AAE 556 Aeroelasticity Lecture 4 Reading: notes assignment from Lecture 3 weisshaar@purdue.edu Armstrong 3329 765-494-5975 Purdue Aeroelasticity

Purdue Aeroelasticity Summary to-date Development of simple models of wing aeroelastic behavior with pitch (torsion) only and pitch and plunge (bending) Models show that torsional deformation creates additional lift, deflection (and stress). Models identify an aeroelastic parameter that defines a dynamic pressure at which lift and torsional deflection approach infinity Models are linear so this will never really happen This special dynamic pressure is called the “divergence dynamic pressure.” Purdue Aeroelasticity

Today and next week’s agenda Define and discuss 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 Purdue Aeroelasticity

The perturbed structure Static stability analysis considers what happens to a flexible system that is in static equilibrium and is then disturbed. If the system tends to come back to its original, undisturbed position, it is stable - if not - it is unstable. We need to apply these above words to equations so that we can put the aeroelastic system to a mathematical test Purdue Aeroelasticity

Stability investigation Given a system that we know is in static equilibrium (forces and moments sum to zero) Add a disturbance to 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? Write static equilibrium equations and see if forces and moments balance Purdue Aeroelasticity

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

Perturbation possibilities KT(Dq)>(DL)e statically stable because it tends to return no static equilibrium in the perturbed state KT(Dq)<(DL)e statically unstable motion away from original position KT(Dq)=(DL)e system stays perturbed but static we have found new static equilibrium point Euler test has found neutral stability Purdue Aeroelasticity

Purdue Aeroelasticity 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 Purdue Aeroelasticity

Static stability investigation is “stiffness based” Neutral stability means this relationship must be zero (2 states) so... Not zero condition at neutral stability static equilibrium displacement (Dq) is not unique Purdue Aeroelasticity

Purdue Aeroelasticity Observations The equation for neutral stability is simply the usual static equilibrium equation with right-hand-side (the input angle ao) 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 Purdue Aeroelasticity

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 Purdue Aeroelasticity

Purdue Aeroelasticity More observations At neutral stability the deformation is not unique (Dq is not zero but can be plus or minus) At neutral static stability the system has many choices (equilibrium states) near its original equilibrium state. airfoil position is uncontrollable - it has no displacement preference when a load is applied. Purdue Aeroelasticity

The 1 DOF divergence condition Neutral stability or Purdue Aeroelasticity

System stiffness, not strength, is important Structural resistance Aero overturning Equilibrium point Slope depends on qSCLa Purdue Aeroelasticity

Stable perturbed system Equilibrium point Purdue Aeroelasticity

Perturbed system-neutral stability Equilibrium point at infinity Lines are parallel Purdue Aeroelasticity

Purdue Aeroelasticity Unstable system Equilibrium point? Purdue Aeroelasticity

Aeroelastic stiffness Aeroelastic stiffness decreases as q increases Purdue Aeroelasticity

Aeroelastic divergence Look at the single degree of freedom typical section and the expression for twist angle with the initial load neglect wing camber previous result "twist amplification" Purdue Aeroelasticity

Purdue Aeroelasticity Twist amplification Purdue Aeroelasticity

Purdue Aeroelasticity Example corrections q bar = 0.5 relative sizes of terms The infinite series in Eqn. 2.47 will converge or diverge depending on the size of the term compared to unity (1). We will have series convergence if , but divergence if . When we have a critical or cross-over condition. The size of the parameter provides a test for the static stability or convergence of the system when it is loaded. the sum of the terms is 2 Purdue Aeroelasticity

Aeroelastic feedback process qo is the twist angle with no aero load/structural response "feedback" Purdue Aeroelasticity

Purdue Aeroelasticity More terms the response to angle of attack qo instead of ao …and, the third term Purdue Aeroelasticity

Purdue Aeroelasticity Conclusion Each term in the series represents a feedback "correction" to the twist created by load interaction We can visualize the load interaction process as a series of corrections to our load given a new load created by the deflection. This process should converge to a final stable answer. If not, we have a mathematically unstable process and a physically unstable process. Series convergence Series divergence Purdue Aeroelasticity

Purdue Aeroelasticity 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 Purdue Aeroelasticity