A general frequency adaptive framework for damped response analysis of wind turbines

https://doi.org/10.1016/j.soildyn.2021.106605Get rights and content

Highlights

  • Use of Dynamic Stiffness method.

  • Exact closed for expression for dynamic damped response of Offshore Wind Turbines.

  • Damping modelled using 7 physically realistic damping factors.

  • Example application of the model shown.

Abstract

Dynamic response analysis of wind turbine towers plays a pivotal role in their analysis, design, stability, performance and safety. Despite extensive research, the quantification of general dynamic response remains challenging due to an inherent lack of the ability to model and incorporate damping from a physical standpoint. This paper develops a frequency adaptive framework for the analysis of the dynamic response of wind turbines under general harmonic forcing with a damped and flexible foundation. The proposed method is founded on an augmented dynamic stiffness formulation based on a Euler-Bernoulli beam-column with elastic end supports along with tip mass and rotary inertia arising from the nacelle of the wind turbine. The dynamic stiffness coefficients are derived from the complex-valued transcendental displacement function which is the exact solution of the governing partial differential equation with appropriate boundary conditions. The closed-form analytical expressions of the dynamic response derived in the paper are exact and valid for higher frequency ranges. The proposed approach avoids the classical modal analysis and consequently the ad-hoc use of the modal damping factors are not necessary. It is shown that the damping in the wind turbine dynamic analysis is completely captured by seven different physically-realistic damping factors. Numerical results shown in the paper quantify the distinctive nature of the impact of the different damping factors. The exact closed-form analytical expressions derived in the paper can be used for benchmarking related experimental and finite element studies and at the initial design/analysis stage.

Introduction

The United Nations has recently declared that we are facing a grave climate emergency and some of the most common manifestations are continuous ocean and atmospheric warming, heat waves and rise in sea level. A practical way to combat climate change and to achieve net-zero emission target to run a country mostly on electricity produced from renewable sources without burning much fossil fuel. Offshore wind turbines have the proven potential for the island and coastal nations and as a result, there is a tremendous rise in the proportion of electricity generation from such sources.

Offshore Wind Turbines are being currently constructed around the world and in extremely challenging sites, see for example deeper water developments and using floating system (Hywind in Scotland, see Ref. [1]), the typhoon and hurricane sites in Japan and China, seismic locations in Taiwan, China, Korea and India [2]. These sites often apply dynamic loading to the structure and the magnitude depends on the location. Due to its shape and form, offshore wind turbine structures are dynamically sensitive as a large rotating mass is applied at the top of the long slender column. Furthermore, the natural frequency of these structures is also close to the forcing frequencies. The typical natural frequency of a 3.6 MW turbine is about 0.33 Hz and that of an 8 MW is 0.22 Hz. As the turbines get larger, the target natural frequency of the overall wind turbine system gets lower and comes near to the wave frequencies. In some offshore development, predicting dynamic responses becomes the main challenge. For example, the predominant wave period in Yellow sea and Bohai sea (Chinese waters) is about 4.8–5 sec [3] and wave loading becomes a critical design consideration for turbines above 8 MW. There are other considerations such as corrosion and fatigue [4] and scour [5]. The readers are referred to studies on dynamics of offshore wind turbine by Zuo et al. [6], Sellami et al. [7], Banerjee at al [8] and Sclavounos et al. [9].

Guided by Limit State philosophy, a design must satisfy the following limit states: ULS (Ultimate Limit State), SLS (Serviceability Limit State), FLS (Fatigue Limit State) and ALS (Accidental Limit State). To evaluate any of the above limit states for different dynamic load scenarios, the response of the structure must be evaluated. A quick method of evaluation of dynamic helps to optimize the design of a given turbine (for a given RNA mass and 1P frequency range) at a given site (wind field and wave/sea states) through the change in physical parameters i.e. foundation stiffness and tower stiffness. In certain challenging sites where the forcing frequency is very close to the natural frequency, damping plays a beneficial role in optimization. There are different sources of damping in an offshore wind turbine: Aerodynamic, hydrodynamic, structural damping, material damping (including the soil). Recently several authors have considered explicit dynamic analysis of wind turbine structures. Bending, axial and torsional vibrations of wind turbines has been considered by Wang et al. [10] and Vitor Chaves et al. [11]. Due to the interest of understanding the performance of offshore wind turbines in seismic areas, dynamic analysis work is being conducted by He et al. [12], Patra and Haldar [13], Zhao M et al. [14] and Jiang W et al. [15]. These studies clearly demonstrate the need for comprehensive dynamic analysis of wind turbine structures.

It has also been established that Soil-Structure Interaction (SSI) is very important for predicting the short term and long-term performance of these structures. For design purposes SSI can be classified as follows: (1) Load transfer mechanism from the foundation to the soil (2) Modes of vibration of the whole system (3) Long term performance in the sense whether or not the foundation will tilt progressively under the combined action of millions of cycles of loads arising from the wind, wave and 1P (rotor frequency) and 2P/3P (blade passing frequency). In a series of previous studies, the authors [[16], [17], [18], [19]] considered the analysis of the first natural frequency of wind turbines taking SSI into account. The recent trend in wind turbine design is towards very large systems. While such large systems give more power output, a potential disadvantage is that they can be susceptible to dynamic loads as the natural frequencies become lower. As a result, many resonance frequencies of the structure will be excited within the operating frequency ranges. Therefore, for a credible dynamic analysis, it is necessary to have a simple approach which can take account of multiple natural frequencies and vibration modes.

It is certainly possible to perform a classical modal analysis [20] for high-frequency vibration problems. However, there are two major issues. Firstly, analytical solutions for the natural frequencies and mode shapes are generally difficult to obtain beyond the first mode. Secondly, simplified proportional modal damping assumptions must be employed for the response analysis. One way these issues can be avoided is by using the dynamic stiffness method [[21], [22], [23], [24], [25]]. This approach can be considered within the broad class of spectral methods [26] for linear dynamical systems. A key feature of the dynamic stiffness method is the use of complex shape functions (due to the presence of damping) which are frequency-dependent [27]. The mass distribution of the element is treated exactly in deriving the element dynamic stiffness matrix. The method does not employ eigenfunction expansions and, consequently, a major step of the traditional finite element analysis, namely, the determination of natural frequencies and mode shapes, is eliminated which automatically avoids the errors due to series truncation [28]. Since the modal expansion is not employed, ad hoc assumptions concerning the damping matrix being proportional to the mass and/or stiffness are not necessary. The dynamic stiffness matrix of one-dimensional structural elements, taking into account the effects of flexure, torsion, axial and shear deformation, and damping, is exactly determinable, which, in turn, enables the exact vibration analysis by an inversion of the global dynamic stiffness matrix [22]. The method is essentially a frequency-domain approach suitable for steady-state harmonic or stationary random excitation problems. The static stiffness matrix and the consistent mass matrix appear as the first two terms in the Taylor expansion [21,29] of the dynamic stiffness matrix in the frequency parameter.

The overview of the paper is as follows. In Section 2 an overview of dynamic stiffness of undamped beam-columns is given. In particular, the equation of motion is discussed in Subsection 2.1, the characteristic equation and essential non-dimensional parameters are explained in Subsection 2.2 and the undamped dynamic stiffness matrix is derived in Subsection 2.3. The dynamic stiffness matrix for damped beam-columns are derived in Section 3. The effect of end restraints and tip mass in considered in Section 4. The consideration of tip mass and rotary inertia in discussed in Subsection 4.1, while the consideration of damped and flexible foundation is proposed in Subsection 4.2. The analysis of dynamic response in the frequency domain is developed in Section 5 where exact closed-form expressions have been derived for systems with fixed foundation (Subsection 5.1) and systems with damped and flexible foundation (Subsection 5.2). The new expressions derived in the paper is summarised in Section 6 and main conclusions are drawn in Section 7.

Section snippets

Overview of dynamic stiffness of undamped beam-columns

In Fig. 1, the schematic diagram of wind turbine tower constrained by flexible springs is shown.

An Euler-Bernoulli beam model is used to mathematically represent the dynamics of the beam. The bending stiffness of the beam is EI and the beam is attached to the foundation. Here x is the spatial coordinate, starting at the bottom and moving along the height of the structure. The interaction of the structure with the foundation is modelled using two springs. The rotational spring with spring

Systems with general damping

The equation of motion of a damped beam-column can be expressed asEI4W(x,t)x4+c15W(x,t)x4t+P2W(x,t)x2+c2W(x,t)t+mW¨(x,t)=F(x,t)

It is assumed that the behaviour of the beam follows the Euler-Bernoulli hypotheses as before. In the above equation c1 is the strain-rate-dependent viscous damping coefficient and c2 is the velocity-dependent viscous damping coefficient. Considering harmonic motion with frequency ω as in Eq. (2) we haveEIL4d4w(ξ)dξ4+iωc1L4d4w(ξ)dξ4+PL2d2w(ξ)dξ2+iωc2w(ξ)mω2w(ξ)

The consideration of the top mass and rotary inertia

The mass of the nacelle and rotor blades are represented by M in Fig. 1. This is very significant and can be more than the mass of the tower. Due to non-negligible geometric dimension, this mass cannot be modelled as a classical point mass. Therefore, rotary inertia of this mass should also be taken into account. We assume that the rotary inertia of the top mass is given by J. From practical experiences it is known that the top mass is subjected to significant aerodynamic damping. Therefore,

System with fixed foundation

For systems with a fixed foundation the dynamic stiffness matrix in (58) is used. The complete equilibrium equation is therefore given byEIL3[Γ6Γ4LΓ5Γ3LΓ4LΓ2L2Γ3LΓ1L2Γ5Γ3L(Γ6Ω2α)Γ4LΓ3LΓ1L2Γ4L(Γ2Ω2β)L2]{w1w2w3w4}={00f3f4}

Here f3 and f4 are amplitudes of the harmonic force and moment applied at the top of the beam. We consider only a transverse force is applied to the top end, therefore, f3=F2 and f4=0. As w1=w2=0 due to the fixed end, eliminating first two rows and columns, and using the

Validation with respect to modal analysis

Modal analysis [20] is the classical approach for dynamic response analysis of complex systems. When used in conjunction with the finite element method, the system is divided into a number of elements. Eigenvalues and eigenvectors are then calculated by solving the generalised eigenvalue problem involving the mass and stiffness matrices of the system. The dynamic response is calculated using the superposition of the eigenmodes. We consider a pinned-pinned beam to compare the results from the

Conclusion

The quantification of the dynamic response of wind turbine towers due to various external forces is of paramount importance. A physics-based analytical approach leading to closed-form expressions of essential dynamic response quantities was presented. The route to this analytical derivation has three key steps. Firstly, noting that the wind turbine tower is a beam-like structure, the dynamic stiffness matrix of a beam with axial compressive force is derived exactly. This is achieved using

Author statement

The two authors have equal contribution to the paper.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References (35)

  • C.S. Manohar et al.

    Dynamic stiffness of randomly parametered beams

    Probabilist Eng Mech

    (1998)
  • J.R. Banerjee

    Dynamic stiffness formulation for structural elements: a general approach

    Comput Struct

    (1997)
  • S. Adhikari et al.

    Dynamic finite element analysis of axially vibrating nonlocal rods

    Finite Elem Anal Des

    (2013)
  • X. Liu et al.

    A spectral dynamic stiffness method for free vibration analysis of plane elastodynamic problems

    Mech Syst Signal Process

    (2017)
  • S. Adhikari et al.

    Dynamic stiffness method for nonlocal damped nano-beams on elastic foundation

    Eur J Mech Solid

    (2021)
  • S. Bisoi et al.

    Dynamic analysis of offshore wind turbine in clay considering soil–monopile–tower interaction

    Soil Dynam Earthq Eng

    (2014)
  • S. Bhattacharya, Design of foundations for offshore wind turbines, Wiley Online...
  • Cited by (8)

    • Seismic design and analysis of offshore wind turbines

      2023, Wind Energy Engineering: A Handbook for Onshore and Offshore Wind Turbines
    • A novel time-variant prediction model for megawatt flexible wind turbines and its application in NTM and ECD conditions

      2022, Renewable Energy
      Citation Excerpt :

      Høeg [5] developed a 17-degree-of-freedom floating offshore wind turbines model using the Euler-Lagrange approach to investigate the influence of gyroscopic couplings on stochastic dynamic responses in both parked and operational conditions. Adhikari [6] developed a frequency adaptive framework for the analysis of the dynamic response of wind turbines under general harmonic force with a damped and flexible foundation. However, he doesn't determine the new damping factors values.

    • Design of monopiles for offshore and nearshore wind turbines in seismically liquefiable soils: Methodology and validation

      2022, Soil Dynamics and Earthquake Engineering
      Citation Excerpt :

      In this analysis, Rayleigh damping was considered. The damping ratio for the structural analysis of the wind turbine is varied between 1% and 20% [57,58] to understand the effect of damping factor variation on the design requirements. The turbine structure and foundation specifications are provided in Table 5.

    View all citing articles on Scopus
    View full text