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

Подпись:— (®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.