The modal dynamics of structures with bladed isotropic rotors is analyzed using Hill's method. First, analytical derivation of the periodic system matrix shows that isotropic rotors with more than two blades can be represented by an exact Fourier series with 3/rev (three per rotor revolution) as the highest order. For two-bladed rotors, the inverse mass matrix has an infinite Fourier series with harmonic components of decreasing norm; thus, the system matrix can be approximated by a truncated Fourier series of predictable accuracy. Second, a novel method for automatically identifying the principal solutions of Hill's eigenvalue problem is introduced. The corresponding periodic eigenvectors can be used to compute symmetric and antisymmetric components of the two-bladed rotor motion, as well as the additional forward and backward whirling components for rotors with more than two blades. To illustrate the use of these generic methods, a simple wind turbine model is set up with three degrees of freedom for each blade and seven degrees of freedom for the nacelle and drivetrain. First, the model parameters are tuned such that the low-order modal dynamics of a three-bladed 10 MW turbine from previous studies is recaptured. Second, one blade is removed, leading to larger and higher harmonic terms in the system matrix. These harmonic terms lead to modal couplings for the two-bladed turbine that do not exist for the three-bladed turbine. A single mode of a two-bladed turbine will also have several resonance frequencies in both the ground-fixed and rotating frames of reference, which complicates the interpretation of simulated or measured turbine responses.

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.

The modal dynamics of three-bladed turbines is well understood, also including
the interaction with aerodynamic forces and a controller. For isotropic
rotors, the Coleman transformation

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

The modal analysis of structures with two-bladed rotors is complicated by the
strong periodicity of the system. The use of Floquet theory or Hill's method
is unavoidable, unless both rotor and support structure are isotropic with
respect to rotation

In this paper, the modal dynamics of structures with a rotor that have two or
more blades are considered, first from a generic model-independent
perspective and then with focus on the differences between the modal
dynamics of two- and three-bladed turbines. In Sect.

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

Let the Lagrangian for a structure with a rotor be written as

Through substitution of the Lagrangian Eq. (

Let

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

Through insertion of Eq. (

The inverse of the mass matrix in Eq. (

For two-bladed rotors,

To solve the equations for the even terms, the mean component

If

Truncation of Eq. (

Insertion of Eq. (

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.

Floquet theory defines that the eigen-solution of the linear periodic system
Eq. (

To numerically solve Hill's eigenvalue problem Eq. (

The identification of the principal solutions can be done
manually for small systems

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

link the remaining eigen-solutions in 2

pick the principal solution in each subset that has the largest mean component

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.

The naming conventions of symmetric and whirling rotor modes of three-bladed
rotors deduced from the modal analysis using the multi-blade coordinates

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

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

The turbine model derived in the next section is based on the simple model
presented in

The convergence of the Fourier series of the system matrix for the two-bladed
turbine is analyzed in Sect.

Figure

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.

The blade motion is described in their own rotating frames (

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

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:

The damping matrices of Rayleigh's dissipation function in Eq. (

The linear equations of motion (Eq.

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. (

Figure

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.

The number of harmonics in the periodic eigenvectors used for Hill's
truncated eigenvalue problem is set to

Figure

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

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.

Figures

The naming of the modes shown in Fig.

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

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.

Figures

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.

The tower side–side mode in Fig.

The symmetric and antisymmetric flapwise modes in Figs.

For the second antisymmetric flapwise mode in Fig.

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.

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.

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

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 repository

This appendix contains a list of the elements of the block matrices in
Eqs. (

The elements of the block matrices for the blade DOFs are

The elements of the block matrices that couple the blade and ground-fixed
DOFs are

The mean and harmonic components of the block matrices Eq. (

The components of the system matrix Eq. (

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