AAE 556 Aeroelasticity Lecture 28, 29,30-Unsteady aerodynamics 1 Purdue Aeroelasticity

What we have Purdue Aeroelasticity 2 … and, what we want … or, better yet

What we will have at the end  A 2x2 set of equations of motion that describe wing response due to a harmonic force input  A set of aerodynamic coefficients that describe the lift and moment and include the lags in the development of the lift and moment  The math necessary to assign some really hairy HW Purdue Aeroelasticity 3

What so hard about that? Some goals  Understand unsteady aerodynamics and the mathematics required to represent it  Understand why we need harmonic motion assumptions make the equations of motion development and the stability analysis easier Purdue Aeroelasticity 4

Point #1 – As an airfoil (a 2D section) rotates 1) a vortex attached to the ¼ chord develops; 2) a counter-vortex builds and is shed downstream; 3) lift is created 5 Purdue Aeroelasticity

For flutter work we conveniently assume that the wing oscillates harmonically – why? Pressures on an oscillating airfoil depend on the strength of the “bound” vortex and the location of shed vortices in the airfoil wake. All of them must be tracked. 6 Purdue Aeroelasticity

What’s the big deal? 1. For general motion, computation of “unsteady” lift requires keeping track of shed vortices position and strength forever 2. Forces and motion depend on the history of the motion itself –Tracking this leads to really complicated math 3. The alternative is to assume that the motion is harmonic 7 Purdue Aeroelasticity The flow will have a memory of how much the structure has deformed and how fast it has deformed. The buzz word is “hereditary.”

Now comes the fun – The mathematics of the problem – Wagner’s function predicts the lift does not after a sudden increase in airfoil angle of attack qSC L   s=Vt/b=Vt/(c/2) V “nondimensional time” 8 Purdue Aeroelasticity

But first – define the lift components and airfoil geometry – positive lift is directed downward 9 Purdue Aeroelasticity e=b/2+baLift per unit length, l Plunge displacement, h

Moment components and airfoil geometry 10 Purdue Aeroelasticity

We will begin our analysis by forcing the airfoil system with a sinusoidal force applied at the shear center 11 Purdue Aeroelasticity Solutions with phase lags

What’s a phase lag?  Because of delays between motion and force there is a phase difference between when deformation amplitude reaches its maximum and when the lift and aerodynamic moment reach theirs  In the case shown,  lags h by 90 degrees. Purdue Aeroelasticity 12

Phase lag definition Phase is the difference in time between two events such as the zero crossing of two waveforms, or the time between a reference and the peak of a waveform. The phase is expressed in degrees Also it is the time between two events divided by the period (also a time), times 360 degrees. 13 Purdue Aeroelasticity

Phase relationships between displacement, velocity, acceleration Cosine (velocity) “leads” sine motion by 90 degrees; it reaches its max before sine does. Acceleration leads displacement by 180 degrees 14 Purdue Aeroelasticity

Complex numbers represent harmonic motion as a rotating vector (a +ib) tt 15 Purdue Aeroelasticity

Displacement, velocity and acceleration represented as rotating vectors tt 16 Purdue Aeroelasticity

Generalized, harmonic aero forces – Theodorsen’s coefficients to describe lift and moment lift expressions - lift/unit length terminology for our airfoil system displacements (b*a=position of the shear center aft of airfoil mid-chord) 17 Purdue Aeroelasticity

There are two lift coefficients they are complex numbers to model phase lags C(k)=Theodorsen’s Circulation Function models the time delays 18 Purdue Aeroelasticity

Theodorsen's circulation function C(k) is a complex number that determines the lag between h and  oscillation and lift development – lift always lags motion 1/k F -G 19 Purdue Aeroelasticity

An approximation to C(k) this was important before MATLAB 1/k F -G 20 Purdue Aeroelasticity

This lift expression looks strange; where is the dynamic pressure? 21 Purdue Aeroelasticity

Insert the k, the reduced frequency, and … 22 Purdue Aeroelasticity

Each lift term has a physical meaning “steady-state lift” - why? 1) inertia term 2) damping term 3) steady state term 23 Purdue Aeroelasticity

Lift terms are classed as either in-phase or out of phase – out of phase terms are called aerodynamic damping in phase terms damping is out of phase 24 Purdue Aeroelasticity

Forced response equations of motion divide by mb 25 Purdue Aeroelasticity

Final equation with harmonic force applied at the shear center 26 Purdue Aeroelasticity

Compute only the steady-state harmonic response- not the transients “known” aero forces due to motion applied externally 27 Purdue Aeroelasticity

Forces and Equations of Motion harmonic, steady state, different phasing between forces, moments and motion 28 Purdue Aeroelasticity

Harmonic aero force 29 Purdue Aeroelasticity xx

Equations Of Motion 30 Purdue Aeroelasticity

Final EOM  is known because we pre-select it 31 Purdue Aeroelasticity

Generate the moment equilibrium equation about the shear center with force applied at the shear center aero moment about shear center Divide by 32 Purdue Aeroelasticity

nondimensionalize 33 Purdue Aeroelasticity

Moment equation 34 Purdue Aeroelasticity

More definitions See slide #8 35 Purdue Aeroelasticity

Moment equilibrium equation 36 Purdue Aeroelasticity

Matrix eq. equations are complex Forced response - find amplitudes in response to input 37 Purdue Aeroelasticity

Matrix eq. equations are complex Forced response - find amplitudes in response to input 38 Purdue Aeroelasticity

Matrix eq. equations are complex Forced response - find displacement amplitudes in response to input 39 Purdue Aeroelasticity

40 Locating resonance points frequency airspeed Purdue Aeroelasticity

The flutter problem – a complex eigenvalue with flutter frequency and airspeed unknown Purdue Aeroelasticity 41

Purdue Aeroelasticity 42 Example calculation  = 20

Purdue Aeroelasticity 43 Elements of the determinant

One way of solving for flutter Theodorsen’s Method Purdue Aeroelasticity 44 Use the quadratic formula to solve for the frequency ratio

Real and imaginary aero Purdue Aeroelasticity 45

Solve for crossing points Purdue Aeroelasticity 46 1/k=V/  b