THE DISTURBED MOTION OF AN AEROPLANE 219 



The free and forced vibrations contain the term t a ~^, being 

 magnified to the (a — /3)th degree, thus confirming the conclusion 

 previously inferred as to the indefinite increase — A(m s ) = o— of 

 the forced vibrations. 



The solution for the symmetrical vibrations is now complete. 

 U^m), U 2 (m), U 3 (m) ; V^m), V 2 (m), V 3 (m) ; and R^m), R 2 (m), 

 R 3 (m) are known, and their derivations with respect to m can 

 be found. 



The reader will do well to refer to the remarks made on 

 page 213 with respect to the minors in the free vibrations, and 

 to read Routh's Advanced Rigid Dynamics — " Forced and Free 

 Oscillations." 



Asymmetrical Vibrations 



The equations of motion are now : 



/wtf „ *\ /w . , 7 \ / 7 wu w a \ 

 \Y + z wJ w + (t cos ^ + z pJ p + l z q ~ ~Y~ ~ T * m6 oh 



- S Z s e" D "' t cos(p s t + B ) 



s i 



L w w + (a| + L p ) p + ( - Fg + L q )q - - S L S2 e~ n V C os(p' S2 t + S^) 

 M w w + (M p - F|)p + (f£ + M )q = - i M s e" n " s 4os(p" s t + & ). 



N.B.— The values n , n' s , n" s ; p s , p' s , p" and e e' 3 , s" s 



1 2 3 * 2 3 '23 



are not necessarily the same as those corresponding to the 

 symmetrical disturbing forces. 

 Consider 



P Sl =Z s e ' 



» E 's 

 P's =L s e s * 



P" s = M s e' £ s * 

 3 3 



and m s = -n s + ip s ; m' = - n' s + ip' s ; and m" s = - n" s -f 



1112 22 «» 3 



ip" s then 



, W ,KJ m st W ,( m 's) m' s t 



w=-therealpartof{s- ? -rTP Si e ' +l-X7 r CT I V 2 + 



% A (m Si ) s . T s 2 A(m' S2 ) 



W,(m" s ) m , st 

 , A(m' s ) s 3 



+ the real part of S{(a s e v )W,(X,)}, 



