216 SCIENCE PROGRESS 



the value of the "m's"(=ip) of the gusts. Stability is here 

 retained. Again, in this case, the forced oscillations on the 

 co-ordinate acted upon will be permanent, and will supersede 

 the free vibrations, which in the case of stability contain a real 

 negative exponential and are therefore evanescent, vanishing 

 ultimately. The free vibrations, of course, decrease among 

 themselves at varying rates depending upon the indices — a a of 

 the exponential. a s , a positive quantity, is the co-efficient of 

 decay or subsidence. 



The Limits of the Forced Vibrations 



If m s = \ s so that U^m^for Ui(Xs) = o} then \ represents a 

 free vibration asU^XjeV which therefore vanishes. The forced 



vibration containing the fraction -ttj — y is finite, however, if there 



are an equal number of roots (m s ) in Ui(m) = o and A(m) = o. 

 Therefore if any free vibration is absent from a co-ordinate — u, 

 say — though present in the other co-ordinates, then a disturbing 

 force of the same period and real exponential will produce a 

 finite forced vibration only. We may then conclude that a dis- 

 turbing force can produce a large vibration in any co-ordinate 

 only if there be present in that co-ordinate a free vibration of 

 nearly the same period and real exponential. 



Again, if the period of a forced vibration is very small " p " 

 in the complex value "m" is very great. There are higher 

 powers of m and therefore of p in A(m) than in U^m), etc. 



A l J , etc., become insignificant. The forced oscillations are now 



of no serious account. 



The forced oscillation on a co-ordinate vanishes when the 

 disturbing force on that co-ordinate — u, say — is of the type 



U 1 (S){|X s e- n s t cos(p s t+E s )> =0. 



1 U^SyV = O is, however, the determinantal equation which 

 gives a free vibration constraining " u " to zero. Therefore 

 when the type of the disturbing force which acts directly on 

 any co-ordinate is the same as that of any mode of free vibration 

 which constrains that co-ordinate to zero the forced vibration 

 will vanish. 



1 See lko\i\.}\ , s\Advanced Rigid Dynamics, vol. ii., chapter on " Forced Oscillations." 



