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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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