Dynamics and Aeroelasticity of Flexible Rotor Blades

Problems of structural dynamics and aeroelasticity of wind turbine rotors are relatively well understood. Major progress in the field of wind turbine rotor dynamics was achieved during last three decades of the twentieth century thanks to improvements in performance of modern computers and application of nonlinear elastic model-based analysis to both helicopter and wind turbine rotors. An extensive summary of the current status of rotorcraft aeroelasticity is given by Friedmann (2003) and Friedmann and Hodges (2003). Friedmann and Hodges (2003) show that many different analytical methods have been developed in the field of helicopter rotor blade dynamics which are also applicable to wind turbines. A range of methods, from the simple but elegant Lagrange’s equations based on an energy approach to computationally intensive but generic and powerful finite element method, have been developed for wind turbine rotors. The problem of rotor blade dynamics can be split into two major phases—formulation of blade equations of motion and their solution. Wind turbine blades, which are subjected to very high oscillatory loads, have dynamic characteristics that are generally com­plicated due to couplings of blade flexible deformations about two orthogonal axes with blade section rotation. The offset of center of gravity (CG) from the elastic axis plays an important role in analysis of rotor blade aeroelastic stability (Friedmann and Hodges 2003). Rotor blades are subjected to harmonic forcing caused by aerodynamic forces and centrifugal forces generated by blade rotation. Centrifugal forces are dependent upon rotor blade radius and cause the blades to stiffen. Coriolis forces are caused by combination of blade rotation and blade deformations and affect dynamic stability of rotor blades. Since rotational effects play important role in rotor blade dynamics and they are not neglected, but taken into consideration. The wind turbine blade degrees of freedom are mutually cou­pled, and the coupling can have a significant effect on blade dynamics and sta­bility. Hence, equations of motion of a rotor blade are expressed in terms of a number of assumed modes which are also mutually coupled. The coupling terms cannot be ignored if one wishes to predict the blade behavior correctly. Houbolt and Brooks (1957) give derivations of combined equations of motion of bending and torsion of a rotor blade modelled as a slender beam. Aeroelastic equations of a helicopter rotor undergoing torsion and both flapwise and chordwise bending can be found in Kaza and Kvaternik (1977, 1979). Ordering schemes have been developed and are applied to equations of motion to retain the most significant terms and ignore terms which will not affect the blade behavior appreciably (Friedmann and Hodges 2003). Although wind turbine rotor blade dynamics can be described with the aid of the Newtonian approach and by the Euler’s equations of motion applied to a lumped mass model of the rotor blade (Bramwell et al. 2001), it is the energy methods that represent the most convenient way of deri­vation of equations of motion. Energy methods are based on the principle of virtual work and Hamilton’s principle. Lagrange’s equations are the result of the appli­cation of these principles. Lagrange’s method allows the derivation of equations of motion via partial differentiation of the Lagrangian which may be expressed in terms of the kinetic and potential energy of a dynamic system. Rosen and Freid – mann (1979) have extended the dynamical equations of motion of a rotating rotor blade to large amplitudes in the nonlinear regime, and Floros (2000) has extended the nonlinear dynamic equations of motion of a rotating rotor blade to layered composite blades with closed cross sections. The integration of all the sectional properties is performed over the closed section using normal and tangential coordinates as illustrated in Fig. 4.7.

Since aspect ratios of wind turbine rotor blades are high, they can be regarded as slender beams. Modelling of rotor blades as flexible, infinitely thin beams is sufficient for most dynamic problems. Although in reality infinite number of modes could be used to describe dynamics of a wind turbine rotor blade, sufficient approximation of the behavior can be made with the aid of the primary dominant modes. The method of assumed modes belongs to the group of global methods and represents a fairly popular way of solving for the motion of rotor blades. A series of functions (mode shapes) are used for the first approximation of blade shape. Lagrange’s method and the Raleigh-Ritz method are some of the most typical of methods of assumed modes. They allow estimation of modal shapes and corre­sponding modal frequencies. However Galerkin’s method which is based on energy considerations represents most popular of these methods. It is widely used as it can be used in the case of nonslinear or non-conservative problems that both Lagrange’s and Raleigh-Ritz methods cannot solve (Bramwell et al. 2001). In Galerkin’s method, an approximation function is substituted into the partial dif­ferential equation of blade motion. The most significant aspect of Galerkin’s method applied to wind turbine rotors is that range of aerodynamic models of the loading and inflow dynamics can be incorporated into the analysis. If n different mode shapes and frequencies are considered, it is transformed into a system of n ordinary differential equations (Bramwell et al. 2001). The blade natural fre­quencies and actual mode shapes are determine from these equations. Thus, the complete set of linear equations may be obtained by combining the linear struc­tural dynamic equations of motion with the expressions for the linearized aero­dynamic loads. One may obtain a typical set of root loci for different rotor speeds (Fig. 4.8) or with different inflow velocities (Fig. 4.9).

Dynamics and Aeroelasticity of Flexible Rotor Blades

Fig. 4.7 Definition of coordinates for thin-walled sections

Fig. 4.8 Blade root locus for changing rotor speeds, with the CG and tension axis offsets equal to zero

 

Blade Flutter Root Locus plot

 

Fig. 4.9 Blade root locus for increasing inflow velocities, with the CG and tension axis offsets equal to zero

 

Blade Flutter Root Locus with variable inflow

 

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Dynamics and Aeroelasticity of Flexible Rotor BladesDynamics and Aeroelasticity of Flexible Rotor Blades

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