1. AAE 556 Aeroelasticity Lecture 21
Modal coordinates and flutter
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2. Modal analysis objectives
Accurate flutter analysis with few degrees of freedom
Identification of interaction in terms of physically meaningful motion – vibration mode shapes obtained from testing or analysis
Key words
Modal orthogonality
generalized mass & stiffness
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3. Reducing the number of system degrees of freedom
Vibration equations with quasi-steady loads
This can be a really big eigenvalue problem with “n” degrees of freedom
Let’s reduce the problem to “m” degrees of freedom and retain accuracy
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4. Purdue Aeroelasticity
Begin with eigenvalue/eigenvector analysis for free vibration frequencies and mode shapes
Solve for eigenvectors (mode shapes)
construct a modal matrix from the eigenvectors
Modal matrix
Eigenvectors (mode shapes)
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5. Purdue Aeroelasticity
Define response in terms of modal amplitude coordinates, h(t), and eigenvectors
System response
Modal matrix
This series can be truncated to reduce DOF number from n to m (m<n)
We go from n actual displacements (the x’s) to a m displacements (the h’s)
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6. Example – String with tension S and three equal masses, m
S=tension
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7. Purdue Aeroelasticity
Any general deflection can be constructed as a summation of normal modes
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8. Use new (modal) coordinates to define motion
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9. Redefine motion in terms of modal coordinates
Pre- and post-multiply by the modal matrix
Why?
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10. Purdue Aeroelasticity
Compute matrix products involving the mode shapes and the mass matrix The generalized masses and orthogonality
Results
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11. Generalized masses (in general) we diagonalize the mass matrix
Generalized mass matrix definition
Diagonal matrix
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12. Stiffness matrix multiplication
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13. Stiffness matrix multiplication showing orthogonality
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14. Generalized stiffness matrix
Diagonal matrix
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15. Purdue Aeroelasticity
Final result - a new set of “modal” coordinates gives a set of totally decoupled equations
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16. System with modal coordinates
The matrix order can be reduced from n degrees of freedom to m degrees of freedom so that the matrix is smaller but still has a great deal of information content
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17. The eigenvalue problem is decoupled into a set of n (or m) equations
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18. The original problem, but this time in modal form
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19. General aeroelastic system with aero displacement dependent loads
The matrix order can be reduced from n degrees of freedom to m degrees of freedom so that the matrix is smaller but still has a great deal of information content
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20. Purdue Aeroelasticity
A problem for homework
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21. Purdue Aeroelasticity
Do the following
Let airspeed be zero
Use MATLAB to compute the natural frequencies and the mode shapes fi
Construct the 2x2 modal displacement matrix Fij
Perform the matrix multiplication to find the modal mass and modal displacement matrices
Identify the modal masses and modal frequencies
Solve for the aerodynamic modal matrix.
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