Introduction and Structural Dynamics

7/28/2019 Introduction and Structural Dynamics

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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

Documents

Introduction and Structural Dynamics