THE DISTURBED MOTION OF AN AEROPLANE 213 



u : v : w can be found from the above determinant ; the ratios 

 being equal to certain first minors and similarly for p, q, w. The 

 equations to find \(on reducing the respective determinants) are: 



Symmetrical oscillations : A Q X 4 + B X 3 + C X 2 + D X + E = O 

 Asymmetrical oscillations : A' \* + B' X 3 + C' X 2 + D' X + E' = O. 



For stability l A , B , C , D , and E must all be positive and 

 B C D — E B 2 — A D 2 > O, and similarly for the values A' etc. 



Forced Oscillations 



Those forces, — X„ — Y lf — Z lt — L*, — M„ — N x representing 

 gusts of wind are periodic when they set up indefinitely 

 increasing oscillations in the aeroplane. As such they may be 

 represented as follows, assuming also that they are continuous : 



3 Any force = f(t) = Pe " kt cos(Xt + a) + P'e - k ' 1 cos(X't + a) 



= lP s e- k s t cos(X s t + a s ),' 



where each term in the summation is called a disturbing force — 

 permanent or evanescent according as K is or is not zero 

 respectively. 



Symmetrical Oscillations 



The equations of motion on substituting values for X lf etc., 

 are now : 



(w| + X u )u + X v v + (X r - ™ cos0 o )r = - IX^e'Vcos (p^t + g^) 



Y u u + (Y v + ^) v + (Y r + ^U + W s . n ^ r = _ ?y ^ - » Vcos^t + ^ 



N u u + N v v + ((£ + N r )r = - f N^e'" "• t cos(p*. $ t + £" Bg ). 



Since B may be computed and operated on as any ordinary 

 algebraic symbol may, we have : 



u = -Vm^ e ""'' cos(p " 1 + ^ + ^ > f Y =. e ~ nVc ° s( pv + £ y + 



^ ? N,e-"".'co s (^t +B y] 



1 Routh's Stability in Motion. 



* See Routh's Advanced Rigid Dynamics, vol. ii., chapter on "Forced 

 Oscillations*" 



