1. AAE 556 Aeroelasticity Lecture 24
Modal coordinates
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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
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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
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10. Purdue Aeroelasticity
Compute matrix products involving the mode shapes and the mass matrix The generalized masses and orthogonality
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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. Final result with new coordinates a set of totally decoupled equations
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15. Purdue Aeroelasticity
One more step
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16. Generalized stiffness matrix
Diagonal matrix
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17. 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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