AAE556 Lectures 34,35 The p-k method, a modern alternative to V-g Purdue Aeroelasticity 1

Genealogy of the V-g or “k” method  Equations of motion for harmonic response (next slide) –Forcing frequency and airspeed are known parameters –Reduced frequency k is determined from  and V –Equations are correct at all values of  and V.  Take away the harmonic applied forcing function –Equations are only true at the flutter point –We have an eigenvalue problem –Frequency and airspeed are unknowns, but we still need k to define the numbers to compute the elements of the eigenvalue problem –We invented V-g artificial damping to create an iterative approach to finding the flutter point Purdue Aeroelasticity 2

Equation #2, moment equilibrium 3 Purdue Aeroelasticity Divide by  2 Include structural damping

The eigenvalue problem Purdue Aeroelasticity 4

Return to the EOM’s before we assumed harmonic motion Purdue Aeroelasticity Here is what we would like to have The first step in solving the general stability problem 25-5

The p-k method casts the flutter problem in the following form Purdue Aeroelasticity …but first, some preliminaries 6

Setting up an alternative solution scheme 7 Purdue Aeroelasticity

The expanded equations 8 Purdue Aeroelasticity

Break into real and imaginary parts 9 Purdue Aeroelasticity

Recognize the mass ratio 10 Purdue Aeroelasticity

Multiply and divide real part by dynamic pressure Multiply imaginary part by p/j  11 Purdue Aeroelasticity

Multiply and divide imaginary part by Vb/Vb

Define A ij and B ij matrices

Place aero parts into EOM’s Note the minus signs

What are the features of the new EOM’s?  We still need k defined before we can evaluate the matrices  Airspeed, V, appears.  The EOM is no longer complex  We can calculate the eigenvalue, p, to determine stability

The p-k problem solution  Choose k=  b/V arbitrarily  Choose altitude ( , and airspeed (V)  Mach number is now known (when appropriate)  Compute AIC’s from Theodorsen formulas or others  Compute aero matrices-B ij and A ij matrices are real  Convert “p-k” equation to first-order state vector form

A state vector contains displacement and velocity “states” State vector = Purdue Aeroelasticity

Relationship between state vector elements An equation of motion with damping becomes Purdue Aeroelasticity

Use an identify relationship for the other equations Purdue Aeroelasticity 19

State vector eigenvalue equation Assume a solution Result Solve for eigenvalues (p) of the [Q] matrix (the plant) Plot results as a function of airspeed Purdue Aeroelasticity

1 st order problem  Mass matrix is diagonal if we use modal approach – so too is structural stiffness matrix  Compute p roots –Roots are either real (positive or negative) –Complex conjugate pairs Purdue Aeroelasticity

Eigenvalue roots   is the estimated system damping  There are “m” computed values of  at the airspeed V  You chose a value of k=  b/V, was it correct? –“line up” the frequencies to make sure k,  and V are consistent Purdue Aeroelasticity

Procedure Input k and V Compute eigenvalues yes Repeat process for each  No, change k Purdue Aeroelasticity

P-k advantages  Lining up frequencies eliminates need for matching flutter speed to Mach number and altitude  p-k approach generates an approximation to the actual system aerodynamic damping near flutter  p-k approach finds flutter speeds of configurations with rigid body modes Purdue Aeroelasticity