Fatigue Estimation for a Rotating Blade of a Wind Turbine

Transcription

1 Rev. Eerg. Re. Vol.5(00)39-47 Fatigue Estimatio for a Rotatig Blade of a Wid Turbie Istitut de Géie Climatique, Uiversity of Costatie Abstract The fatigue estimatio of a rotatig blade ca be useful for prevetig blade breakage, which is a very frequet problem i the use of wid turbie. The estimatio of fatigue for a rotatig blade must go through several steps of calculatio: amog them the calculatio of mode shapes ad frequecies ad the computatio of displacemets ad stresses actig o the blades. I this work two methods are used i the calculatio of the mode shapes, the first oe is based o the discretizatio of the blade elemet, whereas the secod method is based o the resolutio of the differetial equatio of the blade as a cotiuous system. This later method is more advatageous i determiig mode shapes as well as stresses actig o the blades because it does t require the use of a powerful computer. Oce the stresses are calculated, the fatigue is estimated usig the theory of «cumulative damage i fatigue». This estimatio must be based o the statistical distributio of wid speed for the site i use. Résumé Le calcul de la fatigue d ue pale rotative peut ous aider à prévoir les problèmes de structure gééralemet recotrés das l utilisatio des aérogéérateurs (éoliees rapides), où la rupture des pales est très fréquete. Ce calcul doit passer par plusieurs étapes parmi elles le calcul des fréqueces et des modes propres aisi que le calcul des déplacemets et des cotraites qui agisset sur ces pales. Das ce travail deu méthodes différetes sot utilisées pour le calcul des modes propres : la première méthode est basée sur la discrétisatio de la pale, la secode est basée sur la résolutio de l équatio différetielle de la pale comme u système cotiu. Cette derière méthode est plus avatageuse das le calcul des modes propres aisi que das le calcul des cotraites, car elle e écessite pas l utilisatio d u calculateur puissat. Fialemet, la fatigue a été estimée e utilisat la théorie de Mier du dommage cumulé et e se basat sur la distributio statistique des vitesses etrêmes du vet pour le site e questio. Keywords: Wid Eergy - Structural dyamics - Aerodyamics - Numerical Aalysis.. INTRODUCTION The use of the wid eergy to geerate electricity is a twetieth cetury developmet. The first oil price shock, i the early 970s, stimulated iterest i alterative eergy sources, the research ad developmet i wid turbie techology received a substatial boost i several coutries. Also the icreasig cocer for the eviromet provided a further impetus for cleaer sources of eergy, icludig wid eergy. These two factors promoted iterest i the large-scale geeratio of electricity by wid power []. The major cause of wid turbie failure is fatigue, because wid turbie blades eposed to wid loadig are vulerable to cumulative fatigue damage owig to the cyclic ature of the loadig. The difficulty i predictig fatigue is i large part due to a isufficiet kowledge of the dyamic behavior of these machies. Moreover, S-N curve that gives the umber of stress cycles to failure is ecumbered with ucertaity 39

2 40 owig to a limited umber of test specimes as well as variability from oe specime to aother []. The predictio of the dyamic behavior costitutes oe of the most importat processes i the desig of wid turbies, because it ca be useful i estimatig the eergetic performace of the wid turbie as well as the fatigue ad the structural problem of this machie. The study of this dyamic behavior ca be udertake by various aalysis methods [3]. The estimatio of fatigue for a rotatig blade must go through several steps of calculatio. The first step is the calculatio of mode shapes ad frequecies, which is a difficult task due to the complicated ature of the blade rotatig movemet. The approach used i this work is based o the study of the blade movemet. For simplificatio this movemet is decomposed ito two: bedig ad torsio. Each movemet is the studied i a separate maer. For bedig atural frequecies are determied usig a discrete method. This method is based o the equilibrium equatios of the blade elemets. After assemblig of all the elemets the equilibrium equatios ca be trasformed ito a matri relatio. The resolutio of this matri relatio gives the frequecies ad the mode shapes of the blades. The mode shape curves obtaied by this method are ot sufficietly precise, due to the isufficiet umber of poits determied, ad therefore caot be used i the stress calculatio. For this reaso the mode shapes are the recalculated usig aother method, based o the umerical resolutio of the mode shape differetial equatio, despite the difficulties ecoutered i solvig its specific boudary coditios. This later method is more advatageous i determiig mode shapes tha the discrete method (matri method), because it allows us to determie the mode curves i a more precise ad cotiuous maer (large umber of poits are determied) without havig to use a powerful computer. For each method a Fortra computer program is elaborated. For torsio the equatio of the movemet is solved i a similar way as i the case of bedig to determie the torsioal mode shapes ad frequecies. Fially the two movemets are coupled usig the equatio proposed by Brooks. This equatio is resolved iteratively i order to determie the displacemets ad stresses actig o the blade. Aother computer program is writte for the coupled equatio resolutio. If the matri (discrete) method is largely used, the origiality of this work is characterised i the umerical resolutio of the blade equatios, usig it as a cotiuous system. This is doe i the case of the mode shape calculatio, ad also i solvig the coupled equatio (bedig-torsio), i order to determie displacemets ad stresses actig o the blade. Oce the stresses are calculated, the fatigue is estimated usig the mier theory of «cumulative damage i fatigue». This estimatio must be based o the statistical distributio of wid speed for the site i use. I this work oly a horizotal ais wid turbie is cosidered, sice their use ow is almost uiversal.. CALCULATION OF BENDING FREQUENCIES: The equilibrium equatios for a blade elemet havig a legth l, are give [4]: G = G Ω²..m.d = G 0.5.Ω².m.(² - ² ) ()

3 Fatigue Estimatio for a Rotatig Blade of a Wid Turbie 4 V = V - ω².z.m.d = V -.m.ω². l (Z Z ) () Fig. : rotatig blade movemet Ω = X X X X m d X Z Z mdx Z V Z Z G M M.. ) (.. ) (. ) ( ω (3) Fig. : Blade elemet equilibrium Where:

4 4 V is the shear force at the ode M is the bedig momet ode G is the cetrifugal force at the ode Z is the deflectio of the ode m is the liear mass ω is the atural frequecies After assemblig all the blade elemets the sequeces V, G, M ca be trasformed to a matri equatio of the form: ( A - I / ω² ).Z = 0 (4) Where the eigevector Z i are the mode shapes, ad the eige values are proportioal to the frequecies. Figure gives the first mode shape calculated usig this method FIRST MODE SHAPE OF BENDING BY DISCRETE METHOD MODE SHAPE BLADE LENGTH (%) Fig. 3: The first mode shape usig the matri method The use of this method i the mode calculatio is limited by the capacity of the computer used. For this reaso aother method is used, ad which is based o the umerical resolutio of the mode shape differetial equatio, after substitutig i it the frequecies obtaied above. 3. RESOLUTION OF THE MODE SHAPE EQUATION The bedig movemet is epressed by the followig equatio [4]: Where: L G = m. Ω ²..d Z Z Z (E.] ) (G ) t = F (5)

5 Fatigue Estimatio for a Rotatig Blade of a Wid Turbie 43 t is time. F is the aerodyamic load. I case of free vibratios, equatio (5) ca be writte: ² ² (E.I ². Z ² ) (G. Z ) m ². Z t² = 0 (6) Takig: Z = S().ϕ (t) Ad usig the method of variable separatio, the set of two ordiary differetial equatios is obtaied: d² d² (E.I d ². S d² ) - d d (G d S ) - mω ² S = 0 (7) d d²ϕ ω² ϕ = 0 (8) dt² Values of ω are take from the precedet method. The boudary coditios of equatio(7) are: I. Fied ed: S(0) = 0 (Displacemet =0) II. Free ed: ds( 0 ) = 0 (Slope = 0) d d²s(l) d² = 0 (Bedig momet =0) 3 d S (L) 3 = 0 (Shear force = 0) d The difficulty of umerical resolutio of equatio (7) is due to these boudary coditios. To make this resolutio possible, a algorithm is developed [5] i order to estimate the iitial coditios equivalet to (II).

6 FIRST MODE SHAPE MODE SHAPE BLADE LENGTH (%) Fig. 4: The first mode shape usig the secod method This algorithm is ispired from the shootig method, i which iitial value of the fuctio is supposed i order to calculate the fial value, this later value allow the correctio of the iitial value till the right fial value is obtaied. But i our case the problem is more complicated sice it has two iitial values, at the fied ed ad two fial values, at the free ed. The efforts deployed i this resolutio represet the pricipal origial cotributio of this work. The predictor corrector method is the is used to solve this equatio, after the failure of Ruge-Kutta method to reach covergece. The mode shapes obtaied usig this approach are represeted by Figures 4, 5 ad SECOND MODE SHAPE 0.50 THIRD MODE SHAPE MODE SHAPE BLADE LENGTH MODE SHAPE Fig. 5: The secod mode shape usig the secod method BLADE LENGTH (%) Fig.6: The third mode shape usig the secod method This method is more advatageous the the matri method, because it eables us to determie the mode shape curves i a smooth way. This is due to the fact that the umber of the curve poits calculated is ot limited by the memory size, Sice each poit is calculated i fuctio of the four precedet poits (The predictor corrector method Adam s formula [6]). We the oly eed to keep i memory the values of the four last poits.

7 Fatigue Estimatio for a Rotatig Blade of a Wid Turbie TORSIONAL MOVEMENT Torsioal movemet ca be epressed by the followig equatio []: θ (G.I ) C. t C. Ω L. θ = Where: C is the momet of iertia per uit legth. Ω is the agular velocity. θ is the twist agle. L A is the aerodyamic momet. G.J is the torsioal rigidity. Equatio (9) ca be solved i a similar maer as (5) (bedig equatio.) 5. EQUATION OF COUPLED MOVEMENT The two equatio (bedig - torsio) must be coupled i order to determie stresses ad displacemets. This is doe usig the equatio of Brooks [4], which has the form : (0) ² ² ( E.I ². Z ² ) ( G. Z ) m ². Z t² A Ft (, θ,.². θ t² ) =. F Where Ft is a kow fuctio. I this equatio the effect of torsio o bedig is take i accout. The particular difficulty ecoutered i the resolutio of the coupled equatio is due to the fact that the wid load depeds upo the shape of the blade (deflectio), sice this load is a fuctio of the icidece agle, o the other had the load deforms the shape of this blade. This iterdepedece betwee the aerodyamic load ad the blade deflectio is a source of oliearity that complicates the umerical resolutio. To overcome this difficulty a iitial deflectio is supposed, the a iterative method is used to correct this shape. I this resolutio, the computatio time ca cause serious problem if adequate techiques are ot take. Some results obtaied are show i Figure 7 ad i Figure 8. (9) MAXIMUM NORMAL STERSS (Pa) MAXIMUM SHEAR STRESS (Pa) TIME(T/ t) MAXIMUM NORMAL STESS AT THE FIXED END TEMPS(T/ t) MAXIMUM SHEAR STRESS AT THE FIXED END

8 46 Fig. 7: Maimum ormal stress at the fied ed. Fig. 8: Maimum shear stress at the fied ed. Figure 7 represets the maimum ormal stress ad Figure 8 represets the maimum shear stress for a wid speed of 3m/s. 6. FATIGUE CALCULATION FOR THE BLADE The fatigue estimatio for the blade is based o theory of Mier, which ca be applied i the case of a machie part operatig uder alterative stress havig a variable amplitude [7]. This theory is kow as «the liear cumulative damage rule» or «Mier s rule.» This theory assumes that every operatig cycle cosumes a percetage of the part life. Hece the total life of the part ca be estimated by addig up the percetage of life cosumed by each overstress cycle. Mier theory is stated mathematically as follows: If stresses with amplitudes σ, σ,...σ k are applied to a part for a total umber of cycles,,..., k respectively ad suppose that the lives (the allowable umber of cycles) correspodig to these stresses are : N, N,..., N k, the failure may occur if :... k = () N N Nk I a more recet article [8], Mier cites umerous tests that showed if the loadig is i radom, equatio () will usually give coservative predictios (i.e. >.) i Ni I this work a two-blade wid turbie is used with a aca profile. The values of i are calculated for a life period equals to te years ad based o the statistical distributio of wid speed (Figure 9), while the values of N i are take from the edurace limit curve STATISTICAL FREQUENCY MAXIMUM WIND SPEED DISTRIBUTION SITE OF CONSTANTINE (ALGERIA) WIND SPEED (m/s) Fig. 9: etreme wid speed statistical distributio 7. RESULTS The Characteristics of the turbie used are: Number of blades: 0 Profile: NACA00

9 Fatigue Estimatio for a Rotatig Blade of a Wid Turbie 47 Material used: Alumium alloy Blade legth: 8m Average cord: 0.4 m Specific velocity (velocity ratio): 8. The results obtaied usig these blade Characteristics are: The first mode shape calculated by the discrete method is give by Figure 3. The first three mode shapes calculated by the umerical resolutio of the differetial equatio are give by Figures 4 to 6. These curves were compared with those obtaied by referece [9] ad the two results were foud to be very similar. It is oticeable that Figures 4, 5 ad 6 give smoother ad more precise mode shapes tha Figure 3. Figures 8 ad 7 represet the maimum ormal stress ad the maimum shear stress i fuctio of time, for a wid speed of 3m/s. These stresses are determied by the umerical resolutio of the coupled equatio (bedig torsio.) For other rage of wid speed (6, 9, ad 5m/s) similar curves are determied. i To estimate the fatigue, the value i Ni is foud to be equal to 0.8 (less the uity). The values of i are calculated for a life time equals to te years ad determied from the statistical distributio of wid speed (see Fig. 9), while the values of Ni are take from the edurace limit curve. 8. CONCLUSION The fatigue estimatio ca be helpful i prevetig blade breakage, which is a very frequet problem i the use of wid turbies. The stresses calculated by the coupled equatio are used to estimate the fatigue of the blade by the cumulative damage theory. I this theory the wid speed statistical distributio for the regio of Costatie is used, i order to determiate the umber of operatig cycles for each wid speed (stress amplitude.) Accordig to this theory the blade ca resist for a operatig period close to te years, with a maimum operatig wid speed of m/s. Although the characteristics of the blade chose are ot optimal, the above calculatio may be used to develop a mathematical model to optimize these characteristics, i order to miimize its structure problems ad maimize its eergetic performace. REFERENCES [] UNESCO, Wid Eergy Techology, Joh Wiley & sos 997 [] K. O. Rooldk, Reliability-based fatigue desig of wid turbie rotor blades. Egieerig Structures, (999) 0-4 [3] R. Yousi, Dyamic study of wid turbie blade with horizotal ais. Eur. J. Mech A/solids, 0(00)4-5. [4] A. Bramwell, Helicopter dyamics Edward Arold 989.

10 48 [5] Z. Mahri, Etude de vibratios des pales d'u aérogéérateurs, thèse de Magister (uiversité de Costatie.) 99. [6] J. Nougier, Méthode de calcul umérique., Masso 996. [7] M. A. Mier, Cumulative damage i fatigue, Joural of Applied Mechaics, vol 6, Sept 945. [8] R. C. Juviall, Stress strai ad stregth, Mc Graw Hill, 980. [9] A. Bisplig, Aeroelasticity, Addiso Wesley 957.