c
° TU Delft, The Netherlands, 2006
NONLINEAR AEROELASTIC STUDY OF STALL INDUCED
OSCILLATION IN A SYMMETRIC AIRFOIL(ECCOMAS
CFD 2006)
Sunetra Sarkar∗ and Hester Bijl†
∗Delft University of Technology, Aerospace Engineering Kluyverweg 2, 2629 HT, The Netherlands
e-mail: S.Sarkar@tudelft.nl
web page: http://www.lr.tudelft.nl/aerodynamics †Delft University of Technology, Faculty L&R, Kluyverweg 1, 2629 HS Delft, The Netherlands
e-mail: H.Bijl@tudelft.nl
Key words: dynamic stall, aeroelastic stability, nonlinear behavior
Abstract. In this paper the aeroelastic stability of a wind turbine rotor in the dynamic stall regime is investigated. Increased flexibility of modern turbine blades makes them more susceptible to aeroelastic instabilities. Complex oscillation modes like flap/lead-lag are of particular concern, which give way to potential structural damage. We study the stall in-duced oscillations in pitching direction and in combined flapwise-leadlag wise directions. The aerodynamic loads acting on the rotor body in the stall regime are nonlinear. We con-sider a wide ranging parametric variation and investigate their effect on the aeroelastic in-stability and overall nonlinear dynamical response of the system. An engineering dynamic stall model (Onera) has been used to calculate the aerodynamic loads. The aerodynamic loads are given in terms of differential equations which are combined with the governing equations of the aeroelastic system; the resulting system of equations are solved by a 4th
order Runge-Kutta method. In the pitching oscillation study we consider the following parameters: nondimensional airspeed, mean angle of attack, initial condition, structural nonlinearity and reduced frequency and amplitude of external forcing. Quasi-periodic and chaotic response have been observed. The second case of flap/edgewise oscillation in the stall regime identifies nondimensional rotational speed of the rotor along with structural stiffnesses and nonlinearity as most important parameters of the self excited system. How-ever, no chaotic response has been obtained. External forcing shows presence of higher harmonics and quasi-harmonics in the response.
1 INTRODUCTION
oscillation modes like edgewise vibration in the limelight. Chaviaropoulos1 reported
re-cent cases of structural damage to modern wind turbine blades, in particular, occurrence of longitudinal cracks which resulted from severe edgewise vibration. This has typically occurred in stall regulated blades of large size. Wind turbine rotors often have to oper-ate at large angles of attack, in the dynamic stall regime. The resulting flow is largely separated and viscous effects are important. The physical process involves growth and evolution of leading edge vortex structures and their subsequent shedding from the body into the near wake. This largely controls the aerodynamic loads on the airfoil. The flow field involves flow transition and large turbulent regions as well. All these effects make the aerodynamic load prediction during a stall flutter problem much more involved than its traditional bending-torsion counterpart.
The main objective of any computational aeroelastic analysis is to define the instability boundary and identifying the system parameters affecting it. Nonlinearity could play an important part, for it not only could influence the stability but also lead to bifurcations
in the dynamical response and chaos. Dunn and Dugundji5 have presented an
analy-sis and experimental validation of aeroelastic instabilities at the nonlinear dynamic stall
regime for a cantilever plate-like wing structure. Tang and Dowell3;4 have studied
flut-ter and forced response of a helicopflut-ter rotor in bending-torsion mode, using a nonlinear aerodynamical model and considering structural nonlinearities. The dynamic stall model used in the analysis is based on a semi-empirical technique called the Onera dynamic
stall model10;11. Lee et al.6 have presented a review of nonlinear aeroelastic studies
fo-cusing on the pitch-plunge oscillation of an airfoil. They have discussed problems with
both structural and aerodynamic nonlinearities. Price and Fragiskatos9 have presented
a nonlinear stall flutter analysis of a symmetric airfoil, using a nonlinear dynamic stall model by Beddoes-Leishman. In their study, the effect of structural nonlinearity was not
considered. Chaviaropoulos et al.1;2have presented a study on stall induced flap and
edge-wise oscillation in a stall regulated rotor. Nonlinear aerodynamic loads in the dynamic stall regime have been calculated by a quasi-steady Onera model as well as Navier-Stokes solvers. Effects of structural nonlinearity have not been studied.
semi-empirical Onera dynamic stall model, also been used in some earlier works. This dynamic stall model compares well with the experimental results and is computationally much cheaper than a Navier-Stokes solver.
2 EQUATIONS OF MOTION
To derive the equations of motion of a two dimensional blade section, we consider a strip of unit span of a symmetric airfoil. A NACA 0012 profile has been selected. Two oscillation cases have been considered: one is an airfoil oscillating in pitch degree-of-freedom, the second is an airfoil oscillating in the directions of its flap and edge. The first case represents the classical stall flutter case of an airfoil which is predominantly a single degree-of-freedom problem where the torsional motion of the blade prevails. The second case is obtained in stall regulated rotors showing sustained oscillation in the flap and edgewise directions.
The equation of motion for the single degree-of-freedom pitching oscillation is given in
nondimensional form as follows7;8,
α00+ α/(U2) + ¯Knl= 2Cm/(πµrα2) + F0sin(k1τ ). (1)
Here, (0) is denoted as derivative with respect to nondimensional time τ = t ¯V /b; b is the
semi-span; ¯V is the relative wind velocity; Cmis the moment coefficient which is calculated
using the Onera dynamic stall model; U is the nondimensional airspeed defined as U = ¯
V /bωα; µ and rα are nondimensional structural parameters; mass ratio µ = m/(πρb2);
ra-dius of gyration rα= Iα/(mb2); F0 and ¯Knlare nondimensional forcing moment amplitude
and structural nonlinear stiffness respectively; k1 is the reduced frequency of oscillation.
The equations of motion for the combined flap/lead-lag flutter case for a stall regulated
rotor have been derived in the nondimensional form as follows1:
" 1 0 0 1 # ( ¯ y00 ¯ z00 ) + k2 " ¯ ω2 y 0 0 1 + ¯ω2 z # ( ¯ y ¯ z ) = 1 2πµ ( CDcos α − CLsin α CDsin α + CLcos α ) + ( 0 ¯ F0sin(k1τ ) ) (2)
Here, ¯y and ¯z are nondimensional edge and flapwise displacements respectively; k is
the nondimensional rotor speed Ωb/ ¯V ; r is the radial distance of the blade section from
rotor root; b is the semi-span of the airfoil; nondimensional stiffness terms are defined as, ¯
ωy = ωy/Ω and ¯ωz = ωz/Ω.
3 ONERA DYNAMIC STALL MODEL
The aerodynamic loads are computed by the Onera dynamic stall model for a Reynolds
number Re > 106. The elastic axis is at the quarter chord point. For both flutter
of structural damping is insignificant compared to aerodynamic damping below flutter boundary8. 00000000000000 00000000000000 00000000000000 11111111111111 11111111111111 11111111111111 U r V z(flap) Ω pitch α y(lead−lag )
Figure 1: Airfoil coordinate system and oscillation degrees-of-freedom.
The Onera model is constructed in the form of differential equations to model the dy-namic stall process. The physical process of dydy-namic stall involves leading and trailing edge vortex development, their separation and shedding into the wake. Vortex growth increases the aerodynamic loads beyond their stall boundary and separation causes a de-cline in the loads. The loading behavior is dependent on the frequency or rate of airfoil movement. The aerodynamic loads by the Onera model are divided into two equations; equation for the inviscid part, modeled with a single phase lag term to capture the aero-dynamic lag due to the formation of vortex structure; the other is for the viscous part which becomes important above the stall angle. The details of the technique are
dis-cussed elsewhere5;10;11. The coefficients of the inviscid part equation are obtained from
steady state experimental results and are dependent on the airfoil profile. However, the viscous part of the equation uses parameters from unsteady experiments; the coefficient values, therefore, are dependent on dynamic parameters like frequency. The coefficients associated with the appropriate force coefficients, determined empirically by parameter identification. The coefficients used in this study have been taken from the earlier works
of Dunn & Dugundji5 for a NACA 0012 airfoil.
The differential equations of the Onera model are combined with the governing equa-tions of the structural system. The resulting system of equaequa-tions are solved in the time
domain using a numerical integration scheme of 4th order Runge-Kutta technique. The
4 PITCHING OSCILLATION
We assume a symmetric NACA 0012 section to study both flutter cases. The influence of the relevant parameters for individual cases are discussed in the following subsections. The aerodynamic loads are computed by the Onera dynamic stall model. The elastic axis is chosen at the quarter chord point. For both flutter cases, structural damping has been neglected. Previous work suggests that the influence of structural damping is insignificant
compared to aerodynamic damping below flutter boundary8.
The equation of motion for this case has been given in Eq. (1). Previous studies by
Price & Fragiskatos8 have not addressed parameters like initial conditions and structural
nonlinearities which could play significant role in the nonlinear aeroelastic system. These issues are studied in the present work. Multiple quasi-periodic routes have been found for the self excited system with different initial conditions. Geometric structural nonlinearity has also been found to modify the bifurcation pattern of the system response.
4.1 Self Excited System: Effect of Mean Angle Of Attack, Geometric
Non-linearity
In the self excited system, U has been considered as the main bifurcation parameter.
F0 is put to zero. First, effect of different mean angles of attack are considered, with
structural nonlinearity zero. Initial perturbations are given around the mean angle. The
resulting response for a mean angle of 4o is plotted in Fig. 2 for different U. The y-axis
of the plot gives the pitch response at the maximum and minimum points, that is, where
the derivative of the response is zero. The initial perturbation for α is 10o. The response
goes to a limit cycle oscillation (LCO) around U = 16. Below this U, the response is damped. Therefore, from a fixed point, the response becomes a fixed orbit, indicating a
supercritical Hopf bifurcation. Next we consider a different mean angle of attack of 2o.
Critical U at which LCO is obtained is 22 as shown in Fig. 3. Comparing this with the
case of 4o mean angle of attack, we surmise that a smaller initial α
m takes a higher U to
reach the LCO. It is also apparent that, as U increases, the mean angle of incidence αm
about which the oscillation occurs or dies down, increases. LCO occurs only when αm
reaches past the stall angle, at a critical value of U (Ucr). Next, a geometric structural
nonlinearity is added to the system in order to see the effect on the dynamic response.
A cubic nonlinear term of 0.05α3 is introduced into Eq. 1 to this effect. In this case, the
LCO occurs even later at U = 28, much beyond the previous two cases. This is shown in
Fig. 4. Thus, cubic structural nonlinearity has made the Ucr value much higher than its
linear counterpart. This could be attributed to the extra stiffness added to the system.
4.2 Self Excited System: Influence Of Initial Condition
Next we present some more bifurcation routes of the self excited system. A mean angle
of incidence around 0.5o is chosen. Initial angular perturbation is 20◦, much larger than
5 10 15 20 25 30 2 4 6 8 10 12 14 16 U α o at α ’ = 0
Figure 2: Bifurcation plot,αm= 4o,αinit = 10◦.
5 10 15 20 25 30 0 2 4 6 8 10 12 14 U α o at α ’ = 0
Figure 3: Bifurcation plot for αm= 2o,αinit = 10◦.
to period-2 at around U = 6 which leads to aperiodic response at U = 9.7, after following a series of period doubling bifurcations. Fig. 5 shows the aperiodic pattern obtained at U = 9.7. In this plot, the attractor in the phase plane looks similar to a quasi-periodic one; the frequency content shown in Fig. 6 reveals incommensurate behavior. When a Poincare section is taken, the Poincare plot shows a closed loop. Also, simulating with a slightly different initial condition does not make the solution diverge. Hence, we conclude that the response is indeed quasi-periodic and not chaotic.
Next, we consider the mean angle of attack to be 5.5o, again a similar quasi-periodic
5 10 15 20 25 30 35 0 2 4 6 8 10 12 14 U α o at α ’ = 0
Figure 4: Bifurcation plot, αm= 4o and ¯Knl= 0.05α3,αinit = 10◦.
−40 −30 −20 −10 0 10 20 30 −6 −4 −2 0 2 4 6 αo α ’(degrees/nd time
Figure 5: αm= 0.5 and U = 9.7,αinit= 20◦, phase plot.
value and with a larger angular perturbation of 25o. For this case, LCO appears around
U = 3; period doubling is observed at U = 4 and quasi-periodic response is seen at U = 5.1. Though we obtain similar bifurcation plots for both the mean angles of attack cases discussed above, the difference between them is that they occur at different ranges
of U. The bifurcation pattern and the quasi-periodic route for αm = 0.5◦ is presented
in Fig. 7. Now we add a geometric nonlinearity and consider αm = 0.5o case again. A
cubic spring stiffness of 0.01α3 defers the quasi-periodic response slightly to around U =
10.1 from 9.7 of the first case of αm = 0.5o. Otherwise, the bifurcation pattern remains
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 5 10 15 20 nondimensional frequency Psd
Figure 6: αm= 0.5 and U = 9.7,αinit = 20◦,frequency content.
2 3 4 5 6 7 8 9 10 −40 −30 −20 −10 0 10 20 30 U α o, when α ′ =0
Figure 7: Bifurcation plot, αm= 0.5o,αinit = 20◦, ¯Knl=0.
as shown in Fig. 8. For this case, period-2 response appears at U value around 9 while no further period doubling is obtained within the range of U up to 30. No chaotic or quasi-periodic response is found within this range.
4.3 Forced Oscillation
A harmonic forcing couple of the form F0sin(k1τ ) is considered. First, we choose,
F0 = 0.002, U = 15, k1 = 0.2,Knl = 0. A chaotic response is observed. We follow the
chaotic path backward by decreasing F0 and the resulting bifurcation plot is presented in
5 10 15 20 25 30 −40 −30 −20 −10 0 10 20 30 U α o, when α ′ =0
Figure 8: Bifurcation plot,αm= 0.5o,αinit = 20◦, ¯Knl= 0.1α3.
when α is maximum and minimum during one oscillation cycle. The movement of both the extrema points are captured in the bifurcation plane. Next, we vary the oscillation
frequency k1 for F0 = 0.002 and U = 15. The bifurcation diagram is presented in Fig. 10.
This case shows an interesting behavior, a period three response. We have varied k1
between 0.2 and 0.4 with an interval of 0.001. The response starts with chaos and remains so at the lower values of k1. As k1increases to 0.22, a period two response is clearly visible.
These again becomes chaotic through a period doubling cascade at k1 = 0.286. At k1 =
0.295, the chaotic response gives way to a period three response. The response remains
so for the rest of the k1 range.
0 0.5 1 1.5 2 x 10−3 −20 −15 −10 −5 0 5 10 15 20 F0 α o, when α ′ =0
0.2 0.25 0.3 0.35 0.4 −10 −5 0 5 10 15 20 k1 α o, max amplitude
Figure 10: k1as a bifurcation parameter, U = 15, F0 = 0.002, αm= 5.5o.
5 FLAP-EDGEWISE OSCILLATION
The combined flap and edgewise oscillation case in stall regulated rotors has not re-ceived much attention in the nonlinear aeroelastic community. Nevertheless, such
oscil-lations could potentially lead to structural damage1. Chaviaropoulos1;2 has presented a
linear stability analysis for such a system, along with time domain analysis results for a self excited system with different nondimensional rotor speeds. No other parameters have been investigated. The present study considers the influence of the following parameters: nondimensional rotor speed k, linear structural stiffnesses and their ratio; effects of initial conditions; geometric structural nonlinearity. Finally a forced response study is presented. The system shows interesting dynamics like super-harmonic and quasi-harmonic response at different forcing frequencies. However, no chaotic route has been found.
The equation of motion is given in the nondimensional form in Eq. (2). We assume, 1
πµ
= 0.02 and pitch angle θ = 18◦, which is beyond the stall angle of attack. These values are
kept constant throughout the simulations. The total angle of attack α is obtained at each time instant by modifying θ for the instantaneous flapping movement. The structural
stiffnesses are, ¯ωy = 7, ¯ωz = 4 and Reynolds number is assumed to be Re > 106, with
incompressible flow conditions.
It has been shown by Chaviaropoulos et al.2 that aeroelastic instability occurs at low
nondimensional rotor speeds (k = Ωb/ ¯V ). The results were predicted using an eigenvalue
based linear stability model as well as time domain viscous solvers. We consider two test values of k to compare with their results. We consider, k = 0.05 and 0.1 with zero initial conditions. For k = 0.05, an unstable response has been predicted by our time domain
Onera model, as was also reported by Chaviaropoulos et al.2. The time response for both
flap and edgewise oscillations are shown in Fig. 11. For k = 0.1, a stable damped response
effect of structural stiffness. 0 50 100 150 200 250 −2 −1 0 1 2 3 τ Flapwise oscillation (a) 0 50 100 150 200 250 −0.5 0 0.5 τ Edgewise oscillation (b)
Figure 11: Unstable response shown for combined flap/lead-lag oscillation case for k = 0.05, 1
πµ = 0.02,
¯
ωy = 7, ¯ωz = 4.
5.1 Self Excited System: Effect Of Initial Conditions & Structural Stiffness
Continuing with the above mentioned values for θ, ¯ωy, ¯ωz and 1
πµ, we investigate the
effect of varying k between 0.05 and 0.2 in steps of 0.001. The initial perturbation is zero. The system response is unstable at k = 0.05, as mentioned earlier but at all other values above it, only stable period-1 solutions are obtained. However, we observe that adding a nonzero initial condition could change the type of response. This has been presented in Fig. 12 for k = 0.06. With an initial perturbation of 0.5 in the flapping direction the response has become unstable. We also study the influence of linear structural stiffnesses in the flap and edgewise directions, both the effect of their ratio as well as individual values. We increase both the stiffnesses in the edge and flapwise direction in such a way that their ratio ω¯z
¯ ωy
1000 200 300 400 500 600 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 τ Flapwise oscillation (a) 100 200 300 400 500 600 −6 −4 −2 0 2 4 τ Flapwise oscillation (b)
Figure 12: Effect of initial condition, flapwise response shown for k = 0.06, 1
πµ = 0.02, ¯ωy = 7, ¯ωz = 4;
(a) zero initial perturbation , (b) initial perturbation of 0.5 in flap.
0 20 40 60 80 100 0 0.02 0.04 0.06 0.08 0.1 0.12 τ Flapwise oscillation (a) 0 20 40 60 80 100 0 0.01 0.02 0.03 0.04 0.05 τ Flapwise oscillation (b)
Figure 13: Effect of structural stiffness, flapwise oscillation shown for k = 0.1, 1
πµ = 0.02; (a) ¯ωy = 7, ¯ωz
= 4, (b)¯ωy = 7 × 2, ¯ωz = 4 × 2.
5.2 Forced Oscillation
case, we saw chaotic responses appear in the otherwise damped system response after adding forcing term to it. Like in the previous case, we once again vary the forcing
amplitude ( ¯F0) and the nondimensional frequency of the forcing (k1). Other parameters
have been kept at ¯ωy = 7, ¯ωz = 4, k = 0.1. Variation of nondimensional frequency k1 of
the sinusoidal forcing shows interesting dynamical behavior. The system shows presence of two frequencies, one is the forcing frequency of oscillation and the other is the frequency of the unforced system. The nondimensional frequency of the unforced system is found
to be 0.7. The influence of this frequency is more predominant at lower values of k1.
We present the time history and frequency content plots of the response for k1 = 0.05 in
Fig. 14. The response shows two main frequencies, 0.05 and 0.7, that is, a higher harmonic of fourteenth order is present. The time history and frequency content plot for a
quasi-periodic response for k1 = 0.15, have been presented in Fig. 15. The frequencies present
are 0.15 and 0.7 which are incommensurate, hence a quasi-periodic behavior. As the
reduced forcing frequency k1 is increased further, the response gradually shows stronger
influence of k1 only. We present the time history for k1 = 0.2 in Fig. 16(a), where the
presence of higher harmonics is not visible. The frequency content of this response, shown in Fig. 16(b), shows a strong peak around 0.2. Though an extremely weak presence of frequency 0.7 is also observed, this is not reflected on the time history of the response.
4500 4600 4700 4800 4900 5000 −2.3 −2.25 −2.2 −2.15 −2.1 −2.05 −2 −1.95 −1.9 −1.85x 10 −3 τ Edgewise oscillation (a) 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1x 10 −3 nondimensional frequency Psd (b)
Figure 14: Higher harmonic response, k = 0.1, k1 = 0.05, F0 = 0.001, ¯ωy = 7, ¯ωz = 4.(a)edgewise
oscillation time history , (b) edgewise oscillation frequency content.
6 CONCLUSIONS
4500 4600 4700 4800 4900 5000 −2.3 −2.2 −2.1 −2 −1.9 −1.8 x 10−3 τ Edgewise oscillation (a) 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1x 10 −3 nondimensional frequency Psd (b)
Figure 15: Quasi-periodic response, k = 0.1, k1= 0.15, F0= 0.001, ¯ωy= 7, ¯ωz= 4.(a)edgewise oscillation
time history , (b) edgewise oscillation frequency content.
4650 4700 4750 4800 4850 −2.3 −2.2 −2.1 −2 −1.9 −1.8 x 10−3 τ Edgewise oscillation (a) 0.2 0.4 0.6 0.8 1 0 1 2 x 10−4 nondimensional frequency Psd (b)
Figure 16: Predominantly single periodic response with k = 0.1, k1 = 0.2, F0 = 0.001, ¯ωy = 7, ¯ωz =
4.(a)edgewise oscillation time history , (b) edgewise oscillation frequency content.
based on experimental results. Comparison of Onera model with experiments are widely reported in the literature.
A direct numerical integration technique of 4th order Runge-Kutta method has been
used. At each time level the technique uses internal time step sizes smaller than the user prescribed value till convergence is obtained. Nevertheless, we have checked the time step size for convergence by comparing the results with those obtained by smaller time steps. A nonlinear aeroelastic analysis has performed with wide ranging parametric varia-tion and interesting bifurcavaria-tion behavior has been observed. The first system of pitching oscillation has shown the presence of strong nonlinear behavior even without structural nonlinearity and external forcing. For example, quasi-periodic orbits have been found in the self excited system without structural nonlinearity. The bifurcation behavior of the overall nonlinear system is influenced by many parameters: the mean angle of attack, initial conditions and structural nonlinearity. For a forced system, the bifurcation param-eters are the forcing frequency and amplitude. Forcing induces chaotic response in the otherwise damped system. Forcing amplitude as a bifurcation parameter shows chaotic routes through a series of period doubling bifurcations. Forcing frequency shows presence of period-3 oscillation, culminating into chaos.
For the second system, flap-edgewise oscillation, not many previous studies have been available. Therefore, we have taken some test cases from the literature and verified our results with those. We have varied system parameters like structural stiffnesses and their ratio, initial conditions and structural nonlinearity. Initial conditions influence the stability behavior significantly. However, varying structural stiffnesses and their ratio in the flap and edgewise directions do not show any nonlinear dynamical behavior of the system. For the forced system, there are some interesting patterns as the forcing frequency is varied. Super-harmonic and quasi-periodic response are observed at lower values of forcing frequency. However, at higher values the response becomes predominantly single periodic.
It should be noted that, direct integration methods are capable of showing only stable fixed orbits of a nonlinear system. In order to capture the unstable orbits of the sys-tem, some frequency domain techniques should be employed. A nonlinear system with geometric structural nonlinearity often shows fold or jump bifurcations which have not been found in the present analysis. It would be interesting to investigate further for the presence of other bifurcation routes by considering a wider variety of initial conditions. It would also be interesting to see the effect of other classical structural nonlinearities like free-play and hysteresis.
References
[1] P.K.Chaviaropoulos, Flap/lead-lag aeroelastic stability of wind turbine blade sec-tions. Wind Energy, 2, 99-112, (1999).
The VISCEL project. Part II: Aeroelastic stability investigations. Wind Energy, 6, 387-403, (2003).
[3] D.M. Tang and E.H.Dowell, Comparison of theory and experiment for nonlinear flutter and stall response of a helicopter blade. Journal of Sound and Vibration, 165(2),251-276 , (1993).
[4] D.M. Tang and E.H.Dowell, Nonlinear aeroelasticity in rotorcraft . Mathematical and Computer modelling, 18,157-184 , (1993).
[5] P. Dunn and J. Dugundji, Nonlinear stall flutter and divergence analysis of can-tilevered graphite/epoxy wings.AIAA Journal,30(1), 153-162, (1992).
[6] B.H.K.Lee, S.J. Price and Y.S.Wong, Nonlinear aeroelastic analysis of airfoils: bifur-cation and chaos. Progress in Aerospace Sciences, 35, 205-334, (1999).
[7] Y.C Fung, An introduction to the theory of aeroelasticity,John Wiley & Sons, Inc., New York, (1955).
[8] G. Fragiskatos, Nonlinear response and instabilities of a two-degree-of-freedom airfoil oscillating in dynamic stall., M.Eng. Masters Thesis, McGill University, (1999). [9] S. Price and G. Fragiskatos, Nonlinear aeroelastic response of a two-degree-of-freedom
airfoil oscillating in dynamic stall. Proceedings of the 7th International Conference on
Flow Induced Vibration, Ed. S. Ziada and T. Staubli, 437-444, (2000).
[10] C.T. Tran and T. Petot, Semi-empirical model for the dynamic stall of airfoils in view of the application to the calculation of responses of a helicopter blade in forward flight. Vertica, 5, 35-53, (1981).
