WILSON— AEROPLANE ENCOUNTERING GUSTS. 213 



gusts is (p. 64), except for very sharp gusts, essentially a convection 

 of the machine with and by the gust ; for both these reasons 2° may 

 be discarded. This leaves only 1° — periodicity in the head-on 

 gustiness — as likely to be of interest. 

 The gust may be assumed in the form 



u^ = J sin pt or u^=-Je^PK (i) 



The differential equations are (p. 59) 



/(/))« =_ (o.i28Z)3 + i.i6oZ)^ + 3.385D + o.9i7)u„ 



f(D)zv = -D^-(o.ss7D -\-24s8)u„ (2) 



/(/)) ^= — 0.0285 iZ)«i, 



with f(D) =D* + S.49D' + 24.5D'-\- 3-3^50 + 0.917 



= {D^- + 8.359Z} + 23.37) {D' + 0.1308Z) + 0.03924). 



4. In the previous investigation it was found that the short- 

 period heavily damped oscillation was not of much significance 

 except in the case of a sharp up-gust (pp. 62-69), and that its 

 significance in that case was not revealed in the major motion of the 

 machine but in the initial acceleration (or stress) upon it. It may 

 therefore be expected that for periodic head-on gusts the short- 

 period motion will be negligible in its effects. It is consequently 

 desirable to carry out the numerical analysis in such a way as to 

 separate, so far as may be, the short and long natural periods of 

 the machine. 



Let us separate into partial fractions the operator 



I I 



/CD) " (D2 -h 8.359D + 23.37) (D2 + 0.1308D -f 0.03924) ' 

 or 



I 0.016D -|- 0.089 —0.016012)4-0.04263 



/(^ " i"- 4-8.359^ + 23.37 + D2 4. 0.1308D + 0.03924 ■ ^^ 

 The first fraction has to do with the short, the second with the long 

 oscillation. The two operators are to be applied to certain ex- 

 pressions derived from (i) by substitution in (2). 



5. If D = ip, the numerators of (3) have the respective magni- 

 tudes 



(0.089- -|- o.oi6-/?-)i- and (0.0426- -f 0.016-^-)^/-. 



