Along-wind response of a wind turbine tower with blade coupling subjected to rotationally sampled wind loading

Abstract

This paper proposes an approach to investigate the along-wind forced vibration response of a wind turbine tower and rotating blades assembly subjected to rotationally sampled stationary wind loading. The wind turbine assembly consists of three rotating rotor blades connected to the top of a flexible annular tower, constituting a multi-body dynamic entity. The tower and rotating blades are each modelled as discretized multi-degree-of-freedom (MDOF) entities, allowing the free vibration characteristics of each to be obtained using a discrete parameter approach. The free vibration properties of the tower include the effect of a rigid mass at the top, representing the nacelle, and those of the blade include the effects of centrifugal stiffening due to rotation and blade gravity loadings. The blades are excited by drag force time-histories derived from discrete Fourier transform (DFT) representations of rotationally sampled wind turbulence spectra. Blade response time-histories are obtained using the mode acceleration method, which allows for the quantification of base shear forces due to flapping for the three blades to be obtained. This resultant base shear is imparted into the top of the tower. Wind drag loading on the tower is also considered, with a series of spatially correlated nodal force time-histories being derived using DFTs of wind force spectra. The tower/nacelle is then coupled with the rotating blades by combining their equations of motion and solving for the displacement at the top of the tower under compatibility conditions in the frequency domain. An inverse Fourier transform of the frequency domain response yields the response time-history of the coupled system. The response of an equivalent system that does not consider the blade/tower interaction is also investigated, and the results are compared.

Introduction

An understanding of the coupled dynamics of wind turbine towers is of obvious interest to the structural engineer, especially with the potential proliferation of such systems worldwide. As wind turbine towers are being placed in varying global wind environments, knowledge of the dynamic behaviour of the sub-components of the system (the tower and rotor blades), as well as the dynamic interaction of those components with each other, is vital to ensure the serviceably and survivability of such expensive power generating infrastructure. In conventional design analysis, the mass of the components (nacelle and rotor blades) is simply lumped at the top of the tower, and as long as the fundamental frequencies of the tower and blades are far apart, a stochastic forced vibration analysis could be carried out. While the simplicity of this is attractive, it results in economically inefficient designs due to the conservatism required to accommodate the uncertainties of component interaction. This paper aims to illustrate an approach to analytically model the dynamic interaction between the rotor blades and the tower.

Not much published literature is available regarding the dynamic interaction of wind turbine components, especially from the point of view of the structural design of the tower with the interaction of the mechanical rotor blade system. Harrison et al. [1] state that the motion of the tower is strongly connected to the motion of the blades, as the blades transfer an axial force onto the low speed drive shaft which is ultimately transferred into the nacelle base plate at the top of the tower.

The determination of the dynamic characteristics of a multi-body system has traditionally being undertaken by the substructure synthesis or component mode synthesis method. Jang et al. [2] used the finite element method in conjunction with substructure synthesis to estimate the free vibration properties of a spinning flexible disk-spindle system. Other recent work concerned with substructuring analysis includes Jen et al. [3] and Scheble et al. [4]. In coupled analyses such as these, it is first necessary to obtain the free vibration characteristics of all sub-entities, prior to dynamic coupling.

The free vibration properties of a tower carrying a rigid nacelle mass at the top may be evaluated by means such as the discrete parameter method, the finite element method or closed form solutions. The discrete parameter method was used by Wu et al. [5] in a study on the control of transmission towers under the action of stochastic wind loading. Lavassas et al. [6] also used this technique to assess the accuracy and reliability of a more computationally expensive finite element analyses of a wind turbine tower. Recent studies using the finite element technique for free vibration analyses of structures in wind engineering include Bazeos et al. [7] and Dutta et al. [8]. Murtagh et al. [9] derived an expression in closed form to yield the eigenvalues and eigenvectors of a tower/nacelle system comprising a prismatic cantilever beam with a rigid mass at its free end.

The free vibration properties of realistic wind turbine blades are more difficult to obtain, and models with this task are usually mathematically rigorous due to the complex geometry of the blade and the effects of blade rotation. Baumgart [10] used a combination of finite elements and virtual work, allowing for the complex geometry of the blade to obtain the modal parameters of a blade. Naguleswaran [11] presented an approach to determine the free vibration characteristics of a spanwise rotating beam subjected to centrifugal stiffening. This system is utilized in many industrial fields, such as wind turbine tower blades, aircraft rotor blades and turbine rotor blades. Naguleswaran [11] and Banerjee [12] both used the Frobenius method to obtain the natural frequencies of spanwise rotating uniform beams for several cases of boundary conditions. Chung and Yoo [13] used the finite element method to obtain the dynamic properties of a rotating cantilever, whereas Lee et al. [14] carried out experimental studies on the same. All studies showed that the natural frequencies rise as the rotational frequency of the blade increases. A discretized lumped mass version of the continuous method presented by Naguleswaran [11] is used in this paper. Various software codes have been developed by engineers to dynamically analyse the various components of a wind turbine tower. Buhl [15] presented guidelines for the use of the software code ADAMS in free and forced vibrations of wind turbine towers.

Forced vibration analyses of structures may either be carried out in the time or frequency domain, with each having its own distinct merits. Analysis through the time domain allows for the inclusion of behavioural non-linearity and response coupling. Short of possessing actual input time-histories as measured in the field, the engineer is tasked with the artificial generation of relevant time-histories using widely published spectral density functions. The means through which this is possible can be split into three categories, the first based on a fast Fourier transform (FFT) algorithm, the second based on Wavelet Theory and the third based on the Auto-Regressive Moving Average (ARMA) method. Suresh Kumar and Stathopoulos [16] simulated both Gaussian and non-Gaussian wind pressure time-histories based on the fast Fourier transform (FFT) algorithm. Kitagawa and Nomura [17] recently used wavelet theory to generate wind velocity time-histories by assuming that eddies of varying scale and strength may be represented on the time axis by wavelets of corresponding scales. In an investigation on the buffeting of long-span bridges, Minh et al. [18] used the digital filtering ARMA method to numerically generate time-histories of wind turbulence. This paper simulates stochastic drag force time-histories acting on the tower and blades by virtue of the fact that any spectrum may be represented as a Discrete Fourier transform (DFT).

In simulating drag force time-histories on the tower, information on spatial correlation, or coherence is included. Coherence relates the similarity of signals measured over a spatial distance within a random field. Earthquake engineers have studied the relationship between ground accelerations at different points on the earth’s surface; publications in this regard include Hao et al. [19] and Harichandran and Vanmarcke [20]. Coherence is also of great importance to the wind engineer, especially if gust eddies are smaller than the height of a structure. Some of the earliest investigations into the spatial correlation of wind forces were carried out by Panofsky and Singer [21] and Davenport [22] and later augmented by Vickery [23] and Brook [24]. The coherence model proposed by Davenport is adopted for use in this paper. Recent publications involving lateral coherence in wind engineering include Højstrup [25], Sørensen [26] and Minh et al. [18].

In order to simulate the drag force time-histories on the rotating blades, a special type of wind velocity spectrum is needed. Connell [27] reported that a rotating blade is subjected to an atypical fluctuating wind velocity spectrum, known as a rotationally sampled spectrum. Due to the rotation of the blades, the spectral energy distribution is altered, with variance shifting from the lower frequencies to peaks located at integer multiples of the rotational frequency. Kristensen and Frandsen [28], following on from work by Rosenbrock [29], developed a simple model to predict the power spectrum associated with a rotating blade, and this was significantly different to a spectrum where the rotation is not considered. Though literature on this topic is limited, interested parties may refer to work by Madsen and Frandsen [30], Verholek [31], Hardesty et al. [32] and Sørensen [26].

The mode acceleration method is employed to obtain the response time-history of the rotating blades. Williams [33] is credited with the first implementation of the mode acceleration method and Craig [34] concedes that it has superior convergence characteristics compared to the mode-displacement method. Singh [35] presented a method of obtaining the spectral response of a non-classically damped system, based on the mode-acceleration technique. Akgun [36] presented an augmented algorithm based on the mode acceleration method which has improved convergence for computation of stresses in large models. Murtagh et al. [37] estimated the wind induced dynamic response of tapered rotating wind turbine blades using the mode acceleration and traditional fast Fourier techniques, and also considered the rotational sampling of turbulence.

In this paper, the base shear brought about by three blades vibrating due to rotationally sampled turbulence is first computed and then incorporated into the equation of motion for the tower coupled with the blades, where the response at the top of the tower is obtained. The tower is subject to spatially correlated wind drag forces along its height. The coupled system equation of motion is primarily cast in the frequency domain via Fourier transform. This allows the coupling of tower and blades, and the time-domain along-wind response of the coupled assembly is ultimately obtained by inverse Fourier transform. The merits behind this type of approach are numerous. The technique is relatively simple, especially compared with a more computationally expensive finite element formulation. The approach may be used in preliminary quantitative design, which may subsequently be validated by a more rigorous analysis. The dynamic properties of the coupled system are available using the dynamic properties of each of the two subsystems, which is an extension of the substructure synthesis approach.

A system where blade–tower interaction is not included, i.e., the mass of the blades is simply lumped with the mass of the nacelle, is also considered, and the response of this system is compared to the response of the system including blade–tower interaction.