1. Purdue Aeroelasticity 4-1 AAE 556 Aeroelasticity Lecture 4 Reading: 2.8-2.12

2. Purdue Aeroelasticity 4-2 Agenda  Review static stability –Concept of perturbations –Distinguish stability from response  Learn how to do a stability analysis  Find the divergence dynamic pressure using a “perturbation” analysis

3. Purdue Aeroelasticity 4-3 Perturbed airfoil  In flight this airfoil is in static equilibrium at the fixed angle  but what happens if we disturb (perturb) it?  There are three possibilities

4. Purdue Aeroelasticity 4-4 Example  Perturb the airfoil when it is in static equilibrium  To be neutrally stable in this new perturbed position this equation must be an true

5. Purdue Aeroelasticity 4-5 The 1 DOF divergence condition  Neutral stability  or

6. Purdue Aeroelasticity 4-6 Observations  The equation for neutral stability is simply the usual static equilibrium equation with right-hand- side (the input angle  o ) set to zero.  The neutral stability equation describes a special case –only deformation dependent external (aero) and internal (structural) loads are present –these loads are “self-equilibrating” without any other action being taken

7. Purdue Aeroelasticity 7 Stability investigation  Take a system that we know is in static equilibrium (forces and moments sum to zero)  Perturb the system to move it to a different, nearby position (that may or may not be in static equilibrium)  Is this new, nearby state also a static equilibrium point?  Static equilibrium equations for stability are those for a self- equilibrating system

8. Purdue Aeroelasticity 8 Neutral stability  Neutral stability is only possible if the system is “self-equilibrating.”  The internal and external loads created by deformation just balance each other.  The system static stiffness is zero.  We’ll see that this requires that the system aeroelastic matrix become singular (the determinant is zero).

9. Purdue Aeroelasticity 4-9 The deformations at neutral stability are eigenvectors of the problem  At neutral stability the deformation is not unique (  is  not zero - can be plus or minus with indeterminate amplitude)  At neutral static stability the system has many choices (equilibrium states) near its original equilibrium state. –wing position is uncontrollable - it has no displacement preference when a load is applied.

10. Purdue Aeroelasticity 4-10 For stability, only system stiffness is important. This graph shows where the equilibrium point for twist is located Structural resistance Aero overturning Equilibrium point Slope depends on qSC La

11. Purdue Aeroelasticity 4-11 When we perturb the twist angle we move to a different position on the graph. One of the moments will be larger than the other/ Equilibrium point

12. Purdue Aeroelasticity 4-12 The slope of the aero line is a function of dynamic pressure so the line rotates as speed increases. This is a plot of the lines right at divergence. The equilibrium point lies at infinity Lines are parallel

13. Purdue Aeroelasticity 4-13 When the dynamic pressure is larger than the divergence dynamic pressure the crossing point is negative. This is mathematics way of telling you that you are in trouble.

14. Purdue Aeroelasticity 4-14 Let’s examine how aeroelastic stiffness changes with increased dynamic pressure Aeroelastic stiffness decreases as q increases The standard definition of stiffness is as follows

15. Purdue Aeroelasticity 4-15 As we approach aeroelastic divergence we get twist amplification  Consider the single degree of freedom typical section and the expression for twist angle with the initial load due to  o  neglect wing camber

16. Purdue Aeroelasticity 4-16 Write this expression in terms of an infinite series

17. Purdue Aeroelasticity 4-17 The first term is the uncorrected value of twist angle with no aeroelasticity q bar = 0.5 Plot the relative sizes of terms with qbar=0.5 the sum of the infinite series is 2

18. Purdue Aeroelasticity 4-18 Let’s take a look at the series and explain it as an aeroelastic feedback process  o is the twist angle with no aero load/structural response "feedback"

19. Purdue Aeroelasticity 4-19 Write the series slightly differently

20. Purdue Aeroelasticity 4-20 This is the response to angle of attack  o instead of  o …and, the third term The second term is the response to the first term

21. Purdue Aeroelasticity 4-21 Conclusion Each term in the series represents a feedback "correction" to the twist created by load interaction Series convergence Series divergence

22. Purdue Aeroelasticity 4-22 Summary  Divergence condition is a neutral stability condition  Divergence condition can be found using the original equilibrium conditions  Stability does not depend on the value of the applied loads