development of lma models for airfoils in gusty loads and ... · simulate lma process in a simple...

11th International Conference on Vibration ProblemsZ. Dimitrovováet.al. (eds.)

Lisbon, Portugal, 9–12 September 2013

DEVELOPMENT OF LMA MODELS FOR AIRFOILS IN GUSTYLOADS AND RESPONSE ANALYSIS

J.Venkatramani*1, Sandip Chajjed 2, Sayan Gupta3

1Department of Applied Mechanics, Indian Institute of Technology,Chennai, [email protected]

2Department of Aerospace engineering, Indian Institute of Technology,Chennai, India.Sandip.Chhajed @gmail.com

3 Department of Applied Mechanics, Indian Institute of Technology,Chennai, India.gupta.sayan @gmail.com

Keywords: Aeroelasticity, Gusty loads, LMA process, Experimental studies.

Abstract. Wind loads acting on wind turbine blades are inherently random and are oftenmodelled as a Gaussian process. However, studies carried out through real life wind data mea-surement shows that wind loads possess considerable amount of skewness and kurtosis, therebymaking them non-Gaussian. This study focusses on developing load models that capture’s thenon-Gaussianity through a class of processes known as Laplace driven Moving Average (LMA).Measurements obtained from experiments carried out in a wind tunnel under gusty conditionsare used for this load modelling. Subsequently, Response analysis is carried out to study thepropogation of uncertainties into the response due to the random loading.

J.Venkatramani, Sandip Chajjed, Sayan Gupta

1 INTRODUCTION

Wind turbine blades are airfoil like flexible structures andfail primarily due to aeroelasticflutter. Aeroelastic flutter is a phenomenon where the aerodynamic forces overcome the inertialforces and structural forces and thereby resulting in diverging oscillations[1]. The resultantlarge oscillations pose a threat to the structural integrity of airfoil like structures.

Structural failures can be broadly classified into two types[2]; a) Failure due to extremeloading, resulting in resulting in the response crossing a safe threshold called as failure dueto first passage of time. b) Failure due to gradual loading, leading to progressive failure (i.e.)fatigue damage.

Most of the available literature on deterministic flutter, concentrate on systems with structuralor aerodynamic nonlinearities. Numerical and experimental analysis of limit cycle oscillations(LCO) of an airfoil with nonlinear pitching stiffness has been carried out in [3]. A numericalstudy on flutter, is shown in [4], where they observed the onset of LCO, to be well below thelinear flutter speed and is attributed it to the presence of cubic and bilinear nonlinearities.

An innovative method for solving the second-order coupled differential equations describingthe dynamics of a nonlinear airfoil is provided by [5]. For a detailed review on linear and non-linear flutter of airfoils, see [6]. A novel design of a nonlinear flutter apparatus called NonlinearAeroelastic Test Apparatus (NATA) is available in [7]. The mechanism of experimental setupdeveloped here for this current study is similar to NATA. Growth of transient energy leadingto flutter instability of a non-linearly fluttering airfoil has been experimentally shown in [8].Structural nonlinearities aside, airfoils are set into flutter even by the presence of aerodynamicnonlinearity. Stall flutter is an ideal example for the same.Stall flutter phenomenon as anoutcome of Hopf Bifurcation was established experimentally in [9].

As wind is predominantly random in nature, deterministic studies are insufficient to under-stand the real dynamical scenario of airfoils in the field. Uncertainties present in the loads needsto be modeled and the structural response due to these randomloading needs to be studied. Inrecent times, significant attention has been devoted to analyzing the response of airfoil undergusty flow (or) random flutter. The term random flutter impliesthat flutter is investigated withrandomly fluctuating wind speed. The response of a nonlinearly fluttering airfoil in turbulentflow is investigated numerically in [10, 11]. The response has been analyzed in terms of Proba-bility density function (pdf), and Lyapunov exponent. A bifurcation plot with the deterministicand stochastic bifurcation simulated numerically have been shown and argued that the presenceof longitudinal turbulence lowers the flutter limit and alsosignificantly change the stabilitycharacteristics.

The above literature on stochastic flutter, assume wind loads to be stationary and Gaussianrandom process. On the other hand, statistical analysis of real life cyclonic data show that,wind load distributions exhibit significant amount of skewness and kurtosis. Hence, assumingthe loading to be a Gaussian process, introduces significantamount of uncertainties into thereliability analysis. In [12] stochastic loads on a second-order dynamical system were modelledas a Laplace driven Moving Average process (LMA) and it was demonstrated that such a modelcan fit the observed non-Gaussian characteristics from the observed data. The objective of thisstudy is to model the measured uncertainties in wind loads acting on an airfoil as a LMA processand subsequently study the propogation of these uncertainties into the airfoil response.

The paper is organized as follows. In section2, a brief introduction to Laplace MovingAverage (LMA) is presented. The developed experimentalsetup and its associated physicalparameters, along with the equations used to obtain numerical results are given in section3.

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The results obtained by carrying out a deterministic study from experiments and numericalinvestigation are shown in section4. Section5 highlights the development of LMA models forwind loads and presents the non-Gaussian characteristics of the process. The response of theairfoil for random loading is provided in section6. In section7, we conclude by describing thesalient features of this study and the future work that needsto be carried out.

2 LAPLACE MOVING AVERAGE (LMA)

Estimation of failure probabilities for either of failure mode, as mentioned in section1, canbe evaluated using Rice’s formula [13] of the form

µ(u) =∫

∞

0ẋp

XẊ(u, ẋ)dẋ (1)

whereµ(u) is the mean crossing rate,pXẊ

is the joint-pdf of the process and its instantaneousderivative andu is the threshold level. It is obvious that the crux here lies in the knowledge ofthe joint probability density function (j-pdf) of the process. This is not so easily available fornon-Gaussian processes. For structural systems subjectedto non-Gaussian loads, the responseis a non-Gaussian process. Reliability analysis of such systems require computing the crossingstatistics.

Hence, another type of process called Laplace Moving Average (LMA) can be used to modelthe loadings which exhibit non-Gaussian characteristics.The first four statistical moments(mean,variance, skewness and kurtosis) of the marginal distribution is used to characterize thisprocess. LetX(t) be the load, written as a continous time moving average as

X(t) =∫

∞

−∞

f(t− x)d∧(x) (2)

and its characteristic function is given by

φX(t)(ν) = exp(∫

∞

−∞

iζνf(x)−1

νlog(1− iµνf(x) +

σ2f 2(x)ν2

2)dx) (3)

where,f(x) is a kernel function and∧(x) is a random process with stationary and independentincrements having a generalized asymmetric Laplace distribution. The process generated usingEq. 2 is essentially stationary and ergodic. It is to be mentioned here that, if∧(x) is chosen tobe a Brownian motion, the resulting process is Gaussian. Fora LMA process defined by Eq. 2,it can be shown that the first four moments and power spectral density respectively can be givenby,

E[X(t)] = (ζ +µ

ν)∫

∞

−∞

f(x)dx, (4)

V (X(t) =σ2 + µ2

ν, (5)

s = µν0.52µ2 + 3σ2

(µ2 + σ2)1.5

∫∞

−∞

f(x)3dx, (6)

κ = 3ν(2−σ4

(µ2 + σ2)2)∫

∞

−∞

f(x)4dx, (7)

S(ω) = (σ2 + µ2

ν)1

2πFf(ω)2 (8)

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Here,F denotes the Fourier transform andE[X(t)] is the mean,V (X(t) is the variance,s is theskewness andκ is the kurtosis of the process. The choice of kernel functionf(x) is depends onthe PSD and one can choose infinite such functions. However, if one considers only symmetrickernels, a unique kernel can be obtained from Eq. 8. A Laplacemotion can be written as asubordinated Brownian motion as given below,

∧ (x) = ζx+ µΓ(x) + σB(Γ(x)) (9)

where,µ is the asymmetry parameter,ζ is the drift of the process,σ is the scale andΓ(s)represents the independent Gamma process with shape parameter ν. Eq. 9 can be used tosimulate LMA process in a simple and straight forward way. Further details regarding Laplacemotion, parameter extraction and simulation algorithm forLMA process can be found in [12]and [14].

3 EXPERIMENTAL SETUP

A NACA 0012 airfoil made of teak wood is used for this study. The airfoil has a span of 900mm and a chord of 190 mm and is mounted horizontally. The airfoil is mounted horizontallyon its elastic axis, which is located at quarter chord from the leading edge. Figure 1a shows thephotograph of the developed experimental flutter setup. Thesetup has a working mechanismsimilar to that of NATA described in [7]. In this experimental study, the airfoil is consideredrigid and a pair of spring mounted circular discs, with bearings, is used to obtain the pitchmotion of airfoil and a spring mounted transverse aluminiumcarriage, provides the requiredplunge degree of freedom.

The physical parameters associated with the setup and airfoil are provided in table1. A pairof holders were developed to give us the facility of flexible mounting of airfoil at any pointacross its chord. The setup can be used to analyze nonlinear problems, by either using springswith nonlinear stiffnesses in plunge or by changing the shape of disc in pitch.

The setup is placed in a open section,closed loop wind tunnel, here at IIT Madras. Figure 1bshows the photograph of closed wind tunnel. The tunnel has a nozzle of 1600 mm diameter witha 6:1:1 contraction ratio and a collector with 1900 mm diameter. The open distance betweennozzle and collector is about 2000 mm. The tunnel is capable of achieving a maximum windspeed of 45 m/s. Owing to the open working section, the wind tunnel has a slightly higherturbulence intensity of about 1%.

The equations describing the dynamics of the airfoil is provided in [5]. They are as follows ;

ǫ′′ + xαα′′ + 2ζǫ

ω

Uǫ′ + (

ω

U)2(ǫ+ βǫǫ

3) = −1

πµCL(τ) (10)

xαrα2

ǫ′′ + α′′ + 2ζ

Uα′ +

1

U2(α + βαα

3) =2

πµrα2CM(τ). (11)

Here,ǫ = h/b is the non-dimensional heave displacement,α is the pitch angle,m is the totalmass of the frame and airfoil per unit span,rα is the radius of gyration about the elastic axis ofthe total pitching assembly,ζǫ andζα are the damping ratios in plunge and pitch respectively,βǫ is the heaving stiffness co-efficient,βα is the pitching stiffness,ahb denotes the distance ofthe elastic axis from the mid chord andxαb is the distance of the center of mass from the elasticaxis. U or Und is the non-dimensional stream velocity given byU = v/(bωα) ω = (ωǫ/ωα),where,ωǫ andωα are respectively the natural frequencies of the uncoupled plunging and pitch-ing modes andτ = vt/b is the non-dimensional time. The non-homogeneous termsCL(τ) and

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CM(τ) represent the forcing terms and are usually represented as aset of coupled second orderdifferential equations which are functions ofα andǫ and its expressions are available in [5].

S.no Parameter Value1 Total mass in plunge 11.8 Kg2 Total moment of inertia in pitch 0.0886kgm2

3 Radius of gyration 0.5784 Distance between elastic axis and centre of mass 0.335 Damping ratio in plunge 0.156 Damping ratio in pitch 0.057 Coefficient of nonlinearity in heave 5.258 Linear stiffness in plunge 2000 N/m9 Linear pitch stiffness 0.90 Nm/rad

Table 1: Physical parameters of the experimental setup

(a) (b)

Figure 1: (a) Photograph of setup (b) Photograph of wind tunnel

The response measurement is carried out using a pair of lasersensor’s having a range of 600mm and connected to a HBM DAQ. The sampling frequency is takento be 1000 Hz.

4 DETERMINISTIC RESULTS

Tests were initially carried out in a deterministic sense, where the experimental measurementand numerical prediction of response were compared. Initial tests are performed with zerowind velocity, to measure the damping in pitch and plunge mode, by logarthmic decreamenttechnique and these are listed in table 1.

As mentioned in table 1, pitch is linear due to using a circular disc, whereas plunge is non-linear. A curve fitting technique is used to characterize thenonlinearity and is shown in figure2a & 2b.

In the absence of initial perturbations, the onset of flutterLCO was at 15 m/s (Und = 51)and the same was predicted numerically. A bifurcation curveshowing the change of responseof the system with wind speed is shown in figure 3a, for both numerical and experimental

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0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

0

5

10

15

20

25

30

35

40

Deflection (m)

Lo

ad

(N

)

Fitted curve

Data points

(a)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Angle (rad)

Mo

me

nt (N

m)

Data points

(b)

Figure 2: (a) Load vs deflection for plunge spring (b) Moment vs angle for pitch spring

studies. A good match is observed indicating that the numerical model accurately representsthe experimental set up. The numerically simulated phase portrait of the airfoil post bifurcationis shown in figure 3b, indicating limit cycle oscillations (LCO).

0 10 20 30 40 50 60

−0.2

−0.1

0

0.1

0.2

0.3

Non dimensional windspeed (U)No

n d

ime

nsio

na

l a

mp

litu

de

experimental max

experimental min

numerical max

numerical min

(a)

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2−6

−4

−2

0

2

4

6x 10

−3

Plunge amplitude

Plu

ng

e v

elo

city

(b)

Figure 3: (a) Experimental and Numerical bifurcation in plunge mode (b) Phase portrait at 15m/s

5 MODELLING GUSTY WIND LOADS

Uncertainties are introduced in the wind loads by placing a cylindrical pipe vertically on thenozzle cross-section. The pipe has a length of 1600 mm and 3cmdiameter and acts as a bluffbody. Velocity measurements are taken approximately at 500mm from the pipe. Windspeedwas measured using a Delta HD 4V3 TS3 air velocity sensor. Thesensor has a measurementrange of 0-40m/s and a sensitivity of 0.1m/s and is connectedto a NI DAQ. Sampling frequencyis 1000 Hz. The measurements are taken for 300 seconds and is shown in figure 4a. Themeasured velocity has a mean value of 14.2 m/s, variance is 0.025, skewness is -0.35, kurtosisis 3.7.

The existence of skewness and excess kurtosis is seen in figure 4b. Where a histogram of themeasured velocity profile is given. Using WAFO [15], the PSD,kernel and LMA parametersare obtained and shown below in figure 5a. and figure 5b.

6


The LMA parameter’s obtained areν = 13.75, µ = 0.227, σ = 0.7 and ζ (drift)= 13.75.Using Eq. 2 and Eq. 9, the wind load X(t) can be written as a Laplace Moving Average model.

0 50 100 150 200 250 30013.95

14

14.05

14.1

14.15

14.2

14.25

14.3

14.35

14.4

Time(sec)

Ve

locity (

m/s

)

(a)

13.95 14 14.05 14.1 14.15 14.2 14.25 14.3 14.35 14.40

2

4

6

8

10

12x 10

5

Velocity (m/s)

No

. o

f o

ccu

ren

ce

s

(b)

Figure 4: (a) Measured time history of gusty wind load (b) Histogram of the measured velocityshowing non-Gaussian features

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2−2

0

2

4

6

8

10

12

t (sec)

f(t)

(a)

0 50 100 150 200 250 300 350 400 450 5000

0.2

0.4

0.6

0.8

1

1.2

1.4x 10

−5

Frequency

S(w

)

(b)

Figure 5: (a) Normalized Kernel f(t) of X(t) (b) Power spectral density of X(t)

In order to carry out analysis of crossing statistics, the measured wind time history is simu-lated to a longer duration. For details regarding simulation algorithm see [12]. The simulatedvelocity time history is presented below in figure 6.

6 RESPONSE ANALYSIS

In the previous section, random wind loads were modelled as aLMA process. The modelledLMA loads are given as input to Eq. 10 and 11 and response is obtained in terms of phase plot,marginal probability density function and joint probability density function of the response. Forrandom response, one cannot identify the limit cycle amplitudes as there are many cycles withdifferent amplitudes and one needs to look into alternativemetrics to capture the deterministiccharacteristics of the bifurcation. Rather, we now look into the structure of the probabiltydistribution of the response. We consider two cases (a) whenthe non dimensional velocity isbefore the bifruaction point in the deterministic case and (b) a non dimensional velocity which

7


0 500 1000 1500 200010

11

12

13

14

15

16

17

18

time (s)U

(m/s

)

Figure 6: Simulated wind time history

corresponds to post bifurcation in the deterministic case.Figure 7 shows the correspondingphase plot of heave forUm = 50, whereUm is the non dimensional mean wind speed.

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2−8

−6

−4

−2

0

2

4

6

8x 10

−3

Heave amplitude

He

ave

ve

locity

Figure 7: Phase plot forUm= 50

The probability distribution of the response due to LMA loads are shown in figures 8a and8b. It is found that at flutter speed, the plunge response pdf undergoes a qualitative change inshape, from gaussian like unimodal to a bimodal structure, for mean wind speeds before andafter a critical value. This change in response marginal probability density function indicates apossible stochastic bifurcation or P-type bifurcation. For further details, see [10, 11].

The associated joint pdf of the heave response for speeds less than and greater than criticalspeed are shown in figures 9a and 9b.

7 CONCLUDING REMARKS

The problem of modelling the gusty loads, acting on an airfoil as a LMA process, has beencarried out. This enables the non-Gaussian features like skewness and kurtosis, of the marginaldistributions to be retained. Since, direct application ofRice’s formula is impossible undernon-Gaussian loading, crossing intensities of the response is alternatively to be estimated bymodelling the gusty loads as a LMA process. Response analysis in terms of phase plot, marginalpdf and joint pdf of the response are provided.

A qualitative change in the marginal pdf and joint pdf is observed by increasing the meanwind speed close to the deterministic bifurcation speed. This probably is an indication of P-type bifurcation. Rigorous study to characterize the bifurcations require estimation of largestLyapunov exponent and is currently in progress. Futher the effect of non-Gaussian features like

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−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.10

50

100

150

200

250

Ynd

pYY

(a)

−0.2 −0.1 0 0.1 0.2 0.30

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Ynd

pYY

(b)

Figure 8: (a) Marginal pdf of heave forUm = 47 (b) Marginal pdf of heave forUm = 50

−0.1−0.05

00.05

0.1

−4−2

02

4

x 10−3

0

0.5

1

1.5

2x 10

5

Plunge

Estimated Probability Density Function

plunge velocity

de

nsity

(a)

−0.4−0.3

−0.2−0.1

00.1

0.20.3

−0.01−0.005

00.005

0.010

200

400

600

800

1000

Plunge

Estimated Probability Density Function

Plunge velocity

de

nsity

(b)

Figure 9: (a) Joint pdf of heave and heave rate forUm = 47 (b) Joint pdf of heave and heave ratefor Um = 50

skewness and kurtosis into the behavioral response of the airfoil is to be studied experimentallywith appropriate numerical validations.

REFERENCES

[1] Y.C. Fung, An Introduction to the Theory of Aeroelasticity, Wiley, New York, 1955.

[2] Y. Lin, Probabilistic theory of Structural Dynamics. McGraw-Hill Book company, 1967.

[3] Z.C. Yang and L.C. Zhao, Analysis of Limit Cycle Flutter of an Airfoil in IncompressibleFlow. Journal of Sound and Vibrations 123(1), 1-13, 1988.

[4] S.J. Price and H. Alighanbari, The aeroelastic responseof a two-dimensional airfoil withbilinear and cubic structural nonlinearity. Journal of fluids and structures 9, 175-193, 1995.

[5] B. H. K. Lee and L.Y.Jiang, Flutter of an airfoil with cubic restoring force. Journal offluids and structures 13, 75-101, 1999.

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[6] B.H.K. Lee, S.J. Price, Y.S. Wong, Nonlinear aeroelastic analysis of airfoils: bifurcationsand chaos. Progress in Aerospace sciences, 35, 205-334, 1999.

[7] T. O’Neil., T.W. Strganac, Aeroelastic response of a rigid wing supported by nonlinearsprings. Journal of Aircraft 35, 616-622, 1998.

[8] M. Schwartz, S. Manzoor, P. Hemon and E. de Langre, By-pass transition to airfoil flutterby transient growth due to gust impulse. Journal of Fluids and Structures 25, 1272-1281,2009.

[9] G. Dimitriadis and J. Li, Bifurcation Behavior of Airfoil Undergoing Stall Flutter Oscilla-tions in Low-Speed Wind Tunnel. AIAA Journal Vol. 47, No. 11,November 2009.

[10] D. Poirel, S.J Price, Structurally Nonlinear Fluttering Airfoil in Turbulent Flow. AIAAJournal Vol. 39, No. 10, 2001.

[11] D. Poirel, S.J Price, Bifurcation characteristics of atwo-dimensional structurally nonlinearairfoil in turbulent flow. Nonlinear Dynamics. 48: 423-435,2007.

[12] T. Galiter, S. Gupta and I. Rychlik, Crossings of secondorder response processes subjectedto LMA loadings. Journal of Probability and statistics. doi:10.1155/2010/752452, 2010.

[13] S. Rice, Mathematical Analysis of Random Noise. Selected papers on Noise and Stochas-tic Processes. Dover Publications, New York, 1954.

[14] Jithin Jith, Estimation of crossing statistics of second order response of structures sub-jected to LMA loadings. Dual Degree thesis, Department of Applied Mechanics, IndianInstitute of Technology Madras. May 2012.

[15] P.A. Brodtkorb, P. Johannesson, G. Lindgren, I. Rychlik, J.Ryden, E. Sjoe, WAFO - AMatlab toolbox for analysis of random waves and loads. In Proceedings of the 10th Inter-national Offshore and Polar Engineering Conference Seattle. 3:343-350, 2000.

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INTRODUCTIONLAPLACE MOVING AVERAGE (LMA)EXPERIMENTAL SETUPDETERMINISTIC RESULTSMODELLING GUSTY WIND LOADSRESPONSE ANALYSISCONCLUDING REMARKS

development of lma models for airfoils in gusty loads and ... · simulate lma process in a simple...

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