Dynamic Analysis of Wind Turbine Blades Using Radial Basis Functions

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]: 𝐵_𝑖(𝑟)=𝑟−𝑟_𝑖²+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]:𝑢_𝑘(𝑟,𝑡)=^𝑁_𝑖=1𝑣𝑎_𝑘𝑖(𝑡)𝐵_𝑖(𝑟),for𝑘=1,2,3,(2)_𝑘(𝑟,𝑡)=^𝑁_𝑖=1𝑈𝑎_𝑘𝑖(𝑡)𝐵_𝑖(𝑟),for𝑘=1,2,3,(3)_𝑘(𝑟)=^𝑁_𝑖=1𝑉𝑏_𝑘𝑖𝐵_𝑖(𝑟),for𝑘=1,2,3,(4)_𝑘(𝑟)=^𝑁_𝑖=1𝑏_𝑘𝑖𝐵_𝑖(𝑟),for𝑘=1,2,3,(5) where 𝑎_𝑘𝑖, 𝑎_𝑘𝑖, 𝑏_𝑘𝑖, and 𝑏_𝑘𝑖 are coefficients to be determined. Blade deflection 𝑢_𝑘(𝑟,𝑡) is the sum of 𝑁 radial basis functions, each associated with a different center 𝑟_𝑖. Blade deflection 𝑣_𝑘(𝑟,𝑡) denotes the sum of 𝑁 radial basis functions, each associated with a different center 𝑟_𝑖. Blade deflection 𝑈_𝑘(𝑟) is the sum of 𝑁 radial basis functions, each associated with a different center 𝑟_𝑖. Blade deflection 𝑉_𝑘(𝑟) denotes the sum of 𝑁 radial basis functions, each associated with a different center 𝑟_𝑖. The domain contains 𝑁 collocation points.

3. Dynamic Analysis of Wind Turbine Blades

The kinetic energy of rotating wind turbine blades can be derived as𝑇^𝑒=³_𝑘=112^𝐿0𝜌𝐴𝜕𝑢^𝑘𝜕𝑡²+𝜕𝑣^𝑘𝜕𝑡²+Ω𝑣_𝑘cos𝜃²+Ω𝑢_𝑘sin𝜃²𝑑𝑟,(6) where Ω is hub rotational speed, 𝐴 is cross-sectional area, 𝜃 is pitch angle, 𝜌 is blade density, and 𝐿 is blade length. The corresponding strain energy of rotating blades is𝑈^𝑒=³_𝑘=112^𝐿0𝐸𝐼_𝑦𝑦𝜕𝜕𝑢𝑟^𝑘²+2𝐼_𝑥𝑦+𝐼_𝑥𝑥𝜕𝜕𝑢𝑟^𝑘𝜕𝜕𝑣𝑟^𝑘+1𝜕𝜕𝑣𝑟^𝑘²𝑑𝑟2^𝐿0^𝐿𝑟×𝜕𝑢_𝑘𝜌𝐴Ω²𝑟+𝑟₀𝑑𝑟𝜕𝑟²+𝜕𝑣^𝑘𝜕𝑟²𝑑𝑟,(7) where 𝑟₀ is hub radius, 𝑢₁(𝑟,𝑡) is the displacement of the first blade on the 𝑥-axis, 𝑣₁(𝑟,𝑡) is the displacement of the first blade on the 𝑦-axis, 𝑢₂(𝑟,𝑡) is the displacement of the second blade on the 𝑥-axis, 𝑣₂(𝑟,𝑡) is the displacement of the second blade on the 𝑦-axis, 𝑢₃(𝑟,𝑡) is the displacement of the third blade in the 𝑥-axis, 𝑣₃(𝑟,𝑡) 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 𝛿𝑊^𝑒 in wind turbine blades can be derived as𝛿𝑊^𝑒=³_𝑘=1−^𝐿0𝐶₀𝜕𝑢_𝑘𝜕𝑡𝛿𝑢^𝑘𝑑𝑟−^𝐿0𝐶₁𝜕²𝐸𝐼_𝑦𝑦𝜕𝑟²𝜕³𝑢_𝑘𝜕𝑡𝜕𝑟²−2_𝐿0𝛿𝑢_𝑘𝑑𝑟𝐶₁𝜕𝐸𝐼_𝑦𝑦𝜕𝑟𝜕⁴𝑢_𝑘𝜕𝑡𝜕𝑟³−𝛿𝑢_𝑘𝑑𝑟_𝐿0𝐶₁𝐸𝐼_𝑦𝑦𝜕⁵𝑢_𝑘𝜕𝑡𝜕𝑟⁴−𝛿𝑢_𝑘𝑑𝑟_𝐿0𝐶₀𝜕𝑣_𝑘𝜕𝑡𝛿𝑣^𝑘𝑑𝑟−^𝐿0𝐶₁𝜕²𝐸𝐼_𝑥𝑥𝜕𝑟²𝜕³𝑣_𝑘𝜕𝑡𝜕𝑟²−2_𝐿0𝛿𝑣_𝑘𝑑𝑟𝐶₁𝜕𝐸𝐼_𝑥𝑥𝜕𝑟𝜕⁴𝑣_𝑘𝜕𝑡𝜕𝑟³−𝛿𝑣_𝑘𝑑𝑟_𝐿0𝐶₁𝐸𝐼_𝑥𝑥𝜕⁵𝑣_𝑘𝜕𝑡𝜕𝑟⁴𝛿𝑣_𝑘𝑑𝑟,(8) where 𝐶₀ and 𝐶₁ are external and internal damping coefficients of blades, respectively. Equations (6)–(8) are integrated into the Hamilton equation as follows:^𝑡2_𝑡1(𝛿𝑇^𝑒−𝛿𝑈^𝑒+𝛿𝑊^𝑒)𝑑𝑡=0.(9) This leads to the following equations for the motion of the blades:𝜕²𝐸𝐼_𝑦𝑦𝜕𝑟²𝜕²𝑢_𝑘𝜕𝑟²+2𝜕𝐸𝐼^𝑦𝑦𝜕𝑟𝜕³𝑢_𝑘𝜕𝑟³+𝐸𝐼_𝑦𝑦𝜕⁴𝑢_𝑘𝜕𝑟⁴+𝜕²𝐸𝐼_𝑥𝑦𝜕𝑟²𝜕²𝑣_𝑘𝜕𝑟²+2𝜕𝐸𝐼^𝑥𝑦𝜕𝑟𝜕³𝑣_𝑘𝜕𝑟³+𝐸𝐼_𝑥𝑦𝜕⁴𝑣_𝑘𝜕𝑟⁴−𝜌Ω₂−𝜌𝐴Ω²𝑢_𝑘sin²𝜃𝜕𝜕𝑟^𝐿𝑟𝐴𝑟+𝑟₀𝑑𝑟𝜕𝑢^𝑘−𝜌Ω₂𝜕𝑟^𝐿𝑟𝐴𝑟+𝑟₀𝑑𝑟𝜕²𝑢_𝑘𝜕𝑟²+𝐶₀𝜕𝑢_𝑘𝜕𝑡+𝐶¹𝜕²𝐸𝐼_𝑦𝑦𝜕𝑟²𝜕³𝑢_𝑘𝜕𝑟²𝜕𝑡+2𝐶¹𝜕𝐸𝐼_𝑦𝑦𝜕𝑟𝜕⁴𝑢_𝑘+𝐶₁𝜕𝑟³𝜕𝑡𝐸𝐼_𝑦𝑦𝜕⁵𝑢_𝑘𝜕𝑟⁴𝜕𝑡+𝐶¹𝜕²𝐸𝐼_𝑥𝑦𝜕𝑟²𝜕³𝑣_𝑘𝜕𝑟²𝜕𝑡+2𝐶¹𝜕𝐸𝐼_𝑥𝑦𝜕𝑟𝜕⁴𝑣_𝑘+𝐶₁𝜕𝑟³𝜕𝑡𝐸𝐼_𝑥𝑦𝜕⁵𝑣_𝑘𝜕𝑟⁴𝜕𝑡+𝜌𝐴𝜕²𝑢_𝑘𝜕𝑡²𝜕=0for𝑘=1,2,3,₂𝐸𝐼_𝑥𝑥𝜕𝑟²𝜕²𝑣_𝑘𝜕𝑟²+2𝜕𝐸𝐼^𝑥𝑥𝜕𝑟𝜕³𝑣_𝑘𝜕𝑟³+𝐸𝐼_𝑥𝑥𝜕⁴𝑣_𝑘𝜕𝑟⁴+𝜕²𝐸𝐼_𝑥𝑦𝜕𝑟²𝜕²𝑢_𝑘𝜕𝑟²+2𝜕𝐸𝐼^𝑥𝑦𝜕𝑟𝜕³𝑢_𝑘𝜕𝑟³+𝐸𝐼_𝑥𝑦𝜕⁴𝑢_𝑘𝜕𝑟⁴−𝜌Ω₂−𝜌𝐴Ω²𝑣_𝑘cos²𝜃𝜕𝜕𝑟^𝐿𝑟𝐴𝑟+𝑟₀𝑑𝑟𝜕𝑣^𝑘−𝜌Ω₂𝜕𝑟^𝐿𝑟𝐴𝑟+𝑟₀𝑑𝑟𝜕²𝑣_𝑘𝜕𝑟²+𝐶₀𝜕𝑣_𝑘𝜕𝑡+𝐶¹𝜕²𝐸𝐼_𝑥𝑥𝜕𝑟²𝜕³𝑣_𝑘𝜕𝑟²𝜕𝑡+2𝐶¹𝜕𝐸𝐼_𝑥𝑥𝜕𝑟𝜕⁴𝑣_𝑘+𝐶₁𝜕𝑟³𝜕𝑡𝐸𝐼_𝑥𝑥𝜕⁵𝑣_𝑘𝜕𝑟⁴𝜕𝑡+𝐶¹𝜕²𝐸𝐼_𝑥𝑦𝜕𝑟²𝜕³𝑢_𝑘𝜕𝑟²𝜕𝑡+2𝐶¹𝜕𝐸𝐼_𝑥𝑦𝜕𝑟𝜕⁴𝑢_𝑘+𝐶₁𝜕𝑟³𝜕𝑡𝐸𝐼_𝑥𝑦𝜕⁵𝑢_𝑘𝜕𝑟⁴𝜕𝑡+𝜌𝐴𝜕²𝑣_𝑘𝜕𝑡²=0for𝑘=1,2,3.(10) The following equations are the corresponding boundary conditions:𝜕𝑢𝑢_𝑘𝑘(0,𝑡)=0for𝑘=1,2,3,(0,𝑡)𝐸𝐼_𝑦𝑦𝜕𝑟=0for𝑘=1,2,3,𝜕²𝑢_𝑘(𝐿,𝑡)𝜕𝑟²𝜕=0for𝑘=1,2,3,𝜕𝑟𝐸𝐼^𝑦𝑦𝜕²𝑢_𝑘(𝐿,𝑡)𝑣𝜕𝑟²=0for𝑘=1,2,3,_𝑘𝜕𝑣_𝑘(0,𝑡)=0for𝑘=1,2,3,(0,𝑡)𝐸𝐼_𝑥𝑥𝜕𝑟=0for𝑘=1,2,3,𝜕²𝑣_𝑘(𝐿,𝑡)𝜕𝑟²𝜕=0for𝑘=1,2,3,𝜕𝑟𝐸𝐼^𝑥𝑥𝜕²𝑣_𝑘(𝐿,𝑡)𝜕𝑟²=0for𝑘=1,2,3.(11) A solution is thus assumed as𝑢𝑣_𝑘=𝑈_𝑘(𝑟)𝑒^𝜆𝑡for𝑘=1,2,3,_𝑘=𝑉_𝑘(𝑟)𝑒^𝜆𝑡for𝑘=1,2,3.(12) This yields the following equations for the motion of blades: 𝑑²𝐸𝐼_𝑦𝑦𝑑𝑟²𝑑²𝑈_𝑘𝑑𝑟²+2𝑑𝐸𝐼^𝑦𝑦𝑑𝑟𝑑³𝑈_𝑘𝑑𝑟³+𝐸𝐼_𝑦𝑦𝑑⁴𝑈_𝑘𝑑𝑟⁴+𝑑²𝐸𝐼_𝑥𝑦𝑑𝑟²𝑑²𝑉_𝑘𝑑𝑟²+2𝑑𝐸𝐼^𝑥𝑦𝑑𝑟𝑑³𝑉_𝑘𝑑𝑟³+𝐸𝐼_𝑥𝑦𝑑⁴𝑉_𝑘𝑑𝑟⁴−𝜌Ω₂−𝜌𝐴Ω²𝑈_𝑘sin²𝜃𝑑𝑑𝑟^𝐿𝑟𝐴𝑟+𝑟₀𝑑𝑟𝑑𝑈^𝑘−𝜌Ω₂𝑑𝑟^𝐿𝑟𝐴𝑟+𝑟₀𝑑𝑟𝑑²𝑈_𝑘𝑑𝑟²+𝜆𝐶₀𝑈_𝑘+𝜆𝐶₁𝑑²𝐸𝐼_𝑦𝑦𝑑𝑟²𝑑²𝑈_𝑘𝑑𝑟²+2𝜆𝐶₁𝑑𝐸𝐼_𝑦𝑦𝑑𝑟𝑑³𝑈_𝑘𝑑𝑟³+𝜆𝐶₁𝐸𝐼_𝑦𝑦𝑑⁴𝑈_𝑘𝑑𝑟⁴+𝜆𝐶₁𝑑²𝐸𝐼_𝑥𝑦𝑑𝑟²𝑑²𝑉_𝑘𝑑𝑟²+2𝜆𝐶₁𝑑𝐸𝐼_𝑥𝑦𝑑𝑟𝑑³𝑉_𝑘𝑑𝑟³+𝜆𝐶₁𝐸𝐼_𝑥𝑦𝑑⁴𝑉_𝑘𝑑𝑟⁴𝑑+𝜆²𝜌𝐴𝑈_𝑘=0for𝑘=1,2,3,₂𝐸𝐼_𝑥𝑥𝑑𝑟²𝑑²𝑉_𝑘𝑑𝑟²+2𝑑𝐸𝐼^𝑥𝑥𝑑𝑟𝑑³𝑉_𝑘𝑑𝑟³+𝐸𝐼_𝑥𝑥𝑑⁴𝑉_𝑘𝑑𝑟⁴+𝑑²𝐸𝐼_𝑥𝑦𝑑𝑟²𝑑²𝑈_𝑘𝑑𝑟²+2𝑑𝐸𝐼^𝑥𝑦𝑑𝑟𝑑³𝑈_𝑘𝑑𝑟³+𝐸𝐼_𝑥𝑦𝑑⁴𝑈_𝑘𝑑𝑟⁴−𝜌Ω₂−𝜌𝐴Ω²𝑉_𝑘cos²𝜃𝑑𝑑𝑟^𝐿𝑟𝐴𝑟+𝑟₀𝑑𝑟𝑑𝑉^𝑘−𝜌Ω₂𝑑𝑟^𝐿𝑟𝐴𝑟+𝑟₀𝑑𝑟𝑑²𝑉_𝑘𝑑𝑟²+𝜆𝐶₀𝑑𝑉_𝑘𝑑𝑡+𝜆𝐶¹𝑑²𝐸𝐼_𝑥𝑥𝑑𝑟²𝑑²𝑉_𝑘𝑑𝑟²+2𝜆𝐶₁𝑑𝐸𝐼_𝑥𝑥𝑑𝑟𝑑³𝑉_𝑘𝑑𝑟³+𝜆𝐶₁𝐸𝐼_𝑥𝑥𝑑⁴𝑉_𝑘𝑑𝑟⁴+𝜆𝐶₁𝑑²𝐸𝐼_𝑥𝑦𝑑𝑟²𝑑²𝑈_𝑘𝑑𝑟²+2𝜆𝐶₁𝑑𝐸𝐼_𝑥𝑦𝑑𝑟𝑑³𝑈_𝑘𝑑𝑟³+𝜆𝐶₁𝐸𝐼_𝑥𝑦𝑑⁴𝑈_𝑘𝑑𝑟⁴+𝜆²𝜌𝐴𝑉_𝑘=0for𝑘=1,2,3.(13) The corresponding boundary conditions are as follows:𝑑𝑈𝑈_𝑘𝑘(0)=0for𝑘=1,2,3,(0)𝐸𝐼_𝑦𝑦𝑑𝑟=0for𝑘=1,2,3,𝑑²𝑈_𝑘(𝐿)𝑑𝑟²𝑑=0for𝑘=1,2,3,𝑑𝑟𝐸𝐼^𝑦𝑦𝑑²𝑈_𝑘(𝐿)𝑉𝑑𝑟²=0for𝑘=1,2,3,_𝑘𝑑𝑉_𝑘(0)=0for𝑘=1,2,3,(0)𝐸𝐼_𝑥𝑥𝑑𝑟=0for𝑘=1,2,3,𝑑²𝑉_𝑘(𝐿)𝑑𝑟²𝑑=0for𝑘=1,2,3,𝑑𝑟𝐸𝐼^𝑥𝑥𝑑²𝑉_𝑘(𝐿)𝑑𝑟²=0for𝑘=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:^{𝑟𝜌𝐴𝑖𝐵1𝑟𝑖𝑟𝜌𝐴𝑖𝐵2𝑟𝑖𝑟⋯𝜌𝐴𝑖𝐵𝑁𝑟𝑖𝜕2𝑎𝑘1𝜕𝑡2𝜕2𝑎𝑘2𝜕𝑡2⋯𝜕2𝑎𝑘𝑁𝜕𝑡2𝑇+𝐶0𝐵1𝑟𝑖𝐶0𝐵2𝑟𝑖⋯𝐶0𝐵𝑁𝑟𝑖𝜕𝑎𝑘1𝜕𝑡𝜕𝑎𝑘2⋯𝜕𝑡𝜕𝑎𝑘𝑁𝜕𝑡𝑇+𝐶1𝜕2𝐸𝐼𝑦𝑦𝑟𝑖𝜕𝑟2𝜕2𝐵1𝑟𝑖𝜕𝑟2𝐶1𝜕2𝐸𝐼𝑦𝑦𝑟𝑖𝜕𝑟2𝜕2𝐵2𝑟𝑖𝜕𝑟2⋯𝐶1𝜕2𝐸𝐼𝑦𝑦𝑟𝑖𝜕𝑟2𝜕2𝐵𝑁𝑟𝑖𝜕𝑟2𝜕𝑎𝑘1𝜕𝑡𝜕𝑎𝑘2⋯𝜕𝑡𝜕𝑎𝑘𝑁𝜕𝑡𝑇+2𝐶1𝜕𝐸𝐼𝑦𝑦𝑟𝑖𝜕𝜕𝑟3𝐵1𝑟𝑖𝜕𝑟32𝐶1𝜕𝐸𝐼𝑦𝑦𝑟𝑖𝜕𝜕𝑟3𝐵2𝑟𝑖𝜕𝑟3⋯2𝐶1𝜕𝐸𝐼𝑦𝑦𝑟𝑖𝜕𝜕𝑟3𝐵𝑁𝑟𝑖𝜕𝑟3𝜕𝑎𝑘1𝜕𝑡𝜕𝑎𝑘2⋯𝜕𝑡𝜕𝑎𝑘𝑁𝜕𝑡𝑇+𝐶1𝐸𝐼𝑦𝑦𝑟𝑖𝜕4𝐵1𝑟𝑖𝜕𝑟4𝐶1𝐸𝐼𝑦𝑦𝑟𝑖𝜕4𝐵2𝑟𝑖𝜕𝑟4⋯𝐶1𝐸𝐼𝑦𝑦𝑟𝑖𝜕4𝐵𝑁𝑟𝑖𝜕𝑟4𝜕𝑎𝑘1𝜕𝑡𝜕𝑎𝑘2⋯𝜕𝑡𝜕𝑎𝑘𝑁𝜕𝑡𝑇+𝐶1𝜕2𝐸𝐼𝑥𝑦𝑟𝑖𝜕𝑟2𝜕2𝐵1𝑟𝑖𝜕𝑟2𝐶1𝜕2𝐸𝐼𝑥𝑦𝑟𝑖𝜕𝑟2𝜕2𝐵2𝑟𝑖𝜕𝑟2⋯𝐶1𝜕2𝐸𝐼𝑥𝑦𝑟𝑖𝜕𝑟2𝜕2𝐵𝑁𝑟𝑖𝜕𝑟2𝜕𝑎𝑘1𝜕𝜕𝑡𝑎𝑘2⋯𝜕𝜕𝑡𝑎𝑘𝑁𝜕𝑡𝑇+2𝐶1𝜕𝐸𝐼𝑥𝑦𝑟𝑖𝜕𝜕𝑟3𝐵1𝑟𝑖𝜕𝑟32𝐶1𝜕𝐸𝐼𝑥𝑦𝑟𝑖𝜕𝜕𝑟3𝐵2𝑟𝑖𝜕𝑟3⋯2𝐶1𝜕𝐸𝐼𝑥𝑦𝑟𝑖𝜕𝜕𝑟3𝐵𝑁𝑟𝑖𝜕𝑟3𝜕𝑎𝑘1𝜕𝜕𝑡𝑎𝑘2⋯𝜕𝜕𝑡𝑎𝑘𝑁𝜕𝑡𝑇+𝐶1𝐸𝐼𝑥𝑦𝑟𝑖𝜕4𝐵1𝑟𝑖𝜕𝑟4𝐶1𝐸𝐼𝑥𝑦𝑟𝑖𝜕4𝐵2𝑟𝑖𝜕𝑟4⋯𝐶1𝐸𝐼𝑥𝑦𝑟𝑖𝜕4𝐵𝑁𝑟𝑖𝜕𝑟4𝜕𝑎𝑘1𝜕𝜕𝑡𝑎𝑘2⋯𝜕𝜕𝑡𝑎𝑘𝑁𝜕𝑡𝑇+𝜕2𝐸𝐼𝑦𝑦𝑟𝑖𝜕𝑟2𝜕2𝐵1𝑟𝑖𝜕𝑟2𝜕2𝐸𝐼𝑦𝑦𝑟𝑖𝜕𝑟2𝜕2𝐵2𝑟𝑖𝜕𝑟2⋯𝜕2𝐸𝐼𝑦𝑦𝑟𝑖𝜕𝑟2𝜕2𝐵𝑁𝑟𝑖𝜕𝑟2𝑎𝑘1𝑎𝑘2⋯𝑎𝑘𝑁𝑇+2𝜕𝐸𝐼𝑦𝑦𝑟𝑖𝜕𝜕𝑟3𝐵1𝑟𝑖𝜕𝑟32𝜕𝐸𝐼𝑦𝑦𝑟𝑖𝜕𝜕𝑟3𝐵2𝑟𝑖𝜕𝑟3⋯2𝜕𝐸𝐼𝑦𝑦𝑟𝑖𝜕𝜕𝑟3𝐵𝑁𝑟𝑖𝜕𝑟3𝑎𝑘1𝑎𝑘2⋯𝑎𝑘𝑁𝑇+𝐸𝐼𝑦𝑦𝑟𝑖𝜕4𝐵1𝑟𝑖𝜕𝑟4𝐸𝐼𝑦𝑦𝑟𝑖𝜕4𝐵2𝑟𝑖𝜕𝑟4⋯𝐸𝐼𝑦𝑦𝑟𝑖𝜕4𝐵𝑁𝑟𝑖𝜕𝑟4𝑎𝑘1𝑎𝑘2⋯𝑎𝑘𝑁𝑇+𝜕2𝐸𝐼𝑥𝑦𝑟𝑖𝜕𝑟2𝜕2𝐵1𝑟𝑖𝜕𝑟2𝜕2𝐸𝐼𝑥𝑦𝑟𝑖𝜕𝑟2𝜕2𝐵2𝑟𝑖𝜕𝑟2⋯𝜕2𝐸𝐼𝑥𝑦𝑟𝑖𝜕𝑟2𝜕2𝐵𝑁𝑟𝑖𝜕𝑟2𝑎𝑘1𝑎𝑘2⋯𝑎𝑘𝑁𝑇+2𝜕𝐸𝐼𝑥𝑦𝑟𝑖𝜕𝜕𝑟3𝐵1𝑟𝑖𝜕𝑟32𝜕𝐸𝐼𝑥𝑦𝑟𝑖𝜕𝜕𝑟3𝐵2𝑟𝑖𝜕𝑟3⋯2𝜕𝐸𝐼𝑥𝑦𝑟𝑖𝜕𝜕𝑟3𝐵𝑁𝑟𝑖𝜕𝑟3𝑎𝑘1𝑎𝑘2⋯𝑎𝑘𝑁𝑇+𝐸𝐼𝑥𝑦𝑟𝑖𝜕4𝐵1𝑟𝑖𝜕𝑟4𝐸𝐼𝑥𝑦𝑟𝑖𝜕4𝐵2𝑟𝑖𝜕𝑟4⋯𝐸𝐼𝑥𝑦𝑟𝑖𝜕4𝐵𝑁𝑟𝑖𝜕𝑟4𝑎𝑘1𝑎𝑘2⋯𝑎𝑘𝑁𝑇−𝜌𝐴(𝑟)Ω2(sin𝜃)2𝐵1𝑟𝑖𝜌𝐴(𝑟)Ω2(sin𝜃)2𝐵2𝑟𝑖⋯𝜌𝐴(𝑟)Ω2(sin𝜃)2𝐵𝑁𝑟𝑖𝑎𝑘1𝑎𝑘2⋯𝑎𝑘𝑁𝑇−𝜌Ω2𝜕∫𝜕𝑟𝐿𝑟𝑖𝐴(𝑟)𝔄𝑑𝑟𝜕𝐵1𝑟𝑖𝜕𝑟𝜌Ω2𝜕∫𝜕𝑟𝐿𝑟𝑖𝐴(𝑟)𝔄𝑑𝑟𝜕𝐵2𝑟𝑖𝜕𝑟⋯𝜌Ω2𝜕∫𝜕𝑟𝐿𝑟𝑖𝐴(𝑟)𝔄𝑑𝑟𝜕𝐵𝑁𝑟𝑖×𝑎𝜕𝑟𝑘1𝑎𝑘2⋯𝑎𝑘𝑁𝑇−𝜌Ω2∫𝐿𝑟𝑖𝜕𝐴(𝑟)𝔄𝑑𝑟2𝐵1𝑟𝑖𝜕𝑟2𝜌Ω2∫𝐿𝑟𝑖𝜕𝐴(𝑟)𝔄𝑑𝑟2𝐵2𝑟𝑖𝜕𝑟2⋯𝜌Ω2∫𝐿𝑟𝑖𝜕𝐴(𝑟)𝔄𝑑𝑟2𝐵𝑁𝑟𝑖𝜕𝑟2×𝑎𝑘1𝑎𝑘2⋯𝑎𝑘𝑁𝑇=[0],for𝑟𝑖=1,2,…,𝑁,𝑘=1,2,3,𝜌𝐴}_{𝑖𝐵1𝑟𝑖𝑟𝜌𝐴𝑖𝐵2𝑟𝑖𝑟⋯𝜌𝐴𝑖𝐵𝑁𝑟𝑖𝜕2𝑎𝑘1𝜕𝑡2𝜕2𝑎𝑘2𝜕𝑡2⋯𝜕2𝑎𝑘𝑁𝜕𝑡2𝑇+𝐶0𝐵1𝑟𝑖𝐶0𝐵2𝑟𝑖⋯𝐶0𝐵𝑁𝑟𝑖𝜕𝑎𝑘1𝜕𝜕𝑡𝑎𝑘2⋯𝜕𝜕𝑡𝑎𝑘𝑁𝜕𝑡𝑇+𝐶1𝜕2𝐸𝐼𝑥𝑥𝑟𝑖𝜕𝑟2𝜕2𝐵1𝑟𝑖𝜕𝑟2𝐶1𝜕2𝐸𝐼𝑥𝑥𝑟𝑖𝜕𝑟2𝜕2𝐵2𝑟𝑖𝜕𝑟2⋯𝐶1𝜕2𝐸𝐼𝑥𝑥𝑟𝑖𝜕𝑟2𝜕2𝐵𝑁𝑟𝑖𝜕𝑟2𝜕𝑎𝑘1𝜕𝜕𝑡𝑎𝑘2⋯𝜕𝜕𝑡𝑎𝑘𝑁𝜕𝑡𝑇+2𝐶1𝜕𝐸𝐼𝑥𝑥𝑟𝑖𝜕𝜕𝑟3𝐵1𝑟𝑖𝜕𝑟32𝐶1𝜕𝐸𝐼𝑥𝑥𝑟𝑖𝜕𝜕𝑟3𝐵2𝑟𝑖𝜕𝑟3⋯2𝐶1𝜕𝐸𝐼𝑥𝑥𝑟𝑖𝜕𝜕𝑟3𝐵𝑁𝑟𝑖𝜕𝑟3𝜕𝑎𝑘1𝜕𝜕𝑡𝑎𝑘2⋯𝜕𝜕𝑡𝑎𝑘𝑁𝜕𝑡𝑇+𝐶1𝐸𝐼𝑥𝑥𝑟𝑖𝜕4𝐵1𝑟𝑖𝜕𝑟4𝐶1𝐸𝐼𝑥𝑥𝑟𝑖𝜕4𝐵2𝑟𝑖𝜕𝑟4⋯𝐶1𝐸𝐼𝑥𝑥𝑟𝑖𝜕4𝐵𝑁𝑟𝑖𝜕𝑟4𝜕𝑎𝑘1𝜕𝜕𝑡𝑎𝑘2⋯𝜕𝜕𝑡𝑎𝑘𝑁𝜕𝑡𝑇+𝐶1𝜕2𝐸𝐼𝑥𝑦𝑟𝑖𝜕𝑟2𝜕2𝐵1𝑟𝑖𝜕𝑟2𝐶1𝜕2𝐸𝐼𝑥𝑦𝑟𝑖𝜕𝑟2𝜕2𝐵2𝑟𝑖𝜕𝑟2⋯𝐶1𝜕2𝐸𝐼𝑥𝑦𝑟𝑖𝜕𝑟2𝜕2𝐵𝑁𝑟𝑖𝜕𝑟2𝜕𝑎𝑘1𝜕𝑡𝜕𝑎𝑘2⋯𝜕𝑡𝜕𝑎𝑘𝑁𝜕𝑡𝑇+2𝐶1𝜕𝐸𝐼𝑥𝑦𝑟𝑖𝜕𝜕𝑟3𝐵1𝑟𝑖𝜕𝑟32𝐶1𝜕𝐸𝐼𝑥𝑦𝑟𝑖𝜕𝜕𝑟3𝐵2𝑟𝑖𝜕𝑟3⋯2𝐶1𝜕𝐸𝐼𝑥𝑦𝑟𝑖𝜕𝜕𝑟3𝐵𝑁𝑟𝑖𝜕𝑟3𝜕𝑎𝑘1𝜕𝑡𝜕𝑎𝑘2⋯𝜕𝑡𝜕𝑎𝑘𝑁𝜕𝑡𝑇+𝐶1𝐸𝐼𝑥𝑦𝑟𝑖𝜕4𝐵1𝑟𝑖𝜕𝑟4𝐶1𝐸𝐼𝑥𝑦𝑟𝑖𝜕4𝐵2𝑟𝑖𝜕𝑟4⋯𝐶1𝐸𝐼𝑥𝑦𝑟𝑖𝜕4𝐵𝑁𝑟𝑖𝜕𝑟4𝜕𝑎𝑘1𝜕𝑡𝜕𝑎𝑘2⋯𝜕𝑡𝜕𝑎𝑘𝑁𝜕𝑡𝑇+𝜕2𝐸𝐼𝑥𝑥𝑟𝑖𝜕𝑟2𝜕2𝐵1𝑟𝑖𝜕𝑟2𝜕2𝐸𝐼𝑥𝑥𝑟𝑖𝜕𝑟2𝜕2𝐵2𝑟𝑖𝜕𝑟2⋯𝜕2𝐸𝐼𝑥𝑥𝑟𝑖𝜕𝑟2𝜕2𝐵𝑁𝑟𝑖𝜕𝑟2𝑎𝑘1𝑎𝑘2⋯𝑎𝑘𝑁𝑇+2𝜕𝐸𝐼𝑥𝑥𝑟𝑖𝜕𝜕𝑟3𝐵1𝑟𝑖𝜕𝑟32𝜕𝐸𝐼𝑥𝑥𝑟𝑖𝜕𝜕𝑟3𝐵2𝑟𝑖𝜕𝑟3⋯2𝜕𝐸𝐼𝑥𝑥𝑟𝑖𝜕𝜕𝑟3𝐵𝑁𝑟𝑖𝜕𝑟3𝑎𝑘1𝑎𝑘2⋯𝑎𝑘𝑁𝑇+𝐸𝐼𝑥𝑥𝑟𝑖𝜕4𝐵1𝑟𝑖𝜕𝑟4𝐸𝐼𝑥𝑥𝑟𝑖𝜕4𝐵2𝑟𝑖𝜕𝑟4⋯𝐸𝐼𝑥𝑥𝑟𝑖𝜕4𝐵𝑁𝑟𝑖𝜕𝑟4𝑎𝑘1𝑎𝑘2⋯𝑎𝑘𝑁𝑇+𝜕2𝐸𝐼𝑥𝑦𝑟𝑖𝜕𝑟2𝜕2𝐵1𝑟𝑖𝜕𝑟2𝜕2𝐸𝐼𝑥𝑦𝑟𝑖𝜕𝑟2𝜕2𝐵2𝑟𝑖𝜕𝑟2⋯𝜕2𝐸𝐼𝑥𝑦𝑟𝑖𝜕𝑟2𝜕2𝐵𝑁𝑟𝑖𝜕𝑟2𝑎𝑘1𝑎𝑘2⋯𝑎𝑘𝑁𝑇+2𝜕𝐸𝐼𝑥𝑦𝑟𝑖𝜕𝜕𝑟3𝐵1𝑟𝑖𝜕𝑟32𝜕𝐸𝐼𝑥𝑦𝑟𝑖𝜕𝜕𝑟3𝐵2𝑟𝑖𝜕𝑟3⋯2𝜕𝐸𝐼𝑥𝑦𝑟𝑖𝜕𝜕𝑟3𝐵𝑁𝑟𝑖𝜕𝑟3𝑎𝑘1𝑎𝑘2⋯𝑎𝑘𝑁𝑇+𝐸𝐼𝑥𝑦𝑟𝑖𝜕4𝐵1𝑟𝑖𝜕𝑟4𝐸𝐼𝑥𝑦𝑟𝑖𝜕4𝐵2𝑟𝑖𝜕𝑟4⋯𝐸𝐼𝑥𝑦𝑟𝑖𝜕4𝐵𝑁𝑟𝑖𝜕𝑟4𝑎𝑘1𝑎𝑘2⋯𝑎𝑘𝑁𝑇−𝜌𝐴(𝑟)Ω2(cos𝜃)2𝐵1𝑟𝑖𝜌𝐴(𝑟)Ω2(cos𝜃)2𝐵2𝑟𝑖⋯𝜌𝐴(𝑟)Ω2(cos𝜃)2𝐵𝑁𝑟𝑖𝑎𝑘1𝑎𝑘2⋯𝑎𝑘𝑁𝑇−𝜌Ω2𝜕∫𝜕𝑟𝐿𝑟𝑖𝐴(𝑟)𝔄𝑑𝑟𝜕𝐵1𝑟𝑖𝜕𝑟𝜌Ω2𝜕∫𝜕𝑟𝐿𝑟𝑖𝐴(𝑟)𝔄𝑑𝑟𝜕𝐵2𝑟𝑖𝜕𝑟⋯𝜌Ω2𝜕∫𝜕𝑟𝐿𝑟𝑖𝐴(𝑟)𝔄𝑑𝑟𝜕𝐵𝑁𝑟𝑖×𝜕𝑟𝑎𝑘1𝑎𝑘2⋯𝑎𝑘𝑁𝑇−𝜌Ω2∫𝐿𝑟𝑖𝜕𝐴(𝑟)𝔄𝑑𝑟2𝐵1𝑟𝑖𝜕𝑟2𝜌Ω2∫𝐿𝑟𝑖𝜕𝐴(𝑟)𝔄𝑑𝑟2𝐵2𝑟𝑖𝜕𝑟2⋯𝜌Ω2∫𝐿𝑟𝑖𝜕𝐴(𝑟)𝔄𝑑𝑟2𝐵𝑁𝑟𝑖𝜕𝑟2×𝑎𝑘1𝑎𝑘2⋯𝑎𝑘𝑁𝑇=[0]for𝑖=1,2,…,𝑁,𝑘=1,2,3,}(15)where 𝔄 denotes (𝑟+𝑟₀).

Based on the radial basis function technique, (11) take the following discrete forms: ^{𝐵1𝑟1𝐵2𝑟1⋯𝐵𝑁𝑟1𝑎𝑘1𝑎𝑘2⋯𝑎𝑘𝑁𝑇=[0]for𝑘=1,2,3,𝜕𝐵1𝑟1𝜕𝑟𝜕𝐵2𝑟1⋯𝜕𝑟𝜕𝐵𝑁𝑟1𝑎𝜕𝑟𝑘1𝑎𝑘2⋯𝑎𝑘𝑁𝑇=[0]for𝑘=1,2,3,𝐸𝐼𝑦𝑦𝑟𝑁𝜕2𝐵1𝑟𝑁𝜕𝑟2𝐸𝐼𝑦𝑦𝑟𝑁𝜕2𝐵2𝑟𝑁𝜕𝑟2⋯𝐸𝐼𝑦𝑦𝑟𝑁𝜕2𝐵𝑁𝑟𝑁𝜕𝑟2𝑎𝑘1𝑎𝑘2⋯𝑎𝑘𝑁𝑇=[0]for𝑘=1,2,3,𝜕𝐸𝐼𝑦𝑦𝑟𝑁𝜕𝜕𝑟2𝐵1𝑟𝑁𝜕𝑟2𝜕𝐸𝐼𝑦𝑦𝑟𝑁𝜕𝜕𝑟2𝐵2𝑟𝑁𝜕𝑟2⋯𝜕𝐸𝐼𝑦𝑦𝑟𝑁𝜕𝜕𝑟2𝐵𝑁𝑟𝑁𝜕𝑟2𝑎𝑘1𝑎𝑘2⋯𝑎𝑘𝑁𝑇+𝐸𝐼𝑦𝑦𝑟𝑁𝜕3𝐵1𝑟𝑁𝜕𝑟3𝐸𝐼𝑦𝑦𝑟𝑁𝜕3𝐵2𝑟𝑁𝜕𝑟3⋯𝐸𝐼𝑦𝑦𝑟𝑁𝜕3𝐵𝑁𝑟𝑁𝜕𝑟3×𝑎𝑘1𝑎𝑘2⋯𝑎𝑘𝑁𝑇=[0]for𝐵𝑘=1,2,3,}_{1𝑟1𝐵2𝑟1⋯𝐵𝑁𝑟1𝑎𝑘1𝑎𝑘2⋯𝑎𝑘𝑁𝑇=[0]for𝑘=1,2,3,𝜕𝐵1𝑟1𝜕𝑟𝜕𝐵2𝑟1⋯𝜕𝑟𝜕𝐵𝑁𝑟1𝜕𝑟𝑎𝑘1𝑎𝑘2⋯𝑎𝑘𝑁𝑇=[0]for𝑘=1,2,3,𝐸𝐼𝑥𝑥𝑟𝑁𝜕2𝐵1𝑟𝑁𝜕𝑟2𝐸𝐼𝑥𝑥𝑟𝑁𝜕2𝐵2𝑟𝑁𝜕𝑟2⋯𝐸𝐼𝑥𝑥𝑟𝑁𝜕2𝐵𝑁𝑟𝑁𝜕𝑟2𝑎𝑘1𝑎𝑘2⋯𝑎𝑘𝑁𝑇=[0],for𝑘=1,2,3,𝜕𝐸𝐼𝑥𝑥𝑟𝑁𝜕𝜕𝑟2𝐵1𝑟𝑁𝜕𝑟2𝜕𝐸𝐼𝑥𝑥𝑟𝑁𝜕𝜕𝑟2𝐵2𝑟𝑁𝜕𝑟2⋯𝜕𝐸𝐼𝑥𝑥𝑟𝑁𝜕𝜕𝑟2𝐵𝑁𝑟𝑁𝜕𝑟2𝑎𝑘1𝑎𝑘2⋯𝑎𝑘𝑁𝑇+𝐸𝐼𝑥𝑥𝑟𝑁𝜕3𝐵𝑁𝑟𝑁𝜕𝑟3𝐸𝐼𝑥𝑥𝑟𝑁𝜕3𝐵2𝑟𝑁𝜕𝑟3⋯𝐸𝐼𝑥𝑥𝑟𝑁𝜕3𝐵𝑁𝑟𝑁𝜕𝑟3×𝑎𝑘1𝑎𝑘2⋯𝑎𝑘𝑁𝑇=[0]for𝑘=1,2,3.}(16)By applying the radial basis function approach, (4) and (5) are substituted into (13). The following equations are then yielded^{𝜆2𝑟𝜌𝐴𝑖𝐵1𝑟𝑖𝜆2𝑟𝜌𝐴𝑖𝐵2𝑟𝑖⋯𝜆2𝑟𝜌𝐴𝑖𝐵𝑁𝑟𝑖𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇+𝜆𝐶0𝐵1𝑟𝑖𝜆𝐶0𝐵2𝑟𝑖⋯𝜆𝐶0𝐵𝑁𝑟𝑖𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇+𝜆𝐶1𝑑2𝐸𝐼𝑦𝑦𝑟𝑖𝑑𝑟2𝑑2𝐵1𝑟𝑖𝑑𝑟2𝜆𝐶1𝑑2𝐸𝐼𝑦𝑦𝑟𝑖𝑑𝑟2𝑑2𝐵2𝑟𝑖𝑑𝑟2⋯𝜆𝐶1𝑑2𝐸𝐼𝑦𝑦𝑟𝑖𝑑𝑟2𝑑2𝐵𝑁𝑟𝑖𝑑𝑟2𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇+2𝜆𝐶1𝑑𝐸𝐼𝑦𝑦𝑟𝑖𝑑𝑑𝑟3𝐵1𝑟𝑖𝑑𝑟32𝜆𝐶1𝑑𝐸𝐼𝑦𝑦𝑟𝑖𝑑𝑑𝑟3𝐵2𝑟𝑖𝑑𝑟3⋯2𝜆𝐶1𝑑𝐸𝐼𝑦𝑦𝑟𝑖𝑑𝑑𝑟3𝐵𝑁𝑟𝑖𝑑𝑟3𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇+𝜆𝐶1𝐸𝐼𝑦𝑦𝑟𝑖𝑑4𝐵1𝑟𝑖𝑑𝑟4𝜆𝐶1𝐸𝐼𝑦𝑦𝑟𝑖𝑑4𝐵2𝑟𝑖𝑑𝑟4⋯𝜆𝐶1𝐸𝐼𝑦𝑦𝑟𝑖𝑑4𝐵𝑁𝑟𝑖𝑑𝑟4𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇+𝜆𝐶1𝑑2𝐸𝐼𝑥𝑦𝑟𝑖𝑑𝑟2𝑑2𝐵1𝑟𝑖𝑑𝑟2𝜆𝐶1𝑑2𝐸𝐼𝑥𝑦𝑟𝑖𝑑𝑟2𝑑2𝐵2𝑟𝑖𝑑𝑟2⋯𝜆𝐶1𝑑2𝐸𝐼𝑥𝑦𝑟𝑖𝑑𝑟2𝑑2𝐵𝑁𝑟𝑖𝑑𝑟2𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇+2𝜆𝐶1𝑑𝐸𝐼𝑥𝑦𝑟𝑖𝑑𝑑𝑟3𝐵1𝑟𝑖𝑑𝑟32𝜆𝐶1𝑑𝐸𝐼𝑥𝑦𝑟𝑖𝑑𝑑𝑟3𝐵2𝑟𝑖𝑑𝑟3⋯2𝜆𝐶1𝑑𝐸𝐼𝑥𝑦𝑟𝑖𝑑𝑑𝑟3𝐵𝑁𝑟𝑖𝑑𝑟3𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇+𝜆𝐶1𝐸𝐼𝑥𝑦𝑟𝑖𝑑4𝐵1𝑟𝑖𝑑𝑟4𝜆𝐶1𝐸𝐼𝑥𝑦𝑟𝑖𝑑4𝐵2𝑟𝑖𝑑𝑟4⋯𝜆𝐶1𝐸𝐼𝑥𝑦𝑟𝑖𝑑4𝐵𝑁𝑟𝑖𝑑𝑟4𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇+𝑑2𝐸𝐼𝑦𝑦𝑟𝑖𝑑𝑟2𝑑2𝐵1𝑟𝑖𝑑𝑟2𝑑2𝐸𝐼𝑦𝑦𝑟𝑖𝑑𝑟2𝑑2𝐵2𝑟𝑖𝑑𝑟2⋯𝑑2𝐸𝐼𝑦𝑦𝑟𝑖𝑑𝑟2𝑑2𝐵𝑁𝑟𝑖𝑑𝑟2𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇+2𝑑𝐸𝐼𝑦𝑦𝑟𝑖𝑑𝑑𝑟3𝐵1𝑟𝑖𝑑𝑟32𝑑𝐸𝐼𝑦𝑦𝑟𝑖𝑑𝑑𝑟3𝐵2𝑟𝑖𝑑𝑟3⋯2𝑑𝐸𝐼𝑦𝑦𝑟𝑖𝑑𝑑𝑟3𝐵𝑁𝑟𝑖𝑑𝑟3𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇+𝐸𝐼𝑦𝑦𝑟𝑖𝑑4𝐵1𝑟𝑖𝑑𝑟4𝐸𝐼𝑦𝑦𝑟𝑖𝑑4𝐵2𝑟𝑖𝑑𝑟4⋯𝐸𝐼𝑦𝑦𝑟𝑖𝑑4𝐵𝑁𝑟𝑖𝑑𝑟4𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇+𝑑2𝐸𝐼𝑥𝑦𝑟𝑖𝑑𝑟2𝑑2𝐵1𝑟𝑖𝑑𝑟2𝑑2𝐸𝐼𝑥𝑦𝑟𝑖𝑑𝑟2𝑑2𝐵2𝑟𝑖𝑑𝑟2⋯𝑑2𝐸𝐼𝑥𝑦𝑟𝑖𝑑𝑟2𝑑2𝐵𝑁𝑟𝑖𝑑𝑟2𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇+2𝑑𝐸𝐼𝑥𝑦𝑟𝑖𝑑𝑑𝑟3𝐵1𝑟𝑖𝑑𝑟32𝑑𝐸𝐼𝑥𝑦𝑟𝑖𝑑𝑑𝑟3𝐵2𝑟𝑖𝑑𝑟3⋯2𝑑𝐸𝐼𝑥𝑦𝑟𝑖𝑑𝑑𝑟3𝐵𝑁𝑟𝑖𝑑𝑟3𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇+𝐸𝐼𝑥𝑦𝑟𝑖𝑑4𝐵1𝑟𝑖𝑑𝑟4𝐸𝐼𝑥𝑦𝑟𝑖𝑑4𝐵2𝑟𝑖𝑑𝑟4⋯𝐸𝐼𝑥𝑦𝑟𝑖𝑑4𝐵𝑁𝑟𝑖𝑑𝑟4𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇−𝑟𝜌𝐴𝑖Ω2(sin𝜃)2𝐵1𝑟𝑖𝑟𝜌𝐴𝑖Ω2(sin𝜃)2𝐵2𝑟𝑖𝑟⋯𝜌𝐴𝑖Ω2(sin𝜃)2𝐵𝑁𝑟𝑖𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇−𝜌Ω2𝑑∫𝑑𝑟𝐿𝑟𝑖𝐴(𝑟)𝔄𝑑𝑟𝑑𝐵1𝑟𝑖𝑑𝑟𝜌Ω2𝑑∫𝑑𝑟𝐿𝑟𝑖𝐴(𝑟)𝔄𝑑𝑟𝑑𝐵2𝑟𝑖𝑑𝑟⋯𝜌Ω2𝑑∫𝑑𝑟𝐿𝑟𝑖𝐴(𝑟)𝔄𝑑𝑟𝑑𝐵𝑁𝑟𝑖×𝑏𝑑𝑟𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇−𝜌Ω2∫𝐿𝑟𝑖𝑑𝐴(𝑟)𝔄𝑑𝑟2𝐵1𝑟𝑖𝑑𝑟2𝜌Ω2∫𝐿𝑟𝑖𝑑𝐴(𝑟)𝔄𝑑𝑟2𝐵2𝑟𝑖𝑑𝑟2⋯𝜌Ω2∫𝐿𝑟𝑖𝑑𝐴(𝑟)𝔄𝑑𝑟2𝐵𝑁𝑟𝑖𝑑𝑟2×𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇=[0]for𝜆𝑖=1,2,…,𝑁,𝑘=1,2,3,}_{2𝑟𝜌𝐴𝑖𝐵1𝑟𝑖𝜆2𝑟𝜌𝐴𝑖𝐵2𝑟𝑖⋯𝜆2𝑟𝜌𝐴𝑖𝐵𝑁𝑟𝑖𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇+𝜆𝐶0𝐵1𝑟𝑖𝜆𝐶0𝐵2𝑟𝑖⋯𝜆𝐶0𝐵𝑁𝑟𝑖𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇+𝜆𝐶1𝑑2𝐸𝐼𝑥𝑥𝑟𝑖𝑑𝑟2𝑑2𝐵1𝑟𝑖𝑑𝑟2𝜆𝐶1𝑑2𝐸𝐼𝑥𝑥𝑟𝑖𝑑𝑟2𝑑2𝐵2𝑟𝑖𝑑𝑟2⋯𝜆𝐶1𝑑2𝐸𝐼𝑥𝑥𝑟𝑖𝑑𝑟2𝑑2𝐵𝑁𝑟𝑖𝑑𝑟2𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇+2𝜆𝐶1𝑑𝐸𝐼𝑥𝑥𝑟𝑖𝑑𝑑𝑟3𝐵1𝑟𝑖𝑑𝑟32𝜆𝐶1𝑑𝐸𝐼𝑥𝑥𝑟𝑖𝑑𝑑𝑟3𝐵2𝑟𝑖𝑑𝑟3⋯2𝜆𝐶1𝑑𝐸𝐼𝑥𝑥𝑟𝑖𝑑𝑑𝑟3𝐵𝑁𝑟𝑖𝑑𝑟3𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇+𝜆𝐶1𝐸𝐼𝑥𝑥𝑟𝑖𝑑4𝐵1𝑟𝑖𝑑𝑟4𝜆𝐶1𝐸𝐼𝑥𝑥𝑟𝑖𝑑4𝐵2𝑟𝑖𝑑𝑟4⋯𝜆𝐶1𝐸𝐼𝑥𝑥𝑟𝑖𝑑4𝐵𝑁𝑟𝑖𝑑𝑟4𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇+𝜆𝐶1𝑑2𝐸𝐼𝑥𝑦𝑟𝑖𝑑𝑟2𝑑2𝐵1𝑟𝑖𝑑𝑟2𝜆𝐶1𝑑2𝐸𝐼𝑥𝑦𝑟𝑖𝑑𝑟2𝑑2𝐵2𝑟𝑖𝑑𝑟2⋯𝜆𝐶1𝑑2𝐸𝐼𝑥𝑦𝑟𝑖𝑑𝑟2𝑑2𝐵𝑁𝑟𝑖𝑑𝑟2𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇+2𝜆𝐶1𝑑𝐸𝐼𝑥𝑦𝑟𝑖𝑑𝑑𝑟3𝐵1𝑟𝑖𝑑𝑟32𝜆𝐶1𝑑𝐸𝐼𝑥𝑦𝑟𝑖𝑑𝑑𝑟3𝐵2𝑟𝑖𝑑𝑟3⋯2𝜆𝐶1𝑑𝐸𝐼𝑥𝑦𝑟𝑖𝑑𝑑𝑟3𝐵𝑁𝑟𝑖𝑑𝑟3𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇+𝜆𝐶1𝐸𝐼𝑥𝑦𝑟𝑖𝑑4𝐵1𝑟𝑖𝑑𝑟4𝜆𝐶1𝐸𝐼𝑥𝑦𝑟𝑖𝑑4𝐵2𝑟𝑖𝑑𝑟4⋯𝜆𝐶1𝐸𝐼𝑥𝑦𝑟𝑖𝑑4𝐵𝑁𝑟𝑖𝑑𝑟4𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇+𝑑2𝐸𝐼𝑥𝑥𝑟𝑖𝑑𝑟2𝑑2𝐵1𝑟𝑖𝑑𝑟2𝑑2𝐸𝐼𝑥𝑥𝑟𝑖𝑑𝑟2𝑑2𝐵2𝑟𝑖𝑑𝑟2⋯𝑑2𝐸𝐼𝑥𝑥𝑟𝑖𝑑𝑟2𝑑2𝐵𝑁𝑟𝑖𝑑𝑟2𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇+2𝑑𝐸𝐼𝑥𝑥𝑟𝑖𝑑𝑑𝑟3𝐵1𝑟𝑖𝑑𝑟32𝑑𝐸𝐼𝑥𝑥𝑟𝑖𝑑𝑑𝑟3𝐵2𝑟𝑖𝑑𝑟3⋯2𝑑𝐸𝐼𝑥𝑥𝑟𝑖𝑑𝑑𝑟3𝐵𝑁𝑟𝑖𝑑𝑟3𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇+𝐸𝐼𝑥𝑥𝑟𝑖𝑑4𝐵1𝑟𝑖𝑑𝑟4𝐸𝐼𝑥𝑥𝑟𝑖𝑑4𝐵2𝑟𝑖𝑑𝑟4⋯𝐸𝐼𝑥𝑥𝑟𝑖𝑑4𝐵𝑁𝑟𝑖𝑑𝑟4𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇+𝑑2𝐸𝐼𝑥𝑦𝑟𝑖𝑑𝑟2𝑑2𝐵1𝑟𝑖𝑑𝑟2𝑑2𝐸𝐼𝑥𝑦𝑟𝑖𝑑𝑟2𝑑2𝐵2𝑟𝑖𝑑𝑟2⋯𝑑2𝐸𝐼𝑥𝑦𝑟𝑖𝑑𝑟2𝑑2𝐵𝑁𝑟𝑖𝑑𝑟2𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇+2𝑑𝐸𝐼𝑥𝑦𝑟𝑖𝑑𝑑𝑟3𝐵1𝑟𝑖𝑑𝑟32𝑑𝐸𝐼𝑥𝑦𝑟𝑖𝑑𝑑𝑟3𝐵2𝑟𝑖𝑑𝑟3⋯2𝑑𝐸𝐼𝑥𝑦𝑟𝑖𝑑𝑑𝑟3𝐵𝑁𝑟𝑖𝑑𝑟3𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇+𝐸𝐼𝑥𝑦𝑟𝑖𝑑4𝐵1𝑟𝑖𝑑𝑟4𝐸𝐼𝑥𝑦𝑟𝑖𝑑4𝐵2𝑟𝑖𝑑𝑟4⋯𝐸𝐼𝑥𝑦𝑟𝑖𝑑4𝐵𝑁𝑟𝑖𝑑𝑟4𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇−𝑟𝜌𝐴𝑖Ω2(cos𝜃)2𝐵1𝑟𝑖𝑟𝜌𝐴𝑖Ω2(cos𝜃)2𝐵2𝑟𝑖𝑟⋯𝜌𝐴𝑖Ω2(cos𝜃)2𝐵𝑁𝑟𝑖𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇−𝜌Ω2𝑑𝑑𝑟𝐿𝑟𝑖𝐴(𝑟)𝔄𝑑𝑟𝑑𝐵1𝑟𝑖𝑑𝑟𝜌Ω2𝑑𝑑𝑟𝐿𝑟𝑖𝐴(𝑟)𝔄𝑑𝑟𝑑𝐵2𝑟𝑖𝑑𝑟⋯𝜌Ω2𝑑𝑑𝑟𝐿𝑟𝑖𝐴(𝑟)𝔄𝑑𝑟𝑑𝐵𝑁𝑟𝑖×𝑑𝑟𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇−𝜌Ω2∫𝐿𝑟𝑖𝑑𝐴(𝑟)𝔄𝑑𝑟2𝐵1𝑟𝑖𝑑𝑟2𝜌Ω2∫𝐿𝑟𝑖𝑑𝐴(𝑟)𝔄𝑑𝑟2𝐵2𝑟𝑖𝑑𝑟2⋯𝜌Ω2∫𝐿𝑟𝑖𝑑𝐴(𝑟)𝔄𝑑𝑟2𝐵𝑁𝑟𝑖𝑑𝑟2×𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇=[0]for𝑖=1,2,…,𝑁,𝑘=1,2,3.}(17)where 𝔄 denotes (𝑟+𝑟₀).

According to the radial basis function approach, the boundary conditions in (14) have the following discrete forms: ^{𝐵1𝑟1𝐵2𝑟1⋯𝐵𝑁𝑟1𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇=[0]for𝑘=1,2,3,𝑑𝐵1𝑟1𝑑𝑟𝑑𝐵2𝑟1⋯𝑑𝑟𝑑𝐵𝑁𝑟1𝑏𝑑𝑟𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇=[0]for𝑘=1,2,3,𝐸𝐼𝑦𝑦𝑟𝑁𝑑2𝐵1𝑟𝑁𝑑𝑟2𝐸𝐼𝑦𝑦𝑟𝑁𝑑2𝐵2𝑟𝑁𝑑𝑟2⋯𝐸𝐼𝑦𝑦𝑟𝑁𝑑2𝐵𝑁𝑟𝑁𝑑𝑟2𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇=[0]for𝑘=1,2,3,𝑑𝐸𝐼𝑦𝑦𝑟𝑁𝑑𝑑𝑟2𝐵1𝑟𝑁𝑑𝑟2𝑑𝐸𝐼𝑦𝑦𝑟𝑁𝑑𝑑𝑟2𝐵2𝑟𝑁𝑑𝑟2⋯𝑑𝐸𝐼𝑦𝑦𝑟𝑁𝑑𝑑𝑟2𝐵𝑁𝑟𝑁𝑑𝑟2𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇+𝐸𝐼𝑦𝑦𝑟𝑁𝑑3𝐵1𝑟𝑁𝑑𝑟3𝐸𝐼𝑦𝑦𝑟𝑁𝑑3𝐵2𝑟𝑁𝑑𝑟3⋯𝐸𝐼𝑦𝑦𝑟𝑁𝑑3𝐵𝑁𝑟𝑁𝑑𝑟3×𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇=[0]for𝐵𝑘=1,2,3,}_{1𝑟1𝐵2𝑟1⋯𝐵𝑁𝑟1𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁}^𝑇_{=[0]for𝑘=1,2,3,𝑑𝐵1𝑟1𝑑𝑟𝑑𝐵2𝑟1⋯𝑑𝑟𝑑𝐵𝑁𝑟1𝑑𝑟𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇=[0]for𝑘=1,2,3,𝐸𝐼𝑥𝑥𝑟𝑁𝑑2𝐵1𝑟𝑁𝑑𝑟2𝐸𝐼𝑥𝑥𝑟𝑁𝑑2𝐵2𝑟𝑁𝑑𝑟2⋯𝐸𝐼𝑥𝑥𝑟𝑁𝑑2𝐵𝑁𝑟𝑁𝑑𝑟2𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇=[0]for𝑘=1,2,3,𝑑𝐸𝐼𝑥𝑥𝑟𝑁𝑑𝑑𝑟2𝐵1𝑟𝑁𝑑𝑟2𝑑𝐸𝐼𝑥𝑥𝑟𝑁𝑑𝑑𝑟2𝐵2𝑟𝑁𝑑𝑟2⋯𝑑𝐸𝐼𝑥𝑥𝑟𝑁𝑑𝑑𝑟2𝐵𝑁𝑟𝑁𝑑𝑟2𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇+𝐸𝐼𝑥𝑥𝑟𝑁𝑑3𝐵1𝑟𝑁𝑑𝑟3𝐸𝐼𝑥𝑥𝑟𝑁𝑑3𝐵2𝑟𝑁𝑑𝑟3⋯𝐸𝐼𝑥𝑥𝑟𝑁𝑑3𝐵𝑁𝑟𝑁𝑑𝑟3×𝑏𝑘1𝑏𝑘2⋯𝑏𝑘𝑁𝑇=[0]for𝑘=1,2,3.}(18)

4. Results

Figure 1 illustrates the effects of various rotational speeds on calculated frequencies for a wind turbine blade. Computational results suggest that the first frequency of a wind turbine blade is strongly dependent on blade shaft speed. The first frequency of a wind turbine blade increases as rotational speed increases. Numerical results obtained by this study suggest that rotational speed does not significantly affect the second, third, and fourth frequencies of a wind blade. Figure 2 lists the frequencies of a wind turbine blade at different pitch angles. Numerical results demonstrate that pitch angle does not significantly affect the first, second, third, and fourth frequencies of a wind turbine blade.

Figure 1 

The first four frequencies of a wind turbine blade at different rotational speeds.

Figure 2 

The first four frequencies of a wind turbine blade at different pitch angles.

5. Concluding Remarks

The radial basis function scheme has been extensively adopted to solve various problems in different scientific and engineering fields. This study investigated how pitch angles and rotational speeds affect frequencies of a wind generator blade. The simplicity of this formulation makes it a good candidate for very complex applications.