1. AAE 556 Aeroelasticity Lecture 4
Reading: notes assignment from Lecture 3
Armstrong 3329
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2. Purdue Aeroelasticity
Summary to-date
Development of simple models of wing aeroelastic behavior with pitch (torsion) only and pitch and plunge (bending)
Models show that torsional deformation creates additional lift, deflection (and stress).
Models identify an aeroelastic parameter that defines a dynamic pressure at which lift and torsional deflection approach infinity
Models are linear so this will never really happen
This special dynamic pressure is called the “divergence dynamic pressure.”
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3. Today and next week’s agenda
Define and discuss static stability
Concept of perturbations
Distinguish stability from response
Learn how to do a stability analysis
Find the divergence dynamic pressure using a “perturbation” analysis
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4. The perturbed structure
Static stability analysis considers what happens to a flexible system that is in static equilibrium and is then disturbed.
If the system tends to come back to its original, undisturbed position, it is stable - if not - it is unstable.
We need to apply these above words to equations so that we can put the aeroelastic system to a mathematical test
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5. Stability investigation
Given a system that we know is in static equilibrium (forces and moments sum to zero)
Add a disturbance to perturb the system to move it to a different, nearby position (that may or may not be in static equilibrium)
Is this new, nearby state also a static equilibrium point?
Write static equilibrium equations and see if forces and moments balance
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6. Purdue Aeroelasticity
Perturbed airfoil
In flight this airfoil is in static equilibrium at the fixed angle q but what happens if we disturb (perturb) it?
There are three possibilities
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7. Perturbation possibilities
KT(Dq)>(DL)e
statically stable because it tends to return
no static equilibrium in the perturbed state
KT(Dq)<(DL)e
statically unstable
motion away from original position
KT(Dq)=(DL)e
system stays perturbed but static
we have found new static equilibrium point
Euler test has found neutral stability
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8. Purdue Aeroelasticity
Example
Perturb the airfoil when it is in static equilibrium
To be neutrally stable in this new perturbed position this equation must be an true
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9. Static stability investigation is “stiffness based”
Neutral stability
means this relationship must be zero (2 states)
so...
Not zero
condition at neutral stability
static equilibrium displacement (Dq) is not unique
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Observations
The equation for neutral stability is simply the usual static equilibrium equation with right-hand-side (the input angle ao) set to zero.
The neutral stability equation describes a special case
only deformation dependent external (aero) and internal (structural) loads are present
these loads are “self-equilibrating” without any other action being taken
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11. Stability investigation
Take a system that we know is in static equilibrium (forces and moments sum to zero)
Perturb the system to move it to a different, nearby position (that may or may not be in static equilibrium)
Is this new, nearby state also a static equilibrium point?
Static equilibrium equations for stability are those for a self-equilibrating system
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12. Purdue Aeroelasticity
More observations
At neutral stability the deformation is not unique (Dq is not zero but can be plus or minus)
At neutral static stability the system has many choices (equilibrium states) near its original equilibrium state.
airfoil position is uncontrollable - it has no displacement preference when a load is applied.
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13. The 1 DOF divergence condition
Neutral stability or
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14. System stiffness, not strength, is important
Structural
resistance
Aero overturning
Equilibrium point
Slope depends on qSCLa
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15. Stable perturbed system
Equilibrium point
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16. Perturbed system-neutral stability
Equilibrium point at infinity
Lines are parallel
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17. Purdue Aeroelasticity
Unstable system
Equilibrium point?
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18. Aeroelastic stiffness
Aeroelastic stiffness decreases as q increases
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19. Aeroelastic divergence
Look at the single degree of freedom typical section and the expression for twist angle with the initial load
neglect wing camber
previous result
"twist amplification"
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Twist amplification
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Example corrections
q bar = 0.5
relative sizes of terms
The infinite series in Eqn will converge or diverge depending on the size of the term compared to unity (1). We will have series convergence if , but divergence if . When we have a critical or cross-over condition. The size of the parameter provides a test for the static stability or convergence of the system when it is loaded.
the sum of the terms is 2
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22. Aeroelastic feedback process
qo is the twist angle with no aero load/structural response "feedback"
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More terms
the response to angle of attack qo instead of ao
…and, the third term
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Conclusion
Each term in the series represents a feedback "correction" to the twist created by load interaction
We can visualize the load interaction process as a series of corrections to our load given a new load created by the deflection. This process should converge to a final stable answer. If not, we have a mathematically unstable process and a physically unstable process.
Series convergence
Series divergence
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25. Purdue Aeroelasticity
Summary
Divergence condition is a neutral stability condition
Divergence condition can be found using the original equilibrium conditions
Stability does not depend on the value of the applied loads
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