Ahmed H. Bayoumy Graduate Student Mechanical Design & Production, Engineering Department, Faculty of Engineering, Cairo University, Giza, 12613, Egypt e-mail: a.hamdy85@ymail.com Ayman A. Nada 1,2 Assistant Professor Mechanical Engineering Department, Benha Institute of Technology, Benha University, Benha, 13512, Egypt e-mail: arobust@tedata.net.eg Said M. Megahed Professor Mechanical Design & Production, Engineering Department, Faculty of Engineering, Cairo University, Giza, 12613, Egypt e-mail: smegahed@cu.edu.eg A Continuum Based Three-Dimensional Modeling of Wind Turbine Blades Accurate modeling of large wind turbine blades is an extremely challenging problem. This is due to their tremendous geometric complexity and the turbulent and unpredictable conditions in which they operate. In this paper, a continuum based three dimensional fi- nite element model of an elastic wind turbine blade is derived using the absolute nodal coordinates formulation (ANCF). This formulation is very suitable for modeling of large- deformation, large-rotation structures like wind turbine blades. An efficient model of six thin plate elements is proposed for such blades with non-uniform, and twisted nature. Furthermore, a mapping procedure to construct the ANCF model of NACA (National Ad- visory Committee for Aeronautics) wind turbine blades airfoils is established to mesh the geometry of a real turbine blade. The complex shape of such blades is approximated using an absolute nodal coordinate thin plate element, to take the blades tapering and twist into account. Three numerical examples are presented to show the transient response of the wind turbine blades due to gravitational/aerodynamics forces. The simu- lation results are compared with those obtained using ANSYS code with a good agree- ment. [DOI: 10.1115/1.4007798] Keywords: ANCF, wind turbine blade, thin plate element 1 Introduction In recent years the aerodynamic performance of wind turbine blades has been considerably improved. This has contributed to an overall reduction in the cost of wind power produced electricity. The energy capture is approximately proportional to the square of the blade length while the blade weight is approximately propor- tional to the cube of its length [1]. To counteract the weight increase, the development of blades goes towards long and rela- tively flexible structures [1]. The most important aeroelastic components of a wind turbine are the blades. The purpose of the blades is to extract aerodynamic forces from the passing airflow; therefore, they are highly affected by aerodynamic forces. The development of larger wind turbines has resulted in long slender blades with high flexibility. The idea of modeling such flexible multibody systems is to introduce a moving frame of reference to each substructure [2]. Relative to the moving frame, the elastic displacements are relatively small and rendering linear analysis possible [3]. Hence, nonlinearities are confined to the description of the moving frame. The standard formulation of this method presumes that the moving frame is fixed to the rigid body motion of the substructure. The coordinates defining the position and orientation of the moving frame become a part of the degrees of freedom of the multibody system. How- ever, the use of a mixed set of referential and elastic coordinates leads to highly nonlinear inertial couplings between the rigid body motion and elastic deformation. In the case of rotating machinery, the problem of geometric stiffness arises and wrong results should be obtained if the rotating speed reaches the basic natural fre- quency of the flexible blade [4]. To overcome the geometric stiff- ness effect, the internal elastic coupling between different forms of motion should be taken into consideration. Furthermore, real wind turbine blades are made of composite materials, making them anisotropic which increase the internal elastic coupling effect of blade motion. This cannot be described by the moving frame of reference, especially with high rotating speeds. The mod- eling computation problem increases as the rotor blade diameter increases. For instance, the Enercon E-126 is the largest wind tur- bine model built to date, manufactured by the German wind tur- bine producer Enercon. With a hub height of 135 m ð Þ, rotor diameter of 126 m ð Þ and a total height of 198 m ð Þ, this large model can generate up to 7.58 Megawatts of power per turbine [5]. The recently developed ANCF had been used in the analysis of large deformation of flexible multibody systems including belt drives [6–8], rotor blade [9], large deformation piezo-electric lami- nated plates [10], flexible robotic manipulators [4], and cable appli- cations [11]. The important advantage of using this formulation in multibody computer simulations is the constant mass matrix that can be obtained for fully nonlinear dynamic problems. Therefore, this nonlinear finite element formulation can be implemented using nonincremental solution procedures in a general framework of mul- tibody computer algorithms. The elastic forces; in contrast, are cal- culated using a general continuum mechanics approach. This allows for describing the cross-section deformation modes as well as the deformation modes that appear in the existing beam theories (bending, torsion, longitudinal, and shear deformations). In a gen- eral continuum mechanics theory, the deformation and rotation fields within an infinitesimal volume can be uniquely defined using nine components of the displacement gradients in the three- dimensional applications. For this reason, the nine independent gra- dient coordinates are used in the ANCF to describe the large rota- tional motion defined using the three independent rotational parameters as well as the deformation defined using the six strain components [3,12]. Using such gradient coordinates leads to sim- pler expressions of the generalized inertia forces and the exact mod- eling of the rigid body motion [12–14]. Recent advances in the ANCF, involving the method of calcu- lating the strain energy [15], illuminating high frequency modes [16,17] and development of stiff integrators [18,19]; help in reducing the calculation time and enhance the sensitivity of the system equations. Also the formulation of 3D joint constraints is well established and verified [20], which enable constructing the 1 Corresponding author. 2 Present address: College of Engineering, Jazan University, Jazan P.O.706, KSA. Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 5, 2011; final manuscript received August 18, 2012; published online October 30, 2012. Assoc. Editor: Khaled E. Zaazaa. Journal of Computational and Nonlinear Dynamics JULY 2013, Vol. 8 / 031004-1 Copyright V C 2013 by ASME Downloaded From: http://computationalnonlinear.asmedigitalcollection.asme.org/ on 05/10/2013 Terms of Use: http://asme.org/terms