10 flutter
DESCRIPTION
flutterTRANSCRIPT
Vibration
Background
We previously went through a 2-DOF rigid airfoil example and home with steady aerodynamics. We will now look at the use of unsteady aerodynamics in the flutter equations.
2DOF Airfoil
Rigid Airfoil
P reference point
C Center of Mass
Q Aerodynamic center
T chord point
b chord
e and a go from -1 to 1
X = e-a
EOM
EOMs:
Assume harmonic motion:
Substituting into EOMs above:
Where lift and moment are:
Substituting lift and moment in:
Flutter Formulation
Since the EOM are linear and homogenous the determinant must be zero for a nontrivial solution.
How to solve
1)Specify an altitude to fix
2)Specify M
3)Specify K values, start with 0.001 to 1.0
4)At each k, M calculate lh, l, mh, m
Flutter Formulation
How to Solve continued
5)Solve the flutter determinant which is a quadratic equation with complex values for (/)2 that correspond to each of the selected values of k.
6)Interpolate to find the value of k at which the imaginary part of the one of the roots becomes zero. This can be done graphically as well
7)Determine U = b/k and M=U/c
Repeat 3-7 with the value of M obtained in step 7 until Min =Mout and then that is the correct flutter speed.
Repeat process at different values of to determine flutter as a function of altitude.
K-method
EOMs:
Where D=
Nondimensionalizing:
g is equivalent structural damping
Z the unknown is formulated as:
K-Method
Flutter determinant:
Frequency and Damping from the solution comes from eigenvalue of Z:
Where g = 0 is flutter.
The damping value g is an artificial value of structural damping to achieve simple harmonic motion at a frequency i
The k-method is easy to implement but inherently is a mathematically improper formulation. The only place this method is valid is g=0.
P-k method
The p-k method tries to overcome some the mathematical issues that the k-method introduces.
We start by creating a dimensionless time
EOM becomes:
M and are diagonal matrices and:
For a given speed and altitude the flutter determinant is solve for n+1 p values
where k is the reduced frequency and is log dec dam
P-k method
If the aerodynamics are a function of k:
The results have to be iterated on such the k value for the aero is adjusted to match the value from the system eigenvalue for each eigenvalue for each velocity chosen.
Comparison plots
Summary
g=0 results are the same for each method.
At the same velocity, a mode can have two frequencies???
the instability can become more unstable as speed decreases???
NASTRAN K-Method
NASTRAN K-Method
NASTRAN K-Method