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Computational Aeroelasticity The Cultural and Convention Center METU Inonu bulvari Ankara, Turkey Sponsored by: RTA-NATO The Applied Vehicle Technology Panel presented by R.M. Kolonay Ph.D. General Electric Corporate Research & Development Center Ankara, Turkey Oct.. 1-5, 2001
Kolonay
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Presentation Outline
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• Introduction - Fluid-Structure Interactions •Aeroelasticity - Aeroelastic analysis/design in an MDA/MDO Environment Environment
• Static Aeroelasticity • Dynamic Aeroelasticity • Commercial Programs with Aeroelastic Analysis/Design Capabilities
Kolonay
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Introduction
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Fluid Structure Interaction - Any system where the fluid and structure cannot be considered independently to
predict predict the response response of the fluid, fluid, the structure, structure, or both.
Some Fields of Application
• Aerospace Vehicles - Aircraft, Spacecraft, Rotorcraft, Compressors, Combustors, Turbines Turbines
• Utilities - Hydroturbines, Steamturbines, Gasturbines, Piping, Transmission Lines
• Civil Structures - Bridges, Buildings
• Transportations •Trains, Automobiles, Ships
Kolonay
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Introduction
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Fields of Application (Continued)
• Medical - Blood flow in veins, arteries, and heart
• Marine - Submarines, Off-shore Platforms, Docks, Piers
• Computer Technology - High velocity flexible storage devices
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Introduction
Failure Failure to recognize F-S Interaction Tacoma Narrows Bridge #1 (Galloping Girtie) - Chief Designer: Leon Moisseiff - Length: 5,939 ft. - 42 MPH winds induced vortical separated flow that lead to torsional torsion al flutter - Piers used in second bridge - 1992: National Historic Site (natural reef) - Photos taken by Leonard Coatsworth
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Introduction Aeroelasticity (sub-set of FS Int.)
Aeroelasticity (British Engineers Cox and Pugsley credited with term) - Substantial interaction among the aerodynamic, inertial, and structural forces that act upon and within the flight vehicle. Aerodynamic Forces
Static Aero Elasticity
Dynamic Stability Dynamic Aeroelasticity Inertial Forces
Elastic Forces
Kolonay
Mechanical Vibration
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Introduction
Early Aeroelastic Problems
• S. P. Langley’s Aerodome (monoplane) - 1/2 scale flew - October, 1903: Full scale failed, possibly due to wing torsional divergence - 1914 Curtis made some modification and flew successfully.
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Introduction
After Langley’s failure the U.S. War Department reported -
“We are still far from the ultimate goal, and it would seem as if years of constant work ... would still be necessary before we can hope to produce an apparatus of practical utility on these lines.”
9 Days Later ...
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Introduction December 17, 1903
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Introduction
Early Aeroelastic Problems
• Hadley Page 0/400 bomber - Bi-plane tail flutter problems (fuselage torsion coupled with elevators) - DH-9 had similar problems - Solution was to add torsional stiffness between right and left elevators.
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Introduction
Early Aeroelastic Problems
• Fokker D-8 (credited with last official kill of WW I) - D8 had great performance but suffered from wing failures in steep dives - Early monoplanes had insufficient torsional stiffness resulting in: • wing flutter, wing-aileron flutter • loss of aileron effectiveness - Solution: Increase torsional stiffness, mass balancing
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Introduction Computational Aeroelasticity
Early Theoretical Developments[1],[3].
• Wing divergence - Reissner (1926) • Wing flutter - Frazer and Duncan (1929) • Aileron reversal - Cox (1932) • Unsteady aerodynamics and flutter - Glauert, Frazer, Duncan, Kussner, Theodorsen (1935) • 3 DOF wing aileron flutter - Smlig and Wasserman (1942) By Early 1930’s Analytical methods existed to aid designers to consider both static and dynamic aeroelastic phenomena
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Introduction Computational Aeroelasticity
Designs from the 40’s-70’s “designed out” Aeroelastic Effects
• Accomplished by increasing structural stiffness or mass balancing (always at weight cost) 70’s & 80’s brought technology developments in three key areas
• Structures, Controls, and Computational Methods - Advanced composite materials enabled aeroelastic tailoring - Fly By Wire and Digital Control Systems enabled statically unstable aircraft - FEM, CFD, Optimization, Computational Power enabled advanced designs.
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Introduction Aeroelastic Successes • DARPA sponsored X-29 (First flight 1984) - Aeroelastic tailored (graphite epoxy) forward swept wing - Fly By Wire triple redundant digital and analog control system - Germany proposed FSW designs (He 162) in WWII
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Introduction Aeroelastic Successes • Active Aeroelastic Wing USAF/NASA (AAW) - Use control surfaces (leading and trailing edge) as tabs to twist the wing for maneuvers - Use TE surfaces beyond reversal - Produces lighter more maneuverable aircraft
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Introduction
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Product Structural Design in an MDA/MDO Environment o n u t i b i s t r D i
r f e s a n r s T e t l a a e S H s c i t e n g a M o g t n c i t e l e k E r
n a m i c s A e r o d y
M a n uf a ct ur e S t r u c t u r e s
MDA/MDO
a M
t s o C
s s e n t s u b o R
Kolonay
A c o u s t i c s
C o n r t l o s
D y n a m i s c
M a i n t e n a n c e
R e l i a b i l i t y
r P o d u i c b i l t i y
16
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Computational Aeroelasticity
Goal of Computational Aeroelasticity
To accurately predict static and dynamic response/stability so that it can be accounted for (avoided or taken advantage of) early in the design process.
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Computational Aeroelasticity Aeroelastic Equations of Motion
M u˙˙ + Bu˙ + Ku = F ( u, u˙, u˙˙, t ) K – Structural Stiffness B – Structural Damping M – Structural Mass ˙˙, t ) – External Aerodynamic Loads F ( u, u˙, u
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Computational Aeroelasticity Discretization of EOM • Structures K , B, M - Typically, although not necessarily, represented by Finite Elements in either physical or generalized coordinates. Derived in a Lagrangian frame of reference. • External Loads F ( u, u˙, t ) - Aerodynamic loads. Representations range from Prandtl’s lifting line theory to full NavierStokes with turbulence modeling. Represented in physical and generalized coordinates in a (usually) Eulerian frame of reference.
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Computational Aeroelasticity Fluid-Structural Coupling Requirements • Must ensure spatial compatibility - proper energy exchange across the fluid-structural boundary • Time marching solutions require proper time synchronization between fluid and structural systems • For moving CFD meshes GCL[6] must be satisfied If coupling requirements for time-accurate aeroelastic simulation are not met then dynamical equivalence cannot be achieved. That is, regardless of the fineness of the CFD/CSM meshes and the reduction of time step to 0, the scheme may converge to the “wrong” equilibrium/instability point.[5]
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Computational Aeroelasticity General Modeling Comments • Use appropriate theory to capture desired phenomena - Fluids - Navier-Stokes vs. Prandtls’ lifting line theory - Structures - Nonlinear FEM vs. Euler beam theory
• Model the fluid and structure with a consistent fidelity - For a wing don’t model the fluid with NS and the structure with beam theory
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Computational Aeroelasticity Aeroelastic Phenomena
Static Aeroelastic Phenomena
Dynamic Aeroelastic Phenomena
• Lift Effectiveness
• Flutter
• Divergence
• Gust Response
• Control Surface Effectiveness/Reversal
• Buffet
• Aileron Effectiveness/ Reversal
• Panel Flutter
• Limit Cycle Oscillations (LCO)
• Transient Maneuvers • Control Surface Buzz
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Static Aeroelasticity Static Aeroelastic Phenomena
• Lift Effectiveness • Divergence • Control Surface Effectiveness/Reversal • Aileron Effectiveness/Reversal
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Static Aeroelasticity
Static Aeroelastic Effects
• For trimmed flight aeroelastic effects change only load distribution. - Lift - Drag - Pitching Moment - Rolling Moment
• For constrained flight (wind tunnel models) aeroelastic effects change both magnitude and distribution of loads.
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Static Aeroelasticity
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Useful 2-D Section Definitions L
Shear Center/Center of Twist MAC
Aerodynamic Center e
Shear Center/Center of Twist - Applied Shear force results in no moment or twist - Applied moment produces no shear force or bending Aerodynamic Center - Pitching moment independent of angle of attack - 0.25c for subsonic, 0.5c for supersonic Center of Pressure - Total Aerodynamic Moment equal zero (AC=SC for symm. airfoil) e - Eccentricity Kolonay
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Static Aeroelasticity
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Effect of Swept Wing Bending on Streamwise Aerodynamic Incidence “wash out”
“wash in”
A-A
A-A
Flexible Wing
Flexible Wing
Rigid Wing
Rigid Wing
U A
ASW Kolonay
A
A
A
FSW
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Linear Static Aeroelasticity
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EOM
[ K ] { u } + [ M ] { u˙˙}
=
{F (u)}
(1)
{ u˙˙} - rigid body accelerations only, used for inertial relief and trim F ( u ) - Steady aerodynamic forces can be represented as F(u)
= q [ G ] T [ AIC ] [ G
T
] { u } + q [ G ] [ AIRFRC ] { δ }
S
or
(u)
a
= q [ AICS ] { u } + q [ P ] { δ Now (1) can be written as
[ K – q AICS ] { u } + [ M ] { u˙˙}
= q[P
a
]{δ}
(2)
For Linear Aerodynamics [AIC] & [AIRFRC] depend only on Mach Number (M) Kolonay
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Linear Static Aeroelasticity Steady Aerodynamic Loads F ( u ) = q [ G ] T [ AIC ] [ G s ] { u } + q [ G ]
•
T
[ AIRFRC ] { δ }
q = Free stream dynamic pressure
[ G ] T - Spline matrix which transforms forces from Aerodynamic DOF (ADOF) to T Structural DOF (SDOF). { F } = [ G ] { F } s a • [ G ] - Spline matrix which transforms SDOF (displacements) to ADOF (panel slopes) s • {α } = [G ]{u} a s • [ AIC ] - Aerodynamic Influence Coefficient Matrix. Relates forces on ADOF (panels) •
due to unit perturbations of the ADOF (slopes)
[ AIRFRC ] - Unit Rigid body aerodynamic load vectors. One vector for each δ i • { δ } - Vector of aerodynamic configuration parameters (angle of attack, elevator angle, •
aileron deflection, roll rate, pitch rate etc.) Kolonay
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Linear Static Aeroelasticity
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Aeroelastic Effects on Swept Wing Forces and Moments 0.12
0.12
0.11
0.1
0.1
0.09
α
L
C t f i L f o t n e i c fi f e o C
0.08
0.08
0.07 0.06
0.06
0.05
Rigid ASW Flex ASW
0.04
0.04
0.03
Rigid FSW
0.02
0.02
Flex FSW
0.01
2
0 0
2
2
4
6
0 0
2
4
6
8
8
0.002
0.001
0
-0.001
-0.01 -0.02
-0.02
-0.03
Angle of Attack Kolonay
α
Induced Drag C D
α
Pitching Moment C M
α 29
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Linear Static Aeroelasticity Divergence of a Constrained Vehicle • When the aerodynamic stiffness qAICS becomes greater than the structural stiffness K , the structure fails or diverges. • The divergence dynamic pressure for a restrained vehicle can be found by solving the eigenvalue problem (static stability)
[ K – q AICS ] { u }
=
{0}
(3)
• Lowest eigenvalue q D represents the divergence dynamic pressure • The eigenvector { u D } represents the divergent shape • Divergence is independent of initial angle of attack Kolonay
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Linear Static Aeroelasticity
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Affect of Sweep on Lift Effectiveness (M=0.7) ASW
FSW
1 20 0.9 18
qD
0.8 16
) 0 2 ( . q E α
L C
0.7 14 0.6 12 0.5 10 0.4 8 0.3
6
0.2
4
0.1
0
2
0
5
Dynamic Pressure (psi)
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0
0.2
0.4
0.6
0.8
Dynamic Pressure (psi)
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Linear Static Aeroelasticity Static Aeroelastic Trim Equations
Writing equation (2) in the f-set (Reference Appendix A) yields
[ K ff – qAICS ] u f + M ff u˙˙ f
a = P δ f
or
(4)
a a K u + M u˙˙ = P δ ff f ff f f Using the procedure in Appendix A for Guyan reduction equation (4) can be cast in the aset as
a a K u + M u ˙˙ = P δ aa a aa a a with a a a a K = K aa – K G aa a o a a a a –1 a P = P a – K K P a ao oo o
(5)
a T T a T a M M + G M G = M aa + M G + G aa ao o o oa o oo o Kolonay
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Linear Static Aeroelasticity
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Equation (5) can now be partitioned into the r-set and the l-set to
a a K rl K rr a a K ll K lr
ul + u r
M ll M lr M rl M rr
As with the inertial relief formulation
˙˙l u ˙˙r u
=
a P l Pa l
δ
(6)
u˙˙l = Du˙˙r where D is the rigid body transfor-
mation matrix. To produce stability derivatives that are independent of the r-set (i.e. support point) an orthogonality condition is imposed in the form
T
D I
M ll M lr M rl M rr
Using the orthogonality condition and
ul u r
= 0
(7)
u˙˙l = Du ˙˙r equation (6) can be cast in the fol-
lowing form
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Linear Static Aeroelasticity
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a K ll a K rl T D M + M ll rl
a K lr
a K M D + M rr rl rr T D M + M 0 lr rr
l u r ˙˙ u r
M D + M ll lr
u
=
a Pl a δ Pr 0
(8)
T
Equation (8) can be solved by multiplying the first row by D and adding it to the second row. The new second row is interchanged with the third equation to yield the following system of equations.
a K ll
a K lr
M D + M u ll lr l
T T D M + M D M + M ll rl lr rr T a a T a a D K + K D K + K ll rl lr rr
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0 m
r
u r ˙˙ u r
=
a P l δ (9) 0 T a a D P + P l r
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Linear Static Aeroelasticity
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Where m
T
r
T
= D M ll D + D M lr + M rr is defined as the rigid body mass
matrix. Using a simplifying notation equation (9) becomes
R 11 R 12 R 13 R 21 R 22 R 23 R 31 R 32 R 33
ul u r ˙˙r u
Solving the first row of equation (10) for
=
a P l 0 T a a D P + P l r
(10)
u l and substituting in the second and third rows
we obtain the trim equations in the form
K 11 K 12 K 21 K 22
u1 u2
=
P 1 P 2
{ δ }
(11)
with Kolonay
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Linear Static Aeroelasticity – 1 R K 11 = R 22 – R 21 R 11 12 – 1 R K 12 = R 23 – R 21 R 11 13 – 1 R K 21 = R 32 – R 31 R 11 12 – 1 R K 22 = R 33 – R 31 R 11 13
(12)
–1 P a P 1 = – R 21 R 11 l a – R R – 1 P a P 2 = D T P la + P r 31 11 l
u 1 = u r u 2 = u˙˙r Solving equation (11) for u and u 1 2
the rigid body displacements and accelerations
respectively yields Kolonay
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Linear Static Aeroelasticity – 1 [ P δ – K u ] u 1 = K 11 1 12 2 – 1 K ] u [ K 22 – K 21 K 11 12 2
(13)
– 1 P ]δ = [ P 2 – K 21 K 11 1
or
[ LHSA ] { u 2 }
= [ RHSA ] { δ }
(14)
or
[ L ] { u 2 }
= [ R ] { δ }
Equation (14) is the basic equation for static aeroelastic trim analysis. There is one equation for each rigid body degree of freedom (6 DOF trim).
{ u 2 } is the vector of structural
{δ}
accelerations at the support point and is a vector of trim parameters. Partitioning equation (14) into free or unknown (subscripts f,u) values and known or set (subscripts k,s) values and gathering all unknown values to the left yields
Note: System can be over-specified producing trim optimization problem. Kolonay
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Linear Static Aeroelasticity
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L ff – R fu u 2 f – L fk – R fs = L kf – R ku δ u – L kk – R ks Potential values for u 2 k and δ are given in equation (16)
u 2 k δs
(15)
BASE - reference state ALPHA - angle of attack
u
∈ 2
NX - longitudinal acceleration
BETA - yaw angle
NY - lateral acceleration
PRATE - roll rate
NZ - vertical acceleration PACCEL - roll acceleration QACCEL - pitch acceleration RACCEL - yaw acceleration
δ∈
QRATE - pitch rate RRATE - yaw rate
(16)
{ δsym } - symmetric surfaces { δ anti } - antisymmetric surfaces { δ asym } - asymmetric surfaces
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Linear Static Aeroelasticity Rigid Trim Equations
From equation (9) considering only rigid body accelerations and loads yields LHSA
rigid
= R
33
= m
r
T a a RHSA = P2 = D P + P rigid l r and the rigid trim equations as
[ LHSArigid ] { u˙˙r }
Kolonay
=
[ RHSArigid ] { δ }
(17)
(18)
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Linear Static Aeroelasticity
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Stability Derivatives Using equation (14) and using an identity vector for { δ } and employing the rigid body mass matrix m r forces due to unit parameter values can be determined as F =
–1 –1 –1 m r [ K 22 – K 21 K 11 K 12 ] [ P 2 – K 21 K 11 P 1 ] F x F y F z F ∈ M x M y M z
Kolonay
=
Thrust/Drag Side Force Lift Roll Moment Pitch Moment Yaw Moment
(17)
(18)
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Linear Static Aeroelasticity Stability Derivatives
Based on equation (18) non-dimensional stability derivatives are Surface Parameters Rate Parameters F F x x C = ------C = ----- D D qSc qS F F y y C = --------C = -----S S qSb qS F z C = ----- L qS
F z C = -------- L qSc
M x C = --------l qSb
M x C = -----------l 2 qSb
M y C = --------m qSc
M y C = -----------m 2 qSc
M z C = -------- y qSb
(19)
M z C = ----------- y 2 qSb
• Note: These are “unrestrained ” stability derivatives (free-free) Kolonay
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Linear Static Aeroelasticity Example Stability Derivatives for α
From equations (14) and (17)
δ0 = 0 F x δα = 1.0 F y δβ = 0 F z δ –1 = 0 m LHSA RHSA [ ] [ ] [ ] = PRATE r M x = 0 δ QRATE M y δ = 0 RRATE M z α {δ} = 0 surface Yielding C D , C S , C L , C l , C M etc. α α α α α Kolonay
(20)
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Linear Static Aeroelasticity Stability Derivative Types • There are four varieties of flexible stability derivatives - Unrestrained (orthogonality and inertia relief included) - Restrained (orthogonality, no inertial relief) - Supported (no orthogonality, but inertial relief) - Fixed (no orthogonality, no inertial relief)
• For wind tunnel comparison use either Restrained or Fixed
Make sure you know which type of stability derivatives a given program produces
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Linear Static Aeroelasticity Lift Trim Analysis • For straight and level flight i.e. { u 2 } = NZ equation (14) produces a single equation with one free parameter (say α ) LHSA × NZ = RHSA × α
( LHSA × NZ ) = ---------------------------------RHSA or in terms of stability derivatives m × NZ = α qSC r L α
α
α
( mr × NZ )
= -------------------------qSC L
α
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Linear Static Aeroelasticity Aeroelastic and Rigid Trimmed Pressures ( M = 0.7, q = 5.04 psi, nz = 1g )
Aeroelastic Trim ( α = 2.61 ° ) Eq. (14) Kolonay
Rigid Trim ( α = 1.29 ° ) Eq. (18) 45
Linear Static Aeroelasticity
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Rigid and Aeroelastic Trim Pressures vs. Span ( M = 0.7, q = 5.04 psi, nz = 1g ) 2.5
Rigid Trim 0% chord Rigid Trim 50% chord Aeroelastic Trim 0% chord 0% span Aeroelastic Trim 50% chord
2
) i s p 1.5 ( e r u s s 1 e r P 0.5
0
Kolonay
0
0.25
0.5
0.75
Non-Dimensional Semi-Span
1
46
Linear Static Aeroelasticity
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Spanwise Twist Due to Swept Wing Deformations 2
( M = 0.7, q = 5.04 psi, nz = 1g )
1
) . g e 0 d ( 0 e l g n A t -1 s i w T e v i t -2 a l e R -3
0.25
0.5
0.75
1
Flex Trim Rigid Trim Rigid
-4
% Semi-Span Kolonay
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Linear Static Aeroelasticity
Swept Wing Aeroelastic Effects on Trimmed Displacements max z-disp. = 5.4 in.
Aeroelastic Trimmed Displacements Kolonay
max z-disp. = 11.4 in.
Rigid Trimmed Displacements 48
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Static Aeroelasticity Control Surface Effects
β0
Incremental Lift Incremental Moment
β0
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Linear Static Aeroelasticity Roll Trim Analysis (wing with aileron)
Steady state roll (PACCEL = 0) for given LHSA
44
× PACCEL
= RHSA
43
β (aileron deflection)
× β + RHSA 44 × PRATE
RHSA × β 43 PRATE = ------------------------------RHSA 44
or in stability derivative form qSb C l
β
β + C l
PRATE
pb ------2 V
= I PACCEL roll
for steady roll and a given
β
C β l β PRATE = ------------C l pb ------2 V Kolonay
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Linear Static Aeroelasticity
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Roll Rate vs. Dynamic Pressure for
β
= 1.0 °
70
) 50 c e s / g e 30 d ( e t a R 10 l l o 0 R-10
Rigid TECS ASW Flex TECS ASW Rigid TECS FSW Flex TECS FSW Rigid LECS ASW Flex LECS ASW Rigid LECS FSW Flex LECS FSW 0.5
1
1.5
qR ASW_TE -30
qR FSW_TE
-50
Dynamic Pressure (psi) Kolonay
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Static Aeroelasticity
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Aileron Effectiveness Dynamic Pressure (psi) 0
0.5
1
1.5
0.15
f
( C l )
vs. V vs. q
Reversal V
0.1
β
– ------------------------f 0.05 C l pb ------0 2 V
-0.05
Reversal q
-0.1 0
1000
2000
3000
4000
5000
Velocity (in/sec) Kolonay
52
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Linear Static Aeroelasticity
Aeroelastic Effects on Roll Rate Pressures p
q = 0.28 (psi)
p
0.012
0.032
0.010
0.028
0.009
0.024
0.007
0.020
0.006
0.015
0.004
0.011
0.002
0.007
0.001
0.003
-0.001
-0.001
-0.002
-0.005
-0.004
-0.010
-0.006
-0.014
-0.007
-0.018
-0.009
-0.022
-0.010
-0.026
qrigid = 27 (deg/sec)
q = 0.78 (psi)
q = 1.5 (psi) p 0.052 0.046 0.039 0.033 0.026 0.019 0.013 0.006 0.000 -0.007 -0.014 -0.020 -0.027 -0.033 -0.040
qrigid = 46 (deg/sec)
qrigid = 59 (deg/sec)
M=0.7 qrigid = 16 (deg/sec)
Kolonay
qrigid = 0 (deg/sec)
qrigid = -28 (deg/sec)
53
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Linear Static Aeroelasticity Rolling Wing Deformations M = 0.7, q = 1.5 psi
Kolonay
54
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References
1. Bisplinghoff, Ashley and Halfman “Aeroelasticity”, Dover Publications, Addison-Wesley Publishing Company, Inc., 1995. 2. Weisshaar, “Fundamentals of Static and Dynamic Aeroelasticity”, Purdue University School of Aeronautics and Astronautics, West Lafayette, IN 1992. 3. Smilg, B. and Wasserman, L. S., “Application of Three Dimensional Flutter Theory to Aircraft Structures”, USAAF TR 4798, 1942. 4. Neill, D.J., Herendeen, D.L., Venkayya, V.B., “ASTROS Enhancements, Vol IIIASTROS Theoretical Manual”, WL-TR-95-3006. 5. Bendiksen, Oddvar O., “Fluid-Structure Coupling Requirements for Time-Accurate Aeroelastic Simulations”, AD-Vol.53-3, Fluid-Structure Interaction, Aeroelasticity, FlowInduced Vibration and Noise, Volume III ASME, 1997. 6. Farhat, C., “Special course on Parallel Computing in CFD”, AGARD-R807, October 1995. 7. MacNeal, R. H., “The NASTRAN Theoretical Manual,” NASA-SP-221(01), April, 1971. 8. I.E. Garrick and W.H. Reed, III “Historical Development of Aircraft Flutter,” Journal of Aircraft, Vol. 18, No. 11, November 1981.
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