will give rise to relative 

 accelerations of the particles in directions represented by the 

 arrows in Fig. 2. If we compare the arrows in Fig. 1 and Fig. 2 

 we see at once that the effect of these relative accelerations 

 is to cause retrograde motion of the nodes relative to the mass, 

 that is, the nodes will rotate less rapidly than the ring. This 

 explanation is obviously applicable to all the modes of vibration. 



We will now determine the frequency-equations for the two- 

 dimensional vibrations of a thin cylindrical shell or ring of radius 

 a which is rotating about its axis with angular velocity co*. We 

 shall suppose that the cylinder is also acted on by an attractive 

 force /j, times the distance, directed towards the axis. The intro- 

 duction of this attraction will enable us to separate the purely 

 statical effects of centrifugal force, since by taking /j, = co 2 the latter 

 effects will be counteracted. Unless this condition is satisfied, the 

 circumference of the cylinder will be in a state of tension. Let 

 this tension be T (per unit length of generator) ; and let a be the 

 surface density of the cylindrical shell or the line density of the ring. 



When the cylinder is rotating steadily, the condition for relative 

 equilibrium gives by resolving normally 



aco a = — \- aua, 



a 



therefore T = aa 2 (co 2 — fi) (1). 



In order to define the position of any point on the cylinder at 

 any time t, it will be convenient to employ two systems of polar 

 co-ordinates having the centre as pole, in one of which the initial 

 line is fixed while in the other it revolves with angular velocity co. 

 If, in the undisturbed state the polar co-ordinates in the two 

 systems are (a, </>) and (a, 6), we shall have 



cfi = 6 + cot, 



and 6 will be constant for any particle of the ring. 



In the small oscillations, let the small relative tangential and 

 radial displacements of the particle be v and w so that its new polar 

 co-ordinates are (a + w, </> + vja) or {a + w, d + v/a) in the two 

 systems respectively. 



* Compare Lord Eayleigh, Theory of Sound, i. p. 322. 



