Theoretical study to calculate the vibration modes of a wind turbine blade with a new composite material

A precise understanding of the dynamics of a blade is essential for its design, especially in the development of new structures and the resolution of noise and vibration problems. This understanding involves the study of experimental and/or theoretical modal analysis. These latter present effective tools for describing, understanding and modelling the dynamic aspect of each structure, in the present work, we are going to establish the Eigen-mode of a wind turbine blade made by a new composite material ‘ hemp fibre ’ using theoretical calculation for flap-wise, edge-wise and torsional mode using the finite element method applied to a structure consisting of a beam embedded at one end, based on the Euler-Bernoulli hypothesis and the equation of beam’s motion. Fu rthermore; we compare the obtained results with those of composite material made by fibreglass.


Introduction
Energy is essential to the achievement of sustainable development in any country. Unlike fossil fuels, which contain high percentages of carbon (Labdine, s. d.), renewable energies are generated from sources that are obviously inexhaustible-water, sun, biomass, geothermal heat and wind (Ottmar, Edenhofer. Ramon, Pichs-Madruga. Youba, 2011). Among these renewable sources is the wind that is considered one of the best energy sources to reduce imports of fossil fuels and greenhouse gases. By using wind power, we can reduce our dependence on oil and protect the planet.
The blades are the important elements of a wind turbine (Larsen, Hansen, Carl & Baumgart, 2002). The length of the blade determines the amount of energy that can be extracted depending on the wind speed (Shokrieh & Rafiee, 2006). To recover the maximum energy from the wind, the blades must be positioned higher under very variable wind speed conditions which generate vibrations that can cause problems for the structure of the blade (Shokrieh & Rafiee, 2006). The purpose of manufacturing blades with variable lengths is to capture the different wind speeds. The modal analysis provides information on the dynamic properties of wind turbine components and thus leads to an understanding of their dynamic behaviour (Tartibu, Kilfoil & Van Der Merwe, 2012). Many materials have been used, both for the industrial production of wind turbines power and the small domestic applications. Indeed, the prime objective in wind turbine design is to have a great blade in terms of length, width, profile and material (Rabie, Mounir, Boudi & Marjani, 2015;Rabie, Mounir, Boudi & Marjani, 2016).
The rupture of the blade is caused in particular by vibrations (Gherbi, 2011). The study of the vibrations of a blade is one of the most important issues in the design process of a wind turbine. Carbon fibre is widely used in the manufacture of wind turbine blades (Herbert, Iniyan, Sreevalsan & Rajapandian, 2007). The blade of a wind turbine machine has a complex geometry and great bending changes (Labdine, s. d.).
The origin of modal analysis is the study of the characteristics of the vibration (Tartibu, 2013), that is to say, the study of the natural frequencies and the Eigenmodes of a structure or a component of the machine (Cai, Pan, Zhu & Gu, 2013). It can also serve as a starting point for another more dynamic and detailed study, such as a transient dynamic study, a harmonic study or a spectral study (Muzard, 1994). Natural frequencies and mode shapes are necessary parameters in the design of a structure for dynamic loading conditions (Larsen et al., 2002).
In mechanics, the blade is assimilated to a homogeneous beam, in order to simplify the system in order to obtain an estimate of its frequencies and its own modes (Wang, Kolios, Delafin, & Nishino, s. d.). The method consists in associating to the real structure an equivalent structure of the mass-spring to perform the vibration computation, Euler-Bernoulli theorem (Nour, Gherbi, & Chevalier, s. d.).

Theoretical study
For a flush beam subjected to free vibration and the system is considered as a continuous system in which the mass of the beam is considered to be distributed with the rigidity of the shaft (Muzard, 1994) (Banerjee, Su & Jackson, 2006). To determine the behaviour of the wind turbine blade theoretically, the researchers have developed a mathematical model based on the equation of motion of a beam to calculate the frequencies and the Eigen-modes (Liu, 2010) (Bramwell, Done & Balmford, 2000).

Equation of motion for a uniform beam
Consider a free-body diagram of an element of a beam shown in Figure 2. The external force applied to the beam per unit length is denoted ( )   One note;

( )
, The shear force at the right end of the element They considered that the deformation is rather small so the shear deformation will be sufficiently smaller than transverse vibrations. What we in the equation of motion in the y-direction.
, , By simplifying the term V (x, t), we find the following expression The equation of motion around the z-axis passing through the point O leads to the following expression.
The simplification of equation (3.4) gives; ( ) ( ) They considered that the moment of inertia of rotation for the element (dx) is negligible, and if dx is small, then ( ) 2 dx is negligible; therefore equation (5) becomes; By replacing the term V (x, t) in equation (3), we find; By assembling equation (1) and (7)  ,, ,

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( ) ,0 f x t = For there is no external force, Hence, the equation of motion for the free vibration of the beam is given by the expression of a fourth order differential equation; The natural frequency is related to the circular natural frequency as The blade of the wind turbine can be considered as a beam of Euler-Bernoulli (Bot, 1994). For calculating the first natural frequencies and modes of vibration of the beam from equation (11) for a 22 non-rotating state, to validate the theoretical results, these natural frequencies of our composite material will be compared to the results of the fibreglass.
The theoretical modal analysis was performed for a beam of dimension L = 4,200 mm, W = 690 mm, T = 240 mm

Proprietes de materiau
The composite materials used in our study are hemp fibre and fibreglass with Epoxy matrix.

Results and discussion
The results obtained by the two materials are very close which gives validation to our composite material that was used.      We can see that the natural frequencies obtained for the hemp composite material are greater than those of glass composite material which gives more advantage to our material.

Discussion
In this paper, we have presented a numerical modal analysis of a wind turbine blade considered as a Euler-Bernoulli beam. The modal calculations of the fifth order of the hemp composite material in flap-wise, edge-wise and the torsional mode are very closed to these of the fibreglass, especially for the first and the second modes, physically, this result can be explained by the low density of hemp fibre compared to glass fibre and the natural frequency of a structure is calculated without external exciter force or dissipative forces. It proves that the hemp material will be of great value in practical engineering applications of the wind turbine. Indeed, to approve this finding, we are invited to combine the modal test and the numerical calculations, also analysed and investigated frequency variations of the blades with the impact of structural damping. This purpose will be our main interest in future work.

Conclusion
The results give us an approach on the vibration modes of a hemp fibre beam, but the dynamic study of slender and flexible structures like the blades of a wind turbine usually requires resolutions based on the use of finite element models. Because these allow an accurate description of the blade displacement by taking a high number of modes into attention, and this will be our next work carried out in a more detail article on modal analysis of a blade using the element method finished the results of hemp fibre are compared with the results of glass fibre composite, it is shown that the natural frequencies of the hemp fibre composite are higher than those of the fibreglass composite. This result indicates that the hemp fibre composite exhibits advantageous characteristics for future use in the