1. Purdue Aeroelasticity
AAE 556 Aeroelasticity Lecture 5 – 1) Compressibility; 2) Multi-DOF systems
Reading: Sections 2-13 to 2-15
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2. Purdue Aeroelasticity
Homework for Monday?
Prob. 2.1
Uncambered (symmetrical sections) MAC = 0
Lift acts at aero center (AC) a distance e ahead to the shear center
Problem 2.3 – wait to hand in next Friday
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3. Aeroelasticity matters Reflections on the feedback process
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4. Purdue Aeroelasticity
Topic 1 - Flow compressibility (Mach number) has an effect on divergence because the lift-curve slope depends on Mach number
Approximate the effect of Mach number by adding the Prandtl-Glauert correction factor for sub-sonic flow
Plots as a curve vs. M
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5. 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 temperature and temperature depends on altitude
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6. Purdue Aeroelasticity
The divergence equation which contains Mach number must be consistent with the “physics” equation
Choose an altitude
Find the speed of sound
Square both sides of the above equation and solve for MD
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7. Determining MD requires solving a quadratic equation
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8. Purdue Aeroelasticity
If we want to increase the divergence Mach number we must increase stiffness (and weight) to move the math line upward
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9. Purdue Aeroelasticity
Summary
Lift curve slope is one strong factor that determines divergence dynamic pressure
depends on Mach number
Critical Mach number solution for divergence dynamic pressure must be added to the solution process
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10. Topic 2 – Multi-degree-of-freedom (MDOF) systems
Develop process for analyzing MDOF systems
Define theoretical stability conditions for MDOF systems
Reading - Multi-degree-of-freedom systems – Section 2.14
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11. Purdue Aeroelasticity
Here is a 2 DOF, segmented, aeroelastic finite wing model - two discrete aerodynamic surfaces with flexible connections used to represent a finite span wing (page 57)
Torsional springs
fuselage
wing tip
wing root
Torsional degrees of freedom
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12. 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?
q twist angles are measured from a common reference
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13. Purdue Aeroelasticity
The two twist angles are unknowns - we have to construct two free body diagrams to develop equations to find them
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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14. Purdue Aeroelasticity
This is the eventual lift re-distribution equation due to aeroelasticity – let’s see how we find it
Observation - Outer wing panel carries more of the total load than the inner panel as q increases
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15. Torsional static equilibrium is a special case of dynamic equilibrium
Arrange these two simultaneous equations in matrix form
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16. Purdue Aeroelasticity
Summary
The equilibrium equations are written in terms of unknown displacements and known applied loads due to initial angles of attack. These lead to matrix equations.
Matrix equation order, sign convention and ordering of unknown displacements (torsion angles) is important
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17. Problem solution outline
Combine structural and aero stiffness matrices on the left hand side
The aeroelastic stiffness matrix is
Invert matrix and solve for q1 and q2
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18. Purdue Aeroelasticity
The solution for the q’s requires inverting the aeroelastic stiffness matrix
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19. 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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20. 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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21. Solve for the twist angles created by an input angle of attack ao
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22. 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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23. Panel lift computation on each segment gives:
Note that
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24. More algebra - Flexible system lift
Set the wing lift equal to half the airplane weight
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25. Purdue Aeroelasticity
Lift re-distribution due to aeroelasticity (originally presented on slide 13)
Observation - Outer wing panel carries more of the total load than the inner panel as q increases
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