THE DISTURBED MOTION OF AN AEROPLANE 217 



Complete Solutions 



In the general case we consider A(m) {or A(S)} has a roots 

 equal to m { = — n -f- ip } and Ui(8) — taking a type — has /? roots 

 equal to m . a and /3 cannot be greater than 4. 

 [N.B. — Do not confuse a and ft with a s and /3 S in X s = — a s 



± iA.] 



Let m = m + h, where h is ultimately zero. Expanding we 

 have : 



VjP#r* + £ {U,(m )e"»'! . h + . . . g£, JD ^J . e™«' } j* 



— j^ gmt. = - '-= 



M) A(m ) + A'(m ) + A»(m ) £ 



., . & a A(m ) 

 where A°(m ) = -^J-^. 



Since there are a roots equal to m in A(m ) = o, then A(m ) = o 

 = A'(m ) = A"(m ) = A a_I (m ). Similarly, 



U > (m ° ) = °-^rn7 rr ~ 

 and U^-'Cm,,) = U^- 3 (m ) = . . . . U',(m ) = U/mJ = o. 



. . A(8) e - (a + a, t + . . . . a a _^_ x t )e + ^ A<x(mo) 



on putting h equal to zero. 



We also see that the coefficient of e mot is infinite, containing 



powers of [j-j- It may, however, be absorbed into the free 



vibrations — A(x) = o, where X = m — which are 



(a + a 1 t + ...a a _^ I t*-e- I )e m ot ? 



the coefficients being arbitrary constants. 



The forced oscillation on the co-ordinate u is obtained by 

 expanding the last term, the coefficients being similar to those of 

 a binomial expansion. It is 



- real par, offsg {u.tmJ+.U,- W+ . . . . • ( "£^"» U, W} 



The free vibrations are given by single roots of A(X) = o and 

 s' equal roots X and roots m as above. The terms containing 

 X are similar to those containing m and may be included in 

 them. /3 may or may not be equal to o and a = s'. The sign 2 

 denotes terms obtained from values similar to m and X , or X 8> 

 or m s . The roots for the forced vibrations are m s such that 

 A(m s ) £ o and m as above. 



