Manchester Memoirs, Vol. Hi. (1908 J, No. 8. 



VIII. On a New Type of Dynamical Stability. 

 By Andrew Stephenson. 



(Received and read January 28th, igoS.) 



I. A system in a position of equilibrium, and capable 

 of oscillation about that position, may be acted on by 

 periodic force in such a way that no oscillation is generated. 

 If, for example, one end of a vertical stretched string is 

 moved to and fro in the direction of the length, this 

 imposed motion has no tendency to produce lateral 

 vibration. In certain cases, however, it has considerable 

 effect in intensifying an already existing oscillation ; in 

 particular, if the imposed frequency is double that of the 

 lateral motion, a very marked swing is magnified from 

 the slightest initial disturbance. 



Another example of a similar kind is afforded by a 

 pendulum, the point of suspension of which is subject to 

 a vertical periodic motion : if the frequency is double that 

 of the pendulum, any small swing is gradually magnified 

 by cumulative action. 



Various instances of the double frequency effect forced 

 themselves upon the attention of observers, and it appears 

 to have been assumed — possibly from the simplicity of 

 the phenomenon — that it is only in the case of double 

 frequency that this type of disturbance has appreciable 

 influence. Mathematical investigation has shewn, how- 

 ever, that the effect is cumulative in the whole series of 

 cases when the disturbance frequency is approximately 2Jr 

 of the natural frequency of the system, r being any 



March ^th, igo8. 



