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SHORT-PERIOD APPROXIMATION
We found in Chapter 9 that a very good approximation to the short-period mode is given by (9.2,14). We therefore make the same additional assumptions here as led to that approximation—viz. AV = 0, and the speed equation of motion is identically satisfied. The reduced system equation is then found from (10.2,1) to be
— (®ia + GDe + 2/w)
№y8 ^7no)
_ 0 1
The system transfer functions are now most simply found from (10.2,10) by solving for the ratios Aa. jA8e, etc., with the result
|
6* |
Да |
fy&ms – GL/)(Ivs Gmg) |
(a) |
|
1 >1 .4 1 |
(iy-s — Gmil)(GLx + CDe + 2[is) — 2fiGmx |
||
|
&es |
_ A6 _ |
Gms(^/is + GLx + CDr) — GLsGmx |
(b) (10.2,11) |
|
д §e |
s{(fys — Gmq)(GLx + CDe + 2fis) — 2ftGmJ |
||
|
Ggd |
= в®вд |
(«) |
|
When the aerodynamic control transfer functions are replaced by their stability derivative representations, i. e. by (10.2,2) and with |
+ s@m& ’> ®mq = @im
the denominators of the above transfer functions are all polynomials in s that have exactly the same roots as (9.2,14). The numerators become
Of GxS: (GLgGms + 2/лСта) + s(2p, Cms —IyCLi) (a)
Of Ges: Cms(CLx + CD) – CmCLd] +s[(2pCmd + Crr, a(CLa + CD) – GLgGm&]
+ s*(2uCmb) (b) (10.2,12)
Both numerators are of degree one less than their respective denominators.
