Flutter characteristics of cantilever rectangular flexible plate structure under incompressible flow regime are investigated by comparing the results of commercial flutter analysis program ZAERO^{©} with wind tunnel tests conducted in Ankara Wind Tunnel (ART). A rectangular polycarbonate (PC) plate, 5 × 125 × 1000 mm in dimension, is used for both numerical and experimental investigations. Analysis and test results are very compatible with each other. A comparison between two different solution methods (
Interaction of aerodynamic, inertial, and elastic forces may result in instabilities. One of the most important instabilities known in aeroelasticity is flutter. Flutter is an aeroelastic instability which involves one flapwise and one torsional degrees of freedom (DoF). Coupling of the torsional structural mode with the flapwise bending mode results in a flutter mode. Torsion of the structure is the result of the aerodynamic forces. The angle of attack is changed by the torsion. As a result of angle of attack change, the aerodynamic lift force is also changed [
The change of angle of attack due to torsion changes the lift in an unfavorable phase with the flapwise bending which results in flutter. Vibrations grow rapidly at flutter speed. Structural damping cannot compensate the negative damping caused by the flutter mode. Flutter is observed above a certain relative wind speed on the structure; this speed is called the critical flutter speed [
In this study, flutter analyses were realized with ZAERO^{©}, commercial aeroelastic analysis software that uses panel method based on linearized potential flow theory. Modal parameters (natural frequencies and mode shapes) required by ZAERO^{©} were obtained from MSC NASTRAN^{©} solver. All structural FE models were constructed in MSC PATRAN^{©}. Finally, all flutter tests were conducted in Ankara Wind Tunnel (ART).
The study of flutter begins with the research of Lanchester [
The torsion flutter was first found by Glauret in 1929. It is discussed in detail by Smilg [
Dowell [
Libo et al. [
Neal et al. [
Omar and Kurban [
Samikkannu [
Strand and Levinsky [
In this study, ZAERO^{©}, flutter analysis software based on linearized potential flow theory and developed by ZONA^{©} Inc., is used for aeroelastic stability analysis. At this point, it is necessary to explain the aeroelastic theory behind the software. Here, some theoretical background is given. In the Nomenclature, the symbols used in equations and corresponding meanings are tabulated.
The equation of motion of an aeroelastic system can be stated as follows [
Amplitude linearization assumption converts (
The Laplace domain counterpart of (
Solving (
The modal reduction approach provided reduces the size of the eigenvalue problem. Solving this equation is easier than solving (
In order to achieve that conversion, it is desired to obtain aerodynamic transfer function.
ZAERO^{©} obtains unsteady aerodynamics methods in the frequency domain by assuming simple harmonic motion. The obtained aerodynamic transfer function is called the Aerodynamic Influence Coefficient (AIC) matrix.
ZONA6 and ZONA7 are unsteady aerodynamics methods incorporated in ZAERO^{©}. ZONA6 generates AIC matrices for subsonic flow regimes, and ZONA7 generates AIC matrices for supersonic flow regimes. One of the fundamental aerodynamic parameters is the reduced frequency and it is defined as
Unsteady aerodynamics methods are used by ZAERO^{©} to formulate aerodynamic transfer function in frequency domain (
Aircraft are flown to their maximum speeds to show that they are structurally safe at those speeds. After the investigation of the flutter phenomena, flutter tests became an important part of the design and modification of the air vehicles. Figure
Actual verification and validation process of aeroelastic aircraft models [
As it can be seen in that figure, flutter certification is a very complicated task. Every step of this certification procedure requires a large amount of work power, time, and money. Flutter certification procedure can be summarized in five steps sequentially as follows:
Determination of the test configurations.
Ground vibration testing (GVT).
FEmodel updating.
Aeroelastic flutter analysis.
Flight flutter tests.
Passenger planes are designed and they are used. Therefore, their flutter certification is done once. However, fighter aircraft are designed and they are capable of carrying different types and numbers of external stores. Those external stores are not used arbitrarily. They are used in a concept of operation for the aircraft. The concept of operation defines the types and number of munitions that the combat aircraft carries and the location of the munitions on the aircraft stores.
All different external store configurations have their own structural and aerodynamic identity. For each of those configurations, it is necessary to rearrange the flutter certification procedure. Determination of the test configurations is very critical since it is the starting point of the flutter test procedure.
After the test configurations are determined, GVT for each configuration is conducted. GVT is necessary to validate and update the mathematical model of the aircraft by using experimentally determined lowfrequency modes of the whole aircraft structure. This mathematical model of the aircraft is used in flutter analysis for reliable flutter estimations.
The results of the GVT are used for FEmodel updating. Validated FE model is used to predict the critical flutter speeds of the aircraft, and then it is necessary to progress carefully in the model updating stage. Due to this reason, the model updating procedure takes up to several weeks. Another reason for model updating to be done as accurately as possible is that the usage of the updated FE model for the flutter calculations enables covering future modifications on the aircraft without any additional GVT. FE model can be updated for smaller modifications on the structure and it can be used for the flutter calculations of the modified aircraft.
Updated FE model is used in aeroelastic analysis in order to obtain information about the flutter behavior of the aircraft. Computer programs specialized for the flutter analysis, for example, ZAERO^{©}, or some finite element analysis programs such as NASTRAN^{©} can be used for flutter analysis. These results determine the safety limit for the flight flutter tests.
Flutter flight tests are the final step of the flutter certification procedure of the aircraft. As a result of the aeroelastic analyses, most critical configurations in terms of flutter are determined. Flight flutter tests are conducted for those critical configurations. As a result of flight flutter test, flight envelope of the aircraft after the modification is determined. Structural excitation during the flutter test is necessary to detect the impending aeroelastic instabilities. For aircrafts, up to 60 Hz excitation is necessary to excite the selected vibration modes. Lower excitation results in lower aerodynamic damping values than the actual damping levels and large scatter in damping values from the response data. Excitation system should be light enough not to change the modal characteristics of the aircraft.
The most effective way to obtain the desired excitation is to use inertia shakers. Aerodynamic force is also a simple way to obtain the excitation force. In that type of excitation, aerodynamic vanes have a small airfoil mounted at the tip of the wing or stabilizer. Atmospheric turbulence is also used for the excitation during the flight flutter tests [
Accelerometer is used in order to obtain the response of the aircraft during the flight flutter test. The location and the number of measurement points should be chosen carefully in order to get good enough data from the measurements. Pulse code modulation (PCM) or digital telemetry is used in order to transfer the measured responses from the aircraft to the ground station [
Within the scope of this study, platelike structure is analyzed and tested for flutter. In the wind tunnel tests, the rectangular polycarbonate (PC) plate with the dimension of 5 × 125 × 1000 mm is used. In Figure
Experimental setup for PC plate structure: (a) side view and (b) front view.
Wind tunnel tests are conducted at Ankara Wind Tunnel (ART). ART is a subsonic wind tunnel which has a maximum speed of 90 m/s. Firstly, wind tunnel test of the rectangular plate is conducted. Plate is fixed to the floor of the wind tunnel with a fixture as shown in Figure
Material properties of PC obtained from the literature and used in FE model of the plate structures are given in Table
Material properties of (5 × 125 × 1000 mm) PC used in FE model.
Property  Value 

Elastic modulus (GPa) 

Poisson ratio 

Density (kg/mm^{3}) 

Mass (kg) 













Modal analyses of the plate structures have been carried out by MSC NASTRAN^{©}. FE models were constructed for 1/10 scaled cantilever beam plate. Fixed boundary condition and modal parameters of the FE model that correlate best with modal test data were used in flutter analyses.
Aeroelastic model of the rectangular plate obtained with ZAERO^{©} is shown in Figure
Aeroelastic model of the rectangular plate.
Aeroelastic model of the rectangular plate has 40 elements in spanwise direction and 5 elements in chordwise direction. The body in the aeroelastic model has 125 mm diameter in its cylindrical section.
Experimental and numerical results and comparison of them are given under this topic.
Firstly, wind tunnel test of the rectangular plate is conducted. Plate is fixed to the floor of the wind tunnel with a fixture as shown in Figure
Strain gauge placement for the rectangular plate test.
(a) Straintime and (b) PSD graph of the rectangular plate test item during the wind tunnel test.
As it can be seen in straintime data, during the flutter occurrence, strain values change in an uncontrolled manner. After flutter observation in test, the wind tunnel is stopped and the strain data goes back to its normal progress.
Three different (green, red, and blue) strain gauges are attached to the same horizontal line (Figure
Wind tunnel flutter test results for the rectangular plate.
Test #  Test configuration  Flutter speed (m/s)  Flutter frequency (Hz) 

1  Rectangular plate  24.89  8.9 
Modal analyses of the plate structures have been carried out by MSC NASTRAN^{©}. It is seen in Figure
The first 10 natural frequencies of the rectangular plate.
Mode number  1  2  3  4  5  6  7  8  9 

Freq. (Hz)  1.1  6.8  14.9  18.9  26.5  37  44.9  61.4  91.9 
First four (a, b, c, d) modes of the rectangular plate.
The first four mode shapes are shown in Figures
Results of the flutter analysis for the rectangular plate indicate a flutter speed between 22.5 m/s and 23.9 m/s and a flutter frequency between 10.4 Hz and 9.7 Hz for the assumed structural damping between 0% and 4% at the third mode as shown in Table
Flutter speed analysis results of the rectangular plate.
Structural damping, 
0.0%  0.5%  1.0%  1.5%  2.0%  2.5%  3.0%  3.5%  4.0% 

Speed (m/s) ( 
22.5  22.6  22.8  23.0  23.2  23.3  23.5  23.7  23.9 
Frequency (Hz) ( 
10.4  10.3  10.2  10.2  10.1  10.0  9.9  9.8  9.7 
Speed (m/s) ( 
23.8  24.0  24.2  24.4  24.6  24.8  25.0  25.2  25.4 
Frequency (Hz) ( 
9.7  9.6  9.5  9.4  9.3  9.2  9.1  9.0  8.9 
In
When the results for the rectangular plate are investigated, it is seen that both of the results are in good agreement with the test result. But
Flutter speed comparison between test and analysis result versus assumed structural damping.
Flutter frequency comparison between test and analysis result versus speed.
Tables
Comparison of wind tunnel flutter test with flutter analysis for the rectangular plate for
ZAERO^{©} ( 
Test  Speed change (m/s)  Freq. change (Hz)  

Assumed struc. damping (%)  Flutter speed (m/s)  Flutter freq. (Hz)  Flutter speed (m/s)  Flutter freq. (Hz)  
0  22.5  10.4  24.9  8.8  9.6  18.2 
0.5  22.6  10.3  24.9  8.8  9.2  17.0 
1.0  22.8  10.2  24.9  8.8  8.4  15.9 
1.5  23.0  10.2  24.9  8.8  7.6  15.9 
2  23.2  10.1  24.9  8.8  6.8  14.7 
2.5  23.3  10.0  24.9  8.8  6.4  13.6 
3  23.5  9.9  24.9  8.8  5.6  12.5 
3.5  23.7  9.8  24.9  8.8  4.8  11.4 
4  23.9  9.7  24.9  8.8  4.0  10.2 
Comparison of wind tunnel flutter test with flutter analysis for the rectangular plate for
ZAERO^{©} ( 
Test  Speed change (m/s)  Freq. change (Hz)  

Assumed struc. damping (%)  Flutter speed (m/s)  Flutter freq. (Hz)  Flutter speed (m/s)  Flutter freq. (Hz)  
0  23.8  9.7  24.9  8.8  4.4  10.2 
0.5  24.0  9.6  24.9  8.8  3.6  9.1 
1.0  24.2  9.5  24.9  8.8  2.8  8.0 
1.5  24.4  9.4  24.9  8.8  2.0  6.8 
2  24.6  9.3  24.9  8.8  1.2  5.7 
2.5  24.8  9.2  24.9  8.8  0.4  4.5 
3  25.0  9.1  24.9  8.8  0.4  3.4 
3.5  25.2  9.0  24.9  8.8  1.2  2.3 
4  25.4  8.9  24.9  8.8  2.0  1.1 
ZAERO^{©} estimates lower flutter speed than the wind tunnel test results. Therefore, it can be said that analysis results are conservative. It is also concluded that results for the
Flutter frequency estimation is better in
As shown in Tables
Generalized mass matrix generated by structural FE model
Generalized stiffness matrix generated by structural FE model
Structural deformation
Second derivative of the structural deformation
Aerodynamic forces applied on the structure
Aerodynamic forces induced by the structural deformation
External forces
Aerodynamic pressure
Reference length
Reference chord length
Velocity of the undisturbed flow
Truncated modal matrix
Generalized coordinates, that is, modal coordinates
Generalized (modal) mass matrix
Generalized (modal) stiffness matrix
Harmonic oscillatory frequency
Structural deformation defined at the aerodynamic boxes
Resultant aerodynamic forces at the aerodynamic boxes due to
Virtual displacement
Virtual displacement
Transient decay rate coefficient.
The author declares that there are no competing interests regarding the publication of this paper.
The author wishes to thank TÜBİTAKSAGE for their motivations, supports of equipment, and workforce in order to carry out this study.