1. AAE556 – Spring 2008 - 5 5-1 weisshaar@purdue.edu Armstrong 3329 765-494-5975 AAE 556 Aeroelasticity Lecture 5 Reading: text pp. 62-76

2. AAE556 – Spring 2008 - 5 5-2 The Euler divergence condition - when you disturb a static equilibrium state, do you find another equally good static equilibrium state? Assumption – linear system Observation (again) – stiffness, not deflection, is involved Excessive deflection is a consequence, not a cause If so, get off the bus.

3. AAE556 – Spring 2008 - 5 5-3 What does happen to deflection when you get close to divergence? An aeroelastic parameter Expand

4. AAE556 – Spring 2008 - 5 5-4 There is a natural aeroelastic “feedback” process going on here  o is the twist angle with no aero load/structural response "feedback"

5. AAE556 – Spring 2008 - 5 5-5 the response to angle of attack  o instead of  o …and, the third term Interpret the other terms

6. AAE556 – Spring 2008 - 5 5-6 Each term in the expansion is a feedback correction to twist created by the previous term The series should converge – shouldn’t it? Series convergence Series divergence

7. AAE556 – Spring 2008 - 5 5-7 Example - corrections with q bar = 0.5 q bar = 0.5 relative sizes of terms the sum of the terms is 2

8. AAE556 – Spring 2008 - 5 5-8 Flow compressibility has an effect on divergence because it affects the lift-curve slope Approximate the effect of compressibility by adding the Prandtl-Glauert correction factor for sub-sonic flow Plots as a curve vs. M

9. AAE556 – Spring 2008 - 5 5-9 But wait! – there’s more! Mach number depends on altitude and airspeed so two expressions must be reconciled Speed of sound, “a," depends on altitude (because it depends on temperature) M=V/a Physics Math

10. AAE556 – Spring 2008 - 5 5-10 Reconciliation requires solving for M Div by equating the math expression to the physics expression Choose an altitude, find the speed of sound square both sides of the above

11. AAE556 – Spring 2008 - 5 5-11 Determining M D requires solving a quadratic equation

12. AAE556 – Spring 2008 - 5 5-12 If we want to increase the divergence M then we must increase stiffness (and weight) to move the math line upward Even I know that! Just because I don’t care doesn’t mean I don’t understand.

13. AAE556 – Spring 2008 - 5 5-13 Summary  Divergence condition is a stiffness based condition based on the concept of neutral stability and multiple equilibrium states  Stability does not depend on the size of the applied loads, although they may be sizable if you aren’t careful  Lift curve slope is one strong determinator of divergence –depends on Mach number –Critical Mach number solution must be added to the solution process

14. AAE556 – Spring 2008 - 5 5-14 Nonlinear models and finding the effects of aerodynamic stall on stability  Math model is at zero angle of attack when a non-aero load P is applied as shown  Lift is a nonlinear (cubic function of angle of attack)  Structural stiffness reponse is linear Pd is an applied moment

15. AAE556 – Spring 2008 - 5 5-15 Sum the moments about the shear center (the model pin) Define “Effective torsional stiffness” So -

16. AAE556 – Spring 2008 - 5 5-16 Plot effective stiffness for 4 values of q bar

17. AAE556 – Spring 2008 - 5 5-17 Also nondimensionalize the original static equilibrium equation by dividing by K T

18. AAE556 – Spring 2008 - 5 5-18 Plotting displacement angle q shows that multiple equilibrium states are possible above divergence q and the use of nonlinear analysis reveals what the angles are