Asw Theory

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ASWING 5.99 Technical Description — Steady FormulationMark Drela

10 March 2015

SummaryASWING is a program for the prediction of static and quasi-static loads and deformations of air-craft with exible high aspect ratio surfaces and fuselage beams. The key features of the overallstructural/aerodynamic model are sketched in Figure 1. A fully nonlinear Bernoulli-Euler beamrepresentation is used for all the surface and fuselage structures, and an enhanced lifting-line repre-sentation is used to model the aerodynamic surface characteristics. The lifting-line model employswind-aligned trailing vorticity, a Prandtl-Glauert compressibility transformation, and local-stall liftcoefficient limiting to economically predict extreme ight situations with reasonable accuracy.

Fuselage beam

Wind−aligned vortex wake

Unloaded geometry

gravitySurface beam

Surface beam(lifting line)

(slender body)

propulsive force

Angular momentumPoint mass

V gust

gust field

U

Ω

velocities,rotation rates

(x,y,z,t)

Joint

control laws forcontrol surfaces,engines

sensor outputsfor control laws

Figure 1: ASWING conguration representation

A primary intent of ASWING is to enable rapid and extensive exploration of the V-n ight enve-lope to identify potential failure scenarios. Divergence speeds, aileron reversal speeds, deformationeffects on stability derivatives and control effectiveness, along with all the related stress distribu-tions can be readily predicted. Effects of wing exibility on induced drag can also be predicted.The full dynamic extension allows utter predictions.

The interactive user interface allows on-the-spot redesign and resizing, thus permitting quickevaluation of structural and aerodynamic changes on the structural margins and aeroelastic behav-ior. Hence, ASWING is effective for preliminary structural design and sizing. It is also useful forobtaining initial estimates of the airloads on the deformed geometry, so that more realistic loadscan be input into other more detailed structural analyses.

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Contents

1 Coordinate Systems and Structural Equations 4

1.1 Local coordinates and surface-beam transformation tensor . . . . . . . . . . . . . . . 5

1.2 Fuselage-beam transformation tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Beam bending-moment and force resultants . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Moment – beam-curvature relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 Section representation options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5.1 Representation 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5.2 Representation 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5.3 Representation 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5.4 Axis relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.6 Force and moment equilibrium relations . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Discrete Formulation 12

2.1 Interior equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Boundary conditions and interior constraints . . . . . . . . . . . . . . . . . . . . . . 14

3 Applied Loads 15

3.1 Aerodynamic loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Inertial and gravity loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Point-mass loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4 Engine loads and jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4.1 Engine force and moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4.2 Engine axes relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4.3 Engine models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4.4 Propulsive jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.5 Strut loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.6 Joint loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.7 Ground-point loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.8 Beam Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4 Velocity Inuence Coefficients 29

4.1 Prandtl-Glauert Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 Vortex inuence function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3 Volume inuence functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.4 Point-mass inuence functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

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4.5 Ground effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.6 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5 Global Variables and Constraints 37

5.1 Acceleration constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.2 Lift-related constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.3 Moment-related constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.4 Beam joint constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.5 Compliant beam joint constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.6 Circulation coefficient constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.7 Separation and stall modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6 Wing/Beam Description 42

6.1 Point mass description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.2 Engine description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6.3 Strut description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6.4 Ground point description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6.5 Beam joint description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

7 Operating Parameters 47

7.1 Constraint Well-Posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

8 Computed Parameters 498.1 Aerodynamic parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

8.2 Structural parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

8.3 Structural, aerodynamic stability and control derivatives . . . . . . . . . . . . . . . . 50

8.4 Interpretation of Stability Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 51

8.5 Axis transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

8.6 Force and moment derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

8.7 Real and Apparent Masses and Inertias . . . . . . . . . . . . . . . . . . . . . . . . . 54

8.7.1 Real mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

8.7.2 Real-mass moment and mass centroid . . . . . . . . . . . . . . . . . . . . . . 55

8.7.3 Real-mass moment of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

8.7.4 Apparent mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

8.7.5 Apparent-mass moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

8.7.6 Apparent-mass inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

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1 Coordinate Systems and Structural Equations

The aircraft geometry, velocities, accelerations, and structural loads are expressed in a cartesianx,y,z body coordinate system sketched in Figure 2. The axis origin is grounded to the aircraft atsome location. Hence this coordinate frame is not inertial, but instead has some linear acceleration

ao, and also has angular velocity Ω and angular acceleration α o. The relative freestream velocityvector V ∞ , also dened in the x, y, z system, is opposite to the aircraft velocity U and is related tothe velocity magnitude V ∞ and angle of attack α and sideslip β as follows.

V ∞ = V ∞cos α cos β

− sin β sin α cosβ

= − U (1)

x

y

z

V

c

sn

Ω x x

Ω y y

Ω z z

x

y

z

c

sn z

x

y

U U z z

U U y y

U U x x

Ω

Ω

Ω

c

n s

Figure 2: Coordinate systems, velocities, rotations, and accelerations.

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1.1 Local coordinates and surface-beam transformation tensor

For the purpose of dening local stress/strain and aerodynamic force relations, a local beam-element c,s, n cartesian coordinate system is dened, with s nearly parallel with the tension axis.The transformation of any vector A from the x,y,z airplane body axes to the local c,s, n beamelement axes is via the three Euler angles ϕ, ψ,ϑ, applied in that order for a surface beam.

AcAsAn

= ¯̄T AxAyAz

(2)

¯̄T =

cosϑ 0 − sin ϑ

0 1 0

sin ϑ 0 cosϑ

cos ψ sin ψ 0

− sin ψ cosψ 0

0 0 1

1 0 0

0 cosϕ sinϕ

0 − sin ϕ cosϕ

(surface beam) (3)

¯̄T =

cosϑ cosψ cosϑ sin ψ cosϕ + sin ϑ sin ϕ cosϑ sin ψ sinϕ − sin ϑ cosϕ

− sin ψ cosψ cosϕ cosψ sinϕ

sin ϑ cos ψ sinϑ sin ψ cosϕ − cosϑ sin ϕ sinϑ sin ψ sinϕ + cos ϑ cosϕ

(4)

Figure 2 shows the transformation sequence. The beam curvature tensor ¯̄κ is related to the rate of change of the transformation tensor ¯̄T by

d ¯̄T ds

= − ¯̄κ ¯̄T −→ ¯̄κ ≡0 − κn κs

κn 0 − κc− κs κc 0

= −d ¯̄T ds

¯̄T T

(5)

or equivalently

κcκsκn

= d ¯̄T ds

×· ¯̄T (6)

where the tensor cross-dot product ×· is dened as a generalization of the standard vector cross-product rule, with dot products replacing the usual scalar multiplications.

· · · aT 1 · · ·· · · bT 1 · · ·· · · cT 1 · · ·

×·· · · aT 2 · · ·· · · bT 2 · · ·· · · cT 2 · · ·

≡ detı̂ ˆ k̂

a1· b1 · c1·a2 b2 c2

=

b1 · c2 − c1 · b2c1 · a2 − a1 · c2a1 · b2 − b1 · a2

(7)

Applying these operations to the ¯̄T components in (4) gives explicit expressions for the three

curvature components ( κs is actually a twist rate).κcκsκn

= ¯̄K dϕ/dsdϑ/dsdψ/ds

(8)

¯̄K =cos ψ cosϑ 0 − sin ϑ

− sin ψ 1 0cos ψ sinϑ 0 cosϑ

(surface beam) (9)

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1.2 Fuselage-beam transformation tensor

The curvature-denition matrix ¯̄K as given by (9) for a surface beam is singular at a sweep angleof ψ = ± 90◦ . Such a polar singularity always occurs in Euler-angle axis transformations, and willcause numerical solution failure if ψ ≃ ±90◦ is encountered in a calculation. For surface beams,a 90◦ sweep angle is unlikely, but for a typical fuselage beam roughly parallel to the x axis itis inevitable. This problem is eliminated by dening an alternative rotation sequence ψ,ϕ ,ϑ forfuselage beams, with the order of the ϕ and ψ rotations switched from the surface-beam case.

¯̄T =

cosϑ 0 − sin ϑ

0 1 0

sin ϑ 0 cosϑ

1 0 0

0 cosϕ sinϕ

0 − sin ϕ cosϕ

cos ψ sin ψ 0

− sin ψ cosψ 0

0 0 1

(fuselage beam)(10)

¯̄T =

cosϑ cosψ − sin ϑ sin ψ sinϕ cosϑ sin ψ + sin ϑ cosψ sinϕ − sin ϑ cosϕ

− sin ψ cosϕ cosψ cosϕ sinϕ

sin ϑ cos ψ + cos ϑ sin ψ sinϕ sinϑ sin ψ − cosϑ cosψ sinϕ cosϑ cosϕ

(11)

The beam curvatures are then given by the alternative curvature-denition matrix

¯̄K =cosϑ 0 − cosϕ sin ϑ

0 1 sinϕsin ϑ 0 cosϕ cosϑ

(fuselage beam) (12)

which is now singular when the beam is vertical at ϕ = ± 90◦ , which is unlikely for a fuselage. Thesubsequent derivations assume that the appropriate ¯̄T and ¯̄K denitions are used for each type of beam.

1.3 Beam bending-moment and force resultantsStandard beam force-resultant and moment-resultant vectors are dened in the local axes.

F c = τ cs dcdnF s = σss dcdnF n = τ ns dcdn

M c = − σss ndcdnM s = (τ cs n − τ ns c) dcdnM n = σss c dc dn

(13)

These resultants can be expressed in either the local c, s, n axes as above, or in the global x,y,zaxes. These two representations are related by the transformation tensor appropriate for that type

of beam.F cF sF n

= ¯̄T F xF yF z

,M cM sM n

= ¯̄T M xM yM z

(14)

Figure 3 shows the sign conventions for the c,s, n components. Analogous sign conventions are usedin the global x, y,z axes. Note that M s is a torsion load while M c and M n are bending moments.Likewise, F s is an axial load, while F c and F n are shear loads. Such interpretation cannot be madefor the x, y,z components.

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n

s

c

M n

s

M c

n

s

c

F s

F n

F c M

Figure 3: Load resultants on element of beam, in local axes.

1.4 Moment – beam-curvature relations

The beam section of a lifting surface is shown in Figure 4. The elastic and tension axes are offsetfrom the arbitrary s axis by cea ,n ea and cta ,n ta . Assuming the section does not warp out of the c-nplane, the overall extensional strain at some location c, n is

ǫ = ǫs + c(κn − κn 0) − n (κc − κn 0) (15)

where ǫs is the strain at the c, n origin, and κc0, κn 0 are the curvatures of the unloaded beam,calculated from the unloaded ϕ0, ϑ0, and ψ0 distributions (i.e. the jig shape). The angle αA of thezero lift line relative to the chosen c axis will be used later in the aerodynamic formulation.

c

n

c

x o

c

nc s h sh

n

c

c

n cg

cg

mass centroid

, n

tension axis

elastic axis

c ta

n t ae a

ea

axial strain

A

zero lift line

Figure 4: Beam section with curvatures and resulting strain.

Following Minguet [1], the force and moment vectors in the local axes are related to the beam

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strains and curvatures via the stiffness matrix (necessarily symmetric).

F cF sF nM cM sM n

=

E 11 E 12 E 13 E 14 E 15 E 16. E 22 E 23 E 24 E 25 E 26. . E 33 E 34 E 35 E 36. . . E 44 E 45 E 46. . . . E 55 E 56. . . . . E 66

γ cǫsγ nκc − κc0κs − κs 0κn − κn 0

(16)

It is not necessary to include the jig-shape strains such as ǫs 0 since these can be lumped into the jig shape x0,y0,z0 itself. The stiffness matrix has 21 independent elements in general. For beam-bending applications, however, it is assumed that all but the lower right quadrant have a morerestricted form as follows.

E 11 E 12 E 13 E 14 E 15 E 16. E 22 E 23 E 24 E 25 E 26. . E 33 E 34 E 35 E 36. . . E 44 E 45 E 46. . . . E 55 E 56. . . . . E 66

→

GK c 0 0 0 GK c nea 0. EA 0 − EA n ta 0 EA c ta. . GK n 0 − GK n cea 0. . . E 44 E 45 E 46. . . . E 55 E 56. . . . . E 66

(17)

It is convenient to rst dene a moment M ′ which is translated to the tension and elastic axes.

M ′cM ′sM ′n

=M cM sM n

+0 nta 0

− nea 0 cea0 − cta 0

F cF sF n

(18)

The elements of the net moment/curvature stiffness submatrix ¯̄E are dened as follows.

¯̄E ≡EI cc EI cs EI cn

. GJ EI sn

. . EI nn

=E 44 − E 22n2ta E 45 E 46 + E 22cta n ta. E 55 − E 11 n2ea − E 33c2ea E 56. . E 66 − E 22c2ta

The system (16) can then be inverted into the following form.

γ cǫsγ n

=F c/GK cF s /EAF n /GK n

+0 − nea 0

n ta 0 − cta0 cea 0

¯̄E − 1

M ′cM ′sM ′n

(19)

κc − κc0κs − κs 0κn − κn 0

= ¯̄E − 1

M ′cM ′sM ′n

(20)

Normally the stiffnesses GK c, EA, GK n in (19) are sufficiently large so that F c, F s , F n do not signif-icantly contribute to the beam strains γ c, ǫs , γ n in the rst force term in equation (19). The secondmoment term vanishes entirely if the tension and elastic axes coincide with the reference s axis, i.e.cta , n ta , cea , n ea = 0. These assumptions are characteristic of classical Bernoulli-Euler beam theoryformulations. Here, the general form (20) will be retained to make the results independent of therather arbitrary location of the reference s axis, and to give better accuracy for beams which mayhave signicant extensional and shear compliance.

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It is useful to note that for a beam with no twist/bending coupling, this net stiffness matrixtakes the simpler form

¯̄E =EI cc 0 EI cn

. GJ 0

. . EI nn

and if the c, n axes are aligned with the principal bending axes of the beam section, the off-diagonalelements EI cn vanish.

For a rigid beam, the diagonal elements of the stiffness matrix in (16) are effectively innite.In this limiting case, 1 /GK , 1/EA , and ¯̄E

− 1all vanish, automatically giving zero beam strains and

load-induced curvatures in relations (19) and (20).

The extensional stiffness and tension axis locations are dened as

EA = E dcdn (21)cta =

1EA E c dc dn (22)

n ta = 1EA E n dc dn (23)and the net bending stiffnesses are also dened in the usual manner.

EI cc = E (n − n ta )2 dcdn (24)EI nn = E (c − cta )2 dcdn (25)EI cn = − E (c − cta )(n − n ta ) dcdn (26)

Computation of the shear center cea , nea and the stiffnesses GJ , EI cs , EI sn typically requires ashear-ow analysis. For a simple closed shell with enclosed area Ash , perimeter S sh , and uniformwall thickness tsh , the torsional stiffness is

GJ = 4G A2sh t sh /S sh . (27)

The axial material strain at some location can be written as

ǫ = csh (κn − κn 0) − nsh (κc − κn 0) + F s /EA (28)

wherecsh = c − cta , nsh = n − n ta

are distances from the tension axis to a structural location as shown in Figure 4. The shear stressof a uniform hollow torsion shell is given by

τ = M ′s2 Ash tsh

. (29)

where M ′s is the torsion moment about the shear center, dened in equation (18).

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1.5 Section representation options

It’s useful to point out here that the geometric and structural parameters dened above are non-unique, in that any given wing section can be modeled by innitely many alternative but equivalentrepresentations. Three alternative representations are shown in Figure 5. The bottom sketch is theactual wing section to be modeled, specied by the local airfoil incidence α

local, the airfoil’s zero-lift

angle α L=0 (typically negative for cambered airfoils), and the principal-axis bend angle θbend . Thelatter may not be immediately available by inspection.

P r inc i p a l Bend ing Axis

Z e r o L i f t L i n e

x

x

c

n

n

EI = 0cn

Wing Section

Representation 1

c= 0

x

c

n

L=0−

xAircraft Axis

localθ bend

local θ bend=

local=

A L=0 local+= −

A L=0= −

A θ bend L=0= − −

EI = 0cnRepresentation 2

EI = 0cn

Representation 3

+

Figure 5: Three alternative but equivalent representations of a structural wing section. Represen-taion 1 is preferable if the principal axis is obvious by top/bottom structural symmetry.

1.5.1 Representation 1

This is usually the best choice if the principal bending axis angle θbend is known by inspectionfrom symmetry. The c axis is chosen to lie along this principal bending axis, via the twist angleϑ specifying its orientation relative to the airplane x axis. In terms of the section properties, thenecessary dening relations are as follows.

ϑ = αlocal + θbend (30)αA = − α L =0 − θbend (31)

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n ta = 0 (32)

EI cc = E n 2 dcdn (33)EI nn = E (c − cta )2 dcdn (34)EI

cn = 0 (35)

The advantage here is the simplication of the bending stiffnesses, in particular EI cn = 0 bydenition.


This is usually the best choice if the principal axis is not obvious by inspection, so θbend is notimmediately available. The c axis is then chosen to lie along the airfoil’s chord line, giving thefollowing dening relations.

ϑ = αlocal (36)

αA = − α L =0 (37)The bending stiffnesses EI cc, EI nn , EI cn do not simplify in this case, and must be computed alongwith EA, c ta , n ta directly from their full denitions (21)–(26) which are not restated here for brevity.


This choice may be the most convenient if all the section quantities are to be computed in theglobal aircraft axes xyz, e.g. if the section is dened in a CAD model. In this case the c axis ischosen to cooincide with the x axis, giving the following relations for the case of zero sweep anddihedral.

ϑ = 0 (38)αA = − α L =0 + α local (39)

EI cc = EI xx (40)EI nn = EI zz (41)EI cn = EI xz (42)

For nonzero sweep or dihedral, the xz stiffness-calculation plane would need to be rotated perpen-dicular to the spanwise axis.

1.5.4 Axis relations

If specifying θbend is more convenient than specifying EI cn for some reason, standard Mohr’s Circlerelations can be used to switch from Representation 2 to 1.

θbend = 12

arctan − 2 EI cnEI nn − EI cc 2

(43)

(EI cc)1 = 12

(EI nn + EI cc) − 14 (EI nn − EI cc)2 + EI 2cn 2 (44)(EI nn )1 =

12

(EI nn + EI cc) + 14 (EI nn − EI cc)2 + EI 2cn 2 (45)11


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As written, these assume EI nn > EI cc, and are diagrammed in Figure 6.

EI nn( ) EI 1cc( ) EI cc( )2 EI nn( )2 1

EI 2cn

bend

−( )

2θ

Figure 6: Mohr’s circle relations between Representation 1 and 2 stiffnesses.

1.6 Force and moment equilibrium relations

The force and moment balance on a beam element of length ds is expressed in the x, y,z axes as

d F + f ds + ∆ F d(1) = 0 (46)

d M + m ds + ∆ M d(1) + dr × F = 0 (47)

where f and m are the applied distributed force and moment, ∆ F and ∆ M are the appliedconcentrated force and moment, and d(1) is the unit-impulse function.

2 Discrete FormulationThe discrete representation of the beam is given by the twelve nodal variables

ri ≡ {x i yi zi}T θi ≡ {ϕi ϑi ψi}T M i ≡ M x i M y i M z iT F i ≡ F x i F y i F z i

T

where i is the node index along the beam at the prescribed s i locations. This section describes thealgebraic equations governing these variables.

2.1 Interior equations

The displacements and angles are related by three discrete compatibility relations

∆ x∆ y∆ z

= ¯̄T T

a

γ c1 + ǫs

γ n a

∆ s0 (48)

where s0 is the unloaded-beam arc length, and ∆ ( ) is a difference and ( ) a is a simple averagebetween the i and i+1 stations, e.g.

∆ x = xi+1 − x i (49)¯̄T a =

12

¯̄T i+1 + ¯̄T i (50)

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The strains γ c, ǫs ,γ n in the three relations (48) are negligible in most applications. However, if more than one kinematic constraint is applied to the beam, a nonzero ǫs (i.e. a nite EA) mustbe allowed to give the necessary extensional compliance to give a well-posed problem. The shearstrains γ c, γ n in (48) compensate for the case where the s-axis does not cooincide with the elasticaxis, i.e. if cea , n ea = 0 , 0.

The average strains ( γ c,ǫs , γ n )a in (48) are dened from relation (19). This in turn requiresdening the average force and moment vectors in the local csn axes.

F cF sF n a

= ¯̄T aF xF yF z a

,M cM sM n a

= ¯̄T aM xM yM z a

(51)

The translated average moment ( M ′csn )a is then dened using relation (18).

Discretization of the beam curvature equation (16) rst requires a discrete form of the beamcurvatures. These can be dened via either the discrete form of the dening relation (6),

κcκsκn

∆ s = ∆ ¯̄T ×· ¯̄T a (52)

or via the discrete form of the analytically equivalent relation (8).

κcκsκn

∆ s = ¯̄K a

∆ϕ

∆ ϑ

∆ ψ(53)

These two discrete forms are equivalent to second order in ∆ s, ∆ ϕ, ∆ ϑ, ∆ ψ, but (52) becomes singularif the Euler angle changes ∆ϕ, ∆ ϑ, ∆ ψ approach π/ 2, as in a winglet junction, for example. The

second form (53) remains well-conditioned and so is preferable, but care must be taken to initializethe angles so that the differences do not cross a ± π angle branch cut.

The curvature/moment relation is discretized as follows.

¯̄K a

∆ϕ

∆ ϑ

∆ ψ− ¯̄K a 0

∆ϕ0

∆ ϑ0

∆ ψ0= ¯̄E

− 1a

M ′cM ′sM ′n a

∆ s (54)

After discretization, equation (54) above has been multiplied through by the inverse of the discretestiffness matrix to make the Jacobian’s diagonal elements O(1) like the other equations.

The force and moment balance equations (46,47) are easily discretized as follows.

M i+1 − M i + m a ∆ s + ∆ M + ∆ r × F a = 0 (55)

F i+1 − F i + f a ∆ s + ∆ F = 0 (56)

The applied loads f , m, ∆ F , and ∆ M are arbitrary for now, and will be derived in subsequentsections.

The discrete equations (55) and (56) have the attractive property of being strongly conservative,with no net force or moment being “lost” due to discretization errors. The wing root force F , for

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example, is exactly equal to the net integrated loads f and ∆ F along the wing, even for very coarsegrid node spacings.

A perfectly-rigid beam can be easily represented by setting the inverse of the stiffness matrixto zero in equation (54). Equations (55) and (56) will then still properly predict M and F despitethe zero deformation. Of course, a rigid beam is not permissible if a kinematic constraint for any

degree of freedom is imposed at more than one location.Another attractive property of the entire discrete equation system is its ability to admit a

zero-length structural interval where ∆ r = 0, ∆ s = 0, for which case the system reduces to simplesolution-continuity or solution-jump relations.

ri+1 − ri = r0i +1 − r0i θi+1 − θi = θ0i +1 − θ0i

M i+1 − M i = − ∆ M F i+1 − F i = − ∆ F

Normally, r would be continuous at such an interval, but θ might jump as in a dihedral break.Any concentrated load ∆ M or ∆ F applied at a zero-length interval is therefore captured as aperfect discontinuity in the solution with no articial numerical “smearing”, as shown in Figure 7.For maximum accuracy, it is advantageous to intentionally place such zero-length intervals at allconcentrated load locations.

F

F ∆

s

F

F i

i+ 1

i− 2 i− 1 2 4i+3i+i+i i+ 1

Figure 7: Concentrated load on zero-length i . . . i + 1 structural interval captured as perfectdiscontinuity in discrete solution.

2.2 Boundary conditions and interior constraints

The twelve discrete equations (48,54,55,56) for i = 1 . . . I − 1 require twelve appropriate boundaryconditions to form a closed system for the 12 I discrete variables. At each of the two free wing tipsit is appropriate to impose six zero-load conditions.

M = 0 , F = 0 (57)

These might be specied as nonzero if tip loads are applied by tip stores, joints, etc.

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In addition to the twelve load boundary conditions, each beam must have at least six kinematicconditions — three on position ri and three on angle θi — to properly constrain the beam’s rigid-body translation and rotation modes. Such a kinematic constraint location is either a ground ora joint , both described later. The constraint equations are effectively imposed within the discreteinterval i . . . i +1, and replace the moment-balance and force-balance equations (55,56) on that

interval. Because its equilibrium equations are thus displaced, the interval will now admit jumpsin the moment and force resultants, which represent the loads applied by the constraining object.If the grounding constraint is non-physical (e.g. the x,y,z axis origin attachment point) theseloads will end up being computed to be zero within machine precision, since all real external forces,moments, plus inertial-reaction loads must sum up to zero.

3 Applied Loads

The total distributed applied loads consist of lift, drag, acceleration, and apparent-mass forces.

f = f lift + f drag + f acc + f am (58) m = m lift + mdrag + macc + mam (59)

The concentrated applied loads are due to point masses responding to gravity, acceleration, andaero drag, due to engine forces, due to external elastic struts or bracing wires, and due to beam joints.

∆ F = ∆ F pmass + ∆ F eng + ∆ F strut + ∆ F joint (60)∆ M = ∆ M pmass + ∆ M eng + ∆ M strut + ∆ M joint (61)

Expressions for all these loads are derived in the following sections.

3.1 Aerodynamic loads

The circulation distribution Γ( s) for each lifting surface is dened either pointwise at each structuralinterval or as a Fourier series in the Glauert angle θ(s),

Γ(s i ) = Ai (62)

or Γ(s) =K

k=1Ak sin(kθ) , θ = arccos

ssmax

(63)

where smax is the structural semispan, and s has the range − smax ≤ s ≤ smax . The number of Fourier modes needed for good accuracy is typically smaller than the necessary number of structural

intervals, so the Fourier representation is considerably more efficient. The differences are only inthe implementation details, however, and the circulation variables can be either A i or Ak dependingon which approach is chosen. The Fourier approach will be used in the subsequent development.

The local velocity relative to the beam section at location r is

V ( r) = V ∞ − Ω × r + V ind ( r) + V gust ( r) (64)

V ind ( r) =K

k=1vk ( r) Ak + wvol( r) V ∞ (65)

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where Ak are Fourier mode coefficients for the bound circulation and vk are their aerodynamicinuence functions. The induced velocity due to beam volumes is given via the wvol inuencefunction. Their contributions to the overall induced velocity V ind will be treated in a later section.The nonuniform gust perturbation velocity eld V gust , which is prescribed explicitly in subroutineVGUSTE, is included for completeness. Unlike V ∞ , the gust velocity can have a spatial variation and

so can represent tight thermal cores, sharp gusts, etc., which can have signicant variations overthe extent of a large aircraft.

For calculating the various aerodynamic loads, it is useful to dene unit vectors giving thedirections of the local c,s,n unit vectors in airplane axes. These are simply the rows of thetransformation tensor.

− ĉT −− ŝT −− n̂ T −

= ¯̄T (66)

The aerodynamic lift vector then follows from the vector form of the Kutta-Joukowsky theorem,

f lift = ρ Γ V × ŝ (surface beam) (67)

where V is evaluated using (64) at the bound vortex location rh.v. , dened later. The local cir-culation Γ will be determined via a lifting-line formulation which employs the Prandtl-Glauerttransformation, so that this lift force accounts for compressibility. For a fuselage beam with localradius R, the lift force is determined from slender-body theory using the beam-normal velocity V ⊥ .

f lift = ρ V ⊥ ( V · ŝ) 2πRdRds

(fuselage beam) (68)

V ⊥ = V − ( V · ŝ ) ŝ (69)

In this case V is evaluated at ra at the center of the cylindrical beam. The prole drag force isresolved into a friction-drag contribution acting along the local net velocity, and a pressure-dragcontribution acting perpendicular to the beam’s spanwise axis.

f drag = 1

2ρ | V | V c̄ cdf +

12

ρ | V ⊥ | V ⊥ c̄ cdp + 2 ρ V ⊥

| V ⊥ | V · n̂

2

c.p.c̄ (surface beam) (70)

f drag = 1

2ρ | V | V 2R cdf +

12

ρ | V ⊥ | V ⊥ 2R cdp (fuselage beam) (71)

where c̄ is the local chord. For an unswept wing, the sum cdf + cdp constitutes the usual airfoil-section prole drag. For a fuselage, cdf is comparable to the skin friction coefficient C f , and cdp isthe drag coefficient of a circular cylinder, typically 1.2 for subcritical ow, and 0.4 for supercriticalow. The third term in the surface-beam drag force enters in only when the ow-tangency velocity

V · n̂ is nonzero, which occurs when the local cℓ exceeds the stall limits. This constitutes a simplepost-stall model, and gives f drag ≃ρV 2⊥ c̄ when the local incidence angle approaches 90 ◦ .

Excluding the deep-stall condition, the drag forces controlled by cdf and cdp usually contributevery little to structural loads, but are included here since they do not impact the calculation costsignicantly, and can give overall aircraft drag estimates. The fuselage lift forces can have signicantimpact on the overall aircraft pitching moment.

The aerodynamic prole moment vector is approximated by assuming that it is inuenced onlyby the velocity component V ⊥ perpendicular to the wing’s spanwise axis.

m lift = ( c̄/ 4 − x̄o) ĉ × f lift + 12

ρ | V ⊥ |2 c̄2 cm ŝ (surface beam) (72)

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The rst term is simply the contribution of the lift force acting at the quarter-chord point, withx̄o being the chordwise location of the c,s,n origin, as shown in Figure 16. The second terminvolves the 2-D section pitching moment coefficient cm , which is dened as a constant part, pluscontributions from control surface or ap deections δ F . Unsteady Theodorsen-type terms will beincluded in the apparent-mass contributions.

cm = cm o + dcm

dδ F 1δ F 1 + dc

m

dδ F 2δ F 2 . . . 1 1 − M 2⊥ (73)

The Prandtl-Glauert factor uses the local perpendicular Mach number M ⊥ = | V ⊥ |/V sound , whichis consistent with lifting line theory. The contribution of the ap-deection moment derivativedcm /dδ F is very important in most applications, since it is the aerodynamic quantity directlyresponsible for aileron reversal, and often signicantly reduces aileron response in normal ightconditions.

The distributed prole moment force due to friction forces is in general negligible:

mdrag = 0 (74)

3.2 Inertial and gravity loads

Absolute accelerations relative to an inertial frame are dened on the beam axis at ri , and at somenearby offset location r p = ri + ∆ r p.

ai = ao + α o × ri + Ω × Ω × ri (75)

a p = ai + αo × ∆ r p + Ω × Ω × ∆ r p (76)

The coordinate-system origin acceleration ao can be prescribed arbitrarily, or can be related to theoverall external force on the conguration. Both cases will be treated later. It must be stressed thatthe assumption of rigid-body rotations and accelerations made in (75,76) may not be realistic withlarge exible aircraft, since structural dynamics and/or unsteady aerodynamics are being neglected.Nevertheless, the computed results for such cases represent the low-frequency limit, and hence stillgive useful information. The full dynamic formulation will be treated as a separate extension.

The local gravitational and inertial-reaction load acts at the local mass centroid offset from thebeam axis by

∆ rcg = ccg ĉ + ncg n̂ (77)

where ccg , n cg is the location of the mass centroid of the beam cross-section as shown in Figure 4.This ∆ rcg then denes the local acceleration acg via relations (75,76). The corresponding loads are

f acc = µ ( g − acg) (78) macc = ∆ rcg × f acc − ¯̄T

T ¯̄ι ¯̄T α o − Ω × ¯̄T T ¯̄ι ¯̄T Ω (79)

where µ is the mass/length density of the beam, ¯̄ι is the section inertia/length tensor. This tensoris dened to have the form

¯̄ι ≡ιcc 0 00 ιss 00 0 ιnn

+ ∆ s2

12

µ 0 00 0 00 0 µ

(80)

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ιcc = (n − n cg)2 dµ (81)ιnn = (c − ccg)2 dµ (82)ι ss =

[(c − ccg)2 + ( n − ncg)2]dµ = ιnn + ιcc (83)

which reasonably assumes that the tensor’s principal axes are aligned with the local c, n axes. Thesecond term in (80) is formally O(∆ s2) and normally negligible, but is retained to improve accuracyfor very coarse discretizations.

The surface apparent-mass force contribution depends only on the normal component of thelocal accelerations, and the resulting pressure forces can act only normal to the surface. Hence theapparent-mass force is along ˆn and the apparent-mass moment is along ˆ s.

f am = π

4 ρ c̄2 V × Ω · n̂ − ac/ 2 · n̂ n̂ (surface beam) (84)

m am = −π4

ρ c̄2 c̄4

V × Ω · n̂ + c̄8

αo · ŝ ŝ + ∆ rc/ 2 × f am (85)

The midchord acceleration ac/ 2 is evaluated at

∆ rc/ 2 = ( c̄/ 2− x̄o)ĉ

using relations (75,76). All the apparent-mass terms above correspond to the “instantaneous” (non-lag) forces and moments from Theodorsen’s theory. The various vector projections are necessaryto reproduce the theory’s results in the perpendicular c-n airfoil plane. The net velocity V in theCoriolis-like term above is evaluated on the beam itself at ri .

For a fuselage beam, the apparent-mass force is obtained from slender-body theory as follows.

f am = − 2π ρ R 2 ai − ( ai · ŝ) ŝ (fuselage beam) (86)

3.3 Point-mass loads

In addition to the continuous mass/span distribution µ, it is useful to also consider point masses m pwhich are cantilevered from the beam’s structural axis by a rigid pylon with geometric dimensionsc p, s p, n p in the local wing-section coordinates, as shown in Figure 8. These pylon dimensions aredetermined via the transformation tensor.

{ c p s p n p }T = ¯̄T 0 {∆ r p}0 (87)

The ( ) 0 subscript denotes values for the undeformed state which is known a priori, so that c p, s p, n p

are in effect xed constants. Similarly, the undeformed-state angular momentum vector H p0 denesits corresponding xed c, s, n components.

{ H cp H sp H n p }T = ¯̄T 0 H p 0 (88)

The point mass can represent a nacelle, external store, rotor, etc, mounted to the beam. It mustbe noted that the assumption that the pylon is rigid does not eliminate the ability to represent amass which is mounted on an elastic pylon. In this case, the elastic pylon would be represented bya fuselage-type beam, and the point mass would be attached to this exible beam.

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cn

s

p

p

p

m

c

s

n^

^

^

r

r

p

ρ DV V (C A) p p p12

pm (g − a )

p

r p∆i

H

p

p

p I =

,

| |

Figure 8: Point mass m p with inertia ¯̄I p and angular momentum H p, cantilevered from wing byrigid pylon, with various applied loads shown. Pylon endpoint r p is dened by the anchor point ri ,the anchor’s point orientation ( ¯̄T ), and the pylon’s xed c p, s p, n p components.

The location r p of the point mass in airplane axes is given by

∆ r p = ¯̄T T

{ c p s p n p }T = ¯̄T T ¯̄T 0 ∆ r p0 (89)

r p = ri + ∆ r p (90)

where ¯̄T is the transformation tensor at the pylon-attachment location ri on the wing/beam at thelocal c, s, n origin. Although c p, s p, n p are xed, both ri and ¯̄T change as the beam deforms, so thatthe point mass at r p is “waved around” appropriately in space along with the beam. The angularmomentum vector and inertia tensor are rotated in the same manner.

H p = ¯̄T T { H cp H sp H n p }

T = ¯̄T T ¯̄T 0 H p0 (91)¯̄I p = ¯̄T

T ¯̄T 0 ¯̄I p0 ¯̄T T

0¯̄T (92)

The local relative air velocity at the point mass and the mass’s acceleration follow from relations(64) and (75,76).

V p = V ∞ − Ω × r p + V ind ( r p) + V gust ( r p) (93)

a p = ai + α o × ∆ r p + Ω × Ω × ∆ r p (94)

The gravitational, inertial-reaction, and aerodynamic drag loads applied to the beam by the point

mass are

∆ F pmass = m p (g − a p) + 12

ρ | V p| V p (C D A) p (95)

∆ M pmass = ∆ r p × ∆ F pmass − Ω × H p − ¯̄I p αo − Ω × ¯̄I p Ω (96)

with the mass m p, undeformed-state angular momentum H p0 , and the net drag area ( C D A) p of thepoint mass being specied.

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3.4 Engine loads and jet

3.4.1 Engine force and moment

An “Engine” entity is treated very much like a point mass. As shown in Figure 9, it is mounted ona rigid pylon, and has a force F eng and moment M eng acting on it. The loads applied to the beamat the pylon attachment point are

∆ F eng = F eng (97)∆ M eng = M eng + ∆ r p × ∆ F eng (98)

c

sn p

p

p

c

s

n^

^

^

r

F eng

r p∆i

eng M

V

Aeng

V ∆

r jet

jetb

jet x

jet

pr

r

^

specTe0

^e0

^e0

x

y

z

p

Figure 9: Engine cantilevered from wing by rigid pylon, with thrust force and moment. Propulsive jet velocity ∆ V jet affects downstream surfaces on which it impinges.

3.4.2 Engine axes relations

The engine force and moment F eng and M eng in equations (97) and (98) are obtained from anengine “black box” subroutine (treated later), which performs its computations in the engine axesxe ,ye,ze. The transformation from x,y,z to xe ,ye,ze and vice-versa is performed outside the engineblack-box routine. The engine axis directions are specied via the “thrust vector” T spec shown inFigure 9, which will normally correspond to the prop shaft axis for the undeformed geometry. Thisdenes the undeformed-state engine axis unit vectors as follows.

x̂e0 = − T spec / T spec

ŷe0 = k̂ × x̂e0 / k̂ × x̂e0ẑe0 = x̂e0 × ŷe0

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It is assumed that the engine force is unaffected by the rotating the whole powerplant around thex̂e0 axis. This justies the arbitrary placement of the ˆ ye0 axis perpendicular to the k̂-x̂e0 plane.

The rotation of the attachment point and pylon will give the engine axes a new orientation.

x̂e = ¯̄T T ¯̄T 0 x̂e0

ŷe = ¯̄T T

¯̄T 0 ŷe0ẑe = ¯̄T

T ¯̄T 0 ẑe0

These three deformed-state unit vectors dene a convenient engine-axis transformation tensor,

¯̄T e ≡− x̂e −− ŷe −− ẑe −

=− x̂e0 −− ŷe0 −− ẑe0 −

¯̄T T ¯̄T 0

T

(99)

which depends on the specied T spec , and also on the local beam Euler angles ϕi , ϑi , and ψi forthe deformed and undeformed geometry.

An alternative “xed” engine-axis denition is to simply dene

x̂e = x̂e0 (100)ŷe = ŷe0 (101)ẑe = ẑe0 (102)

so that the axes are held xed to their specied orientations, although this would be difficult tophysically implement on a real exible aircraft. In any case, in the program this type of xed engineis specied by simply attaching it at the ground point. The engine location r p will then stay at itsinitial location r p0 .

The engine black-box routine receives the relative velocity V e in the engine axes. This is com-

puted from the body-axes relative velocity V p dened by equation (93), together with the transfor-mation tensor.

V e = ¯̄T e V p (103)

The resulting computed engine force and moment F e , M e are then transormed back to the bodyaxes, again using the transformation tensor.

F eng = ¯̄T T

e F e (104)

M eng = ¯̄T T

e M e (105)

These are then used in (97) and (98) to determine the applied beam loads.

3.4.3 Engine models

Proportional engine modelThe simplest engine model assumes the thrust and moment are directly proportional to the enginepower parameter ∆ e , with specied unit-power force and moment coefficients F e1 and M e1 . The

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thrust and moment are assumed to be in the − x̂e direction (i.e. T spec direction tilted by structuraldeformation).

F e = − ∆ e F e1 x̂e M e = − ∆ e M e1 x̂e (106)

Actuator disk engine modelA more realistic engine model uses actuator disk theory, which can represent effects such as thrustlapse with density and air velocity, e.g. F e(∆ e , ρ, V e), M e(∆ e , ρ, V e). We assume that ∆ e representsthe engine shaft power:

P eng = ∆ e (107)

For constructing the propeller model, we need to dene V eng , which is the axial component of therelative onset ow at the propeller.

V eng = V x e = V p · x̂e (108)

A reasonable estimate for viscous power loss due to prole drag is

P v = B

Reng

0

1

2ρ V 2eng + Ω

2eng r

2 3/ 2 cd c dr

P v ≃ 1

8 ρ V 2eng + Ω

2eng R

2eng

1/ 2V 2eng + 3Ω

2eng R

2eng (C D A)eng (109)

(C D A)eng = BR eng c̄ c̄d (110)

where (C D A)eng is the total effective blade drag area, dened in terms of the number of bladesB , the effective blade chord c̄, and the blade airfoil drag coefficient c̄d . Both c̄ and c̄d are besttaken from the blade radius r ≃ 0.8Reng where the viscous-power integrand is typically close tomaximum. Extended actuator disk theory then gives

F eng = ρAeng V eng + 12

∆V eng ∆V eng (111)

P eng − P v = V eng + 1 + 5 λ2 ∆ V eng2 F eng (112)

P eng − P v = ρAeng V eng + 1 + 5 λ2 ∆ V eng

2 V eng + 12∆V eng ∆V eng (113)λ = V eng / Ωeng Reng (114)

where the empirical factor of 1 + 5 λ2 accounts for induced swirl losses, which become signicantwhen the advance ratio λ becomes large.

If the shaft power P eng is specied, and the viscous power P v is computed using the approxi-mation (109), equation (113) becomes a solvable cubic equation for ∆ V eng . In lieu of using cubicequation formulas, it is simplest to solve equation (113) using Newton iteration. Several precau-

tions must be made for the windmill case where the net inviscid power P i ≡ P eng − P v is negative,which is easily encountered in the power-off case if P eng = 0 and P v is nite. For a given airspeedV eng , the minimum P i corresponds to the Betz limit,

(P i)min = − 827

ρV 3eng Aeng (115)

which occurs when

∆V eng = −23

V eng (at Betz limit) (116)

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Any specied P i below (P i)min must be clipped at ( P i)min , else there will be no physical ∆ V eng rootto the cubic equation (113), possibly resulting in an arithmetic fault.

After ∆ V eng is computed from (113), the thrust force is obtained immediately via (111). Theshaft moment is given simply by a power balance requirement, with Ω eng dened positive about

T spec . The functional dependencies have the following forms.

F eng = F eng (P eng ,ρ ,V eng ) (117)M eng = M eng (P eng , Ωeng ) = − P eng / Ωeng (118)

These are assumed to act along the engine axis direction − x̂e, so that the components of the engineforce and moment vectors F e and M e are then given as follows:

F e = {− F eng , 0 , 0}T (119) M e = {− M eng , 0 , 0}T (120)

Off-axial forces and moments could be incorporated here using P-factor relations.

If the thrust is specied rather than the power, then ∆ V eng does not get involved in the calcula-tions. Since it is still needed for the propulsive jet model developed next, it is in this case computeddirectly from the quadratic thrust relation (111).

∆V eng = V 2eng + 2F engρAeng − V eng (121)3.4.4 Propulsive jet

Engine force will be accompanied by a propulsive jet with velocity increment eld ∆ V jet . The jet isassumed to lie along some unit direction vector ξ̂ , which is also the direction of the trailing vortex

legs in the vortex-lattice system (described in a later section). The spatial variation of this jetvelocity eld is constructed using a combination of actuator disk theory together with jet and wakemixing theory. These will be applied in terms of the lateral distance r jet from the thrust line andthe distance x jet behind the engine, dened by the following vector operations.

x jet = ξ̂ · ( r − r p) (122)r jet = ( r − r p) − ξ̂ x jet (123)r jet = | r jet | (124)

As shown in Figure 10, there will be an initial induced velocity increment ∆ V eng a short distancebehind the engine disk, once ambient pressure is reached. The self-induced velocity increment at

the disk itself will be only ∆ V eng / 2, so that the jet will have a fast initial contraction from enginedisk radius Reng , to R ′eng as given by the mass continuity requirement.

Reng = Aengπ (125)R ′eng = Reng V eng + ∆ V eng / 2V eng + ∆ V eng (126)

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jet x

V ∆ 0 jet

engeng

x mixV ∆

ε

eng

initial contraction

b jetc

b

εc

jet R R’

V eng

Figure 10: Evolution of jet radius and jet prole behind an engine.

Jet spreading and jet velocity proleAs shown in Figure 10, the boundary of the jet will spread laterally due to turbulent mixing at ahalf-angle of arctan( ε). Mixing-layer theory, as described by White [2], assumes the lateral growthrate scales with the lateral turbulent velocity uctuations.

DbDt ∼ v′

dbdx

= K ε∆V

V convectiveApplying this concept to the inner and outer shear layer boundaries gives

εc = K ε|∆V eng |

V eng + 1 .0 ∆V eng(127)

εb = K ε|∆V eng |

V eng + 0 .5 ∆V eng(128)

K ε = 0.11 (129)

The constants 1.0 and 0.5 for the effective convective speeds in the denominators are calibratedto give a nearly constant momentum ux in the resulting jet prole, as will be seen later. TheK ε = 0 .11 value approximately matches observed experimental shear layer spreading rates. Forvery light disk loadings, ∆ V eng ≪V eng , and the spreading angle is close to zero. For very large diskloadings, ∆ V eng ≫ V eng , the outer spreading angle approaches arctan( K ε / 0.5) ≃ 12◦ . The innerspreading angle denes a “mixing distance” xmix where the inner mixing layer boundaries mergeat the centerline, and it also denes the inner and outer mixing layer boundaries c jet and b jet , asshown in Figure 10.

xmix = R′eng /ε c (130)

b jet = R′eng + εb x jet (131)

c jet = R′eng − εc x jet , x jet < x mix

0 , x jet > x mix(132)

The velocity prole over the mixing layer is closely approximated by Schlichting’s asymptotic wakeprole as given by White [2].

∆V jet =

0 , x jet < 0 or r jet > b jet∆V 0jet 1 − [(r jet − c jet )/ (b jet − c jet )]3/ 2

2, x jet > 0 and b jet > r jet > c jet

∆V 0jet , x jet > 0 and c jet > r jet

(133)

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The jet centerline velocity increment ∆ V 0jet used in (133) is obtained by the physical requirementthat the jet’s momentum excess must always be equal to the engine thrust.

F eng = ρ∆V jet (V eng + ∆ V jet ) dA (134)= 2 π

b

0 ρ∆V jet (V eng + ∆ V jet ) r jet dr jet= π ρ k1 V eng ∆V 0jet + k2 ∆V 20jet

k1 = c2 + 910

c(b − c) + 935

(b − c)2

k2 = c2 + 243385

c(b − c) + 2431820

(b − c)2

∆V 0jet = 14 k21k22 V 2eng + F engρπk 2 − 12 k1k2 V eng (135)Figure 11 shows the predicted jet spreading for two different disk loadings, characterized by theinitial jet velocity ∆ V eng relative to V eng . Also shown are the predicted velocity proles, andcenterline velocities ∆ V 0jet . The latter are approximately equal to ∆ V eng until the inner mixinglayer edge reaches the centerline, after which ∆ V 0jet gradually decreases. The total jet momentumexcess dened by (134) is exactly equal to F eng for all x jet locations.

eng

eng

1.5

1.0

0.5

10 20 30 40 50

∆

jet

0 jetV /V

xmix

xmix∆ engV /V

F eng x /R eng

eng∆ engV /V

eng∆ engV /V

eng∆ engV /V

= 1.5

= 0.4

= 1.5

= 0.4

Figure 11: Predicted jet spreading and centerline jet velocities for moderate disk loading and highdisk loading.

Jet swirlIn general, there will also be a swirl velocity ∆ W jet in the prop jet (not shown in the gures). It isassumed that this has the same prole shape as the axial jet velocity.

∆W jet =

0 , x jet < 0 or r jet > b jet∆W 0jet 1 − [(r jet − c jet )/ (b jet − c jet )]3/ 2

2, x jet > 0 and b jet > r jet > c jet

∆W 0jet , x jet > 0 and c jet > r jet

(136)

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The jet centerline velocity increment ∆ W 0jet used in (136) is obtained by the physical requirementthat the jet’s angular momentum must always be equal and opposite to the propeller torque.

− M eng = ρ (V eng + ∆ V jet ) ∆W jet r dA (137)= 2 π

b

0ρ (V eng + ∆ V jet ) ∆W jet r 2

jet dr jet

= πρ (k3 V eng + k4 ∆V 0jet ) ∆ W 0jet

k3 = 2

3c3 +

910

c2(b − c) + 935

c(b − c)2 + 19

(b − c)3

k4 = 2

3c3 +

243385

c2(b − c) + 2431820

c(b − c)2 + 245

(b − c)3

∆W 0jet = − M eng

πρ1

k3V eng + k4∆V 0jet(138)

Jet velocity vectorThe jet axial velocity is assumed to be oriented along the thrust axis direction ˆ xe , and the swirl is

perpendicular to the radial vector.

∆ V jet = x̂e ∆V jet +r jetr jet

× x̂e ∆W jet (139)

3.5 Strut loads

A strut or bracing wire represents another type of load applied to the end of the rigid pyloncantilevered from the beam. One end of the strut is attached to the pylon end at r p, while theother end is grounded (xed in x,y,z axes) at location rw as indicated in Figure 12. The pylonmount for the strut end at r p allows modeling of a typical strut attachment point to a spar ange,in which case the length of the pylon ∆ r p from the tension axis to the spar ange is about half thespar depth.

The strut is assumed to be perfectly exible in bending, but has a nite extensional stiffnessEA w , and thus can change its length in response to extensional loads applied to it by the beam.Hence, it most closely represents a bracing wire, or a bracing strut with pinned endpoints. Notethat such a strut could also be simulated by a fuselage-type beam with negligible bending stiffnesses, joined to the surface at one end, and grounded at the other end. However, the strut is far cheapercomputationally, since it does not introduce additional structural variables. The strut is alsoassumed to be uniform, and hence is simpler to specify in an input le.

Following the notation used earlier to describe the pylon supporting a point mass, the strutlength vector Lw is dened as follows.

Lw = rw − r p (140)

The strut anchor rw is assumed to be at a xed location in airplane axes, while r p changes as thebeam deforms, as with the point mass case. The point load and moment applied to the beam bythe tensile force of the strut is related to its change in length by

∆ F strut = EAw | Lw || Lw |0

− 1 | Lw || Lw |0

(141)

∆ M strut = ∆ r p × ∆ F strut (142)

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c

s

n^

^

^

r p

F strut∆

r

r w

i

Figure 12: External bracing strut or wire attached to wing by rigid pylon, and anchored at xedlocation rw .

where | Lw |0 is the length of the strut or wire in its unloaded state, and ∆ r p is the pylon lengthvector as before. The above expression for ∆ F strut would of course need to be limited if the lengthchange is negative, and the resulting compressive load exceeds the buckling load of the strut.

3.6 Joint loads

Another type of concentrated load on a beam is due to a joint to another beam. Such a joint isimplemented as a rigid pylon connecting the joint locations r1 and r2 on the two beams, as shownin Figure 13. Joint locations which nearly cooincide thus represent a direct structural joint, while joint locations which are relatively distant are in effect connected by a perfectly rigid massless beam(this beam can be given mass by superimposing one or more point masses on it). However, theformulation is the same for any joint distance.

Figure 13 also shows the auxilliary variables ∆ rJ , ∆ θJ , F J , M J , which dene the state of theimaginary link forming the joint, and are used to dene loads and kinematic constraints on the joined beams. The load resultants of the joint are applied as concentrated loads to the beamcontaining joint point #1.

∆

F joint = F J ∆

M joint =

M J (at point #1) (143)

The joint position change and Euler angle change vectors are imposed on the beam at joint point#2 as kinematic constraints in lieu of the moment and force balance equations (55,56) across thestructural interval containing joint point #2.

ri = r0i + ∆ rJ θi = θ0i + ∆ θJ (at point #2) (144)

The “lost” equations (55,56) for the interval containing point #2 will be imposed as global con-straints, as described later.

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c^

s^n^ 2

2

2

c^

s^

n^ 1

1

1

r

c^

s^

n^ 1

1

1

r

n^ 2

c^

s^2

2

1

2

r 1r 2

F J

J

J J

J −F

r

nearby joint

distant joint

jointvariables

2 1 F (r −r ) ∆

#1

#2

M −

−M

Figure 13: Nearby joint simulating a direct structural connection, and a distant joint simulating arigid connecting beam. No distinction is made in the formulation. Variables F J , M J , . . . are loadresultants on the imaginary joint pylon (dashed).

3.7 Ground-point loads

A ground is a joint between a location on a beam and a xed point in the x,y,z axes. A groundmust be present in each beam which has no joints pointing to it, so that the ground serves torestrain the beam’s rigid-body translation and rotation modes. As mentioned earlier, a ground isnot necessary (and should not be used, in fact) if the beam in question already has a joint pointingto it, since the joint kinematic constraints (144) then act as the necessary kinematic constraints.

The ground is implemented very simply by replacing the moment and force equilibrium equa-

tions (55,56) at the interval containing the grounding point with simple kinematic constraints onthe displacements and Euler angles.

ri = r0i , θi = θ0i (at ground) (145)

Note that these are the same as the kinematic joint constraints (144) but without the joint degreesof freedom ∆ rJ , ∆ θJ .

Unlike the joint #2 point, the ground does not have its “lost” equations (55,56) enforced later— they are omitted altogether. The computed reaction loads against the ground point will simply

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be the overall force and moment applied to the conguration, and will produce a discontinuity in F and M . If equilibrium is specied, as it should be in a physically correct representation of an

actual ight condition, these reaction loads and the discontinuities will be computed to be zero.

3.8 Beam Groups

A beam group is a set of beams connected by joints, each joint having #1 and #2 points. Which joint point is labeled #1 or #2 is not physically signicant and does not theoretically inuence thesolution, but it does affect the well-posedness of the numerical problem. The sub-matrix for eachbeam is partially solved independently of the other beams, and the individual partial solutions areappropriately combined via the joint variables and other global variables such as Ω, Ak , etc. Despiteappearances, this approach does not make any approximations to the full Newton method, but itdoes give computational economy compared to using a single matrix for the entire conguration.A ground point and a #2 joint point each receive kinematic constraints during the factorizationof the sub-matrix for that beam. If these constraints are absent, the beam’s rigid body modes areunconstrained, which will produce a singular sub-matrix and numerical solution failure. Every beammust therefore have at least one such kinematically-constrained point. Figure 14 shows instancesof such a “proper” beam group, and an “improper” beam group where this rule is violated. In thelatter, the middle fuselage beam has no ground point or #2 joint point, and hence no kinematicconstraints.

If the beam has nite bending and extensional stiffness, it is permissible to have more than one#2 joint point per beam, but this can give numerical difficulties and should be avoided if at allpossible. In contrast, a beam can have an arbitrary number of #1 joint points with no ill effects,since these merely receive applied forces (143) rather than kinematic constraints (144) or (145).

4 Velocity Inuence Coefficients

4.1 Prandtl-Glauert Transformation

The induced velocity has contributions from the bound and trailing vorticity on the lifting surfacebeams, and source and doublet distributions along the beams with signicant volume. Figure 15shows the quantities involved in the relevant summation integrals. The integration is performed inthe wind-aligned and streamwise-stretched Prandtl-Glauert ξηζ axes to account for compressibility.

ξ ηζ

= ¯̄P xyz

(146)

or rP = ¯̄P r (147)

¯̄P =· · · ξ T · · ·· · · η̂T · · ·

· · · ζ̂ T · · ·

=

1λ cos α cos β − 1λ sin β 1λ sin α cosβ

cos α sin β cosβ sin α sin β − sin α 0 cosα wind

The ξ vector is aligned with the “wind direction”

λ ξ = V ∞

V ∞=

cos α cosβ − sin β

sin α cos β wind

, λ = 1 − M 2PG (148)29


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Ground

Joint

Joint

Ground

Joint

Joint

Overconstrained beam

Unconstrained beam

#1

#2

#1

#2

#1

#2

#1

#2

Properbeam group

Improperbeam group

Figure 14: Proper and improper beam groups. Improper group has unconstrained fuselage beamwith no kinematic constraints, which is not allowed. Overconstrained surface beam is allowed butnot recommended.

as can be seen by comparing with equation (1). The ζ̂ vector is in the xz plane and perpendicularto V ∞ , and hence is antiparallel to the ˆ z vector of standard stability axes. The remaining ˆ η vectoris perpendicular to both ξ and ζ̂ , and is not in general parallel to y axis, except in the zero-sideslipβ =0 case.

As indicated by the subscripts, the implementation actually denes ¯̄P and ξ in terms of the“wind angles” α wind , β wind , which can be set either to the true α , β , or to any other values deemedsuitable for the case at hand. Likewise, the “Prandtl-Glauert” Mach number M PG used to denethe stretching factor λ above can be either the actual Mach number M ∞ , or set to some suitablenominal value. Setting M PG =0 is acceptable for low-speed cases, for which M 2∞ ≪ 1.

The Prandtl-Glauert equation for the perturbation potential φ(x,y,z ) transforms to the Laplaceequation in ξ ,η,ζ space

λ2∂ 2φ∂x 2

+ ∂ 2φ∂y 2

+ ∂ 2φ∂z 2

= 0 → ∂ 2φ

∂ξ 2 +

∂ 2φ∂η2

+ ∂ 2φ∂ζ 2

= 0

whose solution for unit Ak and V ∞ gives the respective inuence functions vk and wvol in termsof appropriate integrals over the singularities. The integrals effectively give the velocities ∂φ/∂ξ ,∂φ/∂η , ∂φ/∂ζ in transformed space. The velocities in physical space are then obtained via thechain rule, which amounts to multiplication by the transpose of the transformation matrix ¯̄P (not

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ind

d

,

( )

d

Slender Body(source+doublet line)

Bound, trailing vorticity(horseshoe vortex filament)

V (r)

Figure 15: Induced velocity contributions from bound vortices and source and doublet distributions.

its inverse).

∂φ/∂x∂φ/∂y∂φ/∂z

=ξ x ηx ζ xξ y ηy ζ yξ z ηz ζ z

∂φ/∂ξ ∂φ/∂η∂φ/∂ζ

= ¯̄P T

∂φ/∂ξ ∂φ/∂η∂φ/∂ζ

(149)

4.2 Vortex inuence function

The vortex inuence function vk ( r) is given by the Biot-Savart integral over the circulation dis-tributions on all the beams. For the Fourier-mode representation (63) on a at unswept wing, asimple approach would be to use the analytical result from Prandtl wing theory

vk = − λπc̄

+ k2b

1sin(θ)

sin(kθ) ẑ (at unswept wing)

where the two terms are from the bound and the trailing vorticity. For the general wing geometriesand ow conditions considered here, a numerical approach must be taken, employing a discretebound/trailing vortex system typically used in vortex-lattice methods. Such a vortex system isshown in Figure 16, with the bound vortices being placed at the quarter-chord point in the localc, n plane, and the trailing vortices being aligned with the relative freestream in the ξ direction. Themodal contribution Γ k to each bound vortex segment, which extends over the structural intervals i . . . s i+1 , is simply the weighted mode sampled at the Glauert-angle midpoint value.

Γk = Ak sin(kθa ) (150)where θa =

12

(θi + θi+1 ) (151)

and also sa = smax cosθa (152)

The alternative “pointwise” circulation discretization (62) assumes a simple piecewise-constantcirculation over each si . . . s i+1 interval.

Γ(sa ) = Ai , si < s a < s i+10 , otherwise (153)

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In this section, the subscript “ a” denotes a midpoint average in θ rather than in physical space.The θ-averaged position vectors are obtained via a weighted interpolation using sa ,

ra = ri + ( ri+1 − ri) sa − s i

∆ s (154)

where ri is simply the beam node position xi , yi , zi . Interpolating in θ gives more accurate resultsfor the predicted aerodynamic properties.

c.p.

s

A sin k k

Γ

c

n

c 4

c

n

Γ

ii+1

r

r

c.p.

h.v.

hc

w i n d d i r e c t i o n

x

A

n

o

Figure 16: Airfoil-plane dimensions, and circulation mode represented by horseshoe vortices.

The location of the bound vortex segment midpoint rh.v. , where the lift force is dened to belocated, is at the quarter-chord point along the ξ axis,

rh.v. = ra + c̄/ 4− x̄o

| ξ × ŝ | ξ (155)

The location of the control point rc.p. is then directly “downstream” along the same ξ direction,

rc.p. = rh.v. + h c̄a| ξ × ŝ |

ξ (156)

where h is chosen so that the local incompressible 2-D section lift-curve slope is reproduced.

h = 14π

dcℓdα

(157)

For a thin airfoil, h = 1/ 2, which is also the value used by the standard vortex-lattice method.

The formulation used here differs from most standard vortex-lattice methods in that the trailingvorticity is always aligned with the freestream velocity direction ξ , as opposed to being predened

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to lie along x̂, say. The control points at rc.p. also “slosh around” as the freestream directionchanges, but always remain at the local 3 / 4 chord line, or slightly farther aft if h > 1/ 2.

Aligning with the freestream naturally gives better accuracy for large angles of attack and largesideslip angles, since this is what the trailing vorticity actually does. On the other hand, it will failif any part of the beam/wing aligns with the freestream, in which case | ξ × ŝ | = 0 in the equations

above. Of course, the physical lifting-line model itself fails in this extreme and rather unlikelysituation, so this may be a moot point.

The quasi-normal vector ˆ n c.p. at the control point is rotated away from the local normal vectorn̂ by the local 2-D section aerodynamic angle of attack αA relative to the local c axis.

n̂c.p. = ¯̄T T

sin αA0

cos αA(158)

This local angle of attack is dened as a constant part, plus contributions from control-surface orap deections δ F .

αA = αA o + dcℓ/dδ F

1dcℓ/dα δ F 1 +

dcℓ/dδ F 2dcℓ/dα δ

F 2 . . . (159)

The control-surface derivatives dcℓ/dδ F are specied properties of the local 2-D section, as aredcm /dδ F and dcℓ/dα .

The overall inuence coefficient function for the k′ th circulation mode is then obtained fromthe Biot-Savart law applied to all the horseshoe vortices with strengths corresponding to unit Ak .

vk = ¯̄P T

∂φ/∂ξ ∂φ/∂η∂φ/∂ζ k

( r) = ¯̄P T 1

4π

I − 1

i=1sin(kθa ) d ℓ × δ δ 3 (160)

The vortex element d ℓ belongs to the horseshoe vortex spanning the i... i+ 1 stations, and δ ex-tends from the vortex element to the eld point, as shown in Figure 15. Both are dened in thetransformed Prandtl-Glauert space.

d ℓ = drP = ¯̄P dr δ = rP − rP (ℓ) = ¯̄P { r − r( ℓ)}

For evaluating the righthand side integrals for surfaces which do not contain the control pointin question, the induced velocity expression for vk is modied by incorporation of a nite vortexcore size ε.

vk = ¯̄P T 1

4π

I − 1

i=1sin(kθa ) d ℓ × δ (δ 2 + ε2)3/ 2 (161)

Choosing ε = max( c̄a / 4 , ∆ s) results in the singular velocity eld of the trailing vortex lamentsbeing smeared into a continuous vortex sheet whose thickness is some reasonable fraction of thechord. This is essential to produce well-behaved lift forces on any surface immersed in the vortexwake of another upstream surface. It also weakens the dependence of the local wake velocity eldon its geometry, justifying the lagging of the wake geometry in the Newton iteration cycle.

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4.3 Volume inuence functions

The volume contribution to the induced velocity is determined via the beam source/length densityσ and doublet/length density ν . Both are set to give ow tangency on the circular beam of cross-sectional area πR 2, immersed in the local unit- V ∞ freestream velocity vector ξ = ¯̄P V ∞ /V ∞ , asshown in Figure 15.

σ(ℓ) = d(πR 2)

dℓ ξ · ℓ̂ ν (ℓ) = 2 πR 2 ξ − ( ξ · ℓ̂)ℓ̂ (162)

These are most conveniently dened in transformed space which then automatically corrects forcompressibility.

ξ =1λ

, 0 , 0T

, ℓ̂ =¯̄P ŝ

| ¯̄P ŝ |, dℓ2 = dξ 2 + dη2 + dζ 2

The integrated effects of the densities then gives the volume inuence function in physical space.

wvol( r) = ¯̄P T 1

4π σ δ (δ 2 + ε2)3/ 2 + νδ 2 − 3(ν · δ ) δ (δ 2 + ε2)5/ 2

dℓ (163)

A suitable desingularizing “core size” is ε = R/ 2.The above expressions for the beam volume inuence function wvol can be used to approximate

the volume inuence of a thick wing. In this case, the circular beam area πR 2 is replaced by

d(πR 2)dℓ

→ dA

dℓ 2πR 2 → 1 +

Ac̄2

A

where A is the airfoil cross-sectional area. The resulting volume inuence coefficient is accurate at

distances greater than the local chord ¯ c.

4.4 Point-mass inuence functions

A point mass with signicant drag and volume contributes to the overall induced velocity in theoweld. This velocity has the same form as equation (163), with the singularity strengths beinggiven in terms of the drag area ( C D A) p and volume V p of the point mass.

σ dℓ = ( C D A) p ν dℓ = V p ξ (164)A suitable core size is ε = max[( C D A/π )1/ 2 , (V /π )1/ 3], which is typically comparable to the radiusof the point-mass object.

4.5 Ground effect

Ground effect is modeled using images of the horseshoe vortices, sources, and doublets representingthe lifting surfaces and fuselage beams. The ground plane is assumed to contain the origin of theinertial (Earth) XYZ coordinate system, and have a unit normal vector ˆ nGE , specied in Earthaxes, denoted by the () E subscript. In the case of level ground, this is simply ˆ nGE = { 0 0 1}T .

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r

r

r’

r’

X

Y

R

x

y z

Z

Image vortex

actual vortex

ground plane

G

G

nG

Γ

Γ ’

Figure 17: Ground effect image vortex and related geometry vectors.

Referring to Figure 17, the position of an aircraft vortex point r relative to the XY Z Earthorigin is

rG = R + r (165)

where R = ¯̄T T

E RE (166)

where the body/earth transformation tensor ¯̄T E will be dened later in terms of the aircraft Eulerangles Φ, Θ, Ψ. The R calculation is required since the aircraft position state is represented in the

Earth axes as RE . We also dene n̂G as the ground unit normal vector in the aircraft axes.

n̂G = ¯̄T T

E n̂G E (167)

The corresponding image’s position in the aircraft frame is nally computed as follows.

r ′G = rG − 2 (rG · n̂G ) n̂G (168)

r ′ = r ′G − R = r − 2 R + r · n̂G n̂G (169)

The image position r ′ is then used to compute the AIC matrices of the images in the sameway as for the actual vortices, sources, and doublets. The image calculations above are actuallyperformed in the wind axes ξηζ , using the transformed vectors

n̂GP = ¯̄P n̂G (170) RP = ¯̄P R (171)

rGP = ¯̄P rG (172)

and the resulting rP is used in the Biot-savart integrations (160) and (161). The image’s circulationstrengths are related to the real strengths as follows.

Γ ′ = − Γ , solid ground planeΓ , free-surface ground plane (173)

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The image’s source and doublet strengths are given as follows.

σ ′ = σ , solid ground plane− σ , free-surface ground plane (174)

ν ′ = ν − 2 (ν · n̂G ) n̂G , solid ground plane

− ν + 2 ( ν · n̂G ) n̂G , free-surface ground plane (175)

4.6 Linearization

Using the wind-aligned Prandtl-Glauert transformation results in overall induced velocity havingthe following dependence:

K

k=1vk Ak + wvol V ∞ = V ind ( r ; Ak V ∞ αwind β wind M PG xi yi zi ) (176)

In practice, only the dependence on Ak needs to be implicitly treated in the coupled struc-

tural/aerodynamic solution procedure for an isolated wing. This is because vk hardly changesbetween Newton iterations, and so all variables besides Ak have a very weak inuence. Theseother variables can then be safely lagged by one Newton iteration with little effect on the overallquadratic convergence. Without the implicit treatment the 12 I × 12I structural Jacobian matrixhas a bandwidth of at most 24, and is very rapidly solved by a block-tridiagonal solver. Withthe exact implicit treatment the matrix would become 1/4 full (the 3 xi , yi , zi variables out of 12for each beam would have global inuence on all beams), resulting in an enormous increase incomputational cost. Hence, a full implicit treatment would be counterproductive.

For multiple-surface congurations, the dependence on r, V ∞ , αwind , β wind , M PG , xi , yi , zi canbe considerably greater, since these quantities then have inuence on the relative spacing betweenthe vortex wake and the downstream surface which is immersed in the wake’s velocity eld. Asa compromise, only the dependence on Ak and on the freestream V ∞ , αwind , β wind is taken intoaccount, which still retains near-quadratic convergence for typical cases. Only the effect of relative deections between aerodynamically interacting surfaces is lagged in the iteration scheme. In themost-simplied case, the vk and wvol inuence functions can be precomputed using the jig geometryx0i , y0i , z0i , and αwind = β wind =0, which is equivalent to the standard prescribed-wake vortex latticemethod, and is appropriate for benign ow conditions and small deformations.

In the presence of ground effect, the overall induced velocity dependence (176) expands to alsoinclude the aircraft position and Euler angles.

K

k=1vk Ak + wvol V ∞ = V ind ( r ; Ak V ∞ αwind β wind M PG xi yi zi R Φ Θ Ψ ) (177)

For near-ground cases the effect of R and the Euler angles can be quite strong. However, for steadycases these are generally prescribed explicitly, and hence can be considered as constants duringlinearization.

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5 Global Variables and Constraints

In addition to the twelve structural nodal unknowns x i , yi , . . . listed earlier, the overall aero/structuralproblem also has the “global” variables

∆ rJ ∆

θJ

F J

M J A1 A2 . . . AK

U

Ω ao α o δ F 1 δ F 2 . . .which require the same number of constraints for a well-posed problem. The surface deections areincluded as unknowns only to allow calculation of control derivatives, developed later.

5.1 Acceleration constraints

The coordinate-origin acceleration ao can be constrained in a number of ways. The simplest ap-proach is the specify an acceleration directly at some convenient reference point rref .

aref ≡ ao + αo × rref + Ω × Ω × rref = aspec (178)

This is appropriate for simulating a wing clamped to a tunnel wall at rref , in which case aspec = 0would be chosen. For simulating an aircraft in ight, it is appropriate to impose a constraint onthe overall force instead.

F ≡ f lift + f drag + f acc + f am ds + ∆ F pmass + ∆ F joint = F spec (179)Concentrated strut loads ∆ F strut have been excluded from the summation, since each such load iscancelled by the opposite load − ∆ F strut on the strut anchor at rw which is obviously attached tothe aircraft somwhere. Note that ∆ F joint similarly cancel among all the beams, and could also beleft out of the summation with no effect. Because the overall force F includes the inertial-reactionloads, F spec must be chosen to be zero for an aircraft in ight.

Algebraic summation of all the discrete structural force equations (56) on all the beams, whichis tantamount to performing the integrations and summations in (179), produces the very simplestatement

F i+1 − F i = F spec (ground interval i . . . i +1) (180)

which simply states that the net applied load shows up as an opposite reaction load at the groundinterval. Because of its simplicity, constraint (180) is actually implemented in lieu of its formalequivalent (179). Note that in the free-ight case the ground interval has no load discontinuity.This is reassuring, since the ground interval in this case is entirely ctitious.

5.2 Lift-related constraints

The components of the aircraft velocity U can be constrained directly, most conveniently via aspecied velocity magnitude V ∞ spec and specied ow angles αspec , β spec . For greatest exibility,these are imposed at some arbitrary reference location rref (e.g. air data port location) xed in thebody axes. The apparent air velocity components at this location are

{ uref vref wref }T = − U − Ω × rref + V gust ( rref ) (181)

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which then allow the specied velocity and ow angles to be imposed by the following equations.

u2ref + v2ref + w2ref = V ∞ spec (182)arctan wref /u ref = αspec (183)

arctan − vref / u2ref + w2ref = β spec (184)

An alternative constraint is to specify the total aerodynamic lift normal to the freestream, witha convenient load factor N included.

L ≡ f lift + f drag ds · {− sin α 0 cosα}T = N Lspec (185)This can replace either the V ∞ spec or the αspec constraint (but not both, obviously). As indi-cated, the total aero lift force L is simply the integral of the distributed aero loading normal tothe freestream along all the lifting surfaces. In most situations, only the contribution of f lift issignicant.

5.3 Moment-related constraints

The angular rates Ω and angular accelerations αo can be constrained directly, e.g.

Ω = Ωspec αo = α spec (186)

or indirectly by specifying moments about the reference point rref .

M ref ≡ M + F × rref = M spec (187)

The total moment vector about the origin is obtained like the force vector, with the inertial loadsincluded.

M = m lift + mdrag + macc + mam + r × f lift + f drag + f acc + f am ds+ ∆ M pmass + ∆ M joint + r × ∆ F pmass + ∆ F joint (188)

As in the case of the total force constraint, the total moment constraint is actually implementedvia the equivalent ground-reaction moment load.

M i+1 − M i = M spec (ground interval i . . . i +1) (189)

Since the inertial reaction loads are also included, all three components of M spec must be chosento be zero if the ASWING model includes all the pieces actually present on the aircraft (i.e. theground is ctitious and must carry no load). In this case F would be zero as well, and so themoment-reference location rref in equation (187) is arbitrary.

If the conguration includes a horizontal stabilizer or equivalent surface which has a pitchresponse, the vertical velocity U z , which is nearly equivalent to α , can be indirectly constrained byspecifying a zero reference pitching moment ( M y)spec = 0. This allows the trim state to be easilycomputed without manual iteration. The sideslip velocity U y , nearly equivalent to the s

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Asw Theory