8 Stephenson, New Type of Dynamical Stability. 



tion of small amplitude a, and frequency ;/ per 27r units 

 of time. The equation of motion is then 



% + —, , an-co% nt 6 = o 



h- + K- 



and the solution* when a]i\{Ji--\-k-) is small, is approxi- 

 mately 



"-^KTi^l^'^') • • • <^* 



It is evident that (i) and (2) are of the same type, a?i 

 in (2) being the maximum velocity of P in its path. 



We have thus proved that when the amplitude of the 

 pivot motion is small, the body swings in a simple vibra- 

 tion of frequency proportional to the frequency and to 

 the amplitude of the applied motion. 



3. Now consider a body free to rotate about a hori- 

 zontal pivot, and set in the position of unstable equilibrium : 

 what will be the effect of a vertical oscillation of the pivot 

 on the stability of the equilibrium ? 



In the position inclined to the vertical at an angle Q, 

 the mean angular acceleration due to the imposed motion 



is \\-7T, — rJ ^ inwards, from (2); while the outward 

 - \h- + li-J 



acceleration due to gravity is ^ B. The resultant is 



therefore 



a-rrh \ h ^ 





and the acceleration is always towards the vertical if 



{anf>2g{lr + k')lli. 



Thus the inverted pendulum is rendered stable by a 



small simple vertical oscillation of the pivot of maximum 



velocity greater than 



j2g{lr + F)llL 



* The investigation is given in a paper "On Induced Stability," /"/<?'/. 

 Mag., February, 1908. 



