1. AAE 556 Aeroelasticity Lecture 6
Multi-degree-of-freedom systems
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2. Nonlinear models and finding the effects of aerodynamic stall on stability
Math model is at zero angle of attack when a non-aero load P is applied as shown
Lift is a nonlinear (cubic function of angle of attack)
Structural stiffness reponse is linear
Pd is an applied moment
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3. Sum the moments about the shear center (the model pin)
Define “Effective torsional stiffness”
So -
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4. Plot effective stiffness for 4 values of q bar
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5. Also nondimensionalize the original static equilibrium equation by dividing by KT
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6. Plotting displacement angle q shows that multiple equilibrium states are possible above divergence q and the use of nonlinear analysis reveals what the angles are
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7. MDOF system study goals
Identify similarities and differences between 1 DOF and MDOF models
Define theoretical stability conditions for MDOF systems
Reading - Multi-degree-of-freedom systems – TAW text p.63-76
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8. Torsional springs wing root
Develop a 2 DOF segmented aeroelastic finite wing model that represents it as two discrete aerodynamic surfaces with flexible connections
Torsional springs
fuselage
wing tip
wing root
Torsional degrees of freedom
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9. Introduce “strip theory” aerodynamic modeling to represent twist dependent airloads
Strip theory assumes that lift depends only on local angle of attack of the strip of aero surface
why is this an assumption?
Notice that q twist angles are measured from a common point
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10. Double arrow vectors are torques
The two twist angles are unknowns and we have to construct two free body diagrams
Structural restoring torques depend on the difference between elastic twist angles
Wing root
Internal shear forces are present, but not drawn
Wing tip
Double arrow vectors are torques
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11. Torsional static equilibrium is a special case of dynamic equilibrium
Arrange these two simultaneous equations in matrix form
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12. Summary
Static equilibrium equations are necessary to solve aeroelastic problems
Solution in terms of unknown displacements and known applied loads
Matrix equation order, sign convention and listed ordering of unknowns is important
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13. Problem solution
Both equations on the left hand side multiply the same vector
The aeroelastic stiffness matrix is
Solve for q1 and q2
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14. The solution for the q’s requires inverting the aeroelastic stiffness matrix
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15. The aeroelastic stiffness matrix determinant is a function of q
The determinant is
where
When dynamic pressure increases, the determinant D tends to zero – what happens to the system then?
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16. Solve for the twist angles created by an initial system input angle of attack ao
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17. Plot the aeroelastic stiffness determinant D against dynamic pressure (parameter)
The determinant of the stiffness matrix is always positive before the air is turned on
Where do the original assumptions about small angles break down?
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18. Twist deformation vs. dynamic pressure parameter
Unstable q region
panel twist, qi/ao
divergence
Outboard
panel (2)
determinant D is zero
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19. Panel lift computation on each segment gives:
Note that
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20. More algebra - Flexible system lift
Set the lift equal to half the airplane weight
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21. Lift re-distribution due to aeroelasticity
Observation - Outer wing panel carries more of the total load than the inner panel as q increases
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22. Question – How do we determine divergence for an MDOF system?
The general form of the aeroelastic static equilibrium equations is
n degrees of freedom
Perturb the system by an amount
Euler question – “Is there an equilibrium solution to the following relationship?”
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23. Static stability
The static stability test reduces to the existence of a homogeneous matrix equation that must hold if the system is neutrally stable ?
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24. Linear algebra says...
only if … or
This is an nth order determinant sometimes called the stability determinant
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25. Summary
System is stable if the aeroelastic stiffness matrix determinant is positive.
When additional loads are applied, the system can absorb energy and go to a unique static deformation mode.
If the stability determinant is negative then the static system, when perturbed, cannot absorb all of the energy due to work done by aeroelastic forces - it must become dynamic and will move, looking for a new equilibrium state.
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