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Notes for Aeroelasticity I
Moti KarpelMoti Karpel
Faculty of aerospace EngineeringFaculty of aerospace Engineering
TechnionTechnionIsrael Institute of Technology,Israel Institute of Technology,
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Introduction
Aeroelasticity deals with the interaction between aerodynamic, elastic and inertial
forces acting on atmospheric flight vehicles. The aerodynamic and inertial loads
deform the structure. The deformations affect the airloads, which closes the
aeroelastic loop.
Static aeroelasticity deals with the effects of structural deformations on steady
aerodynamic load distributions and total force and moment coefficients, and with
static instability (divergence). It is assumed that:The 6 d.o.f. airplane maneuvers are slow compared to the structural
dynamics.
The structure deforms but structural vibrations have negligible effects.
The aerodynamic loads due to change in local angles of attack develop withno delays.
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Dynamic aeroelasticity deals with the interaction between structural dynamicsand unsteady aerodynamics. Delays in the development of aerodynamic loads
are important. The main topics are dynamic instability (flutter) and response to
atmospheric gusts (deterministic and stochastic)
Aeroservoelasticity (ASE) deals with the interaction between aeroelastic and
control systems. The control system reads structural vibrations and activates
aerodynamic control surfaces, which closes the aeroservoelastic loop.
The models in this lecture series assume linearity of the aerodynamic, structural
and control systems.
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ZAERO: A modern Aeroelasticity Package
3D Spline
g-Method Flutter Solution
Aeroservoelasticity
ASTRAN
AERO/UAIC
Deformed
Aero Model
Deformed
FEM Model
Dynamic Loads
First Elastic Modal Acceleration Response
-1.6
-1.2
-0.8
-0.4
0
0.4
0.8
1.21.6
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Time (sec)
Acceleration(g)
MSC/NASTRAN
ZAERO
P-Transform
Pilot Input
-3
-2
-1
0
1
2
3
4
5
0 0 .2 0 .4 0 .6 0 .8 1 1 .2 1 .4 1.6 1 .8 2
Time (sec)
ControlSurface
Deflection(deg)
Maneuver Loads
Ejection Loads
Gust Loads
Nonlinear Flutter
WindTunnelModelASTROS - LIFT TRIMAOA= 1 Deg., M=0.9V=12053 in/sec
-25638.4
-25638.4
-206
30.1
-25638.4
-20630.1
4411.4-20630.1
-20630.1
-5605
.2
-5605.2
-20630.
1
ASTROS RESULTM = 1.2, q= 350 psfAOA = 5 Deg.VSS/ON
Stress Distribution
Static
Aeroelastic
Deformation
Trim/Flight LoadsTrim/Flight LoadsZDMZDMZONA Dynamic Memory &ZONA Dynamic Memory &
Data Management SystemData Management System
Dynamic Pressure (psi)
-0.30
-0.20
-0.10
0.00
0.10
0.20
0.30
0.00 5.00 10.00 15.00 20.00 25.00
5.00
6.00
7.00
8.00
9.00
10.00
0.00 5.00 10.00 15.00 20.00 25.00
Mode 5
Mode 6
Mode 7
Mode 8
True Damping
Matched-Point
Flutter ModeTracking
ZAERO/UAICZAERO/UAIC
ASTRAN
Mach Number Range
Subsonic Transonic Supersonic Hypersonic
ZSAPatM=1.0
ZONA6
DLM
ZTAIC
ZONA7
ZONA51
ZONA7U
Wing/BodywithExternalStores
LiftingSurface
GeometricFidelit
Unsteady Aerodynamics
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Course Outline
1. Structural vibrations and modal coordinates
2. Static aeroelasticity
3. Unsteady aerodynamics
4. Flutter analysis
5. Dynamic response to gust excitation
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Use of Symmetry
Flight vehicles normally have a plane of symmetry. The structural model is constructed for one half only.
Boundary conditions at the plane of symmetry determine whether the model is
symmetric or antisymmetric.
Symmetric and antisymmetric analyses are performed separately.
Unmanned Aerial Vehicle (UAV) model Advanced Fighter Aircraft (AFA)
101
X
YZ
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The Stiffness Matrix
In static equilibrium, the displacement vector is related to the external force vector by
A column {Kj} in [K] is the force vector required to obtain a unit displacement at the
j-th d.o.f. and zero displacements elsewhere.
The stiffness matrix is symmetric.
A single finite element affects only the terms associated with the grid points to which
the element connected. A free-free structure can move as a rigid body with no external forces.
A rigid-body mode {} satisfies
which implies that a free-free stiffness matrix is singular.
A stress model can be normally used for dynamic analysis. Parts which are not
required to be very detailed in the aeroelastic analysis (i.e. fuselage) can be reduced to
a beam-like model.
[ ]{ } { } (1.2)K u P=
[ ]{ } { }0 (1.3)RK =
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x1, y1, z1, x1, y1, z1, x2,...,z2 12
1
2
x1
y1
x1
x
y1 y
z
z1
z1 L
x1, x2
[K] =
EA
L 1 1
1 1
x1, x2
[K] =GJ
L 1 1
1 1
xy
y1, z1, y2, z2
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x1, y1, z1, x1, y1, z1, x2,...,z2 12
[K] =
EA
L
0 12EIzL3
0 0 12EIyL3
sym
0 0 0 GJL
0 0 6EIyL2
0 4EIyL
06EIz
L2 0 0 04EIz
L
EA
L0 0 0 0 0 EA
L
0 12EIzL3
0 0 0 6EIzL2
0 12EIzL3
0 0 12EIyL3
0 6EIyL2
0 0 0 12EIyL3
0 0 0 GJL
0 0 0 0 0 GJL
0 0 6EIzL2
0 2EIyL
0 0 0 6EIyL3
0 4EIyL
0 6EIzL2
0 0 0 2EIzL
0 6EIzL2
0 0 0 4EIzL
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The Mass Matrix
With all stiffness and damping elements ignored,
A column {Mj} in [M] is the force vector required to obtain a unit acceleration
at thej-th d.o.f. and zero accelerations elsewhere.
unit acceleration at the j-th d.o.f. and zero accelerations elsewhere. The mass matrix is symmetric.
A single mass element affects only the terms associated with the grid points towhich the element is connected.
Example: a mass point rigidly connected to a 2 d.o.f. grid point
Mass matrix of a structural element:
Lumped mass matrix: the mass is distributed to the translational d.o.f. Consistent mass matrix: based on a consistent energy formulation
where [Ne
] defines the assumed element inner displacements as function of thegrid displacements
[ ]{ } { } (1.4)M u P=
[ ] [ ] [ ] (1.5)T
e e eVol
M N N dVol=
{ }[ ]{ } (1.6)in e eu N u
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Generalized Coordinates
Any linearly independent set of displacement vectors that satisfy the boundary
conditions can be used as generalized coordinates.
The natural vibration modes are a natural choice because:
they yield a set of uncoupled equations (when the excitation is not a function of theresponse);
they can be (carefully) selected according to the frequency range of interest;
their dynamic properties can be verified in vibration tests.
Do we have to change the generalized coordinates when structural properties
change?
[ ]
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UAV Symmetric Modes
Mode 4: 3.88 Hz, 1st wing bending Mode 6: 15.71 Hz, 2nd wing bendin
ZY
X
101
Z
Y
X
3
Mode 5: 10.47 Hz, wing for & aft
ZY
X
101
Z
Y
X
3
ZY
X
101
Z
Y
X
3
ZY
X
101
Z
Y
X
3
ZY
X
101
Z
Y
X
3
ZY
X
101
Z
Y
X
3
Mode 7: 21.06 Hz, 1st wing torsion Mode 8: 22.76 Hz, 1st fuselage bending Mode 9: 29.44 Hz, aileron rotati
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AFA Anti-Symmetric Normal Modes
Mode 2: missile pitch, 7.37Hz Mode 3: wing bending, 8.96Hz