Solving the boundary value problem of the wind turbine blade equation (calculation of the mode shape functions)
AuthorMahri, Zine Labidine
MetadataShow full item record
CitationMahri, Z. L. (2009). Solving the boundary value problem of the wind turbine blade equation (calculation of the mode shape functions). Maltepe Üniversitesi. s. 404.
Rotor blades are the most flexible part of the wind turbine, and their modal behavior has a great influence on the overall dynamics and energetic performance of the turbine. Consequently, the calculation of mode shapes and frequencies of the blades is essential to predict the structural problem of the rotor such as blade fatigue (which is one of the major concerns of the designers) and to estimate the energetic performances of the turbine as well. This analysis can result in a substantial saving of the system cost of energy. Recently more attention is given to modal analysis and many experimental and numerical studies were carried out. The calculation of mode shapes is in fact a difficult task due to the complex nature of the blade movement. In this work, a numerical approach is used to solve the blade motion equation. The solution of this fourth order differential equation is complicated by its special boundary conditions. This boundary problem is characterized by two initial values (the displacement and the slope are nil, at the fixed end) and two final values (the shear force and the bending moment must be zero at the free end). In order to start any numerical solution of the equation the boundary problem must be converted to an equivalent problem having four known initial values. For this task, an iterative algorithm was developed to estimate the right initial-value problem that matches the speci¯ed boundary problem. This algorithm starts from a first guess of the initial values, to allow the mode equation to be solved in order to obtain the final values (at the free end of the blade). These initial values are then corrected by means of secant formula. This procedure is repeated till the calculated final values coincide with those specified by the original boundary problem. It has been verified that this algorithm converges when the predictor corrector method (Adam's formula) is used to solve the equation, whereas convergence is not achieved when the Runge-Kutta method is employed. A Fortran computer program was implemented to perform these computations. This modal analysis can be used to determine dynamic stresses and to estimate thereafter the fatigue of the blades.
SourceInternational Conference of Mathematical Sciences
- Makale Koleksiyonu 
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