research article numerical study of flutter of a two-dimensional...

Hindawi Publishing CorporationISRNMechanical EngineeringVolume 2013, Article ID 127123, 4 pageshttp://dx.doi.org/10.1155/2013/127123

Research ArticleNumerical Study of Flutter of a Two-DimensionalAeroelastic System

Riccy Kurniawan

Department of Mechanical Engineering, Atma Jaya Catholic University of Indonesia, Jakarta 12930, Indonesia

Correspondence should be addressed to Riccy Kurniawan; [email protected]

Received 15 April 2013; Accepted 14 May 2013

Academic Editors: Y.-H. Lin, Y. H. Park, and X. Yang

Copyright © 2013 Riccy Kurniawan. 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.

This paper deals with the problem of the aeroelastic stability of a typical aerofoil section with two degrees of freedom induced by theunsteady aerodynamic loads. A method is presented to model the unsteady lift and pitching moment acting on a two-dimensionaltypical aerofoil section, operating under attached flow conditions in an incompressible flow. Starting from suitable generalisationsand approximations to aerodynamic indicial functions, the unsteady loads due to an arbitrary forcing are represented in a state-space form. From the resulting equations of motion, the flutter speed is computed through stability analysis of a linear state-spacesystem.

1. Introduction

Flutter is the dynamic aeroelasticity phenomenon wherebythe inertia forces can modify the behaviour of a flexiblesystem so that energy is extracted from the incoming flow.The flutter or critical speed 𝑉

𝐹is defined as the lowest air

speed atwhich a given structurewould exhibit sustained, sim-ple harmonic oscillations. 𝑉

𝐹represents the neutral stability

boundary: oscillations are stable at speeds below it, but theybecome divergent above it.

Theodorsen [1] obtained closed-form solution to theproblem of an unsteady aerodynamic load on an oscillatingaerofoil. This approach assumed the harmonic oscillationsin inviscid and incompressible flow subject to small dis-turbances. Wagner [2] obtained a solution for the so-calledindicial lift on a thin aerofoil undergoing a transient stepchange in angle of attack in an incompressible flow. Theindicial lift response makes a useful starting point for thedevelopment of a general time domain unsteady aerodynam-ics theory. A practical way to tackle the indicial responsemethod is through a state-space formulation in the timedomain, as proposed, for instance, by Leishman and Nguyen[3].

The main objective of this paper is to investigate theaeroelastic stability of a typical aerofoil section with two

degrees of freedom induced by the unsteady aerodynamicloads defined by the Leishman’s state-space model.

2. Aeroelastic Model Formulation

The mechanical model under investigation is a two-dimensional typical aerofoil section in a horizontal flow ofundisturbed speed 𝑉, as shown in Figure 1. Its motion isdefined by two independent degrees of freedom, which areselected to be the vertical displacement (plunge), ℎ, positivedown, and the rotation (pitch), 𝛼. The structural behaviour ismodelled by means of linear bending and torsional springs,which are attached at the elastic axis of the typical aerofoilsection. The springs in the typical aerofoil section can beseen as the restoring forces that the rest of the structureapplies on the section.

The equations of motion for the typical aerofoil sectionhave been derived inmany textbooks of aeroelasticity and canbe expressed in nondimensional form as

𝑟2

𝛼�� +

𝑥𝛼

𝑏ℎ + 𝜔2

𝛼𝑟2

𝛼𝛼 = 2

𝜅

𝜋(𝑉

𝑏)

2

𝐶𝑀 (𝑡)

𝑥𝛼�� +

1

𝑏ℎ +

𝜔2

ℎ

𝑏ℎ =

𝜅

𝜋(𝑉

𝑏)

2

𝐶𝐿 (𝑡) ,

(1)

2 ISRNMechanical Engineering

𝛼

h

Elastic axis

Center ofmass

b

Midchord

ba bx𝛼

Figure 1: A typical aerofoil section with two degrees of freedom.

where 𝐶𝑀(𝑡) and 𝐶

𝐿(𝑡) denote the coefficients of the aero-

dynamic forces corresponding to pitching moment and lift,respectively. For a general motion, where an aerofoil of chord𝑐 = 2𝑏 is undergoing a combination of pitching and plungingmotion in a flowof steady velocity𝑉,Theodorsen [1] obtainedthe aerodynamic coefficients

𝐶𝑀 (𝑡) = −

𝜋

2𝑉2[(1

8+ 𝑎2) �� − 𝑎𝑏ℎ] + 𝜋 (𝑎 +

1

2)𝐶 (𝑘) 𝛼𝑞𝑠

−𝜋

2𝑉2[𝑉(

1

2− 𝑎) ��] ,

(2)

𝐶𝐿 (𝑡) =

𝜋𝑏

𝑉2(𝑉�� + ℎ − 𝑏𝑎��) + 2𝜋𝐶 (𝑘) 𝛼𝑞𝑠. (3)

The first term in (2) and (3) is the noncirculatory orapparent mass part, which results from the flow accelera-tion effect. The second group of terms is the circulatorycomponents arising from the creation of circulation aboutthe aerofoil. Theodorsen’s function 𝐶(𝑘) = 𝐹(𝑘) + 𝑖𝐺(𝑘) isa complex-valued transfer function which depends on thereduced frequency 𝑘, where

𝑘 =𝜔𝑏

𝑉(4)

and 𝛼𝑞𝑠represents a quasisteady aerofoil angle of attack; that

is,

𝛼𝑞𝑠=ℎ

𝑉+ 𝛼 + 𝑏 (

1

2− 𝛼)

��

𝑉. (5)

The indicial response method is the response of theaerodynamic flowfield to a step change in a set of definedboundary conditions such as a step change in aerofoil angleof attack, in pitch rate about some axis, or in a control surfacedeflection (such as a tab of flap). If the indicial aerodynamicresponses can be determined, then the unsteady aerodynamicloads due to arbitrary changes in angle of attack can beobtained through the superposition of indicial aerodynamicresponses using the Duhamel’s integral.

Assuming two-dimensional incompressible potentialflow over a thin aerofoil, the circulatory terms in (2) and (3)can be written as

𝐶 (𝑘) 𝛼𝑞𝑠 = 𝛼𝑞𝑠 (0) 𝜑𝑤 (𝑠) + ∫

𝑠

0

𝑑𝛼𝑞𝑠

𝑑𝑡𝜑𝑤 (𝑠 − 𝑡) 𝑑𝑡,

(6)

where 𝑠 is the nondimensional time, given by

𝑠 =1

𝑏∫

𝑡

0

𝑉𝑑𝑡, (7)

where 𝜑𝑤

is Wagner’s function, which accounts for theinfluence of the shed wake, as does Theodorsen’s function.In fact, bothWagner’s andTheodorsen’s functions represent aFourier transformpair.Wagner’s function is known exactly interms of Bessel functions (see [2] for details), but for practicalimplementation it is useful to represent it approximately. Oneof the most useful expressions is an exponential of the form

𝜑𝑤 (𝑠) ≈ 1 − 𝐴1𝑒

−𝑏1𝑠− 𝐴2𝑒−𝑏2𝑠. (8)

One exponential approximation is given by Jones [4] as

𝜑𝑤 (𝑠) ≈ 1 − 0.165𝑒

−0.0455𝑠− 0.335𝑒

−0.3𝑠. (9)

The state-space equations describing the unsteady aero-dynamics of the typical aerofoil section with two degrees offreedom can be obtained by direct application of Laplacetransforms to the indicial response as

[��1

��2

] = [

[

0 1

−𝑏1𝑏2(𝑉

𝑏)

2

− (𝑏1+ 𝑏2)𝑉

𝑏

]

]

[𝑧1

𝑧2

] + [0

1] 𝛼𝑞𝑠

(10)

with the outputs

𝐶 (𝑘) 𝛼𝑞𝑠 = [𝑏1𝑏2

2(𝑉

𝑏)

2

(𝐴1𝑏1+ 𝐴2𝑏2) (𝑉

𝑏)] [

𝑧1

𝑧2

] +1

2𝛼𝑞𝑠.

(11)

The main benefit of the state-space formulation is thatthe equations can be appended to the equations of motiondirectly, very useful in aeroservoelastic analysis. Further-more, it permits the straightforward addition ofmore featuresto the model, such as gust response and compressibility.

The indicial approach and the state-space formulationlead to a dynamic matrix that governs the behaviour of thesystem and enables future prediction. The analysis of flutterin this case is straightforward and it can be performed inthe frequency domain, since the eigenvalues of the dynamicmatrix directly determine the stability of the system. If, for agiven velocity, any of the eigenvalues has a zero real part, thesystem is neutrally stable, that is, it defines the flutter onset.

3. Results and Discussion

In this section, the stability analysis of the state-space aeroe-lastic equation is presented. The results have been validatedagainst published and experimental results.

ISRNMechanical Engineering 3

Table 1: Aeroelastic parameters for the validation.

Case 𝑥𝛼

𝜅 𝑎 𝑟2

𝛼

a 0.2 1/3 −0.4 0.25b 0.2 1/4 −0.2 0.25c 0 1/5 −0.3 0.25d 0.1 1/10 −0.4 0.25

3

2.5

2

1.5

1

0.5

00 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

𝜔h/𝜔𝛼

Present (a)Present (b)Present (c)Present (d)TG (a)TG (b)TG (c)TG (d)

V∗ F=VF//b𝜔𝛼

Figure 2: Comparison of flutter boundaries from Theodorsen andGarrick [5] with the present computations.

3.1. Validation against Published Results. Theodorsen andGarrick [5] presented a graphical solution of the flutterspeed of the two-dimensional aerofoil for the flexure-torsioncase. In order to validate the present model, a flutter speedcomputation is performed with varying combinations ofaeroelastic parameters, as used by Theodorsen and Garrick,as shown in Table 1.

Figure 2 shows the comparison of the flutter margin fromTheodorsen and Garrick’s work with the present compu-tation. In the graph, nondimensional flutter speed 𝑉∗

𝐹is

presented as a function of the frequency ratio 𝜔ℎ/𝜔𝛼. As can

be seen, the present method provides a good agreement withthe published figures only for low frequency ratios. In fact,as the ratio approaches unit value, the actual curve drifts togenerally lower speeds.

This discrepancy is probably due to numerical inaccu-racies in the curves presented in the original work. Zeiler[6] found a number of erroneous plots in the reports ofTheodorsen and Garrick and provided a few corrected plots.In order to verify the validity of Zeiler’s statement, thenumerical computation of the flutter speed is conductedusing the aeroelastic parameters used by Zeiler.

Figure 3 shows some of the results obtained by Zeiler,compared to the figures obtained byTheodorsen and Garrickand those obtained using the present state-space method. Ascan be observed, the agreement with Zeiler is very good,

6

5

4

3

2

1

00 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

𝜔h/𝜔𝛼

State space (a)Zeiler (a)TG (a)State space (b)Zeiler (b)TG (b)State space (c)Zeiler (c)TG (c)State space (d)Zeiler (d)TG (d)

V∗=V/b𝜔𝛼

Figure 3: Comparisons of flutter boundaries from Zeiler [6] andTheodorsen and Garrick [5] with the present computations. Theparameters used are 𝑎 = −0.3, 𝜅 = 0.05, 𝑟2

𝛼= 0.25, 𝑏 = 0.3 (a)

𝑥𝛼= 0 (b) 𝑥

𝛼= 0.05 (c) 𝑥

𝛼= 0.1, and (d) 𝑥

𝛼= 0.2.

whereas Theodorsen and Garrick’s results deviate consid-erably. This confirms the validity of Zeiler’s statement andprovides evidence of the validity of the results obtained here.

3.2. Validation with Experimental Data. An experiment onflutter speed was performed at 5 × 4 Donald Campbell windtunnels. Pitch and plunge are provided by a set of eight linearsprings. Airspeed was gradually increased until the onset offlutter. The parameter values used in the experimental studyare 𝑥𝛼= 0.00064, 𝜅 = 0.0157, 𝑎 = −0.1443, 𝑟

𝛼= 0.4730,

𝑏 = 0.05, 𝜔𝛼= 61.5637, and 𝜔

ℎ= 8.8468.

The nondimensional flutter speed resulting from thepresent computation flutter analysis is 𝑉∗nom = 4.31 and thatfrom the experimental study is 𝑉∗exp = 4.04. The comparisonshows that the value of the experimental flutter speed istherefore 6.26% smaller than the numerical flutter speed.Thisismay be due to the error anduncertainty that iswell acceptedto occur in experimental studies, and which has affected theflutter speed measurement. Nevertheless, the flutter speedobtained in the experiments agrees with the numerical resultsfairly well.

4. Conclusions

A model to determine the flutter onset of a two-dimensionaltypical aerofoil section has been implemented and thenvalidated. A traditional aerodynamic analysis, based onTheodorsen’s theory and Leishman’s state-space model wasused. The validation was performed, firstly, by solvingTheodorsen and Garrick’s problem for the flexure-torsionflutter of a two-dimensional typical aerofoil section. The sta-bility curves obtained are in close agreement with the resultsreported by more recent solutions of the same problem,whereas the original figures from Theodorsen and Garrickare found to be biased, as was previously reported by Zeiler.

4 ISRNMechanical Engineering

Secondly, validation with experimental data was conductedand the results showed a fairly close agreement.

References

[1] T. Theodorsen, “General Theory of Aerodynamics Instabilityand the Mechanism of Flutter,” NACA Report 496, 1934.

[2] H. Wagner, “Uber die Entstehung des dynamischen Auftriebesvon Tragflugeln,” Zietschrift Fur Angewandte Mathematik UndMechanik, vol. 5, no. 1, pp. 17–35, 1925.

[3] J. G. Leishman and K. Q. Nguyen, “State-space representationof unsteady airfoil behavior,” AIAA Journal, vol. 28, no. 5, pp.836–844, 1990.

[4] R. T. Jones, “The unsteady lift of a wing of finite aspect ratio,”NACA Report 681, 1940.

[5] T. Theodorsen and I. E. Garrick, “Mechanism of flutter: atheoretical and experimental investigation of flutter problem,”NACA Report 685, 1938.

[6] T. A. Zeiler, “Results of Theodorsen and Garrick revisited,”Journal of Aircraft, vol. 37, no. 5, pp. 918–920, 2000.

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