THE DISTURBED MOTION OF AN AEROPLANE 215 



+ the real part of 2{(a s e x s t ). U,(X S )} 



r V,(m s) m t V 2 (m' ) m , t 



v = - the real part of 2 —, V P, e * + a 2 A/ , \ P' e 2 



* Ls, A(m s ) s i s 3 A(m' Sj ) s 2 



V 3( m \) m " s t-1 



+ s A(m" r ) P " s 3 e 3 J 



s .1 



+ the real part of 2{(a s e Xst )V,(X s )} 



R-(m s .) m . t R 2 (m' s ) 



e s 2' 



r = - the real part of [ « ^ P., e . + S ^-y P' s < 



+ the real part of ^{(ase^). R,(X S )} 



where the " Vs " under the summation refer to the free vibra- 

 tions of which values there cannot be more than four — see the 

 free vibrational equation — with the respective four arbitrary- 

 constant values a s . 



Where X s is complex we have the free vibrations given by 



\ = — a* ± i&. 



These are 



(a s cos/3 s t + b s sin/3 s t). 



The ratios u : v : r will possess certain definable determinantal 

 values easily found. The ratios p : q : r possess the same 

 qualities as u : v : r. 



1 Where X s is real we see that the free vibrations are propor- 

 tional to 



U,(X S ) : V^) : R^) or U (X s ) : V (X g ) : R 2 (X S ) or U (X s ) : V (X s ) : R/X,). 



If m s be a root of A(m) = o, the denominators of the forced 

 vibrations become indefinitely small. This gives, however, a 

 value of m s equal to that of a free vibration. We infer, there- 

 fore, that, if any one disturbing force has a period and a real 

 exponential nearly equal to those of any one free vibration, a 

 very large forced oscillation will be produced in the co-ordinates 

 possessing that free vibration. 



Usually the disturbing gusts of wind are of the permanent 

 type Pcos(pt + E). Since resistances — surface friction and head 

 resistances — to motion enter the roots of A(A-) = o giving the 

 free vibrations will all be complex. A real exponential is intro- 

 duced into the values of X, none of which, therefore, can equal 



1 See Routh's Advanced Rigid Dynamics, vol. ii. 



