Advances in Acoustics and Vibration

Advances in Acoustics and Vibration / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 973591 | https://doi.org/10.1155/2011/973591

Ming-Hung Hsu, "Dynamic Analysis of Wind Turbine Blades Using Radial Basis Functions", Advances in Acoustics and Vibration, vol. 2011, Article ID 973591, 11 pages, 2011. https://doi.org/10.1155/2011/973591

Dynamic Analysis of Wind Turbine Blades Using Radial Basis Functions

Academic Editor: K. M. Liew
Received30 Oct 2010
Revised11 Apr 2011
Accepted11 Apr 2011
Published17 Aug 2011

Abstract

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

Wind energy will likely play an important role as a future energy source in countries worldwide. Energy is essential to economic growth. In response to environmental concerns, the amount of energy generated using renewable resources is increasing [1–3]. Herbert et al. [2] reviewed assessment models for wind resources, site selection models, and aerodynamic models, including the wake effect model. Varol et al. [3] demonstrated the positive effect of aerofoils surrounding wind blades. Their notion of surrounding aerofoils resembles that of steering blades in a water turbine. Spee et al. [4] created a novel control strategy for a brushless doubly fed machine applied to variable-speed wind-power generation systems. Vitale and Rossi [5] designed low-power, horizontal-axis wind turbine blades, using an iterative algorithm. With their software, one can determine the optimum blade shape for a wind turbine to satisfy the energy requirements of an electrical system with optimum rotor efficiency. Song and Dhinakaran [6] developed a nonlinear wind turbine control method. Storti and Aboelnaga [7] analyzed 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 tapered and pretwisted turboblade were calculated by Rao [8, 9], Abrate [10], Hodges et al. [11], and Dawson and Carneige [12, 13] using the Rayleigh-Ritz scheme. Gupta and Rao [14] applied the finite element method to identify the frequencies of natural vibration of doubly tapered and twisted beams. Swaminathan and Rao [15] described the vibrations of a rotating pretwisted and tapered blade. Subrahmanyam et al. [16, 17] derived the natural frequencies and mode shapes of a uniform pretwisted cantilever blade using the Reissner approach. Chen and Keer [18] examined the transverse vibrations of a rotating pretwisted Timoshenko beam under axial loading. Storti and Aboelnaga [19] examined transverse deflections of a straight tapered symmetrical beam attached to a rotating hub as a model for the bending vibration of blades. Wagner [20] derived the forced vibration response of subsystems with different natural frequencies and damping attached to a foundation with a finite stiffness and mass. Griffin and Sinha [21–23] determined the dynamic responses of frictionally damped turbine blades. Determining the dynamic features of wind turbine blades is of significant importance. Radial basis functions are important elements of approaches generally called meshless methods. In this study, dynamic problems associated with wind turbine blades are formulated using radial basis functions.

2. Radial Basis Function

A radial basis function is a real-value function whose value depends on distance from an origin. Kansa [24, 25] 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 differential equations using global nodes and collocation procedures. Liu et al. [29] constructed shape functions with the delta function property using radial and polynomial basis functions. Devi and Pepper [30] generated a simplified radial basis function approach for calculating coupled heat transfer of a fluid flow using a local 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 basis function can be expressed as follows [33, 34]: 𝐡𝑖(ξ”π‘Ÿ)=ξ€·π‘Ÿβˆ’π‘Ÿπ‘–ξ€Έ2+𝑐2,(1) where 𝑐 is a shape parameter. The radial basis function is generally utilized to develop functional approximations in the following forms [33, 34]:π‘’π‘˜(π‘Ÿ,𝑑)=𝑁𝑖=1π‘Žπ‘˜π‘–(𝑑)𝐡𝑖𝑣(π‘Ÿ),forπ‘˜=1,2,3,(2)π‘˜(π‘Ÿ,𝑑)=𝑁𝑖=1π‘Žπ‘˜π‘–(𝑑)π΅π‘–π‘ˆ(π‘Ÿ),forπ‘˜=1,2,3,(3)π‘˜(π‘Ÿ)=𝑁𝑖=1π‘π‘˜π‘–π΅π‘–π‘‰(π‘Ÿ),forπ‘˜=1,2,3,(4)π‘˜(π‘Ÿ)=𝑁𝑖=1π‘π‘˜π‘–π΅π‘–(π‘Ÿ),forπ‘˜=1,2,3,(5) where π‘Žπ‘˜π‘–, π‘Žπ‘˜π‘–, π‘π‘˜π‘–, and π‘π‘˜π‘– are coefficients to be determined. Blade deflection π‘’π‘˜(π‘Ÿ,𝑑) is the sum of 𝑁 radial basis functions, each associated with a different center π‘Ÿπ‘–. Blade deflection π‘£π‘˜(π‘Ÿ,𝑑) denotes the sum of 𝑁 radial basis functions, each associated with a different center π‘Ÿπ‘–. Blade deflection π‘ˆπ‘˜(π‘Ÿ) is the sum of 𝑁 radial basis functions, each associated with a different center π‘Ÿπ‘–. Blade deflection π‘‰π‘˜(π‘Ÿ) denotes the sum of 𝑁 radial basis functions, each associated with a different center π‘Ÿπ‘–. The domain contains 𝑁 collocation points.

3. Dynamic Analysis of Wind Turbine Blades

The kinetic energy of rotating wind turbine blades can be derived as𝑇𝑒=3ξ“π‘˜=112ξ€œπΏ0ξƒ©ξ‚΅πœŒπ΄πœ•π‘’π‘˜ξ‚Άπœ•π‘‘2+ξ‚΅πœ•π‘£π‘˜ξ‚Άπœ•π‘‘2+ξ€·Ξ©π‘£π‘˜ξ€Έcosπœƒ2+ξ€·Ξ©π‘’π‘˜ξ€Έsinπœƒ2ξƒͺπ‘‘π‘Ÿ,(6) where Ξ© is hub rotational speed, 𝐴 is cross-sectional area, πœƒ is pitch angle, 𝜌 is blade density, and 𝐿 is blade length. The corresponding strain energy of rotating blades isπ‘ˆπ‘’=3ξ“π‘˜=112ξ€œπΏ0πΈξƒ©πΌπ‘¦π‘¦ξ‚΅πœ•π‘’π‘˜ξ‚Άπœ•π‘Ÿ2+2𝐼π‘₯π‘¦ξ‚΅πœ•π‘’π‘˜πœ•π‘Ÿξ‚Άξ‚΅πœ•π‘£π‘˜ξ‚Άπœ•π‘Ÿ+𝐼π‘₯π‘₯ξ‚΅πœ•π‘£π‘˜ξ‚Άπœ•π‘Ÿ2ξƒͺ+1π‘‘π‘Ÿ2ξ€œπΏ0ξ€œπΏπ‘ŸπœŒπ΄Ξ©2ξ€·π‘Ÿ+π‘Ÿ0ξ€ΈΓ—ξƒ©ξ‚΅π‘‘π‘Ÿπœ•π‘’π‘˜ξ‚Άπœ•π‘Ÿ2+ξ‚΅πœ•π‘£π‘˜ξ‚Άπœ•π‘Ÿ2ξƒͺξƒͺ,π‘‘π‘Ÿ(7) where π‘Ÿ0 is hub radius, 𝑒1(π‘Ÿ,𝑑) is the displacement of the first blade on the π‘₯-axis, 𝑣1(π‘Ÿ,𝑑) is the displacement of the first blade on the 𝑦-axis, 𝑒2(π‘Ÿ,𝑑) is the displacement of the second blade on the π‘₯-axis, 𝑣2(π‘Ÿ,𝑑) is the displacement of the second blade on the 𝑦-axis, 𝑒3(π‘Ÿ,𝑑) is the displacement of the third blade in the π‘₯-axis, 𝑣3(π‘Ÿ,𝑑) is the displacement of the third blade on the 𝑦-axis, and 𝐸 is Young’s modulus. By examining the internal and external damping effects of blades, virtual work π›Ώπ‘Šπ‘’ in wind turbine blades can be derived asπ›Ώπ‘Šπ‘’=3ξ“π‘˜=1ξƒ©βˆ’ξ€œπΏ0𝐢0πœ•π‘’π‘˜πœ•π‘‘π›Ώπ‘’π‘˜ξ€œπ‘‘π‘Ÿβˆ’πΏ0𝐢1πœ•2πΈπΌπ‘¦π‘¦πœ•π‘Ÿ2πœ•3π‘’π‘˜πœ•π‘‘πœ•π‘Ÿ2π›Ώπ‘’π‘˜ξ€œπ‘‘π‘Ÿβˆ’2𝐿0𝐢1πœ•πΈπΌπ‘¦π‘¦πœ•πœ•π‘Ÿ4π‘’π‘˜πœ•π‘‘πœ•π‘Ÿ3π›Ώπ‘’π‘˜βˆ’ξ€œπ‘‘π‘ŸπΏ0𝐢1πΈπΌπ‘¦π‘¦πœ•5π‘’π‘˜πœ•π‘‘πœ•π‘Ÿ4π›Ώπ‘’π‘˜βˆ’ξ€œπ‘‘π‘ŸπΏ0𝐢0πœ•π‘£π‘˜πœ•π‘‘π›Ώπ‘£π‘˜ξ€œπ‘‘π‘Ÿβˆ’πΏ0𝐢1πœ•2𝐸𝐼π‘₯π‘₯πœ•π‘Ÿ2πœ•3π‘£π‘˜πœ•π‘‘πœ•π‘Ÿ2π›Ώπ‘£π‘˜ξ€œπ‘‘π‘Ÿβˆ’2𝐿0𝐢1πœ•πΈπΌπ‘₯π‘₯πœ•πœ•π‘Ÿ4π‘£π‘˜πœ•π‘‘πœ•π‘Ÿ3π›Ώπ‘£π‘˜βˆ’ξ€œπ‘‘π‘ŸπΏ0𝐢1𝐸𝐼π‘₯π‘₯πœ•5π‘£π‘˜πœ•π‘‘πœ•π‘Ÿ4π›Ώπ‘£π‘˜ξƒͺ,π‘‘π‘Ÿ(8) where 𝐢0 and 𝐢1 are external and internal damping coefficients of blades, respectively. Equations (6)–(8) are integrated into the Hamilton equation as follows:ξ€œπ‘‘2𝑑1(π›Ώπ‘‡π‘’βˆ’π›Ώπ‘ˆπ‘’+π›Ώπ‘Šπ‘’)𝑑𝑑=0.(9) This leads to the following equations for the motion of the blades:πœ•2πΈπΌπ‘¦π‘¦πœ•π‘Ÿ2πœ•2π‘’π‘˜πœ•π‘Ÿ2+2πœ•πΈπΌπ‘¦π‘¦πœ•πœ•π‘Ÿ3π‘’π‘˜πœ•π‘Ÿ3+πΈπΌπ‘¦π‘¦πœ•4π‘’π‘˜πœ•π‘Ÿ4+πœ•2𝐸𝐼π‘₯π‘¦πœ•π‘Ÿ2πœ•2π‘£π‘˜πœ•π‘Ÿ2+2πœ•πΈπΌπ‘₯π‘¦πœ•πœ•π‘Ÿ3π‘£π‘˜πœ•π‘Ÿ3+𝐸𝐼π‘₯π‘¦πœ•4π‘£π‘˜πœ•π‘Ÿ4βˆ’πœŒπ΄Ξ©2π‘’π‘˜sin2πœƒβˆ’πœŒΞ©2πœ•ξ‚΅ξ€œπœ•π‘ŸπΏπ‘Ÿπ΄ξ€·π‘Ÿ+π‘Ÿ0ξ€Έξ‚Άπ‘‘π‘Ÿπœ•π‘’π‘˜πœ•π‘Ÿβˆ’πœŒΞ©2ξ‚΅ξ€œπΏπ‘Ÿπ΄ξ€·π‘Ÿ+π‘Ÿ0ξ€Έξ‚Άπœ•π‘‘π‘Ÿ2π‘’π‘˜πœ•π‘Ÿ2+𝐢0πœ•π‘’π‘˜πœ•π‘‘+𝐢1πœ•2πΈπΌπ‘¦π‘¦πœ•π‘Ÿ2πœ•3π‘’π‘˜πœ•π‘Ÿ2πœ•π‘‘+2𝐢1πœ•πΈπΌπ‘¦π‘¦πœ•πœ•π‘Ÿ4π‘’π‘˜πœ•π‘Ÿ3πœ•π‘‘+𝐢1πΈπΌπ‘¦π‘¦πœ•5π‘’π‘˜πœ•π‘Ÿ4πœ•π‘‘+𝐢1πœ•2𝐸𝐼π‘₯π‘¦πœ•π‘Ÿ2πœ•3π‘£π‘˜πœ•π‘Ÿ2πœ•π‘‘+2𝐢1πœ•πΈπΌπ‘₯π‘¦πœ•πœ•π‘Ÿ4π‘£π‘˜πœ•π‘Ÿ3πœ•π‘‘+𝐢1𝐸𝐼π‘₯π‘¦πœ•5π‘£π‘˜πœ•π‘Ÿ4πœ•πœ•π‘‘+𝜌𝐴2π‘’π‘˜πœ•π‘‘2πœ•=0forπ‘˜=1,2,3,2𝐸𝐼π‘₯π‘₯πœ•π‘Ÿ2πœ•2π‘£π‘˜πœ•π‘Ÿ2+2πœ•πΈπΌπ‘₯π‘₯πœ•πœ•π‘Ÿ3π‘£π‘˜πœ•π‘Ÿ3+𝐸𝐼π‘₯π‘₯πœ•4π‘£π‘˜πœ•π‘Ÿ4+πœ•2𝐸𝐼π‘₯π‘¦πœ•π‘Ÿ2πœ•2π‘’π‘˜πœ•π‘Ÿ2+2πœ•πΈπΌπ‘₯π‘¦πœ•πœ•π‘Ÿ3π‘’π‘˜πœ•π‘Ÿ3+𝐸𝐼π‘₯π‘¦πœ•4π‘’π‘˜πœ•π‘Ÿ4βˆ’πœŒπ΄Ξ©2π‘£π‘˜cos2πœƒβˆ’πœŒΞ©2πœ•ξ‚΅ξ€œπœ•π‘ŸπΏπ‘Ÿπ΄ξ€·π‘Ÿ+π‘Ÿ0ξ€Έξ‚Άπ‘‘π‘Ÿπœ•π‘£π‘˜πœ•π‘Ÿβˆ’πœŒΞ©2ξ‚΅ξ€œπΏπ‘Ÿπ΄ξ€·π‘Ÿ+π‘Ÿ0ξ€Έξ‚Άπœ•π‘‘π‘Ÿ2π‘£π‘˜πœ•π‘Ÿ2+𝐢0πœ•π‘£π‘˜πœ•π‘‘+𝐢1πœ•2𝐸𝐼π‘₯π‘₯πœ•π‘Ÿ2πœ•3π‘£π‘˜πœ•π‘Ÿ2πœ•π‘‘+2𝐢1πœ•πΈπΌπ‘₯π‘₯πœ•πœ•π‘Ÿ4π‘£π‘˜πœ•π‘Ÿ3πœ•π‘‘+𝐢1𝐸𝐼π‘₯π‘₯πœ•5π‘£π‘˜πœ•π‘Ÿ4πœ•π‘‘+𝐢1πœ•2𝐸𝐼π‘₯π‘¦πœ•π‘Ÿ2πœ•3π‘’π‘˜πœ•π‘Ÿ2πœ•π‘‘+2𝐢1πœ•πΈπΌπ‘₯π‘¦πœ•πœ•π‘Ÿ4π‘’π‘˜πœ•π‘Ÿ3πœ•π‘‘+𝐢1𝐸𝐼π‘₯π‘¦πœ•5π‘’π‘˜πœ•π‘Ÿ4πœ•πœ•π‘‘+𝜌𝐴2π‘£π‘˜πœ•π‘‘2=0forπ‘˜=1,2,3.(10) The following equations are the corresponding boundary conditions:π‘’π‘˜(0,𝑑)=0forπ‘˜=1,2,3,πœ•π‘’π‘˜(0,𝑑)πœ•π‘Ÿ=0forπ‘˜=1,2,3,πΈπΌπ‘¦π‘¦πœ•2π‘’π‘˜(𝐿,𝑑)πœ•π‘Ÿ2πœ•=0forπ‘˜=1,2,3,ξƒ©πœ•π‘ŸπΈπΌπ‘¦π‘¦πœ•2π‘’π‘˜(𝐿,𝑑)πœ•π‘Ÿ2ξƒͺ𝑣=0forπ‘˜=1,2,3,π‘˜(0,𝑑)=0forπ‘˜=1,2,3,πœ•π‘£π‘˜(0,𝑑)πœ•π‘Ÿ=0forπ‘˜=1,2,3,𝐸𝐼π‘₯π‘₯πœ•2π‘£π‘˜(𝐿,𝑑)πœ•π‘Ÿ2πœ•=0forπ‘˜=1,2,3,ξƒ©πœ•π‘ŸπΈπΌπ‘₯π‘₯πœ•2π‘£π‘˜(𝐿,𝑑)πœ•π‘Ÿ2ξƒͺ=0forπ‘˜=1,2,3.(11) A solution is thus assumed asπ‘’π‘˜=π‘ˆπ‘˜(π‘Ÿ)π‘’πœ†π‘‘π‘£forπ‘˜=1,2,3,π‘˜=π‘‰π‘˜(π‘Ÿ)π‘’πœ†π‘‘forπ‘˜=1,2,3.(12) This yields the following equations for the motion of blades: 𝑑2πΈπΌπ‘¦π‘¦π‘‘π‘Ÿ2𝑑2π‘ˆπ‘˜π‘‘π‘Ÿ2+2π‘‘πΈπΌπ‘¦π‘¦π‘‘π‘‘π‘Ÿ3π‘ˆπ‘˜π‘‘π‘Ÿ3+𝐸𝐼𝑦𝑦𝑑4π‘ˆπ‘˜π‘‘π‘Ÿ4+𝑑2𝐸𝐼π‘₯π‘¦π‘‘π‘Ÿ2𝑑2π‘‰π‘˜π‘‘π‘Ÿ2+2𝑑𝐸𝐼π‘₯π‘¦π‘‘π‘‘π‘Ÿ3π‘‰π‘˜π‘‘π‘Ÿ3+𝐸𝐼π‘₯𝑦𝑑4π‘‰π‘˜π‘‘π‘Ÿ4βˆ’πœŒπ΄Ξ©2π‘ˆπ‘˜sin2πœƒβˆ’πœŒΞ©2π‘‘ξ‚΅ξ€œπ‘‘π‘ŸπΏπ‘Ÿπ΄ξ€·π‘Ÿ+π‘Ÿ0ξ€Έξ‚Άπ‘‘π‘Ÿπ‘‘π‘ˆπ‘˜π‘‘π‘Ÿβˆ’πœŒΞ©2ξ‚΅ξ€œπΏπ‘Ÿπ΄ξ€·π‘Ÿ+π‘Ÿ0ξ€Έξ‚Άπ‘‘π‘‘π‘Ÿ2π‘ˆπ‘˜π‘‘π‘Ÿ2+πœ†πΆ0π‘ˆπ‘˜+πœ†πΆ1𝑑2πΈπΌπ‘¦π‘¦π‘‘π‘Ÿ2𝑑2π‘ˆπ‘˜π‘‘π‘Ÿ2+2πœ†πΆ1π‘‘πΈπΌπ‘¦π‘¦π‘‘π‘‘π‘Ÿ3π‘ˆπ‘˜π‘‘π‘Ÿ3+πœ†πΆ1𝐸𝐼𝑦𝑦𝑑4π‘ˆπ‘˜π‘‘π‘Ÿ4+πœ†πΆ1𝑑2𝐸𝐼π‘₯π‘¦π‘‘π‘Ÿ2𝑑2π‘‰π‘˜π‘‘π‘Ÿ2+2πœ†πΆ1𝑑𝐸𝐼π‘₯π‘¦π‘‘π‘‘π‘Ÿ3π‘‰π‘˜π‘‘π‘Ÿ3+πœ†πΆ1𝐸𝐼π‘₯𝑦𝑑4π‘‰π‘˜π‘‘π‘Ÿ4+πœ†2πœŒπ΄π‘ˆπ‘˜π‘‘=0forπ‘˜=1,2,3,2𝐸𝐼π‘₯π‘₯π‘‘π‘Ÿ2𝑑2π‘‰π‘˜π‘‘π‘Ÿ2+2𝑑𝐸𝐼π‘₯π‘₯π‘‘π‘‘π‘Ÿ3π‘‰π‘˜π‘‘π‘Ÿ3+𝐸𝐼π‘₯π‘₯𝑑4π‘‰π‘˜π‘‘π‘Ÿ4+𝑑2𝐸𝐼π‘₯π‘¦π‘‘π‘Ÿ2𝑑2π‘ˆπ‘˜π‘‘π‘Ÿ2+2𝑑𝐸𝐼π‘₯π‘¦π‘‘π‘‘π‘Ÿ3π‘ˆπ‘˜π‘‘π‘Ÿ3+𝐸𝐼π‘₯𝑦𝑑4π‘ˆπ‘˜π‘‘π‘Ÿ4βˆ’πœŒπ΄Ξ©2π‘‰π‘˜cos2πœƒβˆ’πœŒΞ©2π‘‘ξ‚΅ξ€œπ‘‘π‘ŸπΏπ‘Ÿπ΄ξ€·π‘Ÿ+π‘Ÿ0ξ€Έξ‚Άπ‘‘π‘Ÿπ‘‘π‘‰π‘˜π‘‘π‘Ÿβˆ’πœŒΞ©2ξ‚΅ξ€œπΏπ‘Ÿπ΄ξ€·π‘Ÿ+π‘Ÿ0ξ€Έξ‚Άπ‘‘π‘‘π‘Ÿ2π‘‰π‘˜π‘‘π‘Ÿ2+πœ†πΆ0π‘‘π‘‰π‘˜π‘‘π‘‘+πœ†πΆ1𝑑2𝐸𝐼π‘₯π‘₯π‘‘π‘Ÿ2𝑑2π‘‰π‘˜π‘‘π‘Ÿ2+2πœ†πΆ1𝑑𝐸𝐼π‘₯π‘₯π‘‘π‘‘π‘Ÿ3π‘‰π‘˜π‘‘π‘Ÿ3+πœ†πΆ1𝐸𝐼π‘₯π‘₯𝑑4π‘‰π‘˜π‘‘π‘Ÿ4+πœ†πΆ1𝑑2𝐸𝐼π‘₯π‘¦π‘‘π‘Ÿ2𝑑2π‘ˆπ‘˜π‘‘π‘Ÿ2+2πœ†πΆ1𝑑𝐸𝐼π‘₯π‘¦π‘‘π‘‘π‘Ÿ3π‘ˆπ‘˜π‘‘π‘Ÿ3+πœ†πΆ1𝐸𝐼π‘₯𝑦𝑑4π‘ˆπ‘˜π‘‘π‘Ÿ4+πœ†2πœŒπ΄π‘‰π‘˜=0forπ‘˜=1,2,3.(13) The corresponding boundary conditions are as follows:π‘ˆπ‘˜(0)=0forπ‘˜=1,2,3,π‘‘π‘ˆπ‘˜(0)π‘‘π‘Ÿ=0forπ‘˜=1,2,3,𝐸𝐼𝑦𝑦𝑑2π‘ˆπ‘˜(𝐿)π‘‘π‘Ÿ2𝑑=0forπ‘˜=1,2,3,ξƒ©π‘‘π‘ŸπΈπΌπ‘¦π‘¦π‘‘2π‘ˆπ‘˜(𝐿)π‘‘π‘Ÿ2ξƒͺ𝑉=0forπ‘˜=1,2,3,π‘˜(0)=0forπ‘˜=1,2,3,π‘‘π‘‰π‘˜(0)π‘‘π‘Ÿ=0forπ‘˜=1,2,3,𝐸𝐼π‘₯π‘₯𝑑2π‘‰π‘˜(𝐿)π‘‘π‘Ÿ2𝑑=0forπ‘˜=1,2,3,ξƒ©π‘‘π‘ŸπΈπΌπ‘₯π‘₯𝑑2π‘‰π‘˜(𝐿)π‘‘π‘Ÿ2ξƒͺ=0forπ‘˜=1,2,3.(14) By employing the radial basis function approach, (2) and (3) can be substituted into (10). The equations of motion of wind turbine blades can be rearranged in the following matrix form:ξ€Ίξ€·π‘ŸπœŒπ΄π‘–ξ€Έπ΅1ξ€·π‘Ÿπ‘–ξ€Έξ€·π‘ŸπœŒπ΄π‘–ξ€Έπ΅2ξ€·π‘Ÿπ‘–ξ€Έξ€·π‘Ÿβ‹―πœŒπ΄π‘–ξ€Έπ΅π‘ξ€·π‘Ÿπ‘–ξƒ¬πœ•ξ€Έξ€»2π‘Žπ‘˜1πœ•π‘‘2πœ•2π‘Žπ‘˜2πœ•π‘‘2β‹―πœ•2π‘Žπ‘˜π‘πœ•π‘‘2𝑇+𝐢0𝐡1ξ€·π‘Ÿπ‘–ξ€ΈπΆ0𝐡2ξ€·π‘Ÿπ‘–ξ€Έβ‹―πΆ0π΅π‘ξ€·π‘Ÿπ‘–ξ‚Έξ€Έξ€»πœ•π‘Žπ‘˜1πœ•π‘‘πœ•π‘Žπ‘˜2β‹―πœ•π‘‘πœ•π‘Žπ‘˜π‘ξ‚Ήπœ•π‘‘π‘‡+𝐢1πœ•2πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2πœ•2𝐡1ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2𝐢1πœ•2πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2πœ•2𝐡2ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2⋯𝐢1πœ•2πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2πœ•2π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2ξƒ­ξ‚Έπœ•π‘Žπ‘˜1πœ•π‘‘πœ•π‘Žπ‘˜2β‹―πœ•π‘‘πœ•π‘Žπ‘˜π‘ξ‚Ήπœ•π‘‘π‘‡+2𝐢1πœ•πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•πœ•π‘Ÿ3𝐡1ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ32𝐢1πœ•πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•πœ•π‘Ÿ3𝐡2ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ3β‹―2𝐢1πœ•πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•πœ•π‘Ÿ3π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ3ξƒ­ξ‚Έπœ•π‘Žπ‘˜1πœ•π‘‘πœ•π‘Žπ‘˜2β‹―πœ•π‘‘πœ•π‘Žπ‘˜π‘ξ‚Ήπœ•π‘‘π‘‡+𝐢1πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•4𝐡1ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ4𝐢1πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•4𝐡2ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ4⋯𝐢1πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•4π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ4ξƒ­ξ‚Έπœ•π‘Žπ‘˜1πœ•π‘‘πœ•π‘Žπ‘˜2β‹―πœ•π‘‘πœ•π‘Žπ‘˜π‘ξ‚Ήπœ•π‘‘π‘‡+𝐢1πœ•2𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2πœ•2𝐡1ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2𝐢1πœ•2𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2πœ•2𝐡2ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2⋯𝐢1πœ•2𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2πœ•2π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2ξƒ­ξ‚Έπœ•π‘Žπ‘˜1πœ•πœ•π‘‘π‘Žπ‘˜2β‹―πœ•πœ•π‘‘π‘Žπ‘˜π‘ξ‚Ήπœ•π‘‘π‘‡+2𝐢1πœ•πΈπΌπ‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•πœ•π‘Ÿ3𝐡1ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ32𝐢1πœ•πΈπΌπ‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•πœ•π‘Ÿ3𝐡2ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ3β‹―2𝐢1πœ•πΈπΌπ‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•πœ•π‘Ÿ3π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ3ξƒ­ξ‚Έπœ•π‘Žπ‘˜1πœ•πœ•π‘‘π‘Žπ‘˜2β‹―πœ•πœ•π‘‘π‘Žπ‘˜π‘ξ‚Ήπœ•π‘‘π‘‡+𝐢1𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•4𝐡1ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ4𝐢1𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•4𝐡2ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ4⋯𝐢1𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•4π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ4ξƒ­ξ‚Έπœ•π‘Žπ‘˜1πœ•πœ•π‘‘π‘Žπ‘˜2β‹―πœ•πœ•π‘‘π‘Žπ‘˜π‘ξ‚Ήπœ•π‘‘π‘‡+ξƒ¬πœ•2πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2πœ•2𝐡1ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2πœ•2πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2πœ•2𝐡2ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2β‹―πœ•2πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2πœ•2π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2ξƒ­ξ€Ίπ‘Žπ‘˜1π‘Žπ‘˜2β‹―π‘Žπ‘˜π‘ξ€»π‘‡+2πœ•πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•πœ•π‘Ÿ3𝐡1ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ32πœ•πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•πœ•π‘Ÿ3𝐡2ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ3β‹―2πœ•πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•πœ•π‘Ÿ3π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ3ξƒ­ξ€Ίπ‘Žπ‘˜1π‘Žπ‘˜2β‹―π‘Žπ‘˜π‘ξ€»π‘‡+ξƒ¬πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•4𝐡1ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ4πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•4𝐡2ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ4β‹―πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•4π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ4ξƒ­ξ€Ίπ‘Žπ‘˜1π‘Žπ‘˜2β‹―π‘Žπ‘˜π‘ξ€»π‘‡+ξƒ¬πœ•2𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2πœ•2𝐡1ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2πœ•2𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2πœ•2𝐡2ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2β‹―πœ•2𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2πœ•2π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2ξƒ­ξ€Ίπ‘Žπ‘˜1π‘Žπ‘˜2β‹―π‘Žπ‘˜π‘ξ€»π‘‡+2πœ•πΈπΌπ‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•πœ•π‘Ÿ3𝐡1ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ32πœ•πΈπΌπ‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•πœ•π‘Ÿ3𝐡2ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ3β‹―2πœ•πΈπΌπ‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•πœ•π‘Ÿ3π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ3ξƒ­ξ€Ίπ‘Žπ‘˜1π‘Žπ‘˜2β‹―π‘Žπ‘˜π‘ξ€»π‘‡+𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•4𝐡1ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ4𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•4𝐡2ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ4⋯𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•4π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ4ξƒ­ξ€Ίπ‘Žπ‘˜1π‘Žπ‘˜2β‹―π‘Žπ‘˜π‘ξ€»π‘‡βˆ’ξ€ΊπœŒπ΄(π‘Ÿ)Ξ©2(sinπœƒ)2𝐡1ξ€·π‘Ÿπ‘–ξ€ΈπœŒπ΄(π‘Ÿ)Ξ©2(sinπœƒ)2𝐡2ξ€·π‘Ÿπ‘–ξ€Έβ‹―πœŒπ΄(π‘Ÿ)Ξ©2(sinπœƒ)2π΅π‘ξ€·π‘Ÿπ‘–π‘Žξ€Έξ€»ξ€Ίπ‘˜1π‘Žπ‘˜2β‹―π‘Žπ‘˜π‘ξ€»π‘‡βˆ’ξƒ¬πœŒΞ©2πœ•ξ‚€βˆ«πœ•π‘ŸπΏπ‘Ÿπ‘–ξ‚π΄(π‘Ÿ)π”„π‘‘π‘Ÿπœ•π΅1ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘ŸπœŒΞ©2πœ•ξ‚€βˆ«πœ•π‘ŸπΏπ‘Ÿπ‘–ξ‚π΄(π‘Ÿ)π”„π‘‘π‘Ÿπœ•π΅2ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿβ‹―πœŒΞ©2πœ•ξ‚€βˆ«πœ•π‘ŸπΏπ‘Ÿπ‘–ξ‚π΄(π‘Ÿ)π”„π‘‘π‘Ÿπœ•π΅π‘ξ€·π‘Ÿπ‘–ξ€Έξƒ­Γ—ξ€Ίπ‘Žπœ•π‘Ÿπ‘˜1π‘Žπ‘˜2β‹―π‘Žπ‘˜π‘ξ€»π‘‡βˆ’ξƒ¬πœŒΞ©2βˆ«πΏπ‘Ÿπ‘–πœ•π΄(π‘Ÿ)π”„π‘‘π‘Ÿ2𝐡1ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2𝜌Ω2βˆ«πΏπ‘Ÿπ‘–πœ•π΄(π‘Ÿ)π”„π‘‘π‘Ÿ2𝐡2ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2β‹―πœŒΞ©2βˆ«πΏπ‘Ÿπ‘–πœ•π΄(π‘Ÿ)π”„π‘‘π‘Ÿ2π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2ξƒ­Γ—ξ€Ίπ‘Žπ‘˜1π‘Žπ‘˜2β‹―π‘Žπ‘˜π‘ξ€»π‘‡=[0],forξ€Ίξ€·π‘Ÿπ‘–=1,2,…,𝑁,π‘˜=1,2,3,πœŒπ΄π‘–ξ€Έπ΅1ξ€·π‘Ÿπ‘–ξ€Έξ€·π‘ŸπœŒπ΄π‘–ξ€Έπ΅2ξ€·π‘Ÿπ‘–ξ€Έξ€·π‘Ÿβ‹―πœŒπ΄π‘–ξ€Έπ΅π‘ξ€·π‘Ÿπ‘–ξ‚Έπœ•ξ€Έξ€»2π‘Žπ‘˜1πœ•π‘‘2πœ•2π‘Žπ‘˜2πœ•π‘‘2β‹―πœ•2π‘Žπ‘˜π‘πœ•π‘‘2𝑇+𝐢0𝐡1ξ€·π‘Ÿπ‘–ξ€ΈπΆ0𝐡2ξ€·π‘Ÿπ‘–ξ€Έβ‹―πΆ0π΅π‘ξ€·π‘Ÿπ‘–ξ‚Έπœ•ξ€Έξ€»π‘Žπ‘˜1πœ•πœ•π‘‘π‘Žπ‘˜2β‹―πœ•πœ•π‘‘π‘Žπ‘˜π‘ξ‚Ήπœ•π‘‘π‘‡+𝐢1πœ•2𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2πœ•2𝐡1ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2𝐢1πœ•2𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2πœ•2𝐡2ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2⋯𝐢1πœ•2𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2πœ•2π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2ξƒ­ξ‚Έπœ•π‘Žπ‘˜1πœ•πœ•π‘‘π‘Žπ‘˜2β‹―πœ•πœ•π‘‘π‘Žπ‘˜π‘ξ‚Ήπœ•π‘‘π‘‡+2𝐢1πœ•πΈπΌπ‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπœ•πœ•π‘Ÿ3𝐡1ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ32𝐢1πœ•πΈπΌπ‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπœ•πœ•π‘Ÿ3𝐡2ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ3β‹―2𝐢1πœ•πΈπΌπ‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπœ•πœ•π‘Ÿ3π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ3ξƒ­ξ‚Έπœ•π‘Žπ‘˜1πœ•πœ•π‘‘π‘Žπ‘˜2β‹―πœ•πœ•π‘‘π‘Žπ‘˜π‘ξ‚Ήπœ•π‘‘π‘‡+𝐢1𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπœ•4𝐡1ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ4𝐢1𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπœ•4𝐡2ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ4⋯𝐢1𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπœ•4π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ4ξƒ­ξ‚Έπœ•π‘Žπ‘˜1πœ•πœ•π‘‘π‘Žπ‘˜2β‹―πœ•πœ•π‘‘π‘Žπ‘˜π‘ξ‚Ήπœ•π‘‘π‘‡+𝐢1πœ•2𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2πœ•2𝐡1ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2𝐢1πœ•2𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2πœ•2𝐡2ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2⋯𝐢1πœ•2𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2πœ•2π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2ξƒ­ξ‚Έπœ•π‘Žπ‘˜1πœ•π‘‘πœ•π‘Žπ‘˜2β‹―πœ•π‘‘πœ•π‘Žπ‘˜π‘ξ‚Ήπœ•π‘‘π‘‡+2𝐢1πœ•πΈπΌπ‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•πœ•π‘Ÿ3𝐡1ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ32𝐢1πœ•πΈπΌπ‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•πœ•π‘Ÿ3𝐡2ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ3β‹―2𝐢1πœ•πΈπΌπ‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•πœ•π‘Ÿ3π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ3ξƒ­ξ‚Έπœ•π‘Žπ‘˜1πœ•π‘‘πœ•π‘Žπ‘˜2β‹―πœ•π‘‘πœ•π‘Žπ‘˜π‘ξ‚Ήπœ•π‘‘π‘‡+𝐢1𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•4𝐡1ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ4𝐢1𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•4𝐡2ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ4⋯𝐢1𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•4π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ4ξƒ­ξ‚Έπœ•π‘Žπ‘˜1πœ•π‘‘πœ•π‘Žπ‘˜2β‹―πœ•π‘‘πœ•π‘Žπ‘˜π‘ξ‚Ήπœ•π‘‘π‘‡+ξƒ¬πœ•2𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2πœ•2𝐡1ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2πœ•2𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2πœ•2𝐡2ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2β‹―πœ•2𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2πœ•2π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2ξƒ­ξ€Ίπ‘Žπ‘˜1π‘Žπ‘˜2β‹―π‘Žπ‘˜π‘ξ€»π‘‡+2πœ•πΈπΌπ‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπœ•πœ•π‘Ÿ3𝐡1ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ32πœ•πΈπΌπ‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπœ•πœ•π‘Ÿ3𝐡2ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ3β‹―2πœ•πΈπΌπ‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπœ•πœ•π‘Ÿ3π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ3ξƒ­ξ€Ίπ‘Žπ‘˜1π‘Žπ‘˜2β‹―π‘Žπ‘˜π‘ξ€»π‘‡+𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπœ•4𝐡1ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ4𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπœ•4𝐡2ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ4⋯𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπœ•4π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ4ξƒ­ξ€Ίπ‘Žπ‘˜1π‘Žπ‘˜2β‹―π‘Žπ‘˜π‘ξ€»π‘‡+ξƒ¬πœ•2𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2πœ•2𝐡1ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2πœ•2𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2πœ•2𝐡2ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2β‹―πœ•2𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2πœ•2π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2ξƒ­ξ€Ίπ‘Žπ‘˜1π‘Žπ‘˜2β‹―π‘Žπ‘˜π‘ξ€»π‘‡+2πœ•πΈπΌπ‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•πœ•π‘Ÿ3𝐡1ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ32πœ•πΈπΌπ‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•πœ•π‘Ÿ3𝐡2ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ3β‹―2πœ•πΈπΌπ‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•πœ•π‘Ÿ3π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ3ξƒ­ξ€Ίπ‘Žπ‘˜1π‘Žπ‘˜2β‹―π‘Žπ‘˜π‘ξ€»π‘‡+𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•4𝐡1ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ4𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•4𝐡2ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ4⋯𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπœ•4π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ4ξƒ­ξ€Ίπ‘Žπ‘˜1π‘Žπ‘˜2β‹―π‘Žπ‘˜π‘ξ€»π‘‡βˆ’ξ€ΊπœŒπ΄(π‘Ÿ)Ξ©2(cosπœƒ)2𝐡1ξ€·π‘Ÿπ‘–ξ€ΈπœŒπ΄(π‘Ÿ)Ξ©2(cosπœƒ)2𝐡2ξ€·π‘Ÿπ‘–ξ€Έβ‹―πœŒπ΄(π‘Ÿ)Ξ©2(cosπœƒ)2π΅π‘ξ€·π‘Ÿπ‘–ξ€Έξ€»ξ€Ίπ‘Žπ‘˜1π‘Žπ‘˜2β‹―π‘Žπ‘˜π‘ξ€»π‘‡βˆ’ξƒ¬πœŒΞ©2πœ•ξ‚€βˆ«πœ•π‘ŸπΏπ‘Ÿπ‘–ξ‚π΄(π‘Ÿ)π”„π‘‘π‘Ÿπœ•π΅1ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘ŸπœŒΞ©2πœ•ξ‚€βˆ«πœ•π‘ŸπΏπ‘Ÿπ‘–ξ‚π΄(π‘Ÿ)π”„π‘‘π‘Ÿπœ•π΅2ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿβ‹―πœŒΞ©2πœ•ξ‚€βˆ«πœ•π‘ŸπΏπ‘Ÿπ‘–ξ‚π΄(π‘Ÿ)π”„π‘‘π‘Ÿπœ•π΅π‘ξ€·π‘Ÿπ‘–ξ€Έξƒ­Γ—ξ€Ίπœ•π‘Ÿπ‘Žπ‘˜1π‘Žπ‘˜2β‹―π‘Žπ‘˜π‘ξ€»π‘‡βˆ’ξƒ¬πœŒΞ©2βˆ«πΏπ‘Ÿπ‘–πœ•π΄(π‘Ÿ)π”„π‘‘π‘Ÿ2𝐡1ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2𝜌Ω2βˆ«πΏπ‘Ÿπ‘–πœ•π΄(π‘Ÿ)π”„π‘‘π‘Ÿ2𝐡2ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2β‹―πœŒΞ©2βˆ«πΏπ‘Ÿπ‘–πœ•π΄(π‘Ÿ)π”„π‘‘π‘Ÿ2π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπœ•π‘Ÿ2ξƒ­Γ—ξ€Ίπ‘Žπ‘˜1π‘Žπ‘˜2β‹―π‘Žπ‘˜π‘ξ€»π‘‡=[0]for𝑖=1,2,…,𝑁,π‘˜=1,2,3,(15)where 𝔄 denotes (π‘Ÿ+π‘Ÿ0).

Based on the radial basis function technique, (11) take the following discrete forms: 𝐡1ξ€·π‘Ÿ1𝐡2ξ€·π‘Ÿ1ξ€Έβ‹―π΅π‘ξ€·π‘Ÿ1π‘Žξ€Έξ€»ξ€Ίπ‘˜1π‘Žπ‘˜2β‹―π‘Žπ‘˜π‘ξ€»π‘‡=[0]forξƒ¬π‘˜=1,2,3,πœ•π΅1ξ€·π‘Ÿ1ξ€Έπœ•π‘Ÿπœ•π΅2ξ€·π‘Ÿ1ξ€Έβ‹―πœ•π‘Ÿπœ•π΅π‘ξ€·π‘Ÿ1ξ€Έξƒ­ξ€Ίπ‘Žπœ•π‘Ÿπ‘˜1π‘Žπ‘˜2β‹―π‘Žπ‘˜π‘ξ€»π‘‡=[0]forξƒ¬π‘˜=1,2,3,πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘ξ€Έπœ•2𝐡1ξ€·π‘Ÿπ‘ξ€Έπœ•π‘Ÿ2πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘ξ€Έπœ•2𝐡2ξ€·π‘Ÿπ‘ξ€Έπœ•π‘Ÿ2β‹―πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘ξ€Έπœ•2π΅π‘ξ€·π‘Ÿπ‘ξ€Έπœ•π‘Ÿ2ξƒ­ξ€Ίπ‘Žπ‘˜1π‘Žπ‘˜2β‹―π‘Žπ‘˜π‘ξ€»π‘‡=[0]forξƒ¬π‘˜=1,2,3,πœ•πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘ξ€Έπœ•πœ•π‘Ÿ2𝐡1ξ€·π‘Ÿπ‘ξ€Έπœ•π‘Ÿ2πœ•πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘ξ€Έπœ•πœ•π‘Ÿ2𝐡2ξ€·π‘Ÿπ‘ξ€Έπœ•π‘Ÿ2β‹―πœ•πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘ξ€Έπœ•πœ•π‘Ÿ2π΅π‘ξ€·π‘Ÿπ‘ξ€Έπœ•π‘Ÿ2ξƒ­ξ€Ίπ‘Žπ‘˜1π‘Žπ‘˜2β‹―π‘Žπ‘˜π‘ξ€»π‘‡+ξƒ¬πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘ξ€Έπœ•3𝐡1ξ€·π‘Ÿπ‘ξ€Έπœ•π‘Ÿ3πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘ξ€Έπœ•3𝐡2ξ€·π‘Ÿπ‘ξ€Έπœ•π‘Ÿ3β‹―πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘ξ€Έπœ•3π΅π‘ξ€·π‘Ÿπ‘ξ€Έπœ•π‘Ÿ3ξƒ­Γ—ξ€Ίπ‘Žπ‘˜1π‘Žπ‘˜2β‹―π‘Žπ‘˜π‘ξ€»π‘‡=[0]forξ€Ίπ΅π‘˜=1,2,3,1ξ€·π‘Ÿ1𝐡2ξ€·π‘Ÿ1ξ€Έβ‹―π΅π‘ξ€·π‘Ÿ1ξ€Έξ€»ξ€Ίπ‘Žπ‘˜1π‘Žπ‘˜2β‹―π‘Žπ‘˜π‘ξ€»π‘‡=[0]forξƒ¬π‘˜=1,2,3,πœ•π΅1ξ€·π‘Ÿ1ξ€Έπœ•π‘Ÿπœ•π΅2ξ€·π‘Ÿ1ξ€Έβ‹―πœ•π‘Ÿπœ•π΅π‘ξ€·π‘Ÿ1ξ€Έξƒ­ξ€Ίπœ•π‘Ÿπ‘Žπ‘˜1π‘Žπ‘˜2β‹―π‘Žπ‘˜π‘ξ€»π‘‡=[0]forξƒ¬π‘˜=1,2,3,𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘ξ€Έπœ•2𝐡1ξ€·π‘Ÿπ‘ξ€Έπœ•π‘Ÿ2𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘ξ€Έπœ•2𝐡2ξ€·π‘Ÿπ‘ξ€Έπœ•π‘Ÿ2⋯𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘ξ€Έπœ•2π΅π‘ξ€·π‘Ÿπ‘ξ€Έπœ•π‘Ÿ2ξƒ­ξ€Ίπ‘Žπ‘˜1π‘Žπ‘˜2β‹―π‘Žπ‘˜π‘ξ€»π‘‡=[0],forξƒ¬π‘˜=1,2,3,πœ•πΈπΌπ‘₯π‘₯ξ€·π‘Ÿπ‘ξ€Έπœ•πœ•π‘Ÿ2𝐡1ξ€·π‘Ÿπ‘ξ€Έπœ•π‘Ÿ2πœ•πΈπΌπ‘₯π‘₯ξ€·π‘Ÿπ‘ξ€Έπœ•πœ•π‘Ÿ2𝐡2ξ€·π‘Ÿπ‘ξ€Έπœ•π‘Ÿ2β‹―πœ•πΈπΌπ‘₯π‘₯ξ€·π‘Ÿπ‘ξ€Έπœ•πœ•π‘Ÿ2π΅π‘ξ€·π‘Ÿπ‘ξ€Έπœ•π‘Ÿ2ξƒ­ξ€Ίπ‘Žπ‘˜1π‘Žπ‘˜2β‹―π‘Žπ‘˜π‘ξ€»π‘‡+𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘ξ€Έπœ•3π΅π‘ξ€·π‘Ÿπ‘ξ€Έπœ•π‘Ÿ3𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘ξ€Έπœ•3𝐡2ξ€·π‘Ÿπ‘ξ€Έπœ•π‘Ÿ3⋯𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘ξ€Έπœ•3π΅π‘ξ€·π‘Ÿπ‘ξ€Έπœ•π‘Ÿ3ξƒ­Γ—ξ€Ίπ‘Žπ‘˜1π‘Žπ‘˜2β‹―π‘Žπ‘˜π‘ξ€»π‘‡=[0]forπ‘˜=1,2,3.(16)By applying the radial basis function approach, (4) and (5) are substituted into (13). The following equations are then yieldedξ€Ίπœ†2ξ€·π‘ŸπœŒπ΄π‘–ξ€Έπ΅1ξ€·π‘Ÿπ‘–ξ€Έπœ†2ξ€·π‘ŸπœŒπ΄π‘–ξ€Έπ΅2ξ€·π‘Ÿπ‘–ξ€Έβ‹―πœ†2ξ€·π‘ŸπœŒπ΄π‘–ξ€Έπ΅π‘ξ€·π‘Ÿπ‘–π‘ξ€Έξ€»ξ€Ίπ‘˜1π‘π‘˜2β‹―π‘π‘˜π‘ξ€»π‘‡+ξ€Ίπœ†πΆ0𝐡1ξ€·π‘Ÿπ‘–ξ€Έπœ†πΆ0𝐡2ξ€·π‘Ÿπ‘–ξ€Έβ‹―πœ†πΆ0π΅π‘ξ€·π‘Ÿπ‘–π‘ξ€Έξ€»ξ€Ίπ‘˜1π‘π‘˜2β‹―π‘π‘˜π‘ξ€»π‘‡+ξƒ¬πœ†πΆ1𝑑2πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2𝑑2𝐡1ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2πœ†πΆ1𝑑2πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2𝑑2𝐡2ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2β‹―πœ†πΆ1𝑑2πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2𝑑2π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2ξƒ­ξ€Ίπ‘π‘˜1π‘π‘˜2β‹―π‘π‘˜π‘ξ€»π‘‡+2πœ†πΆ1π‘‘πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘‘π‘Ÿ3𝐡1ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ32πœ†πΆ1π‘‘πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘‘π‘Ÿ3𝐡2ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ3β‹―2πœ†πΆ1π‘‘πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘‘π‘Ÿ3π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ3ξƒ­ξ€Ίπ‘π‘˜1π‘π‘˜2β‹―π‘π‘˜π‘ξ€»π‘‡+ξƒ¬πœ†πΆ1πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘4𝐡1ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ4πœ†πΆ1πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘4𝐡2ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ4β‹―πœ†πΆ1πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘4π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ4ξƒ­ξ€Ίπ‘π‘˜1π‘π‘˜2β‹―π‘π‘˜π‘ξ€»π‘‡+ξƒ¬πœ†πΆ1𝑑2𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2𝑑2𝐡1ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2πœ†πΆ1𝑑2𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2𝑑2𝐡2ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2β‹―πœ†πΆ1𝑑2𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2𝑑2π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2ξƒ­ξ‚ƒπ‘π‘˜1π‘π‘˜2β‹―π‘π‘˜π‘ξ‚„π‘‡+2πœ†πΆ1𝑑𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘‘π‘Ÿ3𝐡1ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ32πœ†πΆ1𝑑𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘‘π‘Ÿ3𝐡2ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ3β‹―2πœ†πΆ1𝑑𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘‘π‘Ÿ3π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ3ξƒ­ξ‚ƒπ‘π‘˜1π‘π‘˜2β‹―π‘π‘˜π‘ξ‚„π‘‡+ξƒ¬πœ†πΆ1𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘4𝐡1ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ4πœ†πΆ1𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘4𝐡2ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ4β‹―πœ†πΆ1𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘4π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ4ξƒ­ξ‚ƒπ‘π‘˜1π‘π‘˜2β‹―π‘π‘˜π‘ξ‚„π‘‡+𝑑2πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2𝑑2𝐡1ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2𝑑2πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2𝑑2𝐡2ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2⋯𝑑2πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2𝑑2π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2ξƒ­ξ€Ίπ‘π‘˜1π‘π‘˜2β‹―π‘π‘˜π‘ξ€»π‘‡+2π‘‘πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘‘π‘Ÿ3𝐡1ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ32π‘‘πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘‘π‘Ÿ3𝐡2ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ3β‹―2π‘‘πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘‘π‘Ÿ3π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ3ξƒ­ξ€Ίπ‘π‘˜1π‘π‘˜2β‹―π‘π‘˜π‘ξ€»π‘‡+ξƒ¬πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘4𝐡1ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ4πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘4𝐡2ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ4β‹―πΈπΌπ‘¦π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘4π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ4ξƒ­ξ€Ίπ‘π‘˜1π‘π‘˜2β‹―π‘π‘˜π‘ξ€»π‘‡+𝑑2𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2𝑑2𝐡1ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2𝑑2𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2𝑑2𝐡2ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2⋯𝑑2𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2𝑑2π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2ξƒ­ξ‚ƒπ‘π‘˜1π‘π‘˜2β‹―π‘π‘˜π‘ξ‚„π‘‡+2𝑑𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘‘π‘Ÿ3𝐡1ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ32𝑑𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘‘π‘Ÿ3𝐡2ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ3β‹―2𝑑𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘‘π‘Ÿ3π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ3ξƒ­ξ‚ƒπ‘π‘˜1π‘π‘˜2β‹―π‘π‘˜π‘ξ‚„π‘‡+𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘4𝐡1ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ4𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘4𝐡2ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ4⋯𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘4π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ4ξƒ­ξ‚ƒπ‘π‘˜1π‘π‘˜2β‹―π‘π‘˜π‘ξ‚„π‘‡βˆ’ξ€Ίξ€·π‘ŸπœŒπ΄π‘–ξ€ΈΞ©2(sinπœƒ)2𝐡1ξ€·π‘Ÿπ‘–ξ€Έξ€·π‘ŸπœŒπ΄π‘–ξ€ΈΞ©2(sinπœƒ)2𝐡2ξ€·π‘Ÿπ‘–ξ€Έξ€·π‘Ÿβ‹―πœŒπ΄π‘–ξ€ΈΞ©2(sinπœƒ)2π΅π‘ξ€·π‘Ÿπ‘–π‘ξ€Έξ€»ξ€Ίπ‘˜1π‘π‘˜2β‹―π‘π‘˜π‘ξ€»π‘‡βˆ’ξƒ¬πœŒΞ©2π‘‘ξ‚€βˆ«π‘‘π‘ŸπΏπ‘Ÿπ‘–ξ‚π΄(π‘Ÿ)π”„π‘‘π‘Ÿπ‘‘π΅1ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘ŸπœŒΞ©2π‘‘ξ‚€βˆ«π‘‘π‘ŸπΏπ‘Ÿπ‘–ξ‚π΄(π‘Ÿ)π”„π‘‘π‘Ÿπ‘‘π΅2ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿβ‹―πœŒΞ©2π‘‘ξ‚€βˆ«π‘‘π‘ŸπΏπ‘Ÿπ‘–ξ‚π΄(π‘Ÿ)π”„π‘‘π‘Ÿπ‘‘π΅π‘ξ€·π‘Ÿπ‘–ξ€Έξƒ­Γ—ξ€Ίπ‘π‘‘π‘Ÿπ‘˜1π‘π‘˜2β‹―π‘π‘˜π‘ξ€»π‘‡βˆ’ξƒ¬πœŒΞ©2βˆ«πΏπ‘Ÿπ‘–π‘‘π΄(π‘Ÿ)π”„π‘‘π‘Ÿ2𝐡1ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2𝜌Ω2βˆ«πΏπ‘Ÿπ‘–π‘‘π΄(π‘Ÿ)π”„π‘‘π‘Ÿ2𝐡2ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2β‹―πœŒΞ©2βˆ«πΏπ‘Ÿπ‘–π‘‘π΄(π‘Ÿ)π”„π‘‘π‘Ÿ2π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2ξƒ­Γ—ξ€Ίπ‘π‘˜1π‘π‘˜2β‹―π‘π‘˜π‘ξ€»π‘‡=[0]forξ€Ίπœ†π‘–=1,2,…,𝑁,π‘˜=1,2,3,2ξ€·π‘ŸπœŒπ΄π‘–ξ€Έπ΅1ξ€·π‘Ÿπ‘–ξ€Έπœ†2ξ€·π‘ŸπœŒπ΄π‘–ξ€Έπ΅2ξ€·π‘Ÿπ‘–ξ€Έβ‹―πœ†2ξ€·π‘ŸπœŒπ΄π‘–ξ€Έπ΅π‘ξ€·π‘Ÿπ‘–ξ‚ƒξ€Έξ€»π‘π‘˜1π‘π‘˜2β‹―π‘π‘˜π‘ξ‚„π‘‡+ξ€Ίπœ†πΆ0𝐡1ξ€·π‘Ÿπ‘–ξ€Έπœ†πΆ0𝐡2ξ€·π‘Ÿπ‘–ξ€Έβ‹―πœ†πΆ0π΅π‘ξ€·π‘Ÿπ‘–ξ€Έξ€»ξ€Ίπ‘π‘˜1π‘π‘˜2β‹―π‘π‘˜π‘ξ€»π‘‡+ξƒ¬πœ†πΆ1𝑑2𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2𝑑2𝐡1ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2πœ†πΆ1𝑑2𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2𝑑2𝐡2ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2β‹―πœ†πΆ1𝑑2𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2𝑑2π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2ξƒ­ξ‚ƒπ‘π‘˜1π‘π‘˜2β‹―π‘π‘˜π‘ξ‚„π‘‡+2πœ†πΆ1𝑑𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘‘π‘Ÿ3𝐡1ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ32πœ†πΆ1𝑑𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘‘π‘Ÿ3𝐡2ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ3β‹―2πœ†πΆ1𝑑𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘‘π‘Ÿ3π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ3ξƒ­ξ‚ƒπ‘π‘˜1π‘π‘˜2β‹―π‘π‘˜π‘ξ‚„π‘‡+ξƒ¬πœ†πΆ1𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπ‘‘4𝐡1ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ4πœ†πΆ1𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπ‘‘4𝐡2ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ4β‹―πœ†πΆ1𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπ‘‘4π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ4ξƒ­ξ‚ƒπ‘π‘˜1π‘π‘˜2β‹―π‘π‘˜π‘ξ‚„π‘‡+ξƒ¬πœ†πΆ1𝑑2𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2𝑑2𝐡1ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2πœ†πΆ1𝑑2𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2𝑑2𝐡2ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2β‹―πœ†πΆ1𝑑2𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2𝑑2π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2ξƒ­ξ€Ίπ‘π‘˜1π‘π‘˜2β‹―π‘π‘˜π‘ξ€»π‘‡+2πœ†πΆ1𝑑𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘‘π‘Ÿ3𝐡1ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ32πœ†πΆ1𝑑𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘‘π‘Ÿ3𝐡2ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ3β‹―2πœ†πΆ1𝑑𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘‘π‘Ÿ3π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ3ξƒ­ξ€Ίπ‘π‘˜1π‘π‘˜2β‹―π‘π‘˜π‘ξ€»π‘‡+ξƒ¬πœ†πΆ1𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘4𝐡1ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ4πœ†πΆ1𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘4𝐡2ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ4β‹―πœ†πΆ1𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘4π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ4ξƒ­ξ€Ίπ‘π‘˜1π‘π‘˜2β‹―π‘π‘˜π‘ξ€»π‘‡+𝑑2𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2𝑑2𝐡1ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2𝑑2𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2𝑑2𝐡2ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2⋯𝑑2𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2𝑑2π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2ξƒ­ξ‚ƒπ‘π‘˜1π‘π‘˜2β‹―π‘π‘˜π‘ξ‚„π‘‡+2𝑑𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘‘π‘Ÿ3𝐡1ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ32𝑑𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘‘π‘Ÿ3𝐡2ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ3β‹―2𝑑𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘‘π‘Ÿ3π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ3ξƒ­ξ‚ƒπ‘π‘˜1π‘π‘˜2β‹―π‘π‘˜π‘ξ‚„π‘‡+𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπ‘‘4𝐡1ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ4𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπ‘‘4𝐡2ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ4⋯𝐸𝐼π‘₯π‘₯ξ€·π‘Ÿπ‘–ξ€Έπ‘‘4π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ4ξƒ­ξ‚ƒπ‘π‘˜1π‘π‘˜2β‹―π‘π‘˜π‘ξ‚„π‘‡+𝑑2𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2𝑑2𝐡1ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2𝑑2𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2𝑑2𝐡2ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2⋯𝑑2𝐸𝐼π‘₯π‘¦ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2𝑑2π΅π‘ξ€·π‘Ÿπ‘–ξ€Έπ‘‘π‘Ÿ2ξƒ­