Flutter Analysis
A flutter analysis determines the
dynamic stability of an aeroelastic system. Three methods of analysis
are available: the American (K) method, a restricted but more
efficient American (KE) method, and the British (PK) method.
The British method not only determines stability boundaries but provides
approximate, but realistic, estimates of system damping at subcritical
speeds that can be used to monitor flight flutter tests. The system
dampings obtained from the K- and KE-methods is a mathematical quantity
not easily related to the physical system damping. As with static aeroelastic
analysis, flutter analysis presupposes a structural model, an aerodynamic
model, and their interconnection by splines. The modal technique is
used to reduce the number of degrees of freedom in the stability analysis.
It should be appreciated by the user that the use of vibration modes
for this purpose constitutes a series solution, and that a sufficient
number of modes must be used to obtain convergence to the required accuracy.
An aspect of the modal method is a transformation of the aerodynamic
influence coefficients into modal coordinates. For computational efficiency,
this transformation is carried out explicitly for only a few Mach numbers
(M) and reduced frequencies (k). These generalized (modal) aerodynamic
force coefficient matrices are then interpolated to any additional Mach
numbers and reduced frequencies required by the flutter analysis. Matrix
interpolation is also performed.Data entries allow the selection of
parameters for the explicit calculations of the aerodynamic matrices.
The flutter analysis is performed in modal coordinates. Because of the
iterative nature of the PK-method, the manner of convergence is frequently
of interest to the Customer and can be seen by specifying the diagnostic.
LDP can also select the real eigenvalue method for use in finding the
vibration modes and frequencies for the modal flutter analysis.
The K-method
The K-method of flutter analysis considers
the aerodynamic loads as complex masses, and the flutter analysis becomes
a vibration analysis using complex arithmetic to determine the frequencies
and artificial dampings required to sustain the assumed harmonic motion.
This is the reason the solution damping is not physical. When a B
matrix is present, complex conjugate pairs of roots are no longer produced.
LDP extracts all requested roots but only selects roots with a positive
imaginary part for the flutter summary output. Usually, one or two of
the parameters will have only a single value. Caution must be used if
a large number of loops are specified; they may take an excessive time
to execute. LDP thus implement multiple subcases can be specified to
pinpoint particular regions for study while controlling CPU resources
of the working machine
The KE-method
The KE-method is ideal when hundreds
of flutter analyses are required. The KE-method is similar to the K-method.
By restricting the functionality, the KE-method is a more efficient
K-method. The two major restrictions are that no damping (B) matrix
is allowed and no eigenvector recovery is made. A complex stiffness
matrix can be used to include the effects of structural damping. The
KE-method therefore cannot consider control systems in which damping
terms are usually essential, but it is a good method for producing a
large number of points for the classical V-g curve of a system without
automatic controls. The KE-method also sorts the data for plotting.
A plot request for one curve gives all of the reduced frequencies for
a mode whereas a similar request in the K-method gives all of the modes
at one k value. Use of the alternative method for the specification
of k is used to produce well-behaved V-g curves for the KE-method.
The PK-method
The PK-method produces results only at
the velocities of interest to the analyst. The PK-method treats the
aerodynamic matrices as real frequency dependent springs and dampers.
A frequency is estimated, and the eigenvalues are found. From an eigenvalue,
a new frequency is found. The convergence to a consistent root is rapid.
Advantages of the method are that it permits control systems analysis
and that the damping values obtained at subcritical flutter conditions
appear to be more representative of the physical damping. Another advantage
occurs when the stability at a specified velocity is required since
many fewer eigenvalue analyses are needed to find the behavior at one
velocity. The input data for the PK-method also allows looping, as in
the K-method. The inner loop of the user data is on velocity, with Mach
number and density on the outer loops. Thus, finding the effects of
variations in one or both of the two parameters in one run is possible.
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