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International Journal of Aerospace Engineering

Volume 2016, Article ID 9212364, 10 pages

http://dx.doi.org/10.1155/2016/9212364

## Subsonic Flutter of Cantilever Rectangular PC Plate Structure

The Scientific and Technical Research Council of Turkey, Defense Industries Research and Development Institute (TÜBİTAK-SAGE), Mamak, 06261 Ankara, Turkey

Received 20 January 2016; Accepted 23 May 2016

Academic Editor: Hamid M. Lankarani

Copyright © 2016 Kemal Yaman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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 (*-method* and *-method*) of ZAERO^{©} is also done. It is seen that the *-method* gives a closer result than the other one. However, -method results are on a conservative side and it is better to use conservative results, namely, -method results. Even if the modal analysis results are used for the flutter analysis for this simple structure, a modal test should be conducted in order to validate the modal analysis results to have accurate flutter analysis results for more complicated structures.

#### 1. Introduction

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 [1].

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 [2]. Design of a new air vehicle or a new external store for an existing air vehicle is a complicated design task. That design task gets more complicated since more complex systems are desired by the consumers as a result of rapid progress in the engineering technology. It is also expected to decrease the design cost of these new products since the cost is always an important design consideration. Another important design consideration is the time. There is no infinite time for research and development of a new product. MIL-HDBK-1763 “Aircraft/Stores Compatibility: Systems Engineering Data Requirements and Test Procedures” explains how to certify a newly developed military aircraft or a new external store for an existing combat aircraft. Some of the tests designated in this standard are expensive and time-consuming, like ground vibration testing (GVT) and flutter test. Therefore, it is expected to decrease time and expenses consumed for these tests. The main objective of this work is to determine degree of accuracy of flutter analysis results of plate-like structures in incompressible flow compared to wind tunnel flutter test results and also compare the *-method* and *-method* solution methods of ZAERO^{©}. It is also objected to compare the different modal analysis results, in which different structural boundary conditions are used, with the modal test result of the plate-like structure. More accurate flutter analyses are going to decrease the number of flutter tests.

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 [3] and Bairstow and Fage [4] in 1916 about the antisymmetrical flutter of a Handley Page bomber. In 1918, Blasius [5] started to make some calculations for the Albatros D3 biplane due to the failure of the lower wing of that plane. The development of the flutter analysis is increased after the development of nonstationary airfoil theory by Kutta and Joukowsky.

The torsion flutter was first found by Glauret in 1929. It is discussed in detail by Smilg [6]. Several types of single degree of freedom flutter involving control surfaces at both subsonic and supersonic speeds have been found [7, 8]. Pure bending flutter of a cantilever swept wing occurs if the wing is heavier than the surrounding air and has a sufficiently large sweep angle [9]. The bending torsion in an incompressible fluid has been studied by Greidanus [10]. Dugundji [11] searched for panel flutter and the rate of damping.

Dowell [12, 13] investigated the two- and three-dimensional plate undergoing cyclic oscillations and aeroelastic instability. Cantilever beam with tip loads having an arbitrary cross section is discussed by Kosmatka [14] using a power series solution technique for the out-of-plane flexure and torsion case. Subsonic flight and supersonic flight are an ordinary event nowadays. Hypersonic flights become more and more popular due to increased needs. As a result, aeroelastic analyses become a more important part of the design of a new aircraft or an external store for an existing aircraft.

Libo et al. [15] designed a wind tunnel test model for the flutter analysis. The model was used in a complete flutter certification procedure. GVT, model updating, and flutter analysis were all done for this model. - solution method was used in flutter analysis.

Neal et al. [16] worked on the design and wind tunnel analysis of a fully adaptive aircraft configuration. The goal of the study was to determine the effect of the sweep, span extension, and tail extension on the aerodynamic characteristics of the model.

Omar and Kurban [17] designed a free-wing unmanned aerial vehicle model and tested it in low speed closed circuit wind tunnel to see the effect of the angle of attack and Reynold’s number.

Samikkannu [18] studied the details of fabrication and ground and wind tunnel testing of a scaled aeroelastic model of T-Tail with a flexible fuselage. Composite materials were used to obtain the required dynamics for the model during the GVT. Wind tunnel test was done in order to see the flutter characteristics of the model.

Strand and Levinsky [19] conducted wind tunnel tests for a free-wing tilt-propeller V/STOL airplane model in order to see aerodynamic characteristics of the model airplane. Lift and drag curves of the airplane have been obtained as a function of propeller tilt angle and thrust coefficient.

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.

#### 2. Methods

##### 2.1. Aeroelastic Stability Equations

The equation of motion of an aeroelastic system can be stated as follows [20]: consists of two parts: Combining (1) and (2) gives If is nonlinear with respect to , flutter analysis is performed by a time-marching procedure solving the following equation: with initial conditions and .

Amplitude linearization assumption converts (4) into an eigenvalue problem for flutter analysis. In this case, the aerodynamic feedback is related to the structural deformation by means of the following convolution integral: where represents the aerodynamic transfer function and is defined as: .

The Laplace domain counterpart of (5) is simply Equation (4) now can readily be transformed into the Laplace domain and results in an eigenvalue problem in terms of given as follows:

###### 2.1.1. Modal Reduction Approach

Solving (7) directly is computationally costly since the FE model of an aircraft contains higher DoF. Therefore, the mass and stiffness matrices are very large in size. Modal reduction approach is used to solve this problem. Structural deformation is expressed in terms of modal coordinates as follows:Substituting (8) into (7) and premultiplying (7) with yield the following flutter equation: Equation (9) can be written as follows:where , , and is the generalized aerodynamic force matrix.

The modal reduction approach provided reduces the size of the eigenvalue problem. Solving this equation is easier than solving (7). Equation (10) is the classical flutter matrix equation.

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.

###### 2.1.2. Unified AIC of ZAERO^{©}

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 ZAERO^{©} uses the panel method which is based on the linearized potential flow theory to solve the integral equations.

###### 2.1.3. Flutter Solution Techniques

*(**1) k-Method*. The classical flutter matrix equation derived in Section 2.1.1 is given as

Unsteady aerodynamics methods are used by ZAERO^{©} to formulate aerodynamic transfer function in frequency domain (-domain): The frequency domain counterpart of the classical flutter matrix equation can be obtained as follows: If we add artificial structural damping to (14), the -method flutter equation is obtained as follows: is the added artificial structural damping.

*(**2) **-Method*. Assume an analytic function in the form of in the domain of and . can be expanded along the imaginary axis (i.e., ) for small by means of damping perturbation method: If is analytic, it must satisfy the Cauchy-Riemann equations such that Combining (17) and (18) yields the following general equation: Thus, the term can be replaced bySubstituting (19) into (20) yields the approximated -domain solution of in terms of for small : Substituting (21) into (12) yields the -method equation as follows:

##### 2.2. Flutter Certification Procedures

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 1 emphasizes the verification and validation steps for an aeroelastic aircraft model.