Aeroservoelastic Analysis Using Matlab
The aeroelastic analysis of the wing relies on an accurate
representation of the unsteady aerodynamic forces. Since the analysis is faced with the problem of coupling the
unsteady aerodynamic representation into the structural response model, this
representation also must be in a convenient computational form. The unsteady aerodynamic model is divided
into two groups which are frequency domain and time domain analyses. The aeroelastic stability analysis is
usually carried out in the frequency domain, such as V-g method and p-k
method. However, in modern aircraft
design, the interaction among the structure, aerodynamics and control system
can not be neglected. The control
system exhibits sufficient gain and phase margins. A difficulty in using modern control theory for the design of systems
to control aeroelastic behavior is the requirement of transforming the unsteady
aerodynamic forces, normally provided in the frequency domain, into the time
domain. The usual procedure for the
time invariant state-space form is to approximate the unsteady aerodynamic
force represented in the frequency domain as a ration function in the Laplace
domain. However, this requires quite
amount of computational time and results in increase in the state due to the
augmented aerodynamic states. A new
formulation for the unsteady aerodynamics in time domain is recently
proposed. In structural dynamic
analysis, the dynamic behavior of the complex structures is often reduced to a
few degrees of freedom using normal mode analysis technique. These techniques could be applied to
unsteady aerodynamic model. With the
reduced order models, it might be possible to predict the unsteady aerodynamic
response of the system over a wide reduced frequency range.
We will present a new aeroelastic model using an aerodynamic
eigen formulation. A reduced-order
model will be obtained by expanding the unsteady vortex into a sum of first few
aerodynamic eigenmodes. To account for
the effects of truncated higher modes, a static correction is included in the
model. To get the system matrix with
the real constant, a modal decomposition technique which transforms the
aerodynamic matrix into a canonical modal form is introduced.
The typical section model is used. A model has two degrees of freedom which are plunge and
pitch. Linear and torsion springs at
the elastic axis act to restrain motion in plunge and twist. The mass ratio µ is 20, the static imbalance
=0.2, the radius of gyration 
is
0.5, and the location of the elastic axis aft of midchord 
is
-0.1. The frequency ratio R of
the uncoupled modes is 0.3.
Figure shows the discrete time eigenvalues and continuous time
eigenvalues for the typical section model, respectively. For this analysis, ten vortex elements are
used to model the airfoil. The wake was
modelled using 100 vortex elements, and the length of the wake was taken to be
10 chord lengths. As shown in Figure,
the magnitude of all the discrete time eigenvalues is less than 1. These eigenvalues are mapped into the left
in the continuous time domain, which means that the system is stable.
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(a) Discrete
Time Eigenvalues, (b) Continuous Time Eigenvalues of the Unsteady Aerodynamics
The eigenmode information is used in order to compute the
flutter speed in the typical section model.
For model reduction, total 20 eigenmodes from 100, which have the
nearest continuous time eigenvalues to the origin, are used in the reduced
order model along with the static correction. Figure shows the frequency(
),
damping of the reduced order model against the reduced velocity(
). For comparison, the results of V-g method
are also plotted. In V-g method,
the unsteady aerodynamics formulated by Theodorsen are used. The flutter speed
calculated using V-g method is about 2.0. Note that the frequency and g from V-g method do not have
the physical meaning except at the flutter speed. The value of g in V-g method denotes the damping
required for the harmonic motion. On
the other hand, the value of g obtained from the current method denotes
the real damping of the system at the specified airspeed. That is identical with the flutter speed
using the current formulation.
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Eigen
Formulation
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Eigen
Formulation
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V-g
Method
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V-g
Method
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Frequency and Damping Plots of the Typical Section Model with
the Variation of the Reduced Velocity
Comparing the V-g method with given typical section model, the
current method has a disadvantage in the computational time for aerodynamic
eigenvalues. However, the situation
will be different when the more realistic wing model is used for the
aeroelastic analysis. The current
formulation can also be directly used for the aeroelastic control system
design.
Download
Matlab files : test.m, eign.m, opnfltr.m, rmcomm2.m, d2cmr.m, flutteru.m, fluttrr.m, outlqr.m, vortex1.m