4 Stephenson, Netv Type of Dynamical Stability. 



of magnitude 2J/F applied in the direction dividing the 

 angle between the rod and AB in the constant ratio k-lJi'. 

 It will be convenient to regard the action at either end of 

 the path as made up of two consecutive impulses, MV, 

 one bringing P to rest, and the second giving it the 

 velocity towards the other end. The angular velocity 

 at the instant between the two impulses is then equal 

 to the mean angular velocity during the cycle of which 

 the instant is the mid point. 



Consider the motion from the instant when P has 

 been brought to rest at A by the action of the first half 

 impulse ; let the inclination of PC \.o AB\i^ 6 and the 

 angular velocity w. The moment of the second half- 

 impulse about C is 



^'^^^■" 



and the resultant change in the angular velocity 



FAd/(/i' + /r). 

 If r is the time of motion to B the inclination of the 

 rod at B is therefore 



e + {w+ y/id/{/i- + k^)]T, 

 and the moment of the impulse at B 



- 2MV-^M + Iw + VhQl{Jr + fe)\T\ 



The angular velocity after this impulse is 



w- Vhdl{lr + k') 



correct to small quantities of the first order, and the 



inclination on reaching A again is 0+2wr. Hence the 



moment of the first half-impulse bringing P to rest at A is 



h- + k 

 By summation the total moment applied during the 

 motion of P from rest to rest at A is 



-2MV ,,,, >■ 

 {Ir + lrf 



