# State-Space Aerodynamic Model for Compressible Flow

By suitably generalizing the indicial response in terms of exponential functions and Mach number as shown previously, the corresponding state-space realization may also be obtained for each component of the loading in a subsonic compressible flow – see Leishman & Nguyen (1990). Firstly, consider the normal force response to continuous forcing in terms of angle of attack. Using the indicial functions given previously, the circulatory normal force response to a variation in AoA can be written in state-space form as

(8.175)

with the output equation for the normal force coefficient given by

where Ъх/fi is the lift-curve-slope for linearized compressible flow and 0:3/4 is the AoA at the 3/4-chord, that is,

о:з/4(0 = a(t) + —.

Similarly, the noncirculatory normal force from AoA can be written in the state-space representation as

x3 = a(t)——— !— x3 = o:(0 + «33X3, (8.177)

каТі

with the output equation for the normal force coefficient given by

C"‘(0 = 4;,. (8.178)

The remaining state equations for the pitching moment and pitch rate terms can be derived in a similar way using all of the other indicial response approximations for pitching moments and pitch rate, as given previously. The individual components of aerodynamic loading are then linearly combined to obtain the overall aerodynamic response. For example, the total normal force coefficient is given by

Cn{t) = Ccn{t) + Cnny) + Cnnc(t)

and an analogous equation holds for the pitching moment about the 1/4-chord. Thus, the overall unsteady aerodynamic response can be described in terms of a two-input, two-output system where the inputs are the airfoil AoA and pitch rate and the outputs are the unsteady normal force (lift) and pitching moment. It can be shown that by rearranging the state equations, the inputs and outputs can be represented in the general form

X |
«п |
0 |
0 |
0 |
0 |
0 |
0 |
0 ‘ |
*1 |
"1 |
0.5" |
||||

X2 |
0 |
«22 |
0 |
0 |
0 |
0 |
0 |
0 |
x2 |
1 |
0.5 |
||||

i3 |
0 |
0 |
«33 |
0 |
0 |
0 |
0 |
0 |
X3 |
1 |
0 |
||||

І4 |
0 |
0 |
0 |
«44 |
0 |
0 |
0 |
0 |
JC4 |
0 |
1 |
||||

*5 |
0 |
0 |
0 |
0 |
«55 |
0 |
0 |
0 |
*5 |
• 1- |
1 |
0 |
|||

І6 |
0 |
0 |
0 |
0 |
0 |
«66 |
0 |
0 |
*6 |
1 |
0 |
||||

І? |
0 |
0 |
0 |
0 |
0 |
0 |
«77 |
0 |
Xl |
0 |
1 |
||||

І8 . |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
«88 _ |
*8 |
0 |
1 I |
||||

Xl |
|||||||||||||||

X2 |
|||||||||||||||

X3 |
4 |
l " |
|||||||||||||

cn |
}_ |
Г «11 |
«12 |
cn |
C14 |
0 |
0 |
0 |
0 |
X4 |
_L |
M |
M |
||

cm |
I" |
|_ c2 |
C22 |
0 |
0 |
C25 |
c26 |
«27 |
«28 _ |
X5 |
-1 |
-7 |
|||

M |
12M – |

*6 x1 *8 |

which involves a bilinear combination of the states xi and X2. Thus, as a byproduct of the above system representation for the unsteady lift, the necessary information may be extracted from the system at a given instant of time to obtain the unsteady axial force component. Finally, the instantaneous pressure drag can be obtained by resolving the components of the normal force and chordwise forces through the geometric AoA a using Eq. 8.174.

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