Purdue Aeroelasticity AAE 556 Aeroelasticity Lecture 5 – 1) Compressibility; 2) Multi-DOF systems Reading: Sections 2-13 to 2-15 Purdue Aeroelasticity

Purdue Aeroelasticity Homework for Monday? Prob. 2.1 Uncambered (symmetrical sections) MAC = 0 Lift acts at aero center (AC) a distance e ahead to the shear center Problem 2.3 – wait to hand in next Friday Purdue Aeroelasticity

Aeroelasticity matters Reflections on the feedback process Purdue Aeroelasticity

Purdue Aeroelasticity Topic 1 - Flow compressibility (Mach number) has an effect on divergence because the lift-curve slope depends on Mach number Approximate the effect of Mach number by adding the Prandtl-Glauert correction factor for sub-sonic flow Plots as a curve vs. M Purdue Aeroelasticity

Purdue Aeroelasticity But wait! – there’s more! Mach number depends on altitude and airspeed so two expressions must be reconciled Physics M=V/a Speed of sound, “a," depends on temperature and temperature depends on altitude Purdue Aeroelasticity

Purdue Aeroelasticity The divergence equation which contains Mach number must be consistent with the “physics” equation Choose an altitude Find the speed of sound Square both sides of the above equation and solve for MD Purdue Aeroelasticity

Determining MD requires solving a quadratic equation Purdue Aeroelasticity

Purdue Aeroelasticity If we want to increase the divergence Mach number we must increase stiffness (and weight) to move the math line upward Purdue Aeroelasticity

Purdue Aeroelasticity Summary Lift curve slope is one strong factor that determines divergence dynamic pressure depends on Mach number Critical Mach number solution for divergence dynamic pressure must be added to the solution process Purdue Aeroelasticity

Topic 2 – Multi-degree-of-freedom (MDOF) systems Develop process for analyzing MDOF systems Define theoretical stability conditions for MDOF systems Reading - Multi-degree-of-freedom systems – Section 2.14 Purdue Aeroelasticity

Purdue Aeroelasticity Here is a 2 DOF, segmented, aeroelastic finite wing model - two discrete aerodynamic surfaces with flexible connections used to represent a finite span wing (page 57) Torsional springs fuselage wing tip wing root Torsional degrees of freedom Purdue Aeroelasticity

Purdue Aeroelasticity Introduce “strip theory” aerodynamic modeling to represent twist dependent airloads Strip theory assumes that lift depends only on local angle of attack of the strip of aero surface why is this an assumption? q twist angles are measured from a common reference Purdue Aeroelasticity

Purdue Aeroelasticity The two twist angles are unknowns - we have to construct two free body diagrams to develop equations to find them Structural restoring torques depend on the difference between elastic twist angles Wing root Internal shear forces are present, but not drawn Wing tip Double arrow vectors are torques Purdue Aeroelasticity

Purdue Aeroelasticity This is the eventual lift re-distribution equation due to aeroelasticity – let’s see how we find it Observation - Outer wing panel carries more of the total load than the inner panel as q increases Purdue Aeroelasticity

Torsional static equilibrium is a special case of dynamic equilibrium Arrange these two simultaneous equations in matrix form Purdue Aeroelasticity

Purdue Aeroelasticity Summary The equilibrium equations are written in terms of unknown displacements and known applied loads due to initial angles of attack. These lead to matrix equations. Matrix equation order, sign convention and ordering of unknown displacements (torsion angles) is important Purdue Aeroelasticity

Problem solution outline Combine structural and aero stiffness matrices on the left hand side The aeroelastic stiffness matrix is Invert matrix and solve for q1 and q2 Purdue Aeroelasticity

Purdue Aeroelasticity The solution for the q’s requires inverting the aeroelastic stiffness matrix Purdue Aeroelasticity

The aeroelastic stiffness matrix determinant is a function of q The determinant is where When dynamic pressure increases, the determinant D tends to zero – what happens to the system then? Purdue Aeroelasticity

Purdue Aeroelasticity Plot the aeroelastic stiffness determinant D against dynamic pressure (parameter) Dynamic pressure parameter determinant The determinant of the stiffness matrix is always positive until the air is turned on Purdue Aeroelasticity

Solve for the twist angles created by an input angle of attack ao Purdue Aeroelasticity

Twist deformation vs. dynamic pressure parameter Unstable q region panel twist, qi/ao divergence Outboard panel (2) determinant D is zero Purdue Aeroelasticity

Panel lift computation on each segment gives: Note that Purdue Aeroelasticity

More algebra - Flexible system lift Set the wing lift equal to half the airplane weight Purdue Aeroelasticity

Purdue Aeroelasticity Lift re-distribution due to aeroelasticity (originally presented on slide 13) Observation - Outer wing panel carries more of the total load than the inner panel as q increases Purdue Aeroelasticity