1. “Adde parvum parvo magnus acervus erir” Ovid Purdue Aeroelasticity
AAE556 Static Stability
“Adde parvum parvo magnus acervus erir” Ovid
(“add little to little and there will be a big pile.”)
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2. Purdue Aeroelasticity
Lecture 3 – summary
Aeroelasticity is concerned with interactions between aerodynamic forces and structural deformation
Develop simple static aeroelastic model with pitch (torsion) and plunge (bending)
Section 2.4
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3. Purdue Aeroelasticity
Reading topics
2.6 Lifting generation-flexible surface
2.7 Example problem – work it through by hand
2.8 Using simple results
2.9 Load factor
2.10 Simple model – 1 degree-of-freedom-emphasis on stiffness, not strength
– Stability definition – essential
2.11 Example problem using perturbation concept
2.12. Analysis example showing when stability is obvious and when it is not
2.13 Compressibility
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4. Aero/structural interaction model
Requirements -
simplicity
manageability
realism
TYPICAL SECTION
airspeed
lift
Notice what this model does and does not do. It ignores the bending deflection and can't predict wing stress or twist anywhere on a real wing. It does approximate the interaction between airloads and the structural twist and point out how and where problems may occur. It is left to the reader how to relate the torsional spring to real life structural features and this is not easy to do. The model does give us a general idea about when we need to crank up the advanced analysis to analyze more accurately the load deflection interaction. It also identifies the importance of the offset between two reference points - the shear center and the aero center and shows us why we need to understand exactly the meaning of these two points.
We can't produce a number for the lift (like pounds) because the angle of attack including flexibility is an unknown in the beginning. The lift will depend on the angle of attack we input, but it also depends on how much the airfoil rotates because of flexibility. This is a statically indeterminate problem and requires that we write an equation for the relationship between loads and deflection.
One degree of freedom
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5. Lift and the aeroelastic parameter
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6. Lift equation with wing flexibility
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7. Two degree of freedom aeroelastic model (Section 2.4)
Goal - add bending deformation (plunge) to the simple 1 dof model
Displacement, h, plunge at the shear center
Airspeed, V
twist, q
Plunge is resisted by spring, Kh
Twist is resisted by spring, KT +h
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8. Sum torsional moments about shear center
Static equilibrium equations Forces, moments and the importance of mechanics to the effort
The problem unknowns are h and q +h
Sum forces
Sum torsional moments about shear center
Structural stiffness matrix
Loads, measured at shear center
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9. Purdue Aeroelasticity
Write the aerodynamic loads in terms of h & q We use matrix methods – that’s our theme
Idealized wing section lift
Twisting moment, at wing shear center, positive nose-up
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10. Purdue Aeroelasticity
Aeroelastic static equilibrium equation Introducing the aeroelastic stiffness matrix constructed out of thin air
Wing static equilibrium written in terms of unknown displacements, h & q
Aeroelastic stiffness matrix
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11. Solution for wing deflections, h & q
Divide by KT to get nondimensional terms
Invert 2x2 matrix
Get BHM
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12. Purdue Aeroelasticity
Wing displacements
plunge +h
twist
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13. Purdue Aeroelasticity
New goals
Define structural static stability
Concept of perturbations
Distinguish stability from response
Learn how to do stability analysis
Find the wing divergence dynamic pressure using a “perturbation” analysis
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14. Purdue Aeroelasticity
Math Summary
Static equilibrium plays an essential role in aeroelastic analysis (surprise, surprise…)
Static equilibrium equations are statically indeterminate (equilibrium depends on knowledge of force/deflection relationship)
Multi-degree-of-freedom systems have as many equations of equilibrium as degrees of freedom
Systems of simultaneous equations can be written (and solved) in matrix form.
Static equilibrium aeroelastic equations yield two important matrices
Structural stiffness matrix – symmetrical if you do it right
Aerodynamic stiffness matrix – aero people will not recognize this term
These matrices are added together to form the aeroelastic stiffness matrix
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15. Euler’s static stability criterion
"A system in static equilibrium is neutrally (statically) stable if there exist nearby static equilibrium states in addition to the original static equilibrium state.”
Stability - the tendency of a system (structural configuration) to return to its original equilibrium state when subjected to a small disturbance (perturbation)
Leonard Euler
Advisor-Bernoulli
Student-LaGrange
“Read Euler, read Euler, he is our master in everything" Laplace
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16. 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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17. 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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18. Purdue Aeroelasticity
Perturbed 1 dof 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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19. 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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20. 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 in perturbed position
new static equilibrium point Dq
Euler test has found a neutral stability condition
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21. 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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22. Purdue Aeroelasticity
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
Absum! (I’m outta here)
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