1. Purdue Aeroelasticity
AAE 556 Aeroelasticity Lecture 5 – 1) Compressibility; 2) Multi-DOF systems
Reading: Sections 2-13 to 2-14
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2. Purdue Aeroelasticity
Homework for Friday?
Uncambered (symmetrical sections) MAC = 0
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3. Purdue Aeroelasticity
Flow compressibility has an effect on divergence because it affects the lift-curve slope
Approximate the effect of compressibility by adding the Prandtl-Glauert correction factor for sub-sonic flow
Plots as a curve vs. M
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4. Purdue Aeroelasticity
But wait! – there’s more! Mach number depends on altitude and airspeed so two expressions must be reconciled
Physics
M=V/a
Speed of sound, “a," depends on altitude (because it depends on temperature)
Math
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5. Choose an altitude, find the speed of sound
Reconciliation requires solving for MDiv by equating the math expression to the physics expression
Choose an altitude, find the speed of sound
square both sides of the above
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6. Determining MD requires solving a quadratic equation
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7. Purdue Aeroelasticity
If we want to increase the divergence Mach number we must increase stiffness (and weight) to move the math line upward
Even I know that! Just because I don’t care doesn’t mean I don’t understand.
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8. Purdue Aeroelasticity
Summary
Divergence condition is a stiffness based condition based on the concept of neutral stability leading to multiple equilibrium states
Stability does not depend on the size of the external loads, although they may be sizable if you aren’t careful
Lift curve slope is one strong determinator of divergence
depends on Mach number
Critical Mach number solution must be added to the solution process
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9. 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 – Section 2.14
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10. Purdue Aeroelasticity
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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11. Purdue Aeroelasticity
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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12. Purdue Aeroelasticity
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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13. Purdue Aeroelasticity
Result to work for with the next few slides– 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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14. Torsional static equilibrium a special case of dynamic equilibrium
Arrange these two simultaneous equations in matrix form
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15. Purdue Aeroelasticity
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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16. Problem solution outline
Combine matrices on the left hand side
The aeroelastic stiffness matrix is
Solve for q1 and q2
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17. Purdue Aeroelasticity
The solution for the q’s requires inverting the aeroelastic stiffness matrix
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18. 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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19. Purdue Aeroelasticity
Plot the aeroelastic stiffness determinant D against dynamic pressure (parameter)
Dynamic pressure parameter
determinant
The determinant of the stiffness matrix is always positive until the air is turned on
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20. Solve for the twist angles created by an input angle of attack ao
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21. 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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22. Panel lift computation on each segment gives:
Note that
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23. More algebra - Flexible system lift
Set the wing lift equal to half the airplane weight
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24. 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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