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Delegates are invited to meet and discuss with the poster presenters in this topic directly after the session 'Aerodynamics and rotor design' taking place on Wednesday, 12 March 2014 at 09:00-10:30. The meet-the-authors will take place in the poster area.


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Traditional dynamic modelling of wind turbine blades either neglects their spin-up motion or uses empirical formulae [1,2]. Here, the equations of motion of a wind turbine (which are highly nonlinear) that consider spin-up motion, accelerating pitch angle, elastic blades made of composite material and also gears, are derived. The resulting model can be used for improved blade design and a better understanding of stress dynamics. Excitation loads are included in the structural equations therefore facilitating their use as a basis for aeroelastic modelling.


In order to model the elastic blades and the main shaft motion, extended Hamilton’s principle has been used. Gears have been incorporated via an external torque using Newton’s laws, and the final equations of motion of the two shafts derived (the second shaft being the generator shaft). Figure 1 depicts: a) the coordinate systems b) the cross section configuration, and c) a schematic of the wind turbine model.

The position vector of the ith blade is defined by eq. 11a, considering rigid body rotation, as follows: a) the main shaft, which is not horizontal and affects the influence of the gravitational force, b) the rotation of the assembly, c) pitch angle, and also the deformation field (eq. 11b-d ) of a Timoshenko composite pre-twisted blade, by considering primary and secondary warping functions (for accurate torsional equations) following Librescu’s approach [3]. Velocity and acceleration vectors are derived considering spin-up motion and accelerating pitch angle in order to determine the variation of kinetic energy. This results in the existence of acceleration terms in the angles, centrifugal and Coriolis forces. Strain energy is defined using the deformation field restricted to linear elastic terms. Finally, for describing nonconservative work, external forces and bending moments are considered, applied in all directions, that can further be extended to provide an aeroelastic model.
Transmission spur gear pairs are modelled considering time variant functions for the teeth mesh stiffness and the static transmission error. The backlash between the meshing teeth is included as a piece-wise linear function. The resisting load in the generator is expressed as a function of the speed of the gearbox output shaft (a more realistic approach than often used), thereby adding a further source of nonlinearity.
Numerical simulations will be conducted as follows: a) discretisation of the equations in space using expansion of the displacements and rotations in series with admissible functions, whilst also incorporating boundary conditions, b) solution of the eigenvalue problem, and c) comparison with other solutions from commercial finite element software.

Main body of abstract

Equations of motion of the full assembly are derived using Timoshenko elastic blades, rigid body motions, acceleration due to gravity and the coupling of the main shaft to the electrical generator shaft via the gears.

The corresponding inertia and stiffness coefficients are defined in Table 2 and Table 3 respectively. For completeness, a nomenclature is given in Table 1.

Modelling of spin-up is described by the equations of motion of the main shaft of the turbine eq. 1, restricted to rigid body motion. It provides a nonlinear equation that couples with the elastic motions of the blades as a smooth nonlinear function, along with a non-smooth term due to the gear coupling. This is an integra-differential equation (eq.1) which can be reducted to an ordinary differential equation by expressing the elastic displacements and rotations of blades in a series of admissible functions. It should be noted that the integra-differential form of the equation is typical in such cases and has been reported in [4], obtained by modelling Euler-Bernoulli rotating isotropic beams, and also in [5] obtained by modelling a rotating composite beam with spin-up motion but with a constant pitch angle. Considering the case of a constant pitch angle, eq. 1 is consistent with that reported in [5]. The non-smooth applied torque, which is part of this equation, occurs due to the gear model and is described by eq. 2, which is similar those reported in [6]. The predicted dynamic transmission error time history highlights single- and double-sided impact between the gear teeth, and identifies system parameters that lead to the generation of the solutions (wind speed, blade characteristics and gear design). Particular emphasis is given to the identification of responses characterized by multiples of the mesh frequency and the potential appearance of chaotic solutions.
The rigid body motion of the second shaft connected to the generator is described by eq.3, and as with eq. 1, provides a non-smooth equation due to gear coupling, and features the impact of the load torque developed by the electrical generator.
Consideration of accelerating pitch angle results in one equation of motion (eq. 4) which is also an integra-differential nonlinear equation of motion, similar to eq. 1 and similarly expresses the elastic motions in series, and can be reduced to an nonlinear ordinary equation of motion. It should be noted, that in the case that accelerating pitch angle and spin-up are not considered, the system is linear and the applied pitch control is restricted to the limitations of the linearization. The consideration of accelerating pitch angle results in a nonlinear equation which can be used for nonlinear control of the pitch angle.
In modelling the elastic dynamics of the blades, the following motions are considered: axial motion which is described by eq. 5, flapwise rotation which is described by eq. 6, chordwise rotation which is described by eq. 7, torsion which is described by eq. 8, chordwise bending which is described by eq. 9, and flapwise bending which is described by eq. 10. All of these are described by nonlinear partial differential equations due to the spin-up motion and accelerating pitch angle.
In [5] the elastic motion of rotating non-pretwisted composite beams with spin-up motion and constant pitch angle, is considered. Through this restriction, it can be seen that the equations derived here (eq.1, eq.5-10) are consistent with those reported in [5].
In [3] the equations of elastic motion of rotating non-pretwisted and pre-twisted composite beams, without considering spin-up motion, are derived. Considering the assumptions therein, the equations derived here (eq.5-10) are consistent with those reported in [3].
Next, the elastic equations of motion of the blades, are discretised, and a modal analysis is performed. Results from finite element simulations using commercial software are used for validation of the model/s.


Equations of motion for a wind turbine, are derived, including consideration of spin-up motion of main shaft coupled by gears to the generator shaft, and also considering accelerating pitch angle, gravitational force, elastic modelling of pre-twisted composite blades, and structural external loads. Explicit equations of motion that describe the spin-up of the main shaft are also derived, which are traditionally described in the literature using an empirical formula, and also the acceleration of the pitch angle. This system can be used for modelling of blades, since the elastic part of Timoshenko beams are modelled by considering either linear and/or nonlinear dynamics. The system of equations is highly nonlinear and the spin-up motion and the accelerating pitch angle must be taken into account in order to perform pitch control without being limited to the region of the validity of the linear system e.g. as in case of constant rotating speed and pitch angle. The system of equations forms a strongly nonlinear and non-smooth system and can be used to simulate, using appropriate aerodynamic forces, the rotation speed in the shaft generator from wind load.
Finally, the equations of motions provided by the new model are in very good agreement with existing similar models reported in literature, under constraints.
The research is particularly timely, since according to reports from the National Renewable Energy Laboratory (USA) [7], advanced drivetrain technology could lead on average to 7% operations/maintenance cost reduction and an increase of 1-2% annual energy production in low windspeed turbines. In addition, gearbox failure in offshore wind turbines can cost around 250 k€ (without considering the accumulating financial losses due to the energy loss). Moreover in [7], it is stated that appropriate modelling of the blades and appropriate control of the dynamics can increase the net annual energy production by 25%, and lower the turbine capital cost by 3%. As an indication, the blade structural repair costs are 87500$ per incident, with around 15 failures over 20 years of operation, being predicted for the considered project (appendix A of [8]).

Learning objectives
The derived equations of motion can be used by commercial organisations for modelling of the elastic blades, applying nonlinear pitch control, and determining the transient rotation speed of the shaft generator from wind forces using the appropriate aerodynamic forces. The time varying gear model parameters and backlash effects can lead to effectively capturing nonlinear behaviour, such as jump phenomena and multiple coexisting underlying dynamic responses.

[1]. Kim, T., Hansen, A.M., Branner, K., 2013, Development of an anisotropic beam finite element for composite wind turbine blades in multibody system, Renewable Energy, 59, pp 172-183.
[2]. Kwon, S., Chung, J., Yoo, H.H., 2013, Structural dynamic modelling and stability of a rotating blade under gravitational force, Journal of Sound and Vibration, 332, pp 2688-2700.
[3]. Librescu, L. and Song, O. (2006), Thin-walled composite beams: Theory and application, Springer, Dordrecht and the Netherlands.
[4]. Warminski, J. and Balthazar, J. M. (2005), Nonlinear vibrations of a beam with a tip mass attached to a rotating hub. In Volume 1: 20th Biennial Conference on Mechanical Vibration and Noise, Parts A, B, and C, pages 1619–1624, ASME.
[5]. Georgiades, F., Latalski, J., Warminski, J., ‘Equations of Motion of Rotating Composite Blades with nonconstant rotating speed’, Meccanica, under review.
[6] Theodossiades, S. and Natsiavas, S. Nonlinear Dynamics of Gear-Pair Systems with Periodic Stiffness and Backlash. Journal of Sound and Vibration, 2000, 229, 287-310.
[7] Cohen, J., Schweizer, T., Laxson, A., Butterfield, S., Schrek, S., Fingersh, L., Veers, P., Ashwill, T., 2008, Technology Improvement Opportunities for Low Wind Speed Turbines and implications for Cost of Energy Reduction, Technical Report NREL/TP-500-41036.
[8]. Martin-Tretton, M., Reha, M., Drunsic, M., Keim, M., 2012, Data Collection for Current U.S. Wind Energy Projects: Component Costs, Financing, Operations, and Maintenance, January 2011-September 2011, National Renewable Energy Laboratory U.S.A, NREL/SR-5000-52707.Link: