1. AAE 556 Aeroelasticity Lectures 22, 23
Typical dynamic instability problems and test review
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2. How to recognize a flutter problem in the making
Given: a 2 DOF system with a parameter Q that creates loads on the system that are linear functions of the displacements
Q=0
Q is a real number
If p12 and p21 have the same sign (both positive or both negative) can flutter occur?
Q not zero
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3. The modified determinant
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4. If flutter occurs two frequencies must merge
FLUTTER – Increasing Q must cause the term under the radical sign to become zero.
For frequency merging flutter to occur, p12 and p21 must have opposite signs.
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5. Purdue Aeroelasticity
If one of the frequencies can be driven to zero then we have divergence
Divergence requires that the cross-coupling terms have the same sign
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6. Aero/structural interaction model TYPICAL SECTION What did we learn?
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7. Divergence-examination vs. perturbation
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8. Perturbations & Euler’s Test
...result - stable - returns -no static equilibrium in perturbed state
...result - unstable -no static equilibrium - motion away from equilibrium state
...result - neutrally stable - system stays - new static equilibrium point
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9. Stability equation is original equilibrium equation with R.H.S.=0.
The stability equation is an equilibrium equation that represents an equilibrium state with no "external loads" –
Only loads that are deformation dependent are included
The neutrally stable state is called self-equilibrating
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10. Multi-degree of freedom systems
From linear algebra, we know that there is a solution to the homogeneous equation only if the determinant of the aeroelastic stiffness matrix is zero
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11. Purdue Aeroelasticity
MDOF stability
Mode shapes? Eigenvectors and eigenvalues.
System is stable if the aeroelastic stiffness matrix determinant is positive. Then the system can absorb energy in a static deformation mode. If the stability determinant is negative then the static system, when perturbed, cannot absorb all of the energy due to work done by aeroelastic forces and must become dynamic.
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12. Three different definitions of roll effectiveness
Generation of lift – unusual but the only game in town for the typical section
Generation of rolling moment –
contrived for the typical section – reduces to lift generation
Multi-dof systems – this is the way to do it
Generation of steady-state rolling rate or velocity-this is the information we really want for airplane performance
Reversal speed is the same no materr which way you do it.
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13. Control effectiveness
reversal is not an instability - large input produces small output
opposite to divergence phenomenon
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14. Steady-state rolling motion
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15. Purdue Aeroelasticity
Swept wings
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16. Purdue Aeroelasticity
Divergence
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17. Purdue Aeroelasticity
Lift effectiveness
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18. Purdue Aeroelasticity
Flexural axis
Flexural axis - locus of points where a concentrated force creates no stream-wise twist (or chordwise aeroelastic angle of attack)
The closer we align the airloads with the flexural axis, the smaller will be aeroelastic effects.
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19. How to recognize a flutter problem in the making
Given: a 2 DOF system with a parameter Q that creates loads on the system that are linear functions of the displacements
Q=0
Q is a real number
If p12 and p21 have the same sign (both positive or both negative) can flutter occur?
Q not zero
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20. If flutter occurs two frequencies must merge
FLUTTER – Increasing Q causes the term under the radical sign to be zero.
For frequency merging flutter to occur, p12 and p21 must have opposite signs.
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21. If one of the frequencies is driven to zero then we have divergence
Divergence requires that the cross-coupling terms are of the same sign
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22. Purdue Aeroelasticity
Fuel line flutter
A hollow, uniform-thickness, flexible tube has a mass per unit length of m slugs/ft. and carries liquid fuel with density r to a rocket engine. The fuel flow rate is U ft/sec. through a pipe cross-section of A. The tube is straight and has supports a distance L apart, the tube bending displacement is approximated to be
Unknown amplitudes of vibrational motion
The free vibration frequencies when the fluid is not flowing are:
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23. Purdue Aeroelasticity
Fluid flow creates system coupling, but through the velocity, not the displacement
Find the divergence speed
Estimate the flow speed that flutter occurs, if it occurs
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24. Purdue Aeroelasticity
Divergence is found by computing the determinant of the aeroelastic stiffness matrix
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25. Purdue Aeroelasticity
Assume that coupling leads to flutter and find an estimate of the merging point
Harmonic motion?
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26. The frequencies are approximated
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