1. AAE 556 Aeroelasticity Lecture 18
Resonance, Mode shapes
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2. Purdue Aeroelasticity
Summary
MDOF systems with n degrees of freedom have n possible “modes of motion”
Mode of motion means a (natural, resonant) frequency with a well-defined mode shape
Eigenvectors – another word for mode shapes – provide information about node lines or node points
Experiments provide node lines and frequencies to compare with analysis
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3. Typical section equations of motion - 2 dof
measured from static equilibrium position
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4. Review - Equations of motion for free vibration
Trial solution
assume harmonic motion
result
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5. Harmonic forcing at or near the natural frequencies
Trial solution
harmonic motion
result
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6. Harmonic forcing at or near the natural frequencies
At or near resonance, the amplitude of the response is large (here it is infinite because we have no damping)
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7. Purdue Aeroelasticity
Defining mode shapes what will the vibrations look like if we force the system at natural frequencies?
(1)
(2)
eigenvalues & eigenvectors
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8. A different expression
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9. Purdue Aeroelasticity
System mode shapes
If q is 1 then how much is h?
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10. Purdue Aeroelasticity
example
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11. Purdue Aeroelasticity
example
reference
bending
actual
reference
torsion
actual
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12. Purdue Aeroelasticity
Mode shape
When we let h/b=1 then we are asking about the amount of q in the plunge mode
1 unit
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13. Shake testing identifies modes and frequencies
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14. Torsional frequency and mode shape
1 radian
node point
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15. Purdue Aeroelasticity
Node point definition
A point in space where there is no displacement, velocity or acceleration when the structure is vibrating at a natural frequency x
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16. Node point depends on eigenvectorx
divide by b
first mode
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17. Node point for second frequency
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18. Purdue Aeroelasticity
Summary
MDOF systems have n modes of motion
Mode of motion means a (natural, resonant) frequency with a mode shape
Eigenvectors – mode shapes – provide node lines or node points
Experiment provides node lines and frequencies
Purdue Aeroelasticity