Introduction

A fundamental understanding of the modal dynamics of structures with bladed rotors is relevant for the design and analysis of wind turbines, helicopters, and other rotating machinery because their vibrational responses are composed of their structural modes. It is important to understand how these modes are excited by resonances or aeroelastic instabilities, i.e., at which frequencies and where on the structure or rotor the individual modes can be excited. Such knowledge is necessary not only for the interpretation of design simulations but also for the understanding of real measurements.

For anisotropic three-bladed rotors, Floquet theory or Hill's method is needed to obtain an eigenvalue problem which leads to the periodic eigenvectors of the principal eigenvalue solutions . To handle the frequency indeterminacy of the periodic eigenvalue solutions from these methods, suggest to select the principal solution such that the harmonic components on the ground-fixed degrees of freedom are minimized. introduce the concept of modal participation factors in which the norm of the individual harmonic components of a periodic eigenvector determines how much the particular component contributes to the response of the particular mode. They also introduce the concept of periodic Campbell diagrams to plot the frequencies of these harmonic components along with the principal frequency. All studies show that periodic mode shapes of turbines with three-bladed anisotropic rotors can contain harmonic components whose frequencies can be shifted more than ±1/rev from the principal modal frequency. The sizes of these higher harmonic components depend on the size of rotor anisotropy.

Analytical system matrix for isotropic rotors

Analytical expressions for the harmonic components of the periodic system matrix for structures with an isotropic rotor are derived in this section. The first-order state-space equation for a periodic system with the period T is given by x˙=A(t)x, where ()˙= d/dt and A(t)=A(t+T) is the T-periodic system matrix of dimension 2ND× 2ND, where ND is the number of degrees of freedom (DOFs). With the state vector ordered as x={u, u˙}T where u and u˙ are the DOFs and their time derivatives, the system matrix can be derived from linear second-order equations of motion as A(t)=0I-M-1(t)K(t)-M-1(t)C(t), where I is a identity matrix and M, C, and K are the T-periodic mass, gyroscopic/damping, and stiffness ND×ND matrices, respectively. Periodicity of these matrices ensures that the system matrix can be written as a Fourier series: A(t)=n=-AneιnΩt, where ι=-1 and Ω 2π/T is the constant mean rotational speed. Note that the mean component of the system matrix A0 is a real matrix, and the complex matrices of the harmonic components come in conjugated pairs: A-n=An,n=1,2,3,, where the bar denotes the complex conjugated operator. The periodic matrices of the second-order equations of motion are derived in the next section. In Sect. , the mass matrix is inverted analytically to obtain its Fourier series. The mean and harmonic component matrices of the periodic system matrix are finally presented in Sect. .

Equations of motion

Let the Lagrangian for a structure with a rotor be written as L=T(t,u,u˙)-V(u), where it is assumed that the potential energy of the conservative forces is time-independent and only depends on the generalized coordinates V=V(u), e.g., the elastic forces. The total kinetic energy is given by an integral of the kinetic energy of each particle over the entire volume V of the structure as T=V12ρr˙Tr˙dV, where ()T denotes to the transpose matrix operator and r˙ is the velocity vector of the particle given as the time derivative of its position vector r=r(t, u), which for the rotor part of the structure will be explicitly time-dependent.

Through substitution of the Lagrangian Eq. () with Eq. () into Lagrange's equations and linearization about a steady-state deflection of the structure u=u0 and u˙=0, the coefficients of the matrices of Eq. () can be written as mij=VρrTuirujdV cij=tmij+VρrTuiujrt-rTujuirtdV+2Du˙iu˙j kij=VρujrTui2rt2dV+2Vuiuj, where all derivatives are evaluated at u=u0 and D is Rayleigh's dissipation function. Note that only the partial time derivatives of the position vector are needed, not the full velocity or acceleration vectors.

Let ug denote the DOFs for the ground-fixed substructure and ubk denote the DOFs for rotating blade number k (blades in this paper are always numbered in the direction of the rotation); then the position vector r in the ground-fixed inertial frame to a particle point on the substructure is written as a function of ug as r=rgug and on the blade number k as r=rcug+TcugR0+R1eιψk+R1e-ιψkrbubk, where the vector rc and the rotation matrix Tc describe the position of the rotor center and the orientation of the rotational axis, respectively, both of which are functions of the ground-fixed DOFs ug. The local position vector rb of a particle on blade number k is a function of ubk, which is the same function for all blades due to the isotropy of the rotor and its discretization. The prescribed rotation of the blade is given by the angle ψk=Ωt+ 2π(k- 1)/B, where B is the number of blades. The rotation matrix is written in exponential form using a real matrix R0 and a complex matrix R1, which are constant and given by the initial orientation of the rotational axis.

Let the conservative and dissipative forces be linear and depend only on the local DOFs and their time derivatives, such that the potential energy and Rayleigh's dissipation function can be written as V=ugTKgug+k=1BubkTKbubkandD=u˙gTCgu˙g+k=1Bu˙bkTCbu˙bk, where Cg and Kg are local damping and stiffness matrices for the ground-fixed substructure, and Cb and Kb are local and identical damping and stiffness matrices for each blade of the isotropic rotor.

Through insertion of Eq. () and the position vectors Eqs. () and () into the coefficients Eq. () and summation over all volumes of the entire structure, the linear equations of motion can be written in a generic matrix form as MrMgrTMgrMgu¨+CrGrgGgrGg+Cgu˙+Kr+Sr0SgrKgu=0, where the ND DOFs are ordered as u=ur,ugTwithur=ub1,,ubbT. Note that ND=Ng+BNb, where Ng and Nb are the number of DOFs on the substructure and each blade, respectively. The block matrices of Eq. () can be written as Mr=diagMb,Mb,,Mb,Cr=diagCb,Cb,,Cb,Kr=diagKb,Kb,,Kb,Sr=diagSb,Sb,,Sb,Mg(t)=Mg,0+Mg,2eι2Ωt+Mg,2e-ι2ΩtGg(t)=Gg,0+Gg,2eι2Ωt+Gg,2e-ι2ΩtMgr(t)=Mgr,0+Mgr,1eιΩt+Mgr,1e-ιΩt,Ggr(t)=Ggr,0+Ggr,1eιΩt+Ggr,1e-ιΩt,Grg(t)=Grg,1eιΩt+Grg,1e-ιΩt,Sgr(t)=Sgr,1eιΩt+Sgr,1e-ιΩt, where the time-dependent coupling matrices can be subdivided further into constant single blade components as Mgr,0=Mgb,0Mgb,0Mgb,0,Mgr,1=Mgb,1Mgb,1ei2π/BMgb,1ei2π(B-1)/B,Ggr,0=Ggb,0Ggb,0Ggb,0,Ggr,1=Ggb,1Ggb,1ei2π/BGgb,1ei2π(B-1)/B,Grg,1=Gbg,1TGbg,1Tei2π/BGbg,1Tei2π(B-1)/BT,Sgr,1=Sgb,1Sgb,1ei2π/BSgb,1ei2π(B-1)/B. The matrices in Eqs. () and () related to inertia forces are listed in Appendix . Note that the 2/rev components of the mass and gyroscopic matrices for the ground-fixed substructure only exist in the case of a two-bladed rotor.

Inversion of mass matrix

The inverse of the mass matrix in Eq. () can be written as M-1=EFTFH, where E=Mr-1+Mr-1MgrTHMgrMr-1,F=-HMgrMr-1,H=Mg-MgrMr-1MgrT-1. Using Eqs. () and (), it can be shown that the inverse of H can be written as H-1=Q0+Q2eι2Ωt+Q2e-ι2Ωt, where the mean and 2/rev components for a B-bladed isotropic rotor are Q0=Mg,0-BMgb,0Mb-1Mgb,0T+Mgb,1Mb-1Mgb,1T+Mgb,1Mb-1Mgb,1T, Q2=Mg,2-2Mgb,1Mb-1Mgb,1TforB=20forB>2. Thus, H=Q0-1 is a real, symmetric, and constant matrix for isotropic rotors with more than two blades. Using Eqs. () and (), this property implies that the inverse mass matrix of such rotors have a finite Fourier series with 2/rev as the highest harmonic component. Note that the second harmonic component for two-bladed rotors is also symmetric, Q2T=Q2.

For two-bladed rotors, H-1 is periodic and H can therefore be written as a Fourier series: H=n=-HneιnΩt, where H-n=Hn. Insertion into the equation H-1H=I and collection of terms of equal harmonics yields Q0H0+Q2H2+Q2H2=Imeanterms,Q0H1+Q2H3+Q2H1=01/revterms,Q0H2m+1+Q2H2m+3+Q2H2m-1=0(2m+1)/revterms,Q0H2m+Q2H2m+2+Q2H2m-2=0(2m)/revterms, where m= 1, 2, …is a positive integer. The equations for odd terms are homogeneous and regular; thus, all odd harmonic components vanish H2m-1=0 for m= 1, 2, ….

To solve the equations for the even terms, the mean component H0 is obtained from Eq. () as a linear function of the second harmonic component H2 and a constant matrix: H0=Q0-1-Q0-1Q2H2+Q2H2. The remaining even equations can be solved recursively for H2m by insertion of the solution for H2m-2 into the 2m/rev equation. It is convenient to split the equations into real and imaginary parts and solve for each part to obtain ReH2mImH2m=k=1m-PkQsTB0-PmQsReH2m+2ImH2m+2form=1,2,, where the following real matrices have been introduced, Qs=ReQ2ImQ2-ImQ2ReQ2,B0=Q0-10, and the recursive matrices Pk=I-Qd-1QsTPk-1Qs-1Qd-1fork=1,2,, where Qd= diag{Q0, Q0} and P0= diag{2Q0-1, 0}, are 2 × 2 block diagonal matrices. Symmetries of Q0 and Q2 and the resulting antisymmetry of Qs causes the matrices Pk to be symmetric. Note that Qd is a regular matrix due to the positive definiteness of the mass matrix.

If ||Qd-1QsTPk-1Qs||< 1 then an inequality for the p-norm of the product PkQsT can be derived as PkQsTQd-1QsT1-Qd-1QsTPk-1Qsfork=1,2,, where here and subsequently denotes any p-norm of a matrix. The condition ||Qd-1QsTPk-1Qs||< 1 is fulfilled for k 1 if ||Qd-1QsTP0Qs||< 1. It is not straightforward to prove this inequality, but its validity for a given model can easily be checked. Intuitively, the p-norm of the constant part Q0 of the mass matrix for the ground-fixed coordinates should be much larger then the p-norm of the second-order harmonic part Q2. It is therefore also assumed that ||PkQsT||< 1 for k= 1, 2, …based on the inequality Eq. (). From the recursive solution Eq. (), this assumption is sufficient (but not necessary) to ensure that the p-norm of harmonic components H2m decreases with their order: H2mk=1mPkQsTB0+PmQsH2m+20form. Thus, closure to the recursive equation (Eq. ) can therefore be obtained by choosing H2Nh+2=0, where 2Nh is the highest harmonic component used in the Fourier series (Eq. ). The recursive solution of Eq. () is computed backward starting with H2Nh and ending with H0 given by Eq. ().

Truncation of Eq. () to 2Nh and insertion into Eq. () shows that the block matrices of the inverse mass matrix are E=n=-(2Nh+2)2Nh+2EneιnΩt,F=n=-(2Nh+1)2Nh+1FneιnΩt,andH=m=-NhNhH2meι2mΩt, where the component matrices En and Fn are written out in Appendix . Thus, the highest order of the harmonics in the inverse mass matrix for a two-bladed rotor is 2Nh+ 2 and involves only the rotor coordinates.

Harmonic components in system matrix

Insertion of Eq. () with Eq. () into the system matrix Eq. () with the gyroscopic/damping and stiffness matrices of Eq. () shows that the Fourier series of the periodic system matrix Eq. () can be truncated to the order N as A(t)=n=-NNAneιnΩt, where N=2Nh+3forB=23forB>2, and the matrices An can be found in Appendix . This analytical derivation of the system matrix is exact for isotropic rotors with more than two blades. In fact, if the anisotropy of the rotor is only related to the stiffnesses of the blades and not the mass distributions or rotor geometry, as in previous studies (e.g., ; ), then the system matrices for such rotors also have finite Fourier series. For two-bladed isotropic rotors, the series converges when the Fourier series of the inverse mass matrix converges; sufficient but not necessary criteria for convergence are given in the previous section.

Modal analysis using Hill's method

This section contains a description of Hill's method and how it can be applied for modal analysis of structures with bladed rotors. First, the concept of periodic mode shapes and Hill's truncated eigenvalue problem is introduced. Then, a novel method for automatic identification of the principal solutions among all eigen-solutions of this eigenvalue problem is briefly presented. The section ends with a description of the modes of bladed rotors, including the identification of the different rotor mode components based on the periodic eigenvectors of Hill's eigenvalue problem.

Periodic mode shapes and Hill's truncated eigenvalue problem

Floquet theory defines that the eigen-solution of the linear periodic system Eq. () consists of an eigenvalue and a corresponding periodic eigenvector. A homogeneous solution to Eq. () can therefore be written as x=m=-vmeιmΩteλt, where vm are harmonic components of the periodic eigenvector and λ is the complex eigenvalue. Insertion into Eq. () with the 2ND× 2ND periodic system matrix written as the infinite Fourier series (Eq. ) and collecting terms of equal harmonics yields an infinite set of 2ND equations: n=-Anvm-n-(λ+ιmΩ)vm=0mZ. These equations constitute the algebraic eigenvalue problem of infinite dimension that forms the basis of Hill's method . The eigenvalue is λ and the eigenvector can written as v={…, v-2T, v-1T, v0T, v1T, v2T, …}T and is of infinite dimension. If the harmonic index in the eigenvector is shifted by an integer s, then the resulting vector v={…, v-2-sT, v-1-sT, v-sT, v1-sT, v2-sT, …}T and the complex number λ+ιsΩ are also an eigen-solution. Thus, although the eigenvalue problem has infinitely many eigen-solutions, there are only 2ND principal solutions from which all other solutions can be constructed. This construction is also clear when inserting the shifted eigen-solution into Eq. (): x=m=-vm-seιmΩte(λ+ιsΩ)t=m=-vmeι(m-s)Ωte(λ+ιsΩ)t=m=-vmeιmΩteλt. This arbitrary s/rev shift between the periodicity of the eigenvector and the frequency of the eigenvalue is identical to the frequency indeterminacy in Floquet analysis due to the logarithm of the complex Floquet multipliers . The advantage of Floquet analysis is that there are only 2ND solutions and the frequency indeterminacy can be chosen arbitrarily for each of them; here the concept of participation factors introduced by is helpful. In Hill's method, the problem is to link all eigen-solutions into 2ND subsets in which a principal eigen-solution can be used to construct all eigen-solutions in the particular subset. A novel method for this linking into subsets and identification of the principal solutions to Hill's eigenvalue problem is presented in the next section.

To numerically solve Hill's eigenvalue problem Eq. () is truncated. When the periodic system matrix has a finite Fourier series (Eq. ), the periodic eigenvectors can also be represented by a finite Fourier series in the homogeneous solution x=m=-MMvmeιmΩteλt, where M is the number of harmonics. The infinite eigenvalue problem Eq. () can then be truncated to finite dimension as n=-NmNmAnvm-n-(λ+ιmΩ)vm=0m[-M:M], where Nm= min(N, |M-m|) is the limit for the summation over the product of the harmonic components of the system matrix and the eigenvector. This limit is lower than the number of harmonics in the system matrix N for the matrix equations where |m|>M-N, showing that truncation errors are introduced in those 2N of the 2M+ 1 matrix equations in Eq. (). The truncation error has been investigated by and , who show a convergence of the principal eigen-solutions of the truncated Hill's matrix for their particular systems when M 2N. This finite matrix can be easily set up from Eq. () by arranging the harmonic components of the periodic eigenvector as v=v-MT,,v-1T,v0T,v1T,,vMTT. The 2ND(2M+ 1) solutions to the truncated eigenvalue problem Eq. () will still follow the above rules for index shift, except that the eigen-solutions with the largest harmonic components of their eigenvectors closest to the “edges” at ±M will be affected by the truncation error. There still exist 2ND subsets of 2M+ 1 eigen-solutions that can be constructed from a principal solution; however, these solutions with periodic eigenvectors of highest harmonic order will be less accurately constructed. Note that the eigenvector Eq. () can be given a norm of 1, such that ||vm|| is the participation factor of the mth harmonic component .

Automatic identification of principal solutions

The identification of the principal solutions can be done manually for small systems . A more systematic approach is to select the 2ND eigen-solutions among the 2ND(2M+ 1) solutions that have periodic eigenvectors where the harmonic components are most centered around the mean component; different methods of this concept can be found in , , and . However, these methods do not always ensure that the identified principal solutions 2ND can construct all 2ND(2M+ 1) solutions, because two selected principal solutions may come from the same subset, whereby one subset is not represented. A novel method is therefore suggested, where the principal solutions are identified in three steps:

remove half the eigen-solutions with eigenvectors dominated by harmonic components of the highest order;

link the remaining eigen-solutions in 2ND subsets based on a modal assurance criterion ;

pick the principal solution in each subset that has the largest mean component v0 in its eigenvector.

Step 1 ensures that the eigen-solutions with the largest truncation errors are removed. Step 2 ensures that all 2ND subsets are represented. The choice of the particular solution in step 3 as the principal one in each subset is less important (similar to Floquet analysis). If the eigenvector with the largest norm of its mean ground-fixed components ||v0,g|| is chosen, then the frequency of the principal eigenvalue will also be the dominant frequency observed in the ground-fixed frame . For drawing of Campbell diagrams, it is convenient to link the subsets across the variation of rotor speed. Step 3 is therefore only done for one rotor speed computation, and the selection of the principal solutions in each subset for subsequent rotor speeds is based on a modal assurance criterion with the previous speed.

Modes of <inline-formula><mml:math display="inline"><mml:mi>B</mml:mi></mml:math></inline-formula>-bladed isotropic rotors

Modes of a structure with a bladed rotor may be dominated by the motion of the substructure and therefore named after its dominant component of the periodic eigenvector, e.g., “tower fore–aft” or “drivetrain torsion” modes of wind turbines. The name of a rotor mode dominated by blade motion will depend on the number of blades.

Each harmonic component of each DOF of a rotor mode may therefore be named symmetric, antisymmetric (exists only for an even number of blades), backward whirling (exists only for B 3), or forward whirling (exists only for B 3), followed by the name of the DOF. For rotors with more than four blades, the whirling components must also have the phase speed index p= 1, 2, … B̃ added to the name of the particular component. The symmetric, antisymmetric, and whirling components of the rotor motion given by Eq. () can be derived by inverting the transformation Eq. (). By inserting the rotor motion ur=vm,reιmΩteλt of the mth harmonic of the periodic eigenvector into Eq. () and setting the time to zero t= 0, the symmetric, antisymmetric, and whirling components of this harmonic can be derived as A0AdABW,1AFW,1ABW,B̃AFW,B̃m=1BIIII-II-IIIIe-ι2π/BIe-ι2π(B-1)/BIIIeι2π/BIeι2π(B-1)/BIIIe-ιB̃2π/BIe-ιB̃2π(B-1)/BIIIeιB̃2π/BIeιB̃2π(B-1)/BIvm,r, where vm,r is the rotor part of a harmonic component vm of the eigenvector v, and the last row of matrices is omitted for rotors with odd number of blades. The harmonic order m of each component is also relevant for its naming, but we again note that it is directly dependent on the choice of the principal solution. Note that two-bladed rotors only have symmetric and antisymmetric components.

Modal analysis of two- and three-bladed wind turbines

The theories presented in the previous sections are applicable to structures with isotropic rotors with any number of blades higher than one. In this section, the modal dynamics of two- and three-bladed turbines are investigated because they are of the highest interest to the wind turbine industry, but also because the finite Fourier series of the system matrix shows that there are no qualitative difference between turbines with three or more identical blades.

The turbine used for the analysis is the DTU 10 MW reference wind turbine (RWT) by with three identical blades. For the sake of comparability of the modal dynamics, the two-bladed version is obtained by reusing the same blade. In reality, the optimal aerodynamic solidity of the rotor would require a redesign ; the blades would get a larger chord and the increased absolute thickness (assuming the same relative thickness of the airfoils) would be used to either decrease the blade mass, increase the blade stiffness (keeping the same bending stresses) or combine these two objectives. In any case, the blade for the two-bladed turbine will have different blade modal frequencies and possible mass distribution, which would complicate the direct comparison of turbine modes to the three-bladed version.

The turbine model derived in the next section is based on the simple model presented in , except that the bending of the main shaft is omitted here and the generator rotation has been added as a new DOF. The model parameters are tuned such that the modal frequencies of the first 11 modes of the three-bladed turbine are as close as possible to the modal frequencies computed for the 10 MW RWT with the high-fidelity linear model of the software HAWCStab2 using beam elements and the method of the Coleman transformation . Minor modifications of the turbine have been introduced such that the center of gravity in the blade cross sections coincides with the pitch axis and the coning is also set to zero.

The convergence of the Fourier series of the system matrix for the two-bladed turbine is analyzed in Sect. . The Campbell diagrams of the principal modal frequencies are presented for 3- and two-bladed turbines in Sect. . The well-known periodic mode shapes of three-bladed turbines are repeated in Sect. , and more complex periodic mode shapes for two-bladed turbines are presented in Sect. . The section ends with a discussion of the differences between the modal dynamics of the two turbine types.

Model kinematics and parameters

Figure  shows an illustration of the structural turbine model. The nacelle and tower motions are described by five DOFs. The nacelle can translate in the two horizontal directions described in the ground-fixed inertial frame (X, Y, Z) by ux (side–side) and uy (fore–aft). It can tilt, roll, and yaw described by the angles θx, θy, and θz, respectively. The azimuthal angle of the blade number one is ψ1=Ωt+ψs+ψg, where Ω is the constant mean speed and ψs and ψg are the torsional and rigid-body rotations of the drivetrain, respectively. The generator is rotating at the speed Ω+ψ˙g.

Illustration of the simple turbine model.

Edgewise (dashed curves) and flapwise (solid curves) deflections in the blade mode shapes (left plots) and the blade mass distribution (right plot) for the blade of the DTU 10 MW RWT. Circles in the left plots are results from the beam element model of the software HAWCStab2 and the curves are polynomial fits used in the present model.

Tuned parameters of simple model to fit the modal properties of the DTU 10 MW RWT up to its 11th mode.

Parameter description Symbol Value Blade length Lb 86.366 m First blade flap frequency ωf1 0.610 Hz First blade edge frequency ωe 0.934 Hz Second blade flap frequency ωf2 1.738 Hz Hub radius Rh 2.8  m Hub mass Mh 105 520 kg Tower to rotor distance Ls 7.1 m Generator inertia on low-speed shaft Ig 3 751 000 kg m2 Drivetrain stiffness Gs 0.668 GNm rad-1 Nacelle/effective tower mass M 446 040 kg Nacelle tilt inertia Ix 4 106 000 kg m2 Nacelle roll inertia Iy 410 600 kg m2 Nacelle yaw inertia Iz 4 106 000 kg m2 Tower top side–side stiffness Kx 7.4 MN m-1 Tower top fore–aft stiffness Ky 7.4 MN m-1 Tower top tilt stiffness Gx 7.462 GNm rad-1 Tower top roll stiffness Gy 7.462 GNm rad-1 Tower top coupling stiffness gxy 0.2035 GN Tower top yaw stiffness Gz 3.5 GNm rad-1

The ground-fixed substructure is modeled as a lumped mass, and the inertia forces from the nacelle and effective tower masses and the generator rotational inertia are derived from the kinetic energy: Tg=12Mu˙x+u˙y2+12Ixθ˙x2+12Iyθ˙y2+12Izθ˙z2+12Igψ˙g2, such that the first term of mg,0,ij in Eq. () is replaced by 2Tg/u˙iu˙j. The potential energy for the linear elastic stiffnesses of the nacelle/tower motion, shaft torsion, and blade deflections is written as V=12Kxux2+12Kyuy2+12Gxθx2+12Gyθy2+12Gzθz2-gxyθxuy+gxyθyux+12Gsψs2+k=1BVbk, where Vbk is the potential energy of blade number k given as Vbk=β=[f1,e,f2]12Mβωβ2qβ,k2+Ω2β=[f1,e,f2]0R2ϕx,βz222ϕy,βz22qβ,k2zRm(s)dsdz, where Mβ and ωβ are the modal mass and frequency of the orthogonal blade modes β= [f1, e, f2]. The last term is the potential energy of the centrifugal forces proportional to the squared rotor speed.

The damping matrices of Rayleigh's dissipation function in Eq. () for the blades Cb and the nacelle/tower motion and shaft torsion Cg are set up using a spectral damping model . The first and second flapwise blade modes are set to have logarithmic decrements of 20 and 10 %, respectively, whereas all other modes are damped 2–5 %. These choices of damping are simply crude approximations to the aeroelastic damping of the turbine in operation , and they have no significant effect on the results of Hill's method. Damping is very important for solutions using Floquet theory where the system is time-integrated.

The linear equations of motion (Eq. ) can be derived from Eq. () and the integrals in Appendix  using the above kinematic description of the blade motion, the kinetic energy of the nacelle/tower and generator inertia, the total potential energy, and the spectral damping matrices. For brevity, the block matrices of Eq. () are not explicitly included. All parameters (except for the blade properties in Fig. ) of the tuned model are listed in Table .

Convergence of Fourier series for two-bladed rotor

Convergence of the Fourier series for the system matrix of two-bladed isotropic rotors is ensured if the constant part of the mass matrix for the ground-fixed DOFs is sufficiently larger then the second-order harmonic part. Using Eqs. () and (), it can be computed that ||Qd-1QsTP0Qs|| 0.22 < 1 and ||PkQsT||< 0.67 for k= 1, 2, …, i.e., the inverse mass matrix and therefore the system matrix have finite Fourier series for the two-bladed rotor.

Figure  shows the 2-norms of the Fourier components of the system matrices for the models with the two- and three-bladed isotropic rotors. The convergence for the two-bladed rotor is observed and for the following analysis the number of harmonics is chosen to be N= 7. The largest norm for the 1/rev component is 46 480 s-1 and the largest norm of an omitted component (9/rev) is 0.03 s-1. For the three-bladed rotor all N= 3 harmonic components are used although the third harmonic is effectively zero.

The 2-norms of the Fourier components of the system matrices for the two- and three-bladed simple models of the DTU 10 MW RWT.

Campbell diagrams of the principal modal frequencies of the three-bladed (left plot) and two-bladed (right plot) version of the DTU 10 MW RWT computed with Hill's method (circles) and with the software HAWCStab2 (crosses) for the three-bladed turbine.

Campbell diagrams of two- and three-bladed turbines

The number of harmonics in the periodic eigenvectors used for Hill's truncated eigenvalue problem is set to M= 2N for both rotors; thus, M= 14 and M= 6 for the two- and three-bladed rotor, respectively. The eigenvalue problem is solved for 33 equidistant rotor speeds ranging from 2 to 10 rpm (nominal speed of the DTU 10 MW RWT is 9.6 rpm). The principal solutions at the lowest speed of 2 rpm are selected among the subsets of solutions as the one that has the largest mean component on the ground-fixed DOFs (see Sect. ). For the remaining rotor speeds, the principal solutions are selected as the ones that have the largest modal assurance criterion relative to the principal solutions selected at the previous rotor speed.

Figure  shows the Campbell diagrams of principal modal frequencies versus rotor speed for the three-bladed (left) and two-bladed (right) versions of the DTU 10 MW RWT. The mode names are deduced from the dominant components in the periodic mode shapes (see Sects.  and ). For the three-bladed rotor, the principal modal frequencies computed with Hill's method (circles) are in close agreement with the modal frequencies computed with the high-fidelity software HAWCStab2 (crosses). This agreement shows that the simple model combined with the automatic identification of the principal solutions is able to predict the same modal frequencies in the ground-fixed frame as computed using the Coleman transformation method.

Comparison of the two Campbell diagrams shows that the tower modes of the two-bladed turbine have slightly higher frequencies due to the lighter rotor. The ±1/rev frequency splitting for the pairs of whirling rotor modes in the ground-fixed frame is clear for the three-bladed turbine (noting that out-of-plane blade deflections are stiffened by the centrifugal forces). For the two-bladed turbine, there are no whirling modes but the antisymmetric modes are still either increasing (first edge and flap) or decreasing (second flap) with 1/rev, because the principal solutions are selected as the solutions with the largest mean components on ground-fixed DOFs. Campbell diagrams containing only the principal frequencies are not very informative for two-bladed turbines. Instead, the periodic mode shapes of the selected principal solutions are now analyzed. Note that first mode of both turbines is the trivial rigid-body rotation of the drivetrain, which is omitted from this analysis.

Harmonic modal components for the three-bladed turbine mode dominated by tower fore–aft motion. Top panel: modal amplitudes plotted versus rotor speed and frequency of the particular harmonic component. Lower left panel: projection onto the plane of modal amplitudes and rotor speed. Lower right panel: projection onto the plane of component frequencies and rotor speed (periodic Campbell diagram). Filled black circles show the principal modal frequencies in the frequencies and rotor speed planes. Only modal components with amplitudes larger than 10 % of the overall maximum amplitude are plotted.

Harmonic modal components for the three-bladed turbine mode dominated by tower side–side motion. Plot layout as in Fig. .

Harmonic modal components for the three-bladed turbine mode dominated by backward whirling (BW) of the rotor in the first flapwise DOF. Plot layout as in Fig. .

Periodic mode shapes of three-bladed turbines

Figures  show the dominant modal components of selected DOFs in the periodic mode shapes of the three-bladed turbine. The modal amplitudes are plotted as functions of both the rotor speed Ω and the frequency of the particular component ω0+mΩ, where ω0 is the principal modal frequency and m is the harmonic order of the component (see Eq. ). Only amplitudes higher than 10 % of the overall maximum amplitude across all rotor speeds are plotted. The generator rotation and shaft torsion angles are multiplied by 10 for better scaling of these small angles compared to blade deflections. Scaling of the tower translations are also applied to show weak couplings for some modes. The plot layout is the same in all figures: a three-dimensional plot in the top and its projections onto the amplitude–rotor speed (left) and frequency–rotor speed (right) planes in the lower plots. The principal frequencies ω0 are always denoted by black filled circles in the frequency–rotor speed planes. The color coding of the modal components is also the same in all figures: black markers denote amplitudes of ground-fixed DOFs. Red, green, and blue markers denote amplitudes of the rotor components of the first flapwise, first edgewise, and second flapwise deflection shapes, respectively, computed using Eq. (). Note that the frequency–rotor speed diagrams (lower right) are similar to the periodic Campbell diagrams in , except that each curve represents the magnitude of the harmonic component for a particular DOF and not the vector norm of all DOFs like the participation factors.

The naming of the modes shown in Fig.  is deduced from the harmonic components that dominate the periodic mode shapes observed in Figs.  across a large part of the rotor speed range. Note that the first flapwise symmetric and forward whirling modes interchange their mode shapes at low rotor speeds. The routine for sorting the modes based on a modal assurance criterion cannot always distinguish strongly coupled modes from each other. This issue is dependent on the resolution of the rotor speed range.

Harmonic modal components for the three-bladed turbine mode dominated by symmetric deflection of the rotor in the first flapwise DOF across the rotor speed range. Plot layout as in Fig. .

Harmonic modal components for the three-bladed turbine mode dominated by forward whirling (FW) of the rotor in the first flapwise DOF across the rotor speed range. Plot layout as in Fig. .

Harmonic modal components for the three-bladed turbine mode dominated by backward whirling (BW) of the rotor in the edgewise DOF. Plot layout as in Fig. .

Harmonic modal components for the three-bladed turbine mode dominated by forward whirling (FW) of the rotor in the edgewise DOF. Plot layout as in Fig. .

Harmonic modal components for the three-bladed turbine mode dominated by backward whirling (BW) of the rotor in the second flapwise DOF. Plot layout as in Fig. .

Harmonic modal components for the mode dominated by forward whirling (FW) of the rotor in the second flapwise DOF. Plot layout as in Fig. .

A general observation for the three-bladed turbine modes is that no harmonic component of the rotor response is more than ±1/rev from the principal frequency. Lowering the threshold for the plotted amplitudes to 0.1 % of the overall maximum amplitude did not change this observation. This agrees with Eq. () and previous studies of isotropic three-bladed turbines using the Coleman transformation method : an inverse Coleman transformation from the ground-fixed frame with a single modal frequency back to the rotating frame shows that symmetric components will have the same frequency as in the ground-fixed frame, but BW and FW components will have frequencies that are +1/rev and -1/rev, respectively, from a principal frequency defined in the ground-fixed frame. If the three-bladed rotor was anisotropic, then secondary harmonics will arise whose magnitudes will depend on the magnitude of the anisotropy .

Looking at the individual mode shapes, there are couplings of the different DOFs in each mode of three-bladed turbines. The tower fore–aft mode (Fig. ) contains symmetric, BW, and FW components in the first flapwise blade mode. Similarly, the tower side–side mode (Fig. ) contains BW and FW components in the first edgewise blade mode and generator rotation due to the nacelle roll. The first BW flapwise mode (Fig. ) is not pure and also contains symmetric and FW components. The strongly coupled first symmetric (Fig. ) and FW (Fig. ) flapwise modes are also not pure and contain other components as well. Note that a BW edgewise component (in green) appears in the first FW flapwise mode at the highest rotor speeds where its BW flapwise component approaches the edgewise blade frequency of about 0.93 Hz. Similarly, the FW flapwise components appear in the first BW edgewise mode (Fig. ) at higher rotor speeds. The first FW edgewise mode (Fig. ) contains a small BW edgewise component and a larger FW flapwise component. Similar couplings can be seen for the remaining modes. The second flapwise whirling modes (Figs.  and ) have the largest blade response at a frequency of about 1.5 Hz (the +1/rev for BW and -1/rev for the FW components). The amplitude ratio of these flapwise DOFs are almost constant, indicating that the rotor modes are constructed as a combination of the two blade modes with the isolated modal frequencies 0.61 and 1.74 Hz. The first symmetric edgewise mode (Fig. ) couples with both the generator rotation and the shaft torsion, but not with any whirling components. The second symmetric flapwise mode (Fig. ) contains whirling components that are just below the selected threshold for the plotted amplitudes and therefore not shown.

Periodic mode shapes of two-bladed turbines

Figures  show the dominant modal components of selected DOFs in the periodic mode shapes of the two-bladed turbine. The scaling of DOF amplitudes, plot layout, and color coding are the same as for the three-bladed turbine. The amplitudes of rotor motion are again computed from the periodic eigenvector using Eq. ().

A general observation for the two-bladed turbine modes is the larger amplitudes and the higher order of the harmonic components compared to the three-bladed turbine. Another observation is that symmetric rotor components always have frequencies that are an even number of 1/rev away from the principal frequency, whereas frequencies for antisymmetric components are shifted an odd number.

Harmonic modal components for the mode dominated by symmetric deflection of the rotor in the second flapwise mode. Plot layout as in Fig. .

Harmonic modal components for the mode dominated by symmetric deflection of the rotor in the edgewise mode. Plot layout as in Fig. .

Harmonic modal components for the two-bladed turbine mode dominated by tower fore–aft motion. Plot layout as in Fig. .

Harmonic modal components for the two-bladed turbine mode dominated by tower side–side motion. Plot layout as in Fig. .

Harmonic modal components for the two-bladed turbine mode dominated by symmetric deflection of the rotor in the first flapwise DOF across the rotor speed range. Plot layout as in Fig. .

Harmonic modal components for the two-bladed turbine mode dominated by antisymmetric deflection of the rotor in the first flapwise DOF across the rotor speed range. Plot layout as in Fig. .

Looking at the individual mode shapes, the increased number of harmonics also increases the number of couplings between the different DOFs in each mode. The tower fore–aft mode in Fig.  couples again with ±1/rev asymmetric rotor motion as for the three-bladed turbine. However, couplings to higher harmonic components of the blade DOFs are much more dominant at rotor speeds where their frequency crosses the corresponding blade frequency. Such resonant couplings occur for the symmetric +4/rev and +6/rev components and antisymmetric +3/rev and +5/rev components of the first flapwise DOF when their frequencies are close to the first flapwise blade frequency, 0.61 Hz, as well as for the antisymmetric +5/rev component of the first edgewise DOF when its frequency crosses 0.93 Hz.

The tower side–side mode in Fig.  is similar to the same mode of the three-bladed turbine with the coupling to the generator rotation and the ±1/rev harmonics of asymmetric rotor responses.

The symmetric and antisymmetric flapwise modes in Figs.  and  interchange mode shapes at low rotor speeds; however, they are here named after their dominant components at the higher rotor speeds. The antisymmetric flapwise mode couples with tower fore–aft motion around 6.5 rpm, where the frequency of the -4/rev component of this DOF is crossing the tower fore–aft frequency of about 0.28 Hz. A similar coupling is seen for the -6/rev component of the tower fore–aft DOF in the antisymmetric edgewise mode in Fig.  at 8.2 rpm. This mode also contains 0/rev and -2/rev components of the tower side–side DOF such that the ±1/rev splitting known from three-bladed rotors appears in the ground-fixed frame.

For the second antisymmetric flapwise mode in Fig. , the antisymmetric +1/rev components of the two flapwise blade DOFs are the two largest components with an almost constant frequency of around 1.46 Hz (a combination of the two flapwise blade modes as for the three-bladed turbine). The blade motion also contains smaller antisymmetric -1/rev and +3/rev components of the two flapwise blade DOFs, whose frequencies are decreasing and increasing with 2/rev in the rotating blade frame. Resonant couplings with symmetric +6, +8, and +10/rev second flapwise DOF and +6/rev first edgewise DOF are observed when their frequencies are crossing the principal frequencies of the first symmetric edgewise and second symmetric flapwise turbine modes around 1.75 Hz. These two symmetric modes in Figs.  and  are similar to the corresponding modes of the three-bladed turbine (see Figs.  and ), except that the second symmetric flapwise mode of the two-bladed turbine contains resonant couplings of the antisymmetric -3/rev and -5/rev components of the two flapwise blade DOFs when their frequencies are crossing the frequency of the second antisymmetric flapwise mode (see Fig. ) around 1.46 Hz.

Harmonic modal components for the two-bladed turbine mode dominated by antisymmetric deflection of the rotor in the edgewise DOF. Plot layout as in Fig. .

Harmonic modal components for the two-bladed turbine mode dominated by antisymmetric deflection of the rotor in the second flapwise DOF. Plot layout as in Fig. .

Harmonic modal components for the two-bladed turbine mode dominated by symmetric deflection of the rotor in the edgewise DOF for the higher rotor speeds. Plot layout as in Fig. .

Harmonic modal components for the two-bladed turbine mode dominated by symmetric deflection of the rotor in the second flapwise DOF. Plot layout as in Fig.  .

Periodic Campbell diagrams of all harmonic components of the tower side–side DOF in all investigated modes for the three-bladed (left panel) and two-bladed (right panel) version of the DTU 10 MW RWT.

Comparison of modal dynamics of two- and three-bladed turbines

The modal dynamics of the turbines with two- and three-bladed rotors show several similarities but also significant differences:

The rigid-body drivetrain rotation mode (trivial and therefore not shown) is identical for the two turbines.

Tower bending modes are similar in frequencies and shapes, except that the fore–aft mode for the two-bladed turbine may contain large components of the first flapwise blade mode when the rotor speed is such that these frequencies of the higher harmonic components are crossing the modal frequency of this blade mode.

The first symmetric edgewise rotor mode is very similar in frequency and shape because its reaction forces do not couple to other DOFs through large periodic terms in the system matrix.

The symmetric flapwise rotor modes are similar in frequency and shape, except that the first symmetric flapwise mode of the two-bladed turbine in Fig.  has a small -2/rev component of the tower fore–aft DOF, and the second symmetric flapwise mode in Fig.  has resonant couplings to antisymmetric flapwise mode.

Asymmetric rotor modes: the antisymmetric modes for the two-bladed turbine and the whirling mode pairs of the three-bladed turbine may seem similar when observed from the ground-fixed frame such as a top tower acceleration signal, where the well-known ±1/rev splitting of the frequency peaks is seen. For example, the tower side–side responses at ±1/rev around the blade edgewise frequency is observed for both the antisymmetric edgewise mode in Fig.  and for the edgewise whirling mode pair in Figs.  and . This similarity has probably caused the misinterpretation of the simulated and measured two-bladed responses in and , but rotor modes of a two-bladed turbine will often have more frequency peaks in both the ground-fixed frame and the rotating blade frame.

The additional modal couplings that exist for two-bladed turbines have the effect that there are additional ways that the modes can be excited by either resonances or interactions with external forces, and that it becomes difficult to interpret frequency spectra from simulations and measurements. To illustrate these effects, the harmonic components of the tower side–side DOF in all modes are plotted in three-dimensional periodic Campbell diagrams for both turbines in Fig. . These plots show qualitatively the response in a tower side–side signal when all modes are excited equally. There are clear similarities between the dominant side–side responses for the two turbines. However, where these responses in the ground-fixed frame for the three-bladed turbine only occur on the principal modal frequencies plotted in its conventional Campbell diagram (see Fig. ), the modes of two-bladed turbine lead to responses in the ground-fixed frame at many other frequencies. The conventional Campbell diagram containing only the principal modal frequencies of a two-bladed turbine therefore has little value, and one should be careful when interpreting frequency response plots for two-bladed turbines.

Conclusions

The modal dynamics of structures with bladed isotropic rotors (identical and equidistantly spaced blades) have been analyzed using the periodic mode shapes obtained using Hill's method on the linear periodic first-order system equation. Analytical derivation of linear second-order equations of motion in a generic form has shown that only 1/rev harmonics occur in the periodic terms when a rotor has more than two blades, whereas a two-bladed rotor also has 2/rev terms. Analytical inversion of the periodic mass matrix has shown that its highest harmonic term 2/rev for an isotropic rotor with more than two blades. The inverse mass matrix for a two-bladed rotor has been shown to have an infinite Fourier series of component matrices whose norm decreases with the harmonic order. The periodic system matrix of isotropic rotors with more than two blades can therefore be represented by an exact Fourier series, with 3/rev being the highest order, whereas for two-bladed rotors it must be approximated by a truncated Fourier series of predictable accuracy.

Through use of the analytic Fourier series of the system matrix to set up Hill's truncated eigenvalue problem, its principal solutions have been automatically identified by a novel method applicable to larger systems. The symmetric and antisymmetric modal components of rotors with two blades and the additional whirling components of rotors with more blades have been extracted directly from the periodic eigenvector corresponding to each principal eigen-solution.

As relevant examples, the generic methods have been used to model and analyze the modal dynamics of both two- and three-bladed versions of a 10 MW turbine. The motion of each blade has been described by its three first mode shapes, and the nacelle motion and drivetrain rotations have been described by seven DOFs. Similarities and significant differences between the modal dynamics of the turbines with two- and three-bladed rotors have been found and summarized in Sect. . A difference is that the larger number and magnitudes of the harmonic components in the system matrix of two-bladed turbines leads to resonant couplings in the mode shapes that have not been observed for the three-bladed turbine. These couplings between DOFs arise when the rotor speed is such that the frequency of a higher harmonic component of one turbine mode is in the vicinity of the frequency of another blade or turbine mode. Another difference is that a single mode of a two-bladed turbine can lead to responses at several frequencies in both the ground-fixed and rotating blade frames of reference, which complicates the interpretation of simulated or measured turbine responses without the theories presented in this paper.

Data availability

A repository contains the model input and Matlab function that generates the system block matrices in Eqs. (13) and (14) for the example with the DTU 10 MW reference wind turbine. Using these matrices and the theory presented in the paper, the reader should be able reproduce the results.

Elements of system matrices

This appendix contains a list of the elements of the block matrices in Eqs. () and () related to the inertia forces and derived by insertion of Eqs. (), (), and () into the coefficients (Eq. ), and summation over all volumes of the structure, the ground-fixed substructure Vg, and each blade Vb, where several properties of the rotation matrix r and its components have been utilized (see, e.g., ). The subscripts “g” and “b” are omitted from the DOFs ui and uj for brevity.

The elements of the block matrices for the blade DOFs are mb,ij=VbρrbTuirbujdVbgb,ij=4ΩVbρrbTuiB1rbujdVbsb,ij=-Ω2Vbρ2rbTuiujA1rb+rbTuiA1rbujdVb, where A1=12(R1-R1) and B1=12(R1+R1) are the symmetric real part and the antisymmetric imaginary part of the matrix R1. Note that R0T=R0 and R1=R1T. The elements of the block matrices for the DOFs of the ground-fixed substructure are mg,0,ij=VgρrgTuirgujdVg+BVbρrcTuircuj+rbTR0TcTuircuj+TcTujrcui+rbTR0TcTuiTcujR0+R1TcTuiTcuj+TcTujTcuiR1rbdVb mg,2,ij=2VbρrbTR1TTcTuiTcujR1rbdVbforB=20otherwisegg,0,ij=ι2ΩBVbρrbTR1TcTuiTcuj-TcTujTcuiR1rbdVbgg,2,ij=ι4ΩVbρrbTR1TTcTuiTcujR1rbdVbforB=20otherwise.

The elements of the block matrices that couple the blade and ground-fixed DOFs are mgb,0,ij=VbρrcTuiTcR0rbujrbTR0TcTuiTcR0+R1TTcTuiTcR1T+R1TcTuiTcR1rbujdVb mgb,1,ij=Vbρ+R1TTcTuiTcR0rbujdVb, ggb,1,ij=ι2ΩVbρrcTuiTcR1+rbTR0TcTuiTcR1rbujdVb gbg,0,ij=ι2ΩVbρrbTR1TTcTujTcR1rbui-rbTR1TcTujTcR1rbuidVb gbg,1,ij=ι2ΩVbρrbTuiR0TcTTcujR1rbdVb sgb,1,ij=Ω2VbρrbTR1TTcTuiTcR0-rcTuiTcR1-rbTR0TcTuiTcR1rbujdVb.

Components of inverted mass matrix

The mean and harmonic components of the block matrices Eq. () of the inverse mass matrix Eq. () can be computed as E0=Mr-1+Mr-1Mgr,1HH2Mgr,1+Mgr,0TH0Mgr,0+Mgr,1HH0Mgr,1+Mgr,1TH0Mgr,1 +Mgr,1TH2Mgr,1Mr-1E2m+1=Mr-1Mgr,0TH2m+2Mgr,1+Mgr,1HH2m+2Mgr,0+Mgr,0TH2mMgr,1+Mgr,1TH2mMgr,0Mr-1 E2m+2=Mr-1Mgr,1HH2m+4Mgr,1+Mgr,0TH2m+2Mgr,0+Mgr,1HH2m+2Mgr,1+Mgr,1TH2m+2Mgr,1+Mgr,1TH2mMgr,1Mr-1 and F2m=-H2mMgr,0Mr-1F2m+1=-H2m+2Mgr,1Mr-1-H2mMgr,1Mr-1, where m= 0, 1, …, Nh, ()H denotes the conjugate transpose matrix operator, the mean and harmonic components of the block matrix H Eq. () can be found in Eqs. () and (), and the conditions H2m=0 for m>Nh must be used. Note that E-n=En and F-n=Fn.

Components of system matrix

The components of the system matrix Eq. () can be written as An=0A12A21,11,nA21,12,nA21,21,nA21,22,nA22,11,nA22,12,nA22,21,nA22,22,n, where A12=I for n= 0 and A12=0 for n> 0, and the block matrices can be computed as A21,11,n=-EnKr+Sr-Fn-1TSgr,1-Fn+1TSgr,1A21,21,n=-FnKr+Sr-Hn-1TSgr,1-Hn+1TSgr,1A21,12,n=-FnTKgA21,22,n=-HnKgA22,11,n=-EnCr-FnTGgr,0-Fn-1TGgr,1-Fn+1TGgr,1A22,21,n=-FnCr-HnGgr,0-Hn-1Ggr,1-Hn+1Ggr,1A22,12,n=-FnTGg,0+Cg-En-1Grg,1-En+1Grg,1-Fn-2TGg,2-Fn+2TGg,2A22,22,n=-HnGg,0+Cg-Fn-1Grg,1-Fn+1Grg,1-Hn-2Gg,2-Hn+2Gg,2 for n= 0, 1, …, 2Nh+ 3 using the following properties of the matrices En, Fn, and Hn: E-n=En,E-n=En,H-n=Hn,andEn+2=Fn+1=Hn=0forn>2Nh. Note that Nh= 0 for isotropic rotors with more than two blades, whereas Nh is selected for two-bladed rotors based on the required accuracy of the system matrix.

Acknowledgements

The support by the Danish Energy Agency through the EUDP-2011 II project “Demonstration of Partial Pitch 2-Bladed Wind Turbine Performance” is gratefully acknowledged. The author would also like to thank Anders M. Hansen, Ilmar F. Santos, Torben J. Larsen, Riccardo Riva, and project partners at Envision Energy for valuable discussions. Edited by: A. Kolios Reviewed by: V. A. Riziotis and D. Ossmann