1. AAE 556 Aeroelasticity Lecture 17
Typical section vibration
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2. Purdue Aeroelasticity
Understanding the origins of flutter Typical section equations of motion - 2 DOF
Plunge displacement h is positive downward & measured at the shear center xq
measured at the shear center from static equilibrium position
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3. A peek ahead at the final result coupled equations of motion dynamically coupled but elastically uncoupled
mg = weight xq
xq is called static unbalance and is the source of dynamic coupling
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Lagrange and analytical dynamics an alternative to FBD’s and Isaac Newton
z(t) is the downward displacement of a small portion of the airfoil at a position x located downstream of the shear center
kinetic energy
strain energy
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5. Expanding the kinetic energy integral
m is the total mass
Sq is called the static unbalance
Iq is called the airfoil mass moment of inertia – it has 2 parts
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6. Equations of motion for the unforced system (Qi = 0)
EOM in matrix form, as promised
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7. Differential equation a trial solution
Goal – frequencies and mode shapes
Substitute this into differential equations
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8. There is a characteristic equation here
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9. The time dependence term is factored out
Determinant of dynamic system matrix
set determinant to zero (characteristic equation)
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10. Nondimensionalize by dividing by m and Iq
Define uncoupled frequency parameters
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11. Solution for natural frequencies
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12. Solutions for exponent s These are complex numbers
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13. solutions for s are complex numbers
and
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14. Example configuration
2b=c
and
and
New terms – the radius of gyration
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15. Natural frequencies change when the wing c.g. or EA positions change
c.g. offset in semi-chords
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16. Purdue Aeroelasticity
Summary?
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