AAE556 – Spring Armstrong AAE 556 Aeroelasticity Lecture 5 Reading: text pp

AAE556 – Spring 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.

AAE556 – Spring What does happen to deflection when you get close to divergence? An aeroelastic parameter Expand

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

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

AAE556 – Spring 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

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

AAE556 – Spring 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

AAE556 – Spring 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

AAE556 – Spring 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

AAE556 – Spring Determining M D requires solving a quadratic equation

AAE556 – Spring 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.

AAE556 – Spring 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

AAE556 – Spring 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

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

AAE556 – Spring Plot effective stiffness for 4 values of q bar

AAE556 – Spring Also nondimensionalize the original static equilibrium equation by dividing by K T

AAE556 – Spring 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