AAE 556 Aeroelasticity Lecture 4 Reading notes assignment

AAE 556 Aeroelasticity Lecture 4 Reading: notes assignment from Lecture 3 weisshaar@purdue. edu Armstrong 3329 765 -494 -5975 Purdue Aeroelasticity 4 -1

Summary to-date i Development of simple models of wing aeroelastic behavior with pitch (torsion) only and pitch and plunge (bending) i Models show that torsional deformation creates additional lift, deflection (and stress). i 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 4 -2

Today and next week’s agenda i Define and discuss static stability – Concept of perturbations – Distinguish stability from response i Learn how to do a stability analysis i Find the divergence dynamic pressure using a “perturbation” analysis Purdue Aeroelasticity 4 -3

The perturbed structure i 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. i We need to apply these above words to equations so that we can put the aeroelastic system to a mathematical test Purdue Aeroelasticity 4 -4

Stability investigation i Given a system that we know is in static equilibrium (forces and moments sum to zero) i Add a disturbance to perturb the system to move it to a different, nearby position (that may or may not be in static equilibrium) i Is this new, nearby state also a static equilibrium point? i Write static equilibrium equations and see if forces and moments balance Purdue Aeroelasticity 4 -5

Perturbed airfoil i In flight this airfoil is in static equilibrium at the fixed angle q but what happens if we disturb (perturb) it? i There are three possibilities Purdue Aeroelasticity 4 -6

Perturbation possibilities i KT(Dq)>(DL)e – statically stable because it tends to return – no static equilibrium in the perturbed state i KT(Dq)<(DL)e – statically unstable – motion away from original position i KT(Dq)=(DL)e – system stays perturbed but static – we have found new static equilibrium point – Euler test has found neutral stability Purdue Aeroelasticity 4 -7

Example i Perturb the airfoil when it is in static equilibrium i To be neutrally stable in this new perturbed position this equation must be an true Purdue Aeroelasticity 4 -8

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 4 -9

Observations i The equation for neutral stability is simply the usual static equilibrium equation with right-handside (the input angle ao) set to zero. i 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 4 -10

Stability investigation i Take a system that we know is in static equilibrium (forces and moments sum to zero) i Perturb the system to move it to a different, nearby position (that may or may not be in static equilibrium) i Is this new, nearby state also a static equilibrium point? i Static equilibrium equations for stability are those for a self-equilibrating 11 system Purdue Aeroelasticity

More observations i At neutral stability the deformation is not unique (Dq is not zero but can be plus or minus) i 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 4 -12

The 1 DOF divergence condition i Neutral stability i or Purdue Aeroelasticity 4 -13

System stiffness, not strength, is important Structural resistance Aero overturning Slope depends on q. SCLa Equilibrium point Purdue Aeroelasticity 4 -14

Stable perturbed system Equilibrium point Purdue Aeroelasticity 4 -15

Perturbed system-neutral stability Lines are parallel Equilibrium point at infinity Purdue Aeroelasticity 4 -16

Unstable system Equilibrium point? Purdue Aeroelasticity 4 -17

Aeroelastic stiffness decreases as q increases Purdue Aeroelasticity 4 -18

Aeroelastic divergence i Look at the single degree of freedom typical section and the expression for twist angle with the initial load i neglect wing camber previous result "twist amplification" Purdue Aeroelasticity 4 -19

Twist amplification Purdue Aeroelasticity 4 -20

Example corrections q bar = 0. 5 relative sizes of terms the sum of the terms is 2 Purdue Aeroelasticity 4 -21

Aeroelastic feedback process qo is the twist angle with no aero load/structural response "feedback" Purdue Aeroelasticity 4 -22

More terms the response to angle of attack qo instead of ao …and, the third term Purdue Aeroelasticity 4 -23

Conclusion Each term in the series represents a feedback "correction" to the twist created by load interaction Series convergence Series divergence Purdue Aeroelasticity 4 -24

Summary i Divergence condition is a neutral stability condition i Divergence condition can be found using the original equilibrium conditions i Stability does not depend on the value of the applied loads Purdue Aeroelasticity 4 -25