Most of the established procedures for analysis of aeroelastic flutter in the development of aircraft are based on frequency domain methods. Proposing new methodologies in this field is always a challenge, because the new methods need to be validated by many experimental procedures. With the interest for new flight control systems and nonlinear behavior of aeroelastic structures, other strategies may be necessary to complete the analysis of such systems. If the aeroelastic model can be written in time domain, using statespace formulation, for instance, then many of the tools used in stability analysis of dynamic systems may be used to help providing an insight into the aeroelastic phenomenon. In this respect, this paper presents a discussion on the use of Gramian matrices to determine conditions of aeroelastic flutter. The main goal of this work is to introduce how observability gramian matrix can be used to identify the system instability. To explain the approach, the theory is outlined and simulations are carried out on two benchmark problems. Results are compared with classical methods to validate the approach and a reduction of computational time is obtained for the second example.
Between various physical phenomena involving fluidstructure interaction, flutter is probably the most representative topic studied in engineering applications such as aircrafts and bridges. The flutter phenomenon is an interaction between structural dynamics and aerodynamics that results in divergent and destructive oscillations of motion [
In 1935, Theodorsen [
These methodologies, which are well established in the research and engineering community, were developed decades ago and have been used in the development of almost all flying commercial and military aircraft.
In this context, this paper proposes an alternative approach for detecting flutter using observability Gramian matrices. The proposed methodology is developed in time domain using statespace representation of the aeroelastic system. The elements of a Gramian matrix are related to the energy of vibration modes and can be seen as an improved observability matrix, introduced by Kalman et al. [
Gramian matrices have been used in the field of control engineering. Their fundamental concepts were proposed by Moore after introducing the balanced reduction for statespace models [
The principal objective of the paper is to show that observability Gramian matrices can be used to detect flutter. The rationale for using this approach is that Gramian matrices contain, in a single index, information about the energy that is transferred from the flux to the structure and then dissipated by any damping mechanism for each flight condition considered in the analysis (flight envelope).
It is shown that the Gramians are sensitive to flutter where energy transferred from the flux in a cycle is larger than the energy dissipated by the damping mechanism.
Part of the procedure to determine the Gramian matrices requires a system defined in time domain, including the aerodynamic forces. If aerodynamic forces are written in terms of reduced frequencies, this can be done using one of the methods such as least square [
The paper illustrate’s the process to obtain the Gramian matrix from an aeroelastic system in time domain. Two numerical examples are used to compare the proposed method with the methods in the literature. The first example is three degrees of freedom typical section airfoil and the second is the AGARD 445.6 wing model developed using finite element method [
The paper shows that it is possible to obtain some advantage in terms of computation time using the proposed methodology.
Assuming a general aeroelastic model that can be written in the statespace form, according to
That stability could be verified by solving an eigenvalue problem for each discrete airspeed point in the flight envelope and checking if the real part of system eigenvalues is negative values. This can be time consuming, specifically for large dimension systems. To overcome this process, the observability Gramian method presented in this paper is based on the solution of a set of linear equations. The next section presents the bases to write the problem in appropriate format.
The concept of observability involves the dynamic matrix
A linear timeinvariant system with
An alternative approach that can be applied for largeorder problems is to use the observability Gramian matrix. The observability Gramian matrix is defined to express quantitative properties of the system, considering it at time
An important property between observability and Gramian matrices is that they share the same Kernel (i.e., the set of all vectors
According to [
Equation (
Using a complex Schur decomposition, the Lyapunov equation can be reshaped as a real linear system of equation and solving it,
The hypothesis introduced in this paper is that observability Gramian matrix contains information which can be used to indicate the amount of energy transferred from the air flow to the structure. In this case, this amount of energy is maximum at the airspeed at which the system becomes unstable.
In order to prove this hypothesis, a Gramian parameter
By computing
The
Gramian parameter used to detect the flutter phenomenon.
In practical implementations the Gramian parameter
The effectiveness of the approach was determined through simulations using two examples. The first was a threedegreeoffreedom typical airfoil section (semichord
Physical and geometric properties of the 2D airfoil.
Aerodynamic semichord 

Airfoil mass 

Plunge frequency 

Pitch frequency 

Control surface rotation frequency 

Air density 

Lag parameters (Roger's method) 

Reduced frequencies 

Figure 

Figure 

Distance between c.e. to c.g. 

Distance between c.e. to c.g. (flap) 

Radius of gyration of the flap referred to 

Radius of gyration of the airfoil referred to 

Elastic center  c.e. 
Center of gravity  c.g. 
Typical section 2D airfoil.
The results of the proposed approach were compared with classical eigenvalues analysis. Figures
Aeroelastic frequency as a function of airspeed (typical section airfoil).
Real part of the eigenvalues as a function of the airspeed (typical section airfoil).
The results of the proposed method are presented in Figure
Gramian parameter computed by the Frobenius norm as a function of the airspeed (typical section airfoil).
The AGARD 445.6 benchmark wing was also used to demonstrate the method. The structural model for the AGARD 445.6 wing was developed using finite element method using the MSC/NASTRAN. The finite element model consisted of plate elements with singlelayer orthotropic material. The model has 231 nodes and 200 elements. Rotation of nodes was neglected, allowing three degrees of freedom per node. See physical properties in Appendix
Aerodynamic and structural matrices were obtained from MSC/NASTRAN program (Aeroelastic Solution 145) for the Mach number 0.50 and
A statespace model was obtained through a MATLAB implementation for which the parameters of lag were chosen as
Aeroelastic frequency as a function of the air density (AGARD wing).
Aeroelastic damping as a function of the air density (AGARD wing).
With the proposed methodology, the Gramian parameter
Gramian parameters computed by the Frobenius norm as a function of the air density (AGARD wing).
Gramian parameters computed by the Frobenius norm as a function of the air density (AGARD wing).
This section shows that it is possible to reduce the computational cost to find the flutter speed using Gramian matrices, especially for industrial applications where the number of nominal and parametric cases of analysis can be very large. Based on the second example (AGARD wing), it is demonstrated that (
According to previous sections, (
Figure
Compute Gramian parameters using observability Gramian matrices obtained from (
Identify the point
Compute crossGramian parameters for points around the maximum.
Comparison of computational time.
Using this procedure, it is possible to obtain some reduction in the computational time to evaluate the system stability. This reduction is represented in Figure
Computational time: pkmethod versus Gramian parameters (AGARD 445.6 wing).
Controllability and observability Gramian matrices have been used extensively in control design and for optimal placement of sensors and actuators in smart structures. They have also been used for solving aeroelastic problems mainly for writing reducedorder models.
This paper has investigated the applicability of observability Gramian matrix to detect the flutter speed in two benchmark structures. It was shown that Gramian matrices represent the transfer of energy from the flow to the structure. That transfer of energy increases when the airspeed is close to the flutter condition. A classical method was used to compare and validate the results obtained by Gramian matrices.
This approach does not compute frequency and damping for the aeroelastic modes. However it allows to determine the aeroelastic modes with largest contribution to the flutter mechanism and the airspeed where the system becomes unstable. Additionally, one advantage is related to the reduction of computational time for analysis when compared to classical pkmethod.
This work is a complementary effort to apply the concepts from control theory in problems involving aeroelasticity in time domain. It can also be used in flight flutter tests using an identification method to obtain the statespace representation instead of identifying aeroelastic frequencies and damping.
An aeroelastic system consisting of structural parameters (mass, damping, and stiffness) subject to the forces from the fluid can be modeled in the Laplace domain using a physical or modal coordinate system. In general, the modal coordinates are used to truncate the system of equations using the eigenvector extracted from structural mass and stiffness matrices. Without loss of generality, both numerical case studies were performed using modal coordinates as shown in the following equations.
The matrices representing the structural parameters are the mass
In this case, the problem that arises from the conversion of (
Equation (
The matrix norms for computing the Gramian parameters are presented as follows.
The
The 2norm is computed based on the largest singular value
The 1norm is the maximum of the columns sums; that is,
This appendix presents complementary information for the AGARD 445.6 wind and more details can be found in [
Figures
AGARD 445.6 wingfirst structural mode.
AGARD 445.6 wingsecond structural mode.
AGARD 445.6 wingthird structural mode.
AGARD 445.6 wingfourth structural mode.
The thickness distribution is governed by the airfoil shape. The material properties used are
There are 10 elements in chordwise and 20 elements in spanwise direction. There are a total of 231 nodes and 200 elements. The boundary conditions of the wing are selected in accordance with the physical model. The root is cantilevered except for the nodes at the leading and trailing edges. Additionally, the rotation around all axis degrees of freedom at all nodes is constrained to zero. In this case, there are 666 degrees of freedom in the model in physical coordinates. This work considered four modes to obtain the model in modal coordinates system.
Table
Reduced frequencies for the case study AGARD wing 445.6.
















The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interests.