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Dynamic Analysis of Wind Turbine Blades Using Radial Basis Functions

Dynamic Analysis of Wind Turbine Blades Using Radial Basis Functions Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2011, Article ID 973591, 11 pages doi:10.1155/2011/973591 Research Article Dynamic Analysis of Wind Turbine Blades Using Radial Basis Functions Ming-Hung Hsu Department of Electrical Engineering, National Penghu University of Science and Technology, Penghu 880, Taiwan Correspondence should be addressed to Ming-Hung Hsu, hsu.mh@msa.hinet.net Received 30 October 2010; Revised 11 April 2011; Accepted 11 April 2011 Academic Editor: K. M. Liew Copyright © 2011 Ming-Hung Hsu. 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. Wind turbine blades play important roles in wind energy generation. The dynamic problems associated with wind turbine blades are formulated using radial basis functions. The radial basis function procedure is used to transform partial differential equations, which represent the dynamic behavior of wind turbine blades, into a discrete eigenvalue problem. Numerical results demonstrate that rotational speed significantly impacts the first frequency of a wind turbine blade. Moreover, the pitch angle does not markedly affect wind turbine blade frequencies. This work examines the radial basis functions for dynamic problems of wind turbine blade. 1. Introduction applied the finite element method to identify the frequencies of natural vibration of doubly tapered and twisted beams. Wind energy will likely play an important role as a future Swaminathan and Rao [15] described the vibrations of a energy source in countries worldwide. Energy is essential rotating pretwisted and tapered blade. Subrahmanyam et al. to economic growth. In response to environmental con- [16, 17] derived the natural frequencies and mode shapes cerns, the amount of energy generated using renewable of a uniform pretwisted cantilever blade using the Reissner resources is increasing [1–3]. Herbert et al. [2]reviewed approach. Chen and Keer [18] examined the transverse assessment models for wind resources, site selection models, vibrations of a rotating pretwisted Timoshenko beam under and aerodynamic models, including the wake effect model. axial loading. Storti and Aboelnaga [19] examined transverse Varol et al. [3] demonstrated the positive effect of aerofoils deflections of a straight tapered symmetrical beam attached surrounding wind blades. Their notion of surrounding to a rotating hub as a model for the bending vibration of aerofoils resembles that of steering blades in a water turbine. blades. Wagner [20] derived the forced vibration response of Spee et al. [4] created a novel control strategy for a brushless subsystems with different natural frequencies and damping doubly fed machine applied to variable-speed wind-power attached to a foundation with a finite stiffness and mass. generation systems. Vitale and Rossi [5] designed low- Griffin and Sinha [21–23] determined the dynamic responses power, horizontal-axis wind turbine blades, using an iterative of frictionally damped turbine blades. Determining the algorithm. With their software, one can determine the dynamic features of wind turbine blades is of significant optimum blade shape for a wind turbine to satisfy the energy importance. Radial basis functions are important elements of requirements of an electrical system with optimum rotor effi- approaches generally called meshless methods. In this study, ciency. Song and Dhinakaran [6] developed a nonlinear wind dynamic problems associated with wind turbine blades are turbine control method. Storti and Aboelnaga [7]analyzed formulated using radial basis functions. transverse deflections of a straight tapered symmetrical beam attached to a rotating hub as a model for bending vibration of blades in turbomachinery. The natural frequencies of a single 2. Radial Basis Function tapered and pretwisted turboblade were calculated by Rao [8, 9], Abrate [10], Hodges et al. [11], and Dawson and Carneige A radial basis function is a real-value function whose [12, 13] using the Rayleigh-Ritz scheme. Gupta and Rao [14] value depends on distance from an origin. Kansa [24, 25] 2 Advances in Acoustics and Vibration investigated a given function or partial derivatives of a function with respect to a coordinate direction expressed as a linear weighted sum of all functional values at all mesh points along the direction initiated based on the concept of the radial basis function. In their algorithm, node distribution was entirely unstructured. Wang and Liu [26] developed a point interpolation meshless method based on radial basis functions that incorporated the Galerkin weak form for solving partial differential equations. Elfelsoufi [27] investigated buckling, flutter, and vibration of beams using radial basis functions. Hon et al. [28] utilized radial basis functions for function fitting and solving partial −100 100 300 500 700 900 differential equations using global nodes and collocation Rotational speed (rpm) procedures. Liu et al. [29] constructed shape functions with Mode 1 Mode 3 the delta function property using radial and polynomial basis Mode 2 Mode 4 functions. Devi and Pepper [30] generated a simplified radial basis function approach for calculating coupled heat transfer Figure 1: The first four frequencies of a wind turbine blade at different rotational speeds. ofafluidflow using alocal pressure correction scheme.In their study, shape functions were constructed using radial basis functions. Vrankar et al. [31] applied a relatively new approach for modeling radionuclide migration through the geosphere using a radial basis function scheme. Qiao and Ernst [32] applied a nonlinear approach for constructing color conversions based on radial basis functions. Their experimental results demonstrated that color conversion can be achieved effectively using the radial basis function approach when constructing nonlinear data maps. A radial 400 basis function can be expressed as follows [33, 34]: B (r) = (r − r ) + c,(1) i i where c is a shape parameter. The radial basis function is generally utilized to develop functional approximations in the following forms [33, 34]: u (r, t) = a (t)B (r),for k = 1, 2, 3, (2) k ki i −10 10 30 50 70 90 110 i=1 Pitch angle (deg) Mode 1 Mode 3 v (r, t) = a (t)B (r),for k = 1, 2, 3, (3) k ki i Mode 2 Mode 4 i=1 N Figure 2: The first four frequencies of a wind turbine blade at different pitch angles. ( ) ( ) U r = b B r ,for k = 1, 2, 3, (4) k ki i i=1 V (r) = b B (r),for k = 1, 2, 3, (5) k ki i i=1 3. Dynamic Analysis of Wind Turbine Blades where a , a , b ,and b are coefficients to be deter- ki ki ki ki The kinetic energy of rotating wind turbine blades can be mined. Blade deflection u (r, t) is the sum of N radial derived as basis functions, each associated with a different center r . Blade deflection v (r, t) denotes the sum of N radial basis functions, each associated with a different center r .Blade 3 L 2 2 1 ∂u ∂v k k 2 ( ) deflection U (r) is the sum of N radial basis functions, each T = ρA + + Ωv cos θ k k 2 ∂t ∂t k=1 associated with a different center r . Blade deflection V (r) i k (6) denotes the sum of N radial basis functions, each associated with a different center r . The domain contains N collocation i ( ) + Ωu sin θ dr, points. Frequency (Hz) Frequency (Hz) Advances in Acoustics and Vibration 3 where Ω is hub rotational speed, A is cross-sectional area, θ This leads to the following equations for the motion of the is pitch angle, ρ is blade density, and L is blade length. The blades: corresponding strain energy of rotating blades is 2 2 2 3 4 2 ∂ EI ∂EI ∂ EI yy ∂ u yy ∂ u ∂ u xy ∂ v k k k k +2 + EI + yy 2 2 3 4 2 2 ∂r ∂r ∂r ∂r ∂r ∂r ∂r L 2 1 ∂u ∂u ∂v k k k U = E I +2I 3 4 yy xy ∂EI ∂ v ∂ v xy k k 2 2 ∂r ∂r ∂r 2 k=1 +2 + EI − ρAΩ u sin θ xy k 3 4 ∂r ∂r ∂r ∂v ∂ ∂u +I dr k xx 2 − ρΩ A(r + r )dr ∂r 0 ∂r ∂r (7) L L L 2 ∂ u + ρAΩ (r + r )dr k 0 2 − ρΩ A(r + r )dr 2 0 0 r ∂r 2 2 ∂u ∂v k k 3 4 × + dr , ∂ EI ∂EI ∂u yy ∂ u yy ∂ u k k k ∂r ∂r + C + C +2C 0 1 1 2 2 3 ∂t ∂r ∂r ∂t ∂r ∂r ∂t 5 3 4 ∂ EI ∂EI ∂ u ∂ v ∂ v xy xy k k k + C EI + C +2C 1 yy 1 1 4 2 2 3 where r is hub radius, u (r, t) is the displacement of the first ∂r ∂t ∂r ∂r ∂t ∂r ∂r ∂t 0 1 blade on the x-axis, v (r, t) is the displacement of the first 5 2 ∂ v ∂ u k k blade on the y-axis, u (r, t) is the displacement of the second + C EI + ρA =0for k = 1, 2, 3, 1 xy 4 2 ∂r ∂t ∂t blade on the x-axis, v (r, t) is the displacement of the second 2 2 3 4 2 blade on the y-axis, u (r, t) is the displacement of the third ∂ EI 3 ∂ EI ∂ v ∂EI ∂ v ∂ v ∂ u xy xx k xx k k k +2 + EI + xx blade in the x-axis, v (r, t) is the displacement of the third 3 2 2 3 4 2 2 ∂r ∂r ∂r ∂r ∂r ∂r ∂r blade on the y-axis, and E is Young’s modulus. By examining 3 4 ∂EI ∂ u ∂ u xy k k the internal and external damping effects of blades, virtual 2 2 +2 + EI − ρAΩ v cos θ xy k 3 4 ∂r ∂r ∂r work δW in wind turbine blades can be derived as ∂ ∂v − ρΩ A(r + r )dr ∂r ∂r e r δW L 2 ∂ v 3 2 L L 2 ∂ EI − ρΩ A(r + r )dr ∂u yy ∂ u 0 k k ∂r = − C δu dr − C δu dr r 0 k 1 k 2 2 ∂t ∂r ∂t∂r 0 0 k=1 2 3 4 ∂v ∂ EI ∂ v ∂EI ∂ v k xx k xx k L + C + C +2C 0 1 1 ∂EI ∂ u 2 2 3 yy ∂t ∂r ∂r ∂t ∂r ∂r ∂t − 2 C δu dr 1 k ∂r ∂t∂r 5 3 4 ∂ EI ∂EI ∂ v ∂ u ∂ u xy xy k k k + C EI + C +2C L 5 1 xx 1 1 4 2 2 3 ∂ u k ∂r ∂t ∂r ∂r ∂t ∂r ∂r ∂t − C EI δu dr 1 yy k ∂t∂r 5 2 ∂ u ∂ v k k + C EI + ρA =0for k = 1, 2, 3. 1 xy L L 2 3 4 2 ∂v ∂ EI ∂ v ∂r ∂t ∂t k xx k − C δv dr − C δv dr 0 k 1 k (10) 2 2 ∂t ∂r ∂t∂r 0 0 L 4 ∂EI ∂ v xx k − 2 C δv dr 1 k The following equations are the corresponding boundary ∂r ∂t∂r conditions: ∂ v − C EI δv dr , 1 xx k ∂t∂r (8) u (0, t) =0for k = 1, 2, 3, ∂u (0, t) =0for k = 1, 2, 3, ∂r where C and C are external and internal damping coeffi- 0 1 cients of blades, respectively. Equations (6)–(8) are inte- ∂ u (L, t) EI =0for k = 1, 2, 3, yy grated into the Hamilton equation as follows: ∂r ∂ ∂ u (L, t) EI =0for k = 1, 2, 3, yy ∂r ∂r e e e (δT − δU + δW )dt = 0. (9) 1 v (0, t) =0for k = 1, 2, 3, k 4 Advances in Acoustics and Vibration ∂v (0, t) k d dV =0for k = 1, 2, 3, − ρΩ A(r + r )dr ∂r dr dr ∂ v (L, t) k L EI =0for k = 1, 2, 3, d V xx 2 − ρΩ A(r + r )dr ∂r 0 r dr ∂ ∂ v (L, t) 2 2 3 EI =0for k = 1, 2, 3. xx dV d EI d V dEI d V k xx k xx k ∂r ∂r + λC + λC +2λC 0 1 1 2 2 3 dt dr dr dr dr (11) 4 2 2 3 d EI dEI d V xy d U xy d U k k k + λC EI + λC +2λC 1 xx 1 1 4 2 2 3 dr dr dr dr dr Asolutionisthusassumed as d U λt 2 u = U (r)e for k = 1, 2, 3, + λC EI + λ ρAV =0for k = 1, 2, 3. k k 1 xy k dr (12) λt v = V (r)e for k = 1, 2, 3. k k (13) This yields the following equations for the motion of blades: The corresponding boundary conditions are as follows: 2 2 2 3 4 2 d EI dEI d EI d U d U d U d V yy yy xy k k k k +2 + EI + yy 2 2 3 4 2 2 U (0) =0for k = 1, 2, 3, dr dr dr dr dr dr dr 3 4 dEI d V d V dU (0) xy k k 2 k +2 + EI − ρAΩ U sin θ =0for k = 1, 2, 3, xy k 3 4 dr dr dr dr L 2 d U (L) d dU k EI =0for k = 1, 2, 3, − ρΩ A(r + r )dr yy dr dr dr L d d U (L) d U 2 EI =0for k = 1, 2, 3, yy ( ) − ρΩ A r + r dr 2 dr dr dr (14) 2 V (0) =0for k = 1, 2, 3, 2 3 k d EI dEI yy d U yy d U k k + λC U + λC +2λC 0 k 1 1 2 2 3 dr dr dr dr dV (0) =0for k = 1, 2, 3, 4 2 dr d EI d U d V k xy k + λC EI + λC 1 yy 1 4 2 2 dr dr dr d V (L) EI =0for k = 1, 2, 3, xx 3 4 dr dEI xy d V d V k k +2λC + λC EI 1 1 xy 3 4 dr dr dr d d V (L) EI =0for k = 1, 2, 3. xx dr dr + λ ρAU =0for k = 1, 2, 3, 2 2 3 4 2 d EI d EI d V dEI d V d V xy d U xx k xx k k k By employing the radial basis function approach, (2)and +2 + EI + xx 2 2 3 4 2 2 dr dr dr dr dr dr dr (3) can be substituted into (10). The equations of motion of wind turbine blades can be rearranged in the following 3 4 dEI xy d U d U k k 2 2 +2 + EI − ρAΩ V cos θ matrix form: xy k 3 4 dr dr dr 2 2 2 ∂ a ∂ a ∂ a k1 k2 kN ρA(r )B (r ) ρA(r )B (r ) ··· ρA(r )B (r ) ··· i 1 i i 2 i i N i 2 2 2 ∂t ∂t ∂t ∂a ∂a ∂a k1 k2 kN + [C B (r ) C B (r ) ··· C B (r )] ··· 0 1 i 0 2 i 0 N i ∂t ∂t ∂t 2 2 2 T 2 2 2 ∂ EI (r ) ∂ EI (r ) ∂ EI (r ) ∂ B (r ) ∂ B (r ) ∂ B (r ) yy i 1 i yy i 2 i yy i N i ∂a ∂a ∂a k1 k2 kN + C C ··· C ··· 1 1 1 2 2 2 2 2 2 ∂r ∂r ∂r ∂r ∂r ∂r ∂t ∂t ∂t 3 3 3 T ∂EI (r ) ∂EI (r ) ∂EI (r ) yy i ∂ B (r ) yy i ∂ B (r ) yy i ∂ B (r ) ∂a ∂a ∂a 1 i 2 i N i k1 k2 kN + 2C 2C ··· 2C 1 1 1 ··· 3 3 3 ∂r ∂r ∂r ∂r ∂r ∂r ∂t ∂t ∂t 4 4 4 ∂ B (r ) ∂ B (r ) ∂ B (r ) ∂a ∂a ∂a 1 i 2 i N i k1 k2 kN ( ) ( ) ( ) + C EI r C EI r ··· C EI r ··· 1 yy i 1 yy i 1 yy i 4 4 4 ∂r ∂r ∂r ∂t ∂t ∂t Advances in Acoustics and Vibration 5 2 2 2 T 2 2 2 ∂ EI (r ) ∂ EI (r ) ∂ EI (r ) ∂ B (r ) ∂ B (r ) ∂ B (r ) xy i xy i xy i ∂a ∂a ∂a 1 i 2 i N i k1 k2 kN + C C ··· C ··· 1 1 1 2 2 2 2 2 2 ∂r ∂r ∂r ∂r ∂r ∂r ∂t ∂t ∂t 3 3 3 ∂EI (r ) ∂EI (r ) ∂EI (r ) xy i ∂ B (r ) xy i ∂ B (r ) xy i ∂ B (r ) ∂a ∂a ∂a 1 i 2 i N i k1 k2 kN + 2C 2C ··· 2C 1 1 1 ··· 3 3 3 ∂r ∂r ∂r ∂r ∂r ∂r ∂t ∂t ∂t 4 4 4 ∂ B (r ) ∂ B (r ) ∂ B (r ) ∂a ∂a ∂a 1 i 2 i N i k1 k2 kN + C EI (r ) C EI (r ) ··· C EI (r ) ··· 1 xy i 1 xy i 1 xy i 4 4 4 ∂r ∂r ∂r ∂t ∂t ∂t 2 2 2 2 2 2 ∂ EI (r ) ∂ EI (r ) ∂ EI (r ) T yy i ∂ B (r ) yy i ∂ B (r ) yy i ∂ B (r ) 1 i 2 i N i + ··· a a ··· a k1 k2 kN 2 2 2 2 2 2 ∂r ∂r ∂r ∂r ∂r ∂r 3 3 3 ( ) ( ) ( ) T ∂EI r ∂ B (r ) ∂EI r ∂ B (r ) ∂EI r ∂ B (r ) yy i yy i yy i 1 i 2 i N i a a ··· a + 2 2 ··· 2 k1 k2 kN 3 3 3 ∂r ∂r ∂r ∂r ∂r ∂r 4 4 4 ∂ B (r ) ∂ B (r ) ∂ B (r ) 1 i 2 i N i + EI (r ) EI (r ) ··· EI (r ) a a ··· a k1 k2 kN yy i yy i yy i 4 4 4 ∂r ∂r ∂r 2 2 2 2 2 2 ∂ EI (r ) ( ) ∂ EI (r ) ( ) ∂ EI (r ) ( ) xy i ∂ B r xy i ∂ B r xy i ∂ B r 1 i 2 i N i a a ··· a + ··· k1 k2 kN 2 2 2 2 2 2 ∂r ∂r ∂r ∂r ∂r ∂r 3 3 3 ∂EI (r ) ∂EI (r ) ∂EI (r ) T ∂ B (r ) ∂ B (r ) ∂ B (r ) xy i 1 i xy i 2 i xy i N i + 2 2 ··· 2 a a ··· a k1 k2 kN 3 3 3 ∂r ∂r ∂r ∂r ∂r ∂r 4 4 4 ∂ B (r ) ∂ B (r ) ∂ B (r ) 1 i 2 i N i a a ··· a + EI (r ) EI (r ) ··· EI (r ) xy i xy i xy i k1 k2 kN 4 4 4 ∂r ∂r ∂r 2 2 2 2 2 2 a a ··· a − ρA(r)Ω (sin θ) B (r ) ρA(r)Ω (sin θ) B (r ) ··· ρA(r)Ω (sin θ) B (r ) k1 k2 kN 1 i 2 i N i L L L ∂ ∂B (r ) ∂ ∂B (r ) ∂ ∂B (r ) 1 i 2 i N i 2 2 2 − ρΩ A(r)Adr ρΩ A(r)Adr ··· ρΩ A(r)Adr ∂r ∂r ∂r ∂r ∂r ∂r r r r i i i a a ··· a k1 k2 kN L L L 2 2 2 ∂ B (r ) ∂ B (r ) ∂ B (r ) 1 i 2 i N i 2 2 2 ( ) ( ) ( ) − ρΩ A r Adr ρΩ A r Adr ··· ρΩ A r Adr 2 2 2 ∂r ∂r ∂r r r r i i i × a a ··· a = [0],for i = 1, 2,... , N, k = 1, 2, 3, k1 k2 kN 2 2 2 ∂ a ∂ a ∂ a k1 k2 kN ρA(r )B (r ) ρA(r )B (r ) ··· ρA(r )B (r ) i 1 i i 2 i i N i ··· 2 2 2 ∂t ∂t ∂t ∂a ∂a ∂a k1 k2 kN + [C B (r ) C B (r ) ··· C B (r )] ··· 0 1 i 0 2 i 0 N i ∂t ∂t ∂t 2 2 2 2 2 2 ∂ EI (r ) ∂ B (r ) ∂ EI (r ) ∂ B (r ) ∂ EI (r ) ∂ B (r ) ∂a ∂a ∂a xx i 1 i xx i 2 i xx i N i k1 k2 kN + C C ··· C ··· 1 1 1 2 2 2 2 2 2 ∂r ∂r ∂r ∂r ∂r ∂r ∂t ∂t ∂t 3 3 3 T ∂EI (r ) ∂ B (r ) ∂EI (r ) ∂ B (r ) ∂EI (r ) ∂ B (r ) ∂a ∂a ∂a xx i 1 i xx i 2 i xx i N i k1 k2 kN + 2C 2C ··· 2C 1 1 1 ··· 3 3 3 ∂r ∂r ∂r ∂r ∂r ∂r ∂t ∂t ∂t 4 4 4 ∂ B (r ) ∂ B (r ) ∂ B (r ) ∂a ∂a ∂a 1 i 2 i N i k1 k2 kN ( ) ( ) ( ) + C EI r C EI r ··· C EI r ··· 1 xx i 1 xx i 1 xx i 4 4 4 ∂r ∂r ∂r ∂t ∂t ∂t 2 2 2 2 2 2 T ∂ EI (r ) ∂ EI (r ) ∂ EI (r ) xy i ∂ B (r ) xy i ∂ B (r ) xy i ∂ B (r ) ∂a ∂a ∂a 1 i 2 i N i k1 k2 kN + C C ··· C 1 1 1 ··· 2 2 2 2 2 2 ∂r ∂r ∂r ∂r ∂r ∂r ∂t ∂t ∂t 6 Advances in Acoustics and Vibration 3 3 3 ∂EI (r ) ∂EI (r ) ∂EI (r ) ∂ B (r ) ∂ B (r ) ∂ B (r ) xy i xy i xy i ∂a ∂a ∂a 1 i 2 i N i k1 k2 kN + 2C 2C ··· 2C ··· 1 1 1 3 3 3 ∂r ∂r ∂r ∂r ∂r ∂r ∂t ∂t ∂t 4 4 4 ∂ B (r ) ∂ B (r ) ∂ B (r ) ∂a ∂a ∂a 1 i 2 i N i k1 k2 kN + C EI (r ) C EI (r ) ··· C EI (r ) 1 xy i 1 xy i 1 xy i ··· 4 4 4 ∂r ∂r ∂r ∂t ∂t ∂t 2 2 2 2 2 2 ∂ EI (r ) ∂ B (r ) ∂ EI (r ) ∂ B (r ) ∂ EI (r ) ∂ B (r ) xx i 1 i xx i 2 i xx i N i a a ··· a + ··· k1 k2 kN 2 2 2 2 2 2 ∂r ∂r ∂r ∂r ∂r ∂r 3 3 3 ∂EI (r ) ∂ B (r ) ∂EI (r ) ∂ B (r ) ∂EI (r ) ∂ B (r ) xx i 1 i xx i 2 i xx i N i + 2 2 ··· 2 a a ··· a k1 k2 kN 3 3 3 ∂r ∂r ∂r ∂r ∂r ∂r 4 4 4 ∂ B (r ) ∂ B (r ) ∂ B (r ) 1 i 2 i N i a a ··· a + EI (r ) EI (r ) ··· EI (r ) xx i xx i xx i k1 k2 kN 4 4 4 ∂r ∂r ∂r 2 2 2 2 2 2 ∂ EI (r ) ∂ EI (r ) ∂ EI (r ) T xy i ∂ B (r ) xy i ∂ B (r ) xy i ∂ B (r ) 1 i 2 i N i + ··· a a ··· a k1 k2 kN 2 2 2 2 2 2 ∂r ∂r ∂r ∂r ∂r ∂r 3 3 3 ∂EI (r ) ( ) ∂EI (r ) ( ) ∂EI (r ) ( ) xy i ∂ B r xy i ∂ B r xy i ∂ B r 1 i 2 i N i a a ··· a + 2 2 ··· 2 k1 k2 kN 3 3 3 ∂r ∂r ∂r ∂r ∂r ∂r 4 4 4 ∂ B (r ) ∂ B (r ) ∂ B (r ) 1 i 2 i N i + EI (r ) EI (r ) ··· EI (r ) a a ··· a k1 k2 kN xy i xy i xy i 4 4 4 ∂r ∂r ∂r 2 2 2 2 2 2 − ρA(r)Ω (cos θ) B (r ) ρA(r)Ω (cos θ) B (r ) ··· ρA(r)Ω (cos θ) B (r ) a a ··· a 1 i 2 i N i k1 k2 kN L L L ∂ ∂B (r ) ∂ ∂B (r ) ∂ ∂B (r ) 1 i 2 i N i 2 2 2 − ρΩ A(r)Adr ρΩ A(r)Adr ··· ρΩ A(r)Adr ∂r ∂r ∂r ∂r ∂r ∂r r r r i i i × a a ··· a k1 k2 kN L 2 L 2 L 2 ∂ B (r ) ∂ B (r ) ∂ B (r ) 1 i 2 i N i 2 2 2 − ρΩ A(r)Adr ρΩ A(r)Adr ··· ρΩ A(r)Adr 2 2 2 r ∂r r ∂r r ∂r i i i a a ··· a × = [0] for i = 1, 2,... , N, k = 1, 2, 3, k1 k2 kN (15) where A denotes (r + r ). Based on the radial basis function technique, (11)take the following discrete forms: a a ··· a [B (r ) B (r ) ··· B (r )] = [0] for k = 1, 2, 3, 1 1 2 1 N 1 k1 k2 kN ∂B (r ) ∂B (r ) ∂B (r ) 1 1 2 1 N 1 T ··· [a a ··· a ] = [0] for k = 1, 2, 3, k1 k2 kN ∂r ∂r ∂r 2 2 2 ∂ B (r ) ∂ B (r ) ∂ B (r ) 1 N 2 N N N T EI (r ) EI (r ) ··· EI (r ) [a a ··· a ] = [0] for k = 1, 2, 3, yy N yy N yy N k1 k2 kN 2 2 2 ∂r ∂r ∂r 2 2 2 ∂EI (r ) ∂EI (r ) ∂EI (r ) ∂ B (r ) ∂ B (r ) ∂ B (r ) yy N yy N yy N 1 N 2 N N N T ··· [a a ··· a ] k1 k2 kN 2 2 2 ∂r ∂r ∂r ∂r ∂r ∂r 3 3 3 ∂ B (r ) ∂ B (r ) ∂ B (r ) 1 N 2 N N N + EI (r ) EI (r ) ··· EI (r ) yy N yy N yy N 3 3 3 ∂r ∂r ∂r ×[a a ··· a ] = [0] for k = 1, 2, 3, k1 k2 kN Advances in Acoustics and Vibration 7 [B (r ) B (r ) ··· B (r )][a a ··· a ] = [0] for k = 1, 2, 3, 1 1 2 1 N 1 k1 k2 kN ∂B (r ) ∂B (r ) ∂B (r ) 1 1 2 1 N 1 ··· [a a ··· a ] = [0] for k = 1, 2, 3, k1 k2 kN ∂r ∂r ∂r 2 2 2 ∂ B (r ) ∂ B (r ) ∂ B (r ) 1 N 2 N N N EI (r ) EI (r ) ··· EI (r ) [a a ··· a ] = [0],for k = 1, 2, 3, xx N xx N xx N k1 k2 kN 2 2 2 ∂r ∂r ∂r 2 2 2 ∂EI (r ) ∂ B (r ) ∂EI (r ) ∂ B (r ) ∂EI (r ) ∂ B (r ) xx N 1 N xx N 2 N xx N N N a a ··· a ··· k1 k2 kN 2 2 2 ∂r ∂r ∂r ∂r ∂r ∂r 3 3 3 ∂ B (r ) ∂ B (r ) ∂ B (r ) N N 2 N N N + EI (r ) EI (r ) ··· EI (r ) xx N xx N xx N 3 3 3 ∂r ∂r ∂r ×[a a ··· a ] = [0] for k = 1, 2, 3. k1 k2 kN (16) By applying the radial basis function approach, (4)and (5) are substituted into (13). The following equations are then yielded 2 2 2 λ ρA(r )B (r ) λ ρA(r )B (r ) ··· λ ρA(r )B (r ) [b b ··· b ] i 1 i i 2 i i N i k1 k2 kN + [λC B (r ) λC B (r ) ··· λC B (r )][b b ··· b ] 0 1 i 0 2 i 0 N i k1 k2 kN 2 2 2 2 2 2 d EI (r ) d EI (r ) d EI (r ) yy i d B (r ) yy i d B (r ) yy i d B (r ) 1 i 2 i N i T + λC λC ··· λC [b b ··· b ] 1 1 1 k1 k2 kN 2 2 2 2 2 2 dr dr dr dr dr dr 3 3 3 dEI (r ) dEI (r ) dEI (r ) d B (r ) d B (r ) d B (r ) yy i yy i yy i 1 i 2 i N i T + 2λC 2λC ··· 2λC [b b ··· b ] 1 1 1 k1 k2 kN 3 3 3 dr dr dr dr dr dr 4 4 4 d B (r ) d B (r ) d B (r ) 1 i 2 i N i + λC EI (r ) λC EI (r ) ··· λC EI (r ) [b b ··· b ] 1 yy i 1 yy i 1 yy i k1 k2 kN 4 4 4 dr dr dr 2 2 2 2 2 2 ( ) ( ) ( ) T d EI r d B (r ) d EI r d B (r ) d EI r d B (r ) xy i xy i xy i 1 i 2 i N i + λC λC ··· λC b b ··· b 1 1 1 k1 k2 kN 2 2 2 2 2 2 dr dr dr dr dr dr 3 3 3 dEI (r ) dEI (r ) dEI (r ) T xy i d B (r ) xy i d B (r ) xy i d B (r ) 1 i 2 i N i + 2λC 2λC ··· 2λC b b ··· b 1 1 1 k1 k2 kN 3 3 3 dr dr dr dr dr dr 4 4 4 ( ) ( ) ( ) d B r d B r d B r 1 i 2 i N i + λC EI (r ) λC EI (r ) ··· λC EI (r ) b b ··· b 1 xy i 1 xy i 1 xy i k1 k2 kN 4 4 4 dr dr dr 2 2 2 2 2 2 d EI (r ) d EI (r ) d EI (r ) yy i d B (r ) yy i d B (r ) yy i d B (r ) 1 i 2 i N i + ··· [b b ··· b ] k1 k2 kN 2 2 2 2 2 2 dr dr dr dr dr dr 3 3 3 dEI (r ) dEI (r ) dEI (r ) yy i d B (r ) yy i d B (r ) yy i d B (r ) 1 i 2 i N i T + 2 2 ··· 2 [b b ··· b ] k1 k2 kN 3 3 3 dr dr dr dr dr dr 4 4 4 d B (r ) d B (r ) d B (r ) 1 i 2 i N i + EI (r ) EI (r ) ··· EI (r ) [b b ··· b ] yy i yy i yy i k1 k2 kN 4 4 4 dr dr dr 2 2 2 2 2 2 d EI (r ) d EI (r ) d EI (r ) xy i d B (r ) xy i d B (r ) xy i d B (r ) 1 i 2 i N i + ··· b b ··· b k1 k2 kN 2 2 2 2 2 2 dr dr dr dr dr dr 3 3 3 dEI (r ) dEI (r ) dEI (r ) T d B (r ) d B (r ) d B (r ) xy i xy i xy i 1 i 2 i N i + 2 2 ··· 2 b b ··· b k1 k2 kN 3 3 3 dr dr dr dr dr dr 8 Advances in Acoustics and Vibration 4 4 4 d B (r ) d B (r ) d B (r ) 1 i 2 i N i + EI (r ) EI (r ) ··· EI (r ) b b ··· b xy i xy i xy i k1 k2 kN 4 4 4 dr dr dr 2 2 2 2 2 2 T − ρA(r )Ω (sin θ) B (r ) ρA(r )Ω (sin θ) B (r ) ··· ρA(r )Ω (sin θ) B (r ) [b b ··· b ] i 1 i i 2 i i N i k1 k2 kN L L L ( ) ( ) ( ) d dB r d dB r d dB r 1 i 2 i N i 2 2 2 − ρΩ A(r)Adr ρΩ A(r)Adr ··· ρΩ A(r)Adr dr dr dr dr dr dr r r r i i i [ ] × b b ··· b k1 k2 kN L 2 L 2 L 2 d B (r ) d B (r ) d B (r ) 1 i 2 i N i 2 2 2 − ρΩ A(r)Adr ρΩ A(r)Adr ··· ρΩ A(r)Adr 2 2 2 dr dr dr r r r i i i × [b b ··· b ] = [0] for i = 1, 2,... , N, k = 1, 2, 3, k1 k2 kN 2 2 2 λ ρA(r )B (r ) λ ρA(r )B (r ) ··· λ ρA(r )B (r ) b b ··· b i 1 i i 2 i i N i k1 k2 kN + [λC B (r ) λC B (r ) ··· λC B (r )] b b ··· b 0 1 i 0 2 i 0 N i k1 k2 kN 2 2 2 2 2 2 d EI (r ) d B (r ) d EI (r ) d B (r ) d EI (r ) d B (r ) xx i 1 i xx i 2 i xx i N i + λC λC ··· λC b b ··· b 1 1 1 k1 k2 kN 2 2 2 2 2 2 dr dr dr dr dr dr 3 3 3 dEI (r ) d B (r ) dEI (r ) d B (r ) dEI (r ) d B (r ) xx i 1 i xx i 2 i xx i N i + 2λC 2λC ··· 2λC b b ··· b 1 1 1 k1 k2 kN 3 3 3 dr dr dr dr dr dr 4 4 4 ( ) ( ) ( ) d B r d B r d B r 1 i 2 i N i + λC EI (r ) λC EI (r ) ··· λC EI (r ) b b ··· b 1 xx i 1 xx i 1 xx i k1 k2 kN 4 4 4 dr dr dr 2 2 2 2 2 2 d EI (r ) d EI (r ) d EI (r ) xy i d B (r ) xy i d B (r ) xy i d B (r ) 1 i 2 i N i + λC λC ··· λC [b b ··· b ] 1 1 1 k1 k2 kN 2 2 2 2 2 2 dr dr dr dr dr dr 3 3 3 dEI (r ) dEI (r ) dEI (r ) xy i d B (r ) xy i d B (r ) xy i d B (r ) 1 i 2 i N i T + 2λC 2λC ··· 2λC [b b ··· b ] 1 1 1 k1 k2 kN 3 3 3 dr dr dr dr dr dr 4 4 4 d B (r ) d B (r ) d B (r ) 1 i 2 i N i + λC EI (r ) λC EI (r ) ··· λC EI (r ) [b b ··· b ] 1 xy i 1 xy i 1 xy i k1 k2 kN 4 4 4 dr dr dr 2 2 2 2 2 2 d EI (r ) d B (r ) d EI (r ) d B (r ) d EI (r ) d B (r ) xx i 1 i xx i 2 i xx i N i + ··· b b ··· b k1 k2 kN 2 2 2 2 2 2 dr dr dr dr dr dr 3 3 3 dEI (r ) d B (r ) dEI (r ) d B (r ) dEI (r ) d B (r ) xx i 1 i xx i 2 i xx i N i + 2 2 ··· 2 b b ··· b k1 k2 kN 3 3 3 dr dr dr dr dr dr 4 4 4 d B (r ) d B (r ) d B (r ) 1 i 2 i N i + EI (r ) EI (r ) ··· EI (r ) b b ··· b xx i xx i xx i k1 k2 kN 4 4 4 dr dr dr 2 2 2 2 2 2 d EI (r ) d EI (r ) d EI (r ) d B (r ) d B (r ) d B (r ) xy i xy i xy i 1 i 2 i N i T [ ] + ··· b b ··· b k1 k2 kN 2 2 2 2 2 2 dr dr dr dr dr dr 3 3 3 dEI (r ) dEI (r ) dEI (r ) xy i d B (r ) xy i d B (r ) xy i d B (r ) 1 i 2 i N i T + 2 2 ··· 2 [b b ··· b ] k1 k2 kN 3 3 3 dr dr dr dr dr dr 4 4 4 d B (r ) d B (r ) d B (r ) 1 i 2 i N i T ( ) ( ) ( ) + EI r EI r ··· EI r [b b ··· b ] xy i xy i xy i k1 k2 kN 4 4 4 dr dr dr 2 2 2 2 2 2 − ρA(r )Ω (cos θ) B (r ) ρA(r )Ω (cos θ) B (r ) ··· ρA(r )Ω (cos θ) B (r ) b b ··· b i 1 i i 2 i i N i k1 k2 kN Advances in Acoustics and Vibration 9 L L L d dB (r ) d dB (r ) d dB (r ) 1 i 2 i N i 2 2 2 − ρΩ A(r)Adr ρΩ A(r)Adr ··· ρΩ A(r)Adr dr dr dr dr dr dr r r r i i i × b b ··· b k1 k2 kN L 2 L 2 L 2 d B (r ) d B (r ) d B (r ) 1 i 2 i N i 2 2 2 − ρΩ A(r)Adr ρΩ A(r)Adr ··· ρΩ A(r)Adr 2 2 2 dr dr dr r r r i i i × b b ··· b = [0] for i = 1, 2,... , N, k = 1, 2, 3. k1 k2 kN (17) where A denotes (r + r ). According to the radial basis function approach, the boundary conditions in (14) have the following discrete forms: [B (r ) B (r ) ··· B (r )][b b ··· b ] = [0] for k = 1, 2, 3, 1 1 2 1 N 1 k1 k2 kN ( ) ( ) ( ) dB r dB r dB r 1 1 2 1 N 1 T ··· [b b ··· b ] = [0] for k = 1, 2, 3, k1 k2 kN dr dr dr 2 2 2 d B (r ) d B (r ) d B (r ) 1 N 2 N N N T EI (r ) EI (r ) ··· EI (r ) [b b ··· b ] = [0] for k = 1, 2, 3, yy N yy N yy N k1 k2 kN 2 2 2 dr dr dr 2 2 2 dEI (r ) dEI (r ) dEI (r ) d B (r ) d B (r ) d B (r ) yy N yy N yy N 1 N 2 N N N T ··· [b b ··· b ] k1 k2 kN 2 2 2 dr dr dr dr dr dr 3 3 3 d B (r ) d B (r ) d B (r ) 1 N 2 N N N + EI (r ) EI (r ) ··· EI (r ) yy N yy N yy N 3 3 3 dr dr dr ×[b b ··· b ] = [0] for k = 1, 2, 3, k1 k2 kN T (18) [B (r ) B (r ) ··· B (r )] b b ··· b = [0] for k = 1, 2, 3, 1 1 2 1 N 1 k1 k2 kN dB (r ) dB (r ) dB (r ) 1 1 2 1 N 1 ··· b b ··· b = [0] for k = 1, 2, 3, k1 k2 kN dr dr dr 2 2 2 d B (r ) d B (r ) d B (r ) 1 N 2 N N N EI (r ) EI (r ) ··· EI (r ) b b ··· b = [0] for k = 1, 2, 3, xx N xx N xx N k1 k2 kN 2 2 2 dr dr dr 2 2 2 dEI (r ) d B (r ) dEI (r ) d B (r ) dEI (r ) d B (r ) xx N 1 N xx N 2 N xx N N N ··· b b ··· b k1 k2 kN 2 2 2 dr dr dr dr dr dr 3 3 3 d B (r ) d B (r ) d B (r ) 1 N 2 N N N + EI (r ) EI (r ) ··· EI (r ) xx N xx N xx N 3 3 3 dr dr dr × b b ··· b = [0] for k = 1, 2, 3. k1 k2 kN 4. Results rotational speed increases. Numerical results obtained by this study suggest that rotational speed does not significantly Figure 1 illustrates the effects of various rotational speeds affect the second, third, and fourth frequencies of a wind on calculated frequencies for a wind turbine blade. Com- blade. Figure 2 lists the frequencies of a wind turbine blade putational results suggest that the first frequency of a wind at different pitch angles. Numerical results demonstrate that turbine blade is strongly dependent on blade shaft speed. pitch angle does not significantly affect the first, second, third, and fourth frequencies of a wind turbine blade. The first frequency of a wind turbine blade increases as 10 Advances in Acoustics and Vibration 5. Concluding Remarks allowing for shear deflection and rotary inertia by the Reissner method,” International Journal of Mechanical Sciences, vol. 23, The radial basis function scheme has been extensively no. 9, pp. 517–530, 1981. adopted to solve various problems in different scientific and [17] K. B. Subrahmanyam and J. S. Rao, “Coupled bending- engineering fields. This study investigated how pitch angles bending vibrations of pretwisted tapered cantilever beams and rotational speeds affect frequencies of a wind generator treated by the Reissner method,” Journal of Sound and Vibration, vol. 82, no. 4, pp. 577–592, 1982. blade. The simplicity of this formulation makes it a good candidate for very complex applications. [18] W. R. Chen and L. M. Keer, “Transverse vibrations of a rotating twisted Timshenko beam under axial loading,” ASME Journal of Vibration and Acoustics, vol. 115, pp. 285–294, 1993. References [19] D. Storti and Y. Aboelnaga, “Bending vibrations of a class [1] T.K.Fung,R.L.Scheffler, and J. Stolpe, “Wind energy-a of rotating beams with hypergeometric solutions,” Journal of utility perspective,” IEEE Transactions on Power Apparatus and Applied Mechanics, vol. 54, no. 2, pp. 311–314, 1987. Systems, vol. 100, no. 3, pp. 1176–1182, 1981. [20] J. T. Wagner, “Coupling of turbomachine blade vibrations [2] G. M. J. Herbert, S. Iniyan, E. Sreevalsan, and S. Rajapandian, through the rotor,” ASME Journal of Engineering for Power, vol. “A review of wind energy technologies,” Renewable and 89, pp. 502–512, 1967. Sustainable Energy Reviews, vol. 11, no. 6, pp. 1117–1145, [21] J. H. Griffin, “Friction damping of resonant stresses in gas turbine engine airfoils,” ASME Journal of Engineering for [3] A. Varol, C. Ilkilic, and Y. Varol, “Increasing the efficiency Power, vol. 102, no. 2, pp. 329–333, 1980. of wind turbines,” Journal of Wind Engineering and Industrial [22] J. H. Griffin and A. Sinha, “The interaction between mistuning Aerodynamics, vol. 89, no. 9, pp. 809–815, 2001. and friction in the forced response of bladed disk assemblies,” [4] R. Spee, S. Bhowmik, and J. H. R. Enslin, “Novel control ASME Journal of Engineering for Gas Turbines and Power, vol. strategies for variable-speed doubly fed wind power genera- 107, no. 1, pp. 107–205, 1985. tion systems,” Renewable Energy, vol. 6, no. 8, pp. 907–915, [23] A. Sinha and J. H. Griffin, “Effects of static friction on the forced response of frictionally damped turbine blades,” ASME [5] A. J. Vitale and A. P. Rossi, “Computational method for Journal of Engineering for Gas Turbines and Power, vol. 106, no. the design of wind turbine blades,” International Journal of 1, pp. 65–69, 1984. Hydrogen Energy, vol. 33, no. 13, pp. 3466–3470, 2008. [24] E. J. Kansa, “Multiquadrics-a scattered data approximation [6] Y. D. Song and B. Dhinakaran, “Nonlinear variable speed con- scheme with applications to computational fluid-dynamics- trol of wind turbines,” in Proceedings of the IEEE International I surface approximations and partial derivative estimates,” Conference on Control Applications (CCA ’99), pp. 814–819, Computers and Mathematics with Applications, vol. 19, no. 8-9, August 1999. pp. 127–145, 1990. [7] D. Storti and Y. Aboelnaga, “Bending vibrations of a class [25] E. J. Kansa, “Multiquadrics-a scattered data approximation of rotating beams with hypergeometric solutions,” Journal of scheme with applications to computational fluid-dynamics-II Applied Mechanics, vol. 54, no. 2, pp. 311–314, 1987. solutions to parabolic, hyperbolic and elliptic partial differen- [8] J.S.Rao,“Flexuralvibration of pretwisted taperedcantilever tial equations,” Computers and Mathematics with Applications, blades,” ASME Journal of Engineering for Industry, vol. 94, no. vol. 19, no. 8-9, pp. 147–161, 1990. 1, pp. 343–346, 1972. [26] J. G. Wang and G. R. Liu, “A point interpolation meshless [9] J. S. Rao, “Vibrations of rotating, pretwisted and tapered method based on radial basis functions,” International Journal blades,” Mechanism and Machine Theory,vol. 12, no.4,pp. for Numerical Methods in Engineering, vol. 54, no. 11, pp. 331–337, 1977. 1623–1648, 2002. [10] S. Abrate, “Vibration of non-uniform rods and beams,” [27] L. A. Elfelsoufi, “Buckling, flutter and vibration analyses of Journal of Sound and Vibration, vol. 185, no. 4, pp. 703–716, beams by integral equation formulations,” Computers and Structures, vol. 83, no. 31-32, pp. 2632–2649, 2005. [11] D. H. Hodges, Y. Y. Chung, and X. Y. Shang, “Discrete transfer [28] Y. C. Hon, M. W. Lu, W. M. Xue, and Y. M. Zhu, “Multiquadric matric method for non-uniform rotating beams,” Journal of method for the numerical solution of a biphasic mixture Sound and Vibration, vol. 169, no. 2, pp. 276–283, 1994. model,” Applied Mathematics and Computation,vol. 88, no.2- [12] B. Dawson, “Coupled bending-bending vibrations of pre- 3, pp. 153–175, 1997. twisted cantilever blading treated by Rayleigh-Ritz energy method,” Journal of Mechanical Engineering Science, vol. 10, [29] G. R. Liu, X. Zhao,K.Y.Dai,Z.H.Zhong,G.Y.Li, andX. Han, “Static and free vibration analysis of laminated com- pp. 381–386, 1968. posite plates using the conforming radial point interpolation [13] B. Dawson and W. Carneige, “Model curves of pretwisted method,” Composites Science and Technology, vol. 68, no. 2, pp. beams of rectangular cross-section,” Journal of Mechanical 354–366, 2008. Engineering Science, vol. 11, pp. 1–13, 1969. [30] K. N. Devi and D. W. Pepper, “A meshless radial basis function [14] R. S. Gupta and S. S. Rao, “Finite element eigenvalue analysis of tapered and twisted Timoshenko beams,” JournalofSound method for fluid flow with heat transfer,” International Conference on Computational and Experimental Engineering and Vibration, vol. 56, no. 2, pp. 187–200, 1978. and Sciences , vol. 6, no. 1, pp. 13–18, 2008. [15] M. Swaminathan and J. S. Rao, “Vibrations of rotating, pretwisted and tapered blades,” Mechanism and Machine [31] L. Vrankar, G. Turk, and F. Runovc, “Modelling of radionu- Theory, vol. 12, no. 4, pp. 331–337, 1977. clide migration through the geosphere with radial basis function method and geostatistics,” Journal of the Chinese [16] K. B. Subrahmanyam, S. V. Kulkarni, and J. S. Rao, “Coupled bending-bending vibrations of pre-twisted cantilever blading Institute of Engineers, vol. 27, no. 4, pp. 455–462, 2004. Advances in Acoustics and Vibration 11 [32] Y. Qiao and L. 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Dynamic Analysis of Wind Turbine Blades Using Radial Basis Functions

Advances in Acoustics and Vibration , Volume 2011 – Aug 17, 2011

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Hindawi Publishing Corporation
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Copyright © 2011 Ming-Hung Hsu. 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.
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10.1155/2011/973591
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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2011, Article ID 973591, 11 pages doi:10.1155/2011/973591 Research Article Dynamic Analysis of Wind Turbine Blades Using Radial Basis Functions Ming-Hung Hsu Department of Electrical Engineering, National Penghu University of Science and Technology, Penghu 880, Taiwan Correspondence should be addressed to Ming-Hung Hsu, hsu.mh@msa.hinet.net Received 30 October 2010; Revised 11 April 2011; Accepted 11 April 2011 Academic Editor: K. M. Liew Copyright © 2011 Ming-Hung Hsu. 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. Wind turbine blades play important roles in wind energy generation. The dynamic problems associated with wind turbine blades are formulated using radial basis functions. The radial basis function procedure is used to transform partial differential equations, which represent the dynamic behavior of wind turbine blades, into a discrete eigenvalue problem. Numerical results demonstrate that rotational speed significantly impacts the first frequency of a wind turbine blade. Moreover, the pitch angle does not markedly affect wind turbine blade frequencies. This work examines the radial basis functions for dynamic problems of wind turbine blade. 1. Introduction applied the finite element method to identify the frequencies of natural vibration of doubly tapered and twisted beams. Wind energy will likely play an important role as a future Swaminathan and Rao [15] described the vibrations of a energy source in countries worldwide. Energy is essential rotating pretwisted and tapered blade. Subrahmanyam et al. to economic growth. In response to environmental con- [16, 17] derived the natural frequencies and mode shapes cerns, the amount of energy generated using renewable of a uniform pretwisted cantilever blade using the Reissner resources is increasing [1–3]. Herbert et al. [2]reviewed approach. Chen and Keer [18] examined the transverse assessment models for wind resources, site selection models, vibrations of a rotating pretwisted Timoshenko beam under and aerodynamic models, including the wake effect model. axial loading. Storti and Aboelnaga [19] examined transverse Varol et al. [3] demonstrated the positive effect of aerofoils deflections of a straight tapered symmetrical beam attached surrounding wind blades. Their notion of surrounding to a rotating hub as a model for the bending vibration of aerofoils resembles that of steering blades in a water turbine. blades. Wagner [20] derived the forced vibration response of Spee et al. [4] created a novel control strategy for a brushless subsystems with different natural frequencies and damping doubly fed machine applied to variable-speed wind-power attached to a foundation with a finite stiffness and mass. generation systems. Vitale and Rossi [5] designed low- Griffin and Sinha [21–23] determined the dynamic responses power, horizontal-axis wind turbine blades, using an iterative of frictionally damped turbine blades. Determining the algorithm. With their software, one can determine the dynamic features of wind turbine blades is of significant optimum blade shape for a wind turbine to satisfy the energy importance. Radial basis functions are important elements of requirements of an electrical system with optimum rotor effi- approaches generally called meshless methods. In this study, ciency. Song and Dhinakaran [6] developed a nonlinear wind dynamic problems associated with wind turbine blades are turbine control method. Storti and Aboelnaga [7]analyzed formulated using radial basis functions. transverse deflections of a straight tapered symmetrical beam attached to a rotating hub as a model for bending vibration of blades in turbomachinery. The natural frequencies of a single 2. Radial Basis Function tapered and pretwisted turboblade were calculated by Rao [8, 9], Abrate [10], Hodges et al. [11], and Dawson and Carneige A radial basis function is a real-value function whose [12, 13] using the Rayleigh-Ritz scheme. Gupta and Rao [14] value depends on distance from an origin. Kansa [24, 25] 2 Advances in Acoustics and Vibration investigated a given function or partial derivatives of a function with respect to a coordinate direction expressed as a linear weighted sum of all functional values at all mesh points along the direction initiated based on the concept of the radial basis function. In their algorithm, node distribution was entirely unstructured. Wang and Liu [26] developed a point interpolation meshless method based on radial basis functions that incorporated the Galerkin weak form for solving partial differential equations. Elfelsoufi [27] investigated buckling, flutter, and vibration of beams using radial basis functions. Hon et al. [28] utilized radial basis functions for function fitting and solving partial −100 100 300 500 700 900 differential equations using global nodes and collocation Rotational speed (rpm) procedures. Liu et al. [29] constructed shape functions with Mode 1 Mode 3 the delta function property using radial and polynomial basis Mode 2 Mode 4 functions. Devi and Pepper [30] generated a simplified radial basis function approach for calculating coupled heat transfer Figure 1: The first four frequencies of a wind turbine blade at different rotational speeds. ofafluidflow using alocal pressure correction scheme.In their study, shape functions were constructed using radial basis functions. Vrankar et al. [31] applied a relatively new approach for modeling radionuclide migration through the geosphere using a radial basis function scheme. Qiao and Ernst [32] applied a nonlinear approach for constructing color conversions based on radial basis functions. Their experimental results demonstrated that color conversion can be achieved effectively using the radial basis function approach when constructing nonlinear data maps. A radial 400 basis function can be expressed as follows [33, 34]: B (r) = (r − r ) + c,(1) i i where c is a shape parameter. The radial basis function is generally utilized to develop functional approximations in the following forms [33, 34]: u (r, t) = a (t)B (r),for k = 1, 2, 3, (2) k ki i −10 10 30 50 70 90 110 i=1 Pitch angle (deg) Mode 1 Mode 3 v (r, t) = a (t)B (r),for k = 1, 2, 3, (3) k ki i Mode 2 Mode 4 i=1 N Figure 2: The first four frequencies of a wind turbine blade at different pitch angles. ( ) ( ) U r = b B r ,for k = 1, 2, 3, (4) k ki i i=1 V (r) = b B (r),for k = 1, 2, 3, (5) k ki i i=1 3. Dynamic Analysis of Wind Turbine Blades where a , a , b ,and b are coefficients to be deter- ki ki ki ki The kinetic energy of rotating wind turbine blades can be mined. Blade deflection u (r, t) is the sum of N radial derived as basis functions, each associated with a different center r . Blade deflection v (r, t) denotes the sum of N radial basis functions, each associated with a different center r .Blade 3 L 2 2 1 ∂u ∂v k k 2 ( ) deflection U (r) is the sum of N radial basis functions, each T = ρA + + Ωv cos θ k k 2 ∂t ∂t k=1 associated with a different center r . Blade deflection V (r) i k (6) denotes the sum of N radial basis functions, each associated with a different center r . The domain contains N collocation i ( ) + Ωu sin θ dr, points. Frequency (Hz) Frequency (Hz) Advances in Acoustics and Vibration 3 where Ω is hub rotational speed, A is cross-sectional area, θ This leads to the following equations for the motion of the is pitch angle, ρ is blade density, and L is blade length. The blades: corresponding strain energy of rotating blades is 2 2 2 3 4 2 ∂ EI ∂EI ∂ EI yy ∂ u yy ∂ u ∂ u xy ∂ v k k k k +2 + EI + yy 2 2 3 4 2 2 ∂r ∂r ∂r ∂r ∂r ∂r ∂r L 2 1 ∂u ∂u ∂v k k k U = E I +2I 3 4 yy xy ∂EI ∂ v ∂ v xy k k 2 2 ∂r ∂r ∂r 2 k=1 +2 + EI − ρAΩ u sin θ xy k 3 4 ∂r ∂r ∂r ∂v ∂ ∂u +I dr k xx 2 − ρΩ A(r + r )dr ∂r 0 ∂r ∂r (7) L L L 2 ∂ u + ρAΩ (r + r )dr k 0 2 − ρΩ A(r + r )dr 2 0 0 r ∂r 2 2 ∂u ∂v k k 3 4 × + dr , ∂ EI ∂EI ∂u yy ∂ u yy ∂ u k k k ∂r ∂r + C + C +2C 0 1 1 2 2 3 ∂t ∂r ∂r ∂t ∂r ∂r ∂t 5 3 4 ∂ EI ∂EI ∂ u ∂ v ∂ v xy xy k k k + C EI + C +2C 1 yy 1 1 4 2 2 3 where r is hub radius, u (r, t) is the displacement of the first ∂r ∂t ∂r ∂r ∂t ∂r ∂r ∂t 0 1 blade on the x-axis, v (r, t) is the displacement of the first 5 2 ∂ v ∂ u k k blade on the y-axis, u (r, t) is the displacement of the second + C EI + ρA =0for k = 1, 2, 3, 1 xy 4 2 ∂r ∂t ∂t blade on the x-axis, v (r, t) is the displacement of the second 2 2 3 4 2 blade on the y-axis, u (r, t) is the displacement of the third ∂ EI 3 ∂ EI ∂ v ∂EI ∂ v ∂ v ∂ u xy xx k xx k k k +2 + EI + xx blade in the x-axis, v (r, t) is the displacement of the third 3 2 2 3 4 2 2 ∂r ∂r ∂r ∂r ∂r ∂r ∂r blade on the y-axis, and E is Young’s modulus. By examining 3 4 ∂EI ∂ u ∂ u xy k k the internal and external damping effects of blades, virtual 2 2 +2 + EI − ρAΩ v cos θ xy k 3 4 ∂r ∂r ∂r work δW in wind turbine blades can be derived as ∂ ∂v − ρΩ A(r + r )dr ∂r ∂r e r δW L 2 ∂ v 3 2 L L 2 ∂ EI − ρΩ A(r + r )dr ∂u yy ∂ u 0 k k ∂r = − C δu dr − C δu dr r 0 k 1 k 2 2 ∂t ∂r ∂t∂r 0 0 k=1 2 3 4 ∂v ∂ EI ∂ v ∂EI ∂ v k xx k xx k L + C + C +2C 0 1 1 ∂EI ∂ u 2 2 3 yy ∂t ∂r ∂r ∂t ∂r ∂r ∂t − 2 C δu dr 1 k ∂r ∂t∂r 5 3 4 ∂ EI ∂EI ∂ v ∂ u ∂ u xy xy k k k + C EI + C +2C L 5 1 xx 1 1 4 2 2 3 ∂ u k ∂r ∂t ∂r ∂r ∂t ∂r ∂r ∂t − C EI δu dr 1 yy k ∂t∂r 5 2 ∂ u ∂ v k k + C EI + ρA =0for k = 1, 2, 3. 1 xy L L 2 3 4 2 ∂v ∂ EI ∂ v ∂r ∂t ∂t k xx k − C δv dr − C δv dr 0 k 1 k (10) 2 2 ∂t ∂r ∂t∂r 0 0 L 4 ∂EI ∂ v xx k − 2 C δv dr 1 k The following equations are the corresponding boundary ∂r ∂t∂r conditions: ∂ v − C EI δv dr , 1 xx k ∂t∂r (8) u (0, t) =0for k = 1, 2, 3, ∂u (0, t) =0for k = 1, 2, 3, ∂r where C and C are external and internal damping coeffi- 0 1 cients of blades, respectively. Equations (6)–(8) are inte- ∂ u (L, t) EI =0for k = 1, 2, 3, yy grated into the Hamilton equation as follows: ∂r ∂ ∂ u (L, t) EI =0for k = 1, 2, 3, yy ∂r ∂r e e e (δT − δU + δW )dt = 0. (9) 1 v (0, t) =0for k = 1, 2, 3, k 4 Advances in Acoustics and Vibration ∂v (0, t) k d dV =0for k = 1, 2, 3, − ρΩ A(r + r )dr ∂r dr dr ∂ v (L, t) k L EI =0for k = 1, 2, 3, d V xx 2 − ρΩ A(r + r )dr ∂r 0 r dr ∂ ∂ v (L, t) 2 2 3 EI =0for k = 1, 2, 3. xx dV d EI d V dEI d V k xx k xx k ∂r ∂r + λC + λC +2λC 0 1 1 2 2 3 dt dr dr dr dr (11) 4 2 2 3 d EI dEI d V xy d U xy d U k k k + λC EI + λC +2λC 1 xx 1 1 4 2 2 3 dr dr dr dr dr Asolutionisthusassumed as d U λt 2 u = U (r)e for k = 1, 2, 3, + λC EI + λ ρAV =0for k = 1, 2, 3. k k 1 xy k dr (12) λt v = V (r)e for k = 1, 2, 3. k k (13) This yields the following equations for the motion of blades: The corresponding boundary conditions are as follows: 2 2 2 3 4 2 d EI dEI d EI d U d U d U d V yy yy xy k k k k +2 + EI + yy 2 2 3 4 2 2 U (0) =0for k = 1, 2, 3, dr dr dr dr dr dr dr 3 4 dEI d V d V dU (0) xy k k 2 k +2 + EI − ρAΩ U sin θ =0for k = 1, 2, 3, xy k 3 4 dr dr dr dr L 2 d U (L) d dU k EI =0for k = 1, 2, 3, − ρΩ A(r + r )dr yy dr dr dr L d d U (L) d U 2 EI =0for k = 1, 2, 3, yy ( ) − ρΩ A r + r dr 2 dr dr dr (14) 2 V (0) =0for k = 1, 2, 3, 2 3 k d EI dEI yy d U yy d U k k + λC U + λC +2λC 0 k 1 1 2 2 3 dr dr dr dr dV (0) =0for k = 1, 2, 3, 4 2 dr d EI d U d V k xy k + λC EI + λC 1 yy 1 4 2 2 dr dr dr d V (L) EI =0for k = 1, 2, 3, xx 3 4 dr dEI xy d V d V k k +2λC + λC EI 1 1 xy 3 4 dr dr dr d d V (L) EI =0for k = 1, 2, 3. xx dr dr + λ ρAU =0for k = 1, 2, 3, 2 2 3 4 2 d EI d EI d V dEI d V d V xy d U xx k xx k k k By employing the radial basis function approach, (2)and +2 + EI + xx 2 2 3 4 2 2 dr dr dr dr dr dr dr (3) can be substituted into (10). The equations of motion of wind turbine blades can be rearranged in the following 3 4 dEI xy d U d U k k 2 2 +2 + EI − ρAΩ V cos θ matrix form: xy k 3 4 dr dr dr 2 2 2 ∂ a ∂ a ∂ a k1 k2 kN ρA(r )B (r ) ρA(r )B (r ) ··· ρA(r )B (r ) ··· i 1 i i 2 i i N i 2 2 2 ∂t ∂t ∂t ∂a ∂a ∂a k1 k2 kN + [C B (r ) C B (r ) ··· C B (r )] ··· 0 1 i 0 2 i 0 N i ∂t ∂t ∂t 2 2 2 T 2 2 2 ∂ EI (r ) ∂ EI (r ) ∂ EI (r ) ∂ B (r ) ∂ B (r ) ∂ B (r ) yy i 1 i yy i 2 i yy i N i ∂a ∂a ∂a k1 k2 kN + C C ··· C ··· 1 1 1 2 2 2 2 2 2 ∂r ∂r ∂r ∂r ∂r ∂r ∂t ∂t ∂t 3 3 3 T ∂EI (r ) ∂EI (r ) ∂EI (r ) yy i ∂ B (r ) yy i ∂ B (r ) yy i ∂ B (r ) ∂a ∂a ∂a 1 i 2 i N i k1 k2 kN + 2C 2C ··· 2C 1 1 1 ··· 3 3 3 ∂r ∂r ∂r ∂r ∂r ∂r ∂t ∂t ∂t 4 4 4 ∂ B (r ) ∂ B (r ) ∂ B (r ) ∂a ∂a ∂a 1 i 2 i N i k1 k2 kN ( ) ( ) ( ) + C EI r C EI r ··· C EI r ··· 1 yy i 1 yy i 1 yy i 4 4 4 ∂r ∂r ∂r ∂t ∂t ∂t Advances in Acoustics and Vibration 5 2 2 2 T 2 2 2 ∂ EI (r ) ∂ EI (r ) ∂ EI (r ) ∂ B (r ) ∂ B (r ) ∂ B (r ) xy i xy i xy i ∂a ∂a ∂a 1 i 2 i N i k1 k2 kN + C C ··· C ··· 1 1 1 2 2 2 2 2 2 ∂r ∂r ∂r ∂r ∂r ∂r ∂t ∂t ∂t 3 3 3 ∂EI (r ) ∂EI (r ) ∂EI (r ) xy i ∂ B (r ) xy i ∂ B (r ) xy i ∂ B (r ) ∂a ∂a ∂a 1 i 2 i N i k1 k2 kN + 2C 2C ··· 2C 1 1 1 ··· 3 3 3 ∂r ∂r ∂r ∂r ∂r ∂r ∂t ∂t ∂t 4 4 4 ∂ B (r ) ∂ B (r ) ∂ B (r ) ∂a ∂a ∂a 1 i 2 i N i k1 k2 kN + C EI (r ) C EI (r ) ··· C EI (r ) ··· 1 xy i 1 xy i 1 xy i 4 4 4 ∂r ∂r ∂r ∂t ∂t ∂t 2 2 2 2 2 2 ∂ EI (r ) ∂ EI (r ) ∂ EI (r ) T yy i ∂ B (r ) yy i ∂ B (r ) yy i ∂ B (r ) 1 i 2 i N i + ··· a a ··· a k1 k2 kN 2 2 2 2 2 2 ∂r ∂r ∂r ∂r ∂r ∂r 3 3 3 ( ) ( ) ( ) T ∂EI r ∂ B (r ) ∂EI r ∂ B (r ) ∂EI r ∂ B (r ) yy i yy i yy i 1 i 2 i N i a a ··· a + 2 2 ··· 2 k1 k2 kN 3 3 3 ∂r ∂r ∂r ∂r ∂r ∂r 4 4 4 ∂ B (r ) ∂ B (r ) ∂ B (r ) 1 i 2 i N i + EI (r ) EI (r ) ··· EI (r ) a a ··· a k1 k2 kN yy i yy i yy i 4 4 4 ∂r ∂r ∂r 2 2 2 2 2 2 ∂ EI (r ) ( ) ∂ EI (r ) ( ) ∂ EI (r ) ( ) xy i ∂ B r xy i ∂ B r xy i ∂ B r 1 i 2 i N i a a ··· a + ··· k1 k2 kN 2 2 2 2 2 2 ∂r ∂r ∂r ∂r ∂r ∂r 3 3 3 ∂EI (r ) ∂EI (r ) ∂EI (r ) T ∂ B (r ) ∂ B (r ) ∂ B (r ) xy i 1 i xy i 2 i xy i N i + 2 2 ··· 2 a a ··· a k1 k2 kN 3 3 3 ∂r ∂r ∂r ∂r ∂r ∂r 4 4 4 ∂ B (r ) ∂ B (r ) ∂ B (r ) 1 i 2 i N i a a ··· a + EI (r ) EI (r ) ··· EI (r ) xy i xy i xy i k1 k2 kN 4 4 4 ∂r ∂r ∂r 2 2 2 2 2 2 a a ··· a − ρA(r)Ω (sin θ) B (r ) ρA(r)Ω (sin θ) B (r ) ··· ρA(r)Ω (sin θ) B (r ) k1 k2 kN 1 i 2 i N i L L L ∂ ∂B (r ) ∂ ∂B (r ) ∂ ∂B (r ) 1 i 2 i N i 2 2 2 − ρΩ A(r)Adr ρΩ A(r)Adr ··· ρΩ A(r)Adr ∂r ∂r ∂r ∂r ∂r ∂r r r r i i i a a ··· a k1 k2 kN L L L 2 2 2 ∂ B (r ) ∂ B (r ) ∂ B (r ) 1 i 2 i N i 2 2 2 ( ) ( ) ( ) − ρΩ A r Adr ρΩ A r Adr ··· ρΩ A r Adr 2 2 2 ∂r ∂r ∂r r r r i i i × a a ··· a = [0],for i = 1, 2,... , N, k = 1, 2, 3, k1 k2 kN 2 2 2 ∂ a ∂ a ∂ a k1 k2 kN ρA(r )B (r ) ρA(r )B (r ) ··· ρA(r )B (r ) i 1 i i 2 i i N i ··· 2 2 2 ∂t ∂t ∂t ∂a ∂a ∂a k1 k2 kN + [C B (r ) C B (r ) ··· C B (r )] ··· 0 1 i 0 2 i 0 N i ∂t ∂t ∂t 2 2 2 2 2 2 ∂ EI (r ) ∂ B (r ) ∂ EI (r ) ∂ B (r ) ∂ EI (r ) ∂ B (r ) ∂a ∂a ∂a xx i 1 i xx i 2 i xx i N i k1 k2 kN + C C ··· C ··· 1 1 1 2 2 2 2 2 2 ∂r ∂r ∂r ∂r ∂r ∂r ∂t ∂t ∂t 3 3 3 T ∂EI (r ) ∂ B (r ) ∂EI (r ) ∂ B (r ) ∂EI (r ) ∂ B (r ) ∂a ∂a ∂a xx i 1 i xx i 2 i xx i N i k1 k2 kN + 2C 2C ··· 2C 1 1 1 ··· 3 3 3 ∂r ∂r ∂r ∂r ∂r ∂r ∂t ∂t ∂t 4 4 4 ∂ B (r ) ∂ B (r ) ∂ B (r ) ∂a ∂a ∂a 1 i 2 i N i k1 k2 kN ( ) ( ) ( ) + C EI r C EI r ··· C EI r ··· 1 xx i 1 xx i 1 xx i 4 4 4 ∂r ∂r ∂r ∂t ∂t ∂t 2 2 2 2 2 2 T ∂ EI (r ) ∂ EI (r ) ∂ EI (r ) xy i ∂ B (r ) xy i ∂ B (r ) xy i ∂ B (r ) ∂a ∂a ∂a 1 i 2 i N i k1 k2 kN + C C ··· C 1 1 1 ··· 2 2 2 2 2 2 ∂r ∂r ∂r ∂r ∂r ∂r ∂t ∂t ∂t 6 Advances in Acoustics and Vibration 3 3 3 ∂EI (r ) ∂EI (r ) ∂EI (r ) ∂ B (r ) ∂ B (r ) ∂ B (r ) xy i xy i xy i ∂a ∂a ∂a 1 i 2 i N i k1 k2 kN + 2C 2C ··· 2C ··· 1 1 1 3 3 3 ∂r ∂r ∂r ∂r ∂r ∂r ∂t ∂t ∂t 4 4 4 ∂ B (r ) ∂ B (r ) ∂ B (r ) ∂a ∂a ∂a 1 i 2 i N i k1 k2 kN + C EI (r ) C EI (r ) ··· C EI (r ) 1 xy i 1 xy i 1 xy i ··· 4 4 4 ∂r ∂r ∂r ∂t ∂t ∂t 2 2 2 2 2 2 ∂ EI (r ) ∂ B (r ) ∂ EI (r ) ∂ B (r ) ∂ EI (r ) ∂ B (r ) xx i 1 i xx i 2 i xx i N i a a ··· a + ··· k1 k2 kN 2 2 2 2 2 2 ∂r ∂r ∂r ∂r ∂r ∂r 3 3 3 ∂EI (r ) ∂ B (r ) ∂EI (r ) ∂ B (r ) ∂EI (r ) ∂ B (r ) xx i 1 i xx i 2 i xx i N i + 2 2 ··· 2 a a ··· a k1 k2 kN 3 3 3 ∂r ∂r ∂r ∂r ∂r ∂r 4 4 4 ∂ B (r ) ∂ B (r ) ∂ B (r ) 1 i 2 i N i a a ··· a + EI (r ) EI (r ) ··· EI (r ) xx i xx i xx i k1 k2 kN 4 4 4 ∂r ∂r ∂r 2 2 2 2 2 2 ∂ EI (r ) ∂ EI (r ) ∂ EI (r ) T xy i ∂ B (r ) xy i ∂ B (r ) xy i ∂ B (r ) 1 i 2 i N i + ··· a a ··· a k1 k2 kN 2 2 2 2 2 2 ∂r ∂r ∂r ∂r ∂r ∂r 3 3 3 ∂EI (r ) ( ) ∂EI (r ) ( ) ∂EI (r ) ( ) xy i ∂ B r xy i ∂ B r xy i ∂ B r 1 i 2 i N i a a ··· a + 2 2 ··· 2 k1 k2 kN 3 3 3 ∂r ∂r ∂r ∂r ∂r ∂r 4 4 4 ∂ B (r ) ∂ B (r ) ∂ B (r ) 1 i 2 i N i + EI (r ) EI (r ) ··· EI (r ) a a ··· a k1 k2 kN xy i xy i xy i 4 4 4 ∂r ∂r ∂r 2 2 2 2 2 2 − ρA(r)Ω (cos θ) B (r ) ρA(r)Ω (cos θ) B (r ) ··· ρA(r)Ω (cos θ) B (r ) a a ··· a 1 i 2 i N i k1 k2 kN L L L ∂ ∂B (r ) ∂ ∂B (r ) ∂ ∂B (r ) 1 i 2 i N i 2 2 2 − ρΩ A(r)Adr ρΩ A(r)Adr ··· ρΩ A(r)Adr ∂r ∂r ∂r ∂r ∂r ∂r r r r i i i × a a ··· a k1 k2 kN L 2 L 2 L 2 ∂ B (r ) ∂ B (r ) ∂ B (r ) 1 i 2 i N i 2 2 2 − ρΩ A(r)Adr ρΩ A(r)Adr ··· ρΩ A(r)Adr 2 2 2 r ∂r r ∂r r ∂r i i i a a ··· a × = [0] for i = 1, 2,... , N, k = 1, 2, 3, k1 k2 kN (15) where A denotes (r + r ). Based on the radial basis function technique, (11)take the following discrete forms: a a ··· a [B (r ) B (r ) ··· B (r )] = [0] for k = 1, 2, 3, 1 1 2 1 N 1 k1 k2 kN ∂B (r ) ∂B (r ) ∂B (r ) 1 1 2 1 N 1 T ··· [a a ··· a ] = [0] for k = 1, 2, 3, k1 k2 kN ∂r ∂r ∂r 2 2 2 ∂ B (r ) ∂ B (r ) ∂ B (r ) 1 N 2 N N N T EI (r ) EI (r ) ··· EI (r ) [a a ··· a ] = [0] for k = 1, 2, 3, yy N yy N yy N k1 k2 kN 2 2 2 ∂r ∂r ∂r 2 2 2 ∂EI (r ) ∂EI (r ) ∂EI (r ) ∂ B (r ) ∂ B (r ) ∂ B (r ) yy N yy N yy N 1 N 2 N N N T ··· [a a ··· a ] k1 k2 kN 2 2 2 ∂r ∂r ∂r ∂r ∂r ∂r 3 3 3 ∂ B (r ) ∂ B (r ) ∂ B (r ) 1 N 2 N N N + EI (r ) EI (r ) ··· EI (r ) yy N yy N yy N 3 3 3 ∂r ∂r ∂r ×[a a ··· a ] = [0] for k = 1, 2, 3, k1 k2 kN Advances in Acoustics and Vibration 7 [B (r ) B (r ) ··· B (r )][a a ··· a ] = [0] for k = 1, 2, 3, 1 1 2 1 N 1 k1 k2 kN ∂B (r ) ∂B (r ) ∂B (r ) 1 1 2 1 N 1 ··· [a a ··· a ] = [0] for k = 1, 2, 3, k1 k2 kN ∂r ∂r ∂r 2 2 2 ∂ B (r ) ∂ B (r ) ∂ B (r ) 1 N 2 N N N EI (r ) EI (r ) ··· EI (r ) [a a ··· a ] = [0],for k = 1, 2, 3, xx N xx N xx N k1 k2 kN 2 2 2 ∂r ∂r ∂r 2 2 2 ∂EI (r ) ∂ B (r ) ∂EI (r ) ∂ B (r ) ∂EI (r ) ∂ B (r ) xx N 1 N xx N 2 N xx N N N a a ··· a ··· k1 k2 kN 2 2 2 ∂r ∂r ∂r ∂r ∂r ∂r 3 3 3 ∂ B (r ) ∂ B (r ) ∂ B (r ) N N 2 N N N + EI (r ) EI (r ) ··· EI (r ) xx N xx N xx N 3 3 3 ∂r ∂r ∂r ×[a a ··· a ] = [0] for k = 1, 2, 3. k1 k2 kN (16) By applying the radial basis function approach, (4)and (5) are substituted into (13). The following equations are then yielded 2 2 2 λ ρA(r )B (r ) λ ρA(r )B (r ) ··· λ ρA(r )B (r ) [b b ··· b ] i 1 i i 2 i i N i k1 k2 kN + [λC B (r ) λC B (r ) ··· λC B (r )][b b ··· b ] 0 1 i 0 2 i 0 N i k1 k2 kN 2 2 2 2 2 2 d EI (r ) d EI (r ) d EI (r ) yy i d B (r ) yy i d B (r ) yy i d B (r ) 1 i 2 i N i T + λC λC ··· λC [b b ··· b ] 1 1 1 k1 k2 kN 2 2 2 2 2 2 dr dr dr dr dr dr 3 3 3 dEI (r ) dEI (r ) dEI (r ) d B (r ) d B (r ) d B (r ) yy i yy i yy i 1 i 2 i N i T + 2λC 2λC ··· 2λC [b b ··· b ] 1 1 1 k1 k2 kN 3 3 3 dr dr dr dr dr dr 4 4 4 d B (r ) d B (r ) d B (r ) 1 i 2 i N i + λC EI (r ) λC EI (r ) ··· λC EI (r ) [b b ··· b ] 1 yy i 1 yy i 1 yy i k1 k2 kN 4 4 4 dr dr dr 2 2 2 2 2 2 ( ) ( ) ( ) T d EI r d B (r ) d EI r d B (r ) d EI r d B (r ) xy i xy i xy i 1 i 2 i N i + λC λC ··· λC b b ··· b 1 1 1 k1 k2 kN 2 2 2 2 2 2 dr dr dr dr dr dr 3 3 3 dEI (r ) dEI (r ) dEI (r ) T xy i d B (r ) xy i d B (r ) xy i d B (r ) 1 i 2 i N i + 2λC 2λC ··· 2λC b b ··· b 1 1 1 k1 k2 kN 3 3 3 dr dr dr dr dr dr 4 4 4 ( ) ( ) ( ) d B r d B r d B r 1 i 2 i N i + λC EI (r ) λC EI (r ) ··· λC EI (r ) b b ··· b 1 xy i 1 xy i 1 xy i k1 k2 kN 4 4 4 dr dr dr 2 2 2 2 2 2 d EI (r ) d EI (r ) d EI (r ) yy i d B (r ) yy i d B (r ) yy i d B (r ) 1 i 2 i N i + ··· [b b ··· b ] k1 k2 kN 2 2 2 2 2 2 dr dr dr dr dr dr 3 3 3 dEI (r ) dEI (r ) dEI (r ) yy i d B (r ) yy i d B (r ) yy i d B (r ) 1 i 2 i N i T + 2 2 ··· 2 [b b ··· b ] k1 k2 kN 3 3 3 dr dr dr dr dr dr 4 4 4 d B (r ) d B (r ) d B (r ) 1 i 2 i N i + EI (r ) EI (r ) ··· EI (r ) [b b ··· b ] yy i yy i yy i k1 k2 kN 4 4 4 dr dr dr 2 2 2 2 2 2 d EI (r ) d EI (r ) d EI (r ) xy i d B (r ) xy i d B (r ) xy i d B (r ) 1 i 2 i N i + ··· b b ··· b k1 k2 kN 2 2 2 2 2 2 dr dr dr dr dr dr 3 3 3 dEI (r ) dEI (r ) dEI (r ) T d B (r ) d B (r ) d B (r ) xy i xy i xy i 1 i 2 i N i + 2 2 ··· 2 b b ··· b k1 k2 kN 3 3 3 dr dr dr dr dr dr 8 Advances in Acoustics and Vibration 4 4 4 d B (r ) d B (r ) d B (r ) 1 i 2 i N i + EI (r ) EI (r ) ··· EI (r ) b b ··· b xy i xy i xy i k1 k2 kN 4 4 4 dr dr dr 2 2 2 2 2 2 T − ρA(r )Ω (sin θ) B (r ) ρA(r )Ω (sin θ) B (r ) ··· ρA(r )Ω (sin θ) B (r ) [b b ··· b ] i 1 i i 2 i i N i k1 k2 kN L L L ( ) ( ) ( ) d dB r d dB r d dB r 1 i 2 i N i 2 2 2 − ρΩ A(r)Adr ρΩ A(r)Adr ··· ρΩ A(r)Adr dr dr dr dr dr dr r r r i i i [ ] × b b ··· b k1 k2 kN L 2 L 2 L 2 d B (r ) d B (r ) d B (r ) 1 i 2 i N i 2 2 2 − ρΩ A(r)Adr ρΩ A(r)Adr ··· ρΩ A(r)Adr 2 2 2 dr dr dr r r r i i i × [b b ··· b ] = [0] for i = 1, 2,... , N, k = 1, 2, 3, k1 k2 kN 2 2 2 λ ρA(r )B (r ) λ ρA(r )B (r ) ··· λ ρA(r )B (r ) b b ··· b i 1 i i 2 i i N i k1 k2 kN + [λC B (r ) λC B (r ) ··· λC B (r )] b b ··· b 0 1 i 0 2 i 0 N i k1 k2 kN 2 2 2 2 2 2 d EI (r ) d B (r ) d EI (r ) d B (r ) d EI (r ) d B (r ) xx i 1 i xx i 2 i xx i N i + λC λC ··· λC b b ··· b 1 1 1 k1 k2 kN 2 2 2 2 2 2 dr dr dr dr dr dr 3 3 3 dEI (r ) d B (r ) dEI (r ) d B (r ) dEI (r ) d B (r ) xx i 1 i xx i 2 i xx i N i + 2λC 2λC ··· 2λC b b ··· b 1 1 1 k1 k2 kN 3 3 3 dr dr dr dr dr dr 4 4 4 ( ) ( ) ( ) d B r d B r d B r 1 i 2 i N i + λC EI (r ) λC EI (r ) ··· λC EI (r ) b b ··· b 1 xx i 1 xx i 1 xx i k1 k2 kN 4 4 4 dr dr dr 2 2 2 2 2 2 d EI (r ) d EI (r ) d EI (r ) xy i d B (r ) xy i d B (r ) xy i d B (r ) 1 i 2 i N i + λC λC ··· λC [b b ··· b ] 1 1 1 k1 k2 kN 2 2 2 2 2 2 dr dr dr dr dr dr 3 3 3 dEI (r ) dEI (r ) dEI (r ) xy i d B (r ) xy i d B (r ) xy i d B (r ) 1 i 2 i N i T + 2λC 2λC ··· 2λC [b b ··· b ] 1 1 1 k1 k2 kN 3 3 3 dr dr dr dr dr dr 4 4 4 d B (r ) d B (r ) d B (r ) 1 i 2 i N i + λC EI (r ) λC EI (r ) ··· λC EI (r ) [b b ··· b ] 1 xy i 1 xy i 1 xy i k1 k2 kN 4 4 4 dr dr dr 2 2 2 2 2 2 d EI (r ) d B (r ) d EI (r ) d B (r ) d EI (r ) d B (r ) xx i 1 i xx i 2 i xx i N i + ··· b b ··· b k1 k2 kN 2 2 2 2 2 2 dr dr dr dr dr dr 3 3 3 dEI (r ) d B (r ) dEI (r ) d B (r ) dEI (r ) d B (r ) xx i 1 i xx i 2 i xx i N i + 2 2 ··· 2 b b ··· b k1 k2 kN 3 3 3 dr dr dr dr dr dr 4 4 4 d B (r ) d B (r ) d B (r ) 1 i 2 i N i + EI (r ) EI (r ) ··· EI (r ) b b ··· b xx i xx i xx i k1 k2 kN 4 4 4 dr dr dr 2 2 2 2 2 2 d EI (r ) d EI (r ) d EI (r ) d B (r ) d B (r ) d B (r ) xy i xy i xy i 1 i 2 i N i T [ ] + ··· b b ··· b k1 k2 kN 2 2 2 2 2 2 dr dr dr dr dr dr 3 3 3 dEI (r ) dEI (r ) dEI (r ) xy i d B (r ) xy i d B (r ) xy i d B (r ) 1 i 2 i N i T + 2 2 ··· 2 [b b ··· b ] k1 k2 kN 3 3 3 dr dr dr dr dr dr 4 4 4 d B (r ) d B (r ) d B (r ) 1 i 2 i N i T ( ) ( ) ( ) + EI r EI r ··· EI r [b b ··· b ] xy i xy i xy i k1 k2 kN 4 4 4 dr dr dr 2 2 2 2 2 2 − ρA(r )Ω (cos θ) B (r ) ρA(r )Ω (cos θ) B (r ) ··· ρA(r )Ω (cos θ) B (r ) b b ··· b i 1 i i 2 i i N i k1 k2 kN Advances in Acoustics and Vibration 9 L L L d dB (r ) d dB (r ) d dB (r ) 1 i 2 i N i 2 2 2 − ρΩ A(r)Adr ρΩ A(r)Adr ··· ρΩ A(r)Adr dr dr dr dr dr dr r r r i i i × b b ··· b k1 k2 kN L 2 L 2 L 2 d B (r ) d B (r ) d B (r ) 1 i 2 i N i 2 2 2 − ρΩ A(r)Adr ρΩ A(r)Adr ··· ρΩ A(r)Adr 2 2 2 dr dr dr r r r i i i × b b ··· b = [0] for i = 1, 2,... , N, k = 1, 2, 3. k1 k2 kN (17) where A denotes (r + r ). According to the radial basis function approach, the boundary conditions in (14) have the following discrete forms: [B (r ) B (r ) ··· B (r )][b b ··· b ] = [0] for k = 1, 2, 3, 1 1 2 1 N 1 k1 k2 kN ( ) ( ) ( ) dB r dB r dB r 1 1 2 1 N 1 T ··· [b b ··· b ] = [0] for k = 1, 2, 3, k1 k2 kN dr dr dr 2 2 2 d B (r ) d B (r ) d B (r ) 1 N 2 N N N T EI (r ) EI (r ) ··· EI (r ) [b b ··· b ] = [0] for k = 1, 2, 3, yy N yy N yy N k1 k2 kN 2 2 2 dr dr dr 2 2 2 dEI (r ) dEI (r ) dEI (r ) d B (r ) d B (r ) d B (r ) yy N yy N yy N 1 N 2 N N N T ··· [b b ··· b ] k1 k2 kN 2 2 2 dr dr dr dr dr dr 3 3 3 d B (r ) d B (r ) d B (r ) 1 N 2 N N N + EI (r ) EI (r ) ··· EI (r ) yy N yy N yy N 3 3 3 dr dr dr ×[b b ··· b ] = [0] for k = 1, 2, 3, k1 k2 kN T (18) [B (r ) B (r ) ··· B (r )] b b ··· b = [0] for k = 1, 2, 3, 1 1 2 1 N 1 k1 k2 kN dB (r ) dB (r ) dB (r ) 1 1 2 1 N 1 ··· b b ··· b = [0] for k = 1, 2, 3, k1 k2 kN dr dr dr 2 2 2 d B (r ) d B (r ) d B (r ) 1 N 2 N N N EI (r ) EI (r ) ··· EI (r ) b b ··· b = [0] for k = 1, 2, 3, xx N xx N xx N k1 k2 kN 2 2 2 dr dr dr 2 2 2 dEI (r ) d B (r ) dEI (r ) d B (r ) dEI (r ) d B (r ) xx N 1 N xx N 2 N xx N N N ··· b b ··· b k1 k2 kN 2 2 2 dr dr dr dr dr dr 3 3 3 d B (r ) d B (r ) d B (r ) 1 N 2 N N N + EI (r ) EI (r ) ··· EI (r ) xx N xx N xx N 3 3 3 dr dr dr × b b ··· b = [0] for k = 1, 2, 3. k1 k2 kN 4. Results rotational speed increases. Numerical results obtained by this study suggest that rotational speed does not significantly Figure 1 illustrates the effects of various rotational speeds affect the second, third, and fourth frequencies of a wind on calculated frequencies for a wind turbine blade. Com- blade. Figure 2 lists the frequencies of a wind turbine blade putational results suggest that the first frequency of a wind at different pitch angles. Numerical results demonstrate that turbine blade is strongly dependent on blade shaft speed. pitch angle does not significantly affect the first, second, third, and fourth frequencies of a wind turbine blade. The first frequency of a wind turbine blade increases as 10 Advances in Acoustics and Vibration 5. 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