1. AAE 556 Aeroelasticity Lecture 6 – Multi-DOF systems
Reading: Sections 2-13 to 2-15
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2. Purdue Aeroelasticity
Homework for Friday?
Problem 2.3
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3. 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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4. 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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5. 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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6. Torsional static equilibrium is a special case of dynamic equilibrium
Arrange these two simultaneous equations in matrix form
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7. Purdue Aeroelasticity
Strain energy
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8. 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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9. 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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10. Purdue Aeroelasticity
The solution for the q’s requires inverting the aeroelastic stiffness matrix
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11. 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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12. 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 turning the air on reduces its size
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13. Solve for the twist angles created by an input angle of attack ao
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14. 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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15. More algebra - Flexible system lift
Set the wing lift equal to half the airplane weight
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16. 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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17. Purdue Aeroelasticity
MDOF Divergence
In general we have matrix relationships developed from EOM’s
These can be converted into perturbation relationships
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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 – divergence occurs
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19. Determinant D plotted against dynamic pressure parameter
This nth order determinant is called the stability determinant or the characteristic equation
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20. Purdue Aeroelasticity
Eigenvectors 1 2
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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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