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Stability derivatives calculations on An F/A-18 Aircraft Flight Model Conference Paper · March 2007

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Stability derivatives calculations on An F/A-18 Aircraft Flight Model by R. M. Botez, A. Mamert, P. Dionne, B. Fayard, A. Bodron, D. Popescu 1.

Introduction

Stability and control derivatives calculations play a crucial role in the simulation of an aircraft since they determine the accuracy of the flight model as they can duplicate the actual aircraft behavior in response to random perturbations (such as gusts and turbulence) and/or control inputs from pilots. Concerning the calculation of these derivatives, investigations are conducted on an aircraft expressed as a linear time invariant state space system. The main values of stability derivatives (lift, drag and their derivatives with respect to the angle of attack) are validated on an aircraft. The derivatives are calculated for three different configurations: 1. wing, 2. wing and body configuration and 3. wing and body and tail configuration. for subsonic, transonic and supersonic speeds regimes and for four types of wings configurations : Straight and Non-Straight-Tapered, Double-Delta, Cranked and Curved. These derivatives are calculated from geometrical aircraft data. In this paper, we present the main F/A-18 aircraft geometry data. In most of cases, these data are calculated from references and were compared, when information existed, with data given by NASA Dryden Flight Research Laboratories. An automated computer program is conceived and validated in Matlab, as Matlab software is used in aircraft control and simulation today in a high number of aerospace companies. This computer program is presented in this paper, and its structure breaks down into 3 categories: 1. 2. 3.

The input function, which records the needed geometry for the calculations. The user built functions, which find specific graphical values based on predefined input interpolated or not. The stability derivatives, which are calculated using the methods for subsonic, transonic and supersonic speed regimes in which the user built functions and the inputs variables, are incorporated.

In the next section, an example for the calculation of the pitching moment coefficient derivative with respect to the angle of attack CLα is shown. All other coefficients were calculated by methods described [2], and we will not describe here all their calculations methods in details.

1

2.

Geometrical parameters needed to calculate the stability and control derivatives

Following geometrical parameters are needed: 2.1

Cr =

Surface root chord

S

(1 + λ ) b / 2

=

2b (1 + λ ) A

(1)

where S is the reference area, b is the wing span, A is the wing area, l = Ct / Cr . 2.2 C=

2.3

Mean aerodynamic chord 2 S

b/2

∫ C dy 2

(2)

0

Root chord of exposed surface

The root chord of exposed wing surface Cre is defined with the leading edge angles (ΛLE) and trailing edge angles (ΛTE) for any configuration: d Cre = Cr − [tan(Λ LE ) − tan(ΛTE )] 2

(3)

where d is the fuselage diameter and Cr is the root chord. 2.4

Surface of exposed wing

Is the surface of two half wings without fuselage contribution: Se =

(Ct + Cre )(b − d ) 2

(4)

where Cre is the exposed root chord, Ct is the tip chord, b is the span, d is the fuselage diameter. 2.5

Wetted area of exposed wing

t⎞ ⎛ SWe = 2 S e ⎜ 1 + 0.2 ⎟ c⎠ ⎝

(5)

2

where t is the airfoil thickness, c is the airfoil chord, Se is the exposed surface of the wing. In this paper, the report between the thickness and the chord is constant. 2.6 Ae =

Aspect ratio of exposed panel

(b − d )

2

(6)

Se

where Se is the exposed wing surface and d is the fuselage diameter. 2.7

Sweepback angle of trailing edge

The leading edge and the trailing edge angles are given by the following equations: X b/2 Y tan(ΛTE ) = b/2 tan(Λ LE ) =

X Cr

Ct

Y

We introduce the following notation Y = X + Ct - Cr from where: 2 (C t − C r ) b 2(C t − C r )(C t + C r ) tan ( Δ TE ) = tan( Δ LE ) + b (C t + C r ) tan( Δ TE ) = tan( Δ LE ) +

(7)

2( λ − 1)(C t + C r ) b ( λ + 1) 4( λ − 1) tan ( Δ TE ) = tan( Δ LE ) + A ( λ + 1)

tan ( Δ TE ) = tan( Δ LE ) +

2.8

Sweepback angle of 50% and 25% chord line

We need to determine the sweep angles at 25% and 50% from the chord. We generalize the equations (7): tan( Λ n ) = tan(Λ m ) −

4 1− λ [( n − m) ] A 1+ λ

(8)

where m = 0 and n are equal to 0.5 and 0.25, we obtain :

3

tan( Λ c / 2 ) = tan(Λ LE ) −

4 1− λ [(0.5) ] 1 + λ at 50% from the chord. A

tan( Λ c / 4 ) = tan(Λ LE ) −

4 1− λ [(0.25) ] 1 + λ at 25% from the chord. A

2.9

λe =

2.10

Taper ratio of exposed panel

(9)

Ct Cre

Semi wedge angle measured perpendicular to leading edge Δy ⎛ Δy⊥ ⎞ , where Δy⊥ = ⎟ cos( Λ LE ) ⎝ 5,85 ⎠

δ ⊥ = arctan ⎜

(10)

For the wing, we should still determine the following quantities: • • • • • • •

Dihedral angle with positive wing tips up, Freestream Mach number Trailing edge angle based on airfoil coordinates at 95% and 99% chord Stream-wise trailing edge angle Leading edge radius Slope of airfoil surface at leading edge Amount of camber (% de chord)

For the wing, horizontal and vertical tail, we need to calculate: • • • •

Angle of attack at which section lift curve begins to deviate from linear variation Section zero lift angle of attack Angle of attack for design lift coefficient Section design lift coefficient

We further explain the method to calculate the lift coefficient derivative with the angle of attack C Lα .

4

Lift coefficient derivative with the angle of attack C Lα

3.

For a conventional, tail-aft airplane with subsonic Mach number, this derivative is combination of wing W, fuselage B, and tail T contributions. The aircraft derivative can be estimated [1]: CLα = CLαWB + CLα H η H

SH S

⎛ dε ⎞ ⎟ ⎜1 − ⎝ dα ⎠

(11)

where ε is the downwash angle at the horizontal tail, ηH is the ratio of dynamic pressure S at the tail to the free stream dynamic pressure and the area ratio is denoted by H . Since S the ratio of wing span b to fuselage diameter d is reasonably large, following approximation may be done: CLαWB ≈ KWB C LαW 2

where

KWB

⎛d ⎞ ⎛d⎞ = 1 − 0.25 ⎜ ⎟ + 0.25 ⎜ ⎟ ⎝b⎠ ⎝b⎠

(12)

Compressibility correction factor is written as β = 1 − M 2 . The average wing section lift curve slope is: Clα =

π a∞ A a∞ + π A2 + 4

cos Λ LE

(13)

Ratio of actual wing section lift curve slope to 2π ,

2π AW

CLαW = 2+

2+

=

dε dα

(15)

tan 2 Λ c / 2 H AH 2 β 2 )+4 + (1 β2 k2

k=

where

M

2

2π AH

CLα H =

dε dα

(14)

tan 2 Λ c / 2W AW β (1 )+4 + β2 k2 2

CLαW

, 2π CLαW

M

CLαW

M =0

M =0

the

downwash

ratio

at

the

horizontal

tail

is

where ε is the downwash angle, i.e. the mean angle at the tail

5

dε is a dα function of the wing geometry and position of the tailplane relative to the wing. The downwash gradient at low speed,

plane that the airflow is deflected through by wing. The downwash derivative

dε dα

1.19

M =0

= 4.44 ⎡⎣ K A K λ K H cos Λ c / 4 ⎤⎦

where:

hH 1 1 10 − 3λ b , Kλ = and K H = KA = − 1.7 A 1+ A 7 2lH 3 b CLαW at Mach number equal to zero M = 0 is given by the following equation: 1−

CLαW

M =0

(16)

2π AW

= 2+

AW 2 (1 + tan 2 Λ c / 2W ) + 4 k2

Other equations are derived for the same coefficient from [2], by use of interpolation tables and figures. Results expressed in terms of stability derivatives for four flight cases are represented in the next section. 4.

Results

A number of four flight conditions dependent on angles of attack, altitudes, Mach numbers and pitch angles were considered in this paper for the F/A-18 aircraft. We compared the values for stability derivatives obtained experimentally by NASA DFRC with the values of the same stability derivatives obtained in this paper semi-empirically based on interpolations in Tables and Figures [2]: Flight condition 1 • • • •

Angle of attack α = 1.63110 Altitude H = 1000 pi Mach number M = 0.7 Pitch angle θ = 1.63110

NASA DFRC stability derivatives CD = 0.027751 (Drag) CL = 0.12162 (Lift) CDα = 0.095 (Drag vs angle of attack)

Stability derivatives here calculated CD = 0.026157 (Drag) CL = 0.136 (Lift) CDα = 0.08387 (Drag vs angle of attack)

CLα = 5.5055 (Lift vs angle of attack)

CLα = 4.546687 (Lift vs angle of attack)

6

Flight condition 2 • • • •

Angle of attack α = 100 Altitude H = 20,000 pi Mach angle M = 0.7 Pitch angle θ = 100

NASA DFRC stability derivatives CD = 0.15398 (Drag) CL = 0.96719 (Lift) CDα = 1.5473 (Drag vs angle of attack)

Stability derivatives here calculated CD = 0.12775 (Drag) CL = 0.96123 (Lift) CDα = 1.13613 (Drag vs angle of attack)

CLα = 3.8061 (Lift vs angle of attack)

CLα = 4.50696 (Lift vs angle of attack)

Flight condition 3 • • • •

Angle of attack α = 1.3660 Altitude H = 5,000 pi Mach angle M = 0.9 Pitch angle θ = 1.3660

NASA DFRC stability derivatives CD = 0.032161 (Drag) CL = 0.1029 (Lift) CDα = 0.1554 (Drag vs angle of attack)

Stability derivatives here calculated CD = 0.0218622 (Drag) CL = 0.116873 (Lift) CDα = 1.19664 (Drag vs angle of attack)

CLα = 7.4078 (Lift vs angle of attack)

CLα = 5.477677 (Lift vs angle of attack)

Flight condition 4 • • • •

Angle of attack α = 100 Altitude H = 10,000 pi Mach angle M = 0.8 Pitch angle θ = 100

NASA DFRC stability derivatives CD = 0.15215 (Drag) CL = 0.96 (Lift) CDα = 1.5374 (Drag vs angle of attack)

Stability derivatives here calculated CD = 0.1379 (Drag) CL = 1.0623 (Lift) CDα = 1.1359 (Drag vs angle of attack)

CLα = 3.7554 (Lift vs angle of attack)

CLα = 4.9738 (Lift vs angle of attack)

7

5.

Conclusions

Main stability derivatives (lift, drag and their derivatives with respect to the angle of attack) were calculated for an F/A-18 aircraft for a number of four flight conditions. These values were compared with the values given by the NASA DFRC researchers. This work is applied and validated on the F/A-18 aircraft and is useful for future aircraft simulation and control in aerospace industry. One advantage of this work is that the execution time needed for the lift and drag coefficients calculations is very small compared to the time needed to calculate these coefficients based on Computational Fluid Dynamics (CFD) computer knowledge. Acknowledgments The authors would like to thank Mr. Marty Brenner at NASA Dryden Research Flight Centre for his collaboration. References [1] Roskam, J., “Methods for Estimating Stability and Control Derivatives of Conventional Subsonic Airplanes”, University of Kansas, Lawrence, Kansas. [2] Hoak, D.E., Finck, R.D., “USAF Stability and Control Derivatives DATCOM”, 1978.

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