EQUATIONS OF THE OSCILLATORY MOTION. 



CHAPTER II. 

 MEASUREMENT OF OSCILLATIONS. 



675. EQUATIONS OF THE OSCILLATORY MOTION. We shall 

 frequently have to consider the oscillatory motion of a solid about 

 an axis, on each side of the position of equilibrium. Before touching 

 on the methods of observation, we shall examine generally the 

 mechanical conditions of this motion. 



When the movable system is displaced from its position of equi- 

 librium, from the very conditions of the experiment it is acted on by 

 a couple, which tends to bring it back, and the moment of which is a 

 function of the angle of deflection. This directive couple is due 

 either to the elastic reaction of the system, or to an external force, 

 or to a concurrence of several given causes. Besides this directive 

 action, the value of which only depends on the position of the 

 system, there are retarding forces analogous to friction, due to the 

 motion itself, and which depend on the velocity. 

 Let us call 



K the moment of inertia of the movable system ; 



x the angle of deflection at the time /, counted from the 



position of equilibrium ; 



/ (x) the moment of the couple, which tends to bring the 

 system into the position of rest ; 



o> = - the angular velocity of rotation ; 

 at 



<j> (w) the moment of the retarding forces. 

 It is known that, in the rotation of a solid about an axis, the 

 product of the moment of inertia by the angular acceleration is 

 equal to the sum of the moments of the forces in reference to the 

 axis. The equation of motion is then 



We must first observe that if the function < is zero, the movable 

 body acquires the same velocity every time it passes the position of 



VOL. II. C 



