Eigenvalue analysis

This page describes how Ashes computes the undamped natural frequencies and mode shapes of a model at rest. The user interface is described in the Eigenmodes page of the User Manual. For the sweep over rotor speed, see Campbell diagram.

1 The eigenvalue problem

A structure with no damping and no external load obeys

$$\mathbf{M}\ddot{\mathbf{u}} + \mathbf{K}\mathbf{u} = \mathbf{0}$$

where
$$\mathbf{M}$$
is the mass matrix,
$$\mathbf{K}$$
the stiffness matrix and
$$\mathbf{u}$$
the vector of nodal degrees of freedom. Looking for solutions that oscillate in a fixed shape,
$$\mathbf{u}(t) = \boldsymbol{\phi}\sin(\omega t)$$
, turns this into the generalised eigenvalue problem

$$\mathbf{K}\boldsymbol{\phi} = \lambda\mathbf{M}\boldsymbol{\phi}, \qquad \lambda = \omega^{2}$$
Equation 1 The undamped generalised eigenvalue problem

Each solution is a pair: an eigenvalue
$$\lambda$$
giving the natural angular frequency
$$\omega = \sqrt{\lambda}$$
in rad·s -1, reported as
$$f = \omega/2\pi$$
in Hz, and an eigenvector
$$\boldsymbol{\phi}$$
giving the mode shape.

Four consequences of this formulation are worth spelling out, because they explain most of the questions that come up about eigenmode results.
  • The analysis is undamped. Damping does not appear in Equation 1, so the frequencies are undamped natural frequencies and the mode shapes are real: every point of the structure passes through its neutral position at the same instant. For the light structural damping typical of a wind turbine the difference from the damped frequency is very small (see Damping).
  • The analysis is linear, about the initial position. The two matrices are assembled once, from the model as built. Effects that depend on the deflected shape, and loads that depend on the motion — aerodynamic and hydrodynamic damping in particular — are not part of the problem.
  • There is no scale. If
    $$\boldsymbol{\phi}$$
    is a solution then so is any multiple of it. A mode shape therefore carries only relative displacements, and the shapes Ashes exports are normalised against the largest displacement in the model.
  • Constrained degrees of freedom are removed from the system before it is solved, so a fixed component of a node contributes nothing to any mode shape.

Note: hydrodynamic added mass is taken into account. It is assembled into the mass matrix rather than applied as a load, precisely so that it affects the eigenmodes of an offshore model (see Morison equation).

2 The solver

Ashes does not form and factorise the whole eigenproblem at once. It uses shifted inverse iteration, which finds one mode at a time and needs nothing more than the Gauss elimination the static solver already provides. The method is standard; a textbook treatment is given in Bathe (1996).

2.1 Finding one mode

For a trial value
$$\mu$$
called the shift, the stiffness is shifted by a multiple of the mass and the system is iterated:

$$\left(\mathbf{K} - \mu\mathbf{M}\right)\mathbf{v}_{k+1} = \mathbf{M}\mathbf{v}_{k}$$
Equation 2 One inverse-iteration step

Each step is a single linear solve whose right-hand side is an inertia load built from the previous shape estimate. The iteration converges towards the mode whose eigenvalue is closest to the shift, and it converges fast: at every step, the contribution of each other mode is suppressed in proportion to how much further from the shift it lies.

The iteration starts from a random vector, so that no mode is missed through a start vector that happens to be orthogonal to it, and the estimate is normalised against its largest component after each step to keep it bounded. The eigenvalue estimate is the Rayleigh quotient, evaluated with the unshifted matrices:

$$\lambda = \frac{\mathbf{v}^{\mathrm{T}}\mathbf{K}\mathbf{v}}{\mathbf{v}^{\mathrm{T}}\mathbf{M}\mathbf{v}}$$

This is the best eigenvalue a given shape can produce, and its error is second order in the error of the shape — so the frequency is accurate well before the shape is. The iteration stops when the relative change in the Rayleigh quotient between two successive steps falls below 10 -8. An eigenvalue below 10 -6 is treated as zero, that is, as a rigid-body or otherwise unconstrained direction, and is not reported as a mode.

2.2 Walking up the spectrum

Because inverse iteration returns the mode nearest the shift, the modes are found in order by moving the shift upwards. Once a mode has been found at
$$\lambda_{n}$$
, the next shift is placed just above it:

$$\mu = \lambda_{n} + \Delta\lambda, \qquad \Delta\lambda = 2\sqrt{\lambda_{n}}\,\delta\omega$$
Equation 3 Advancing the shift by one search resolution

Here
$$\delta\omega$$
is the search resolution. The factor
$$2\sqrt{\lambda_{n}}$$
follows from
$$\lambda = \omega^{2}$$
: a step of
$$\delta\omega$$
in frequency is a step of
$$2\omega\,\delta\omega$$
in eigenvalue. The search resolution is therefore a spacing in frequency, and stays one all the way up the spectrum.

A converged result is accepted as a new mode only if it lies at least
$$\Delta\lambda$$
above the previous one. If it does not — that is, if the iteration fell back onto the mode already found — the step is enlarged by a factor of 1.99 and the shift retried, as often as needed until the search escapes. This is what lets the search cross a wide gap in the spectrum without requiring a coarse resolution everywhere else.

The search ends when the requested number of modes has been found, or when the shift passes the top of the search range.

3 What the settings control

The three fields of the Modes box in the Eigenmodes window map directly onto the algorithm above.

Setting Role in the algorithm
Modes to find How many times the shift is advanced and a new mode accepted. Fewer may be returned if the top of the range is reached first.
Search range [Hz] The lower bound is where the first shift is placed; the upper bound is where the search stops. Narrowing the range does not make an individual mode cheaper to find, but it stops the search spending modes on frequencies of no interest.
Search resolution [Hz]
$$\delta\omega$$
in Equation 3, converted internally to rad·s -1. It is both the initial step between shifts and the minimum separation at which two modes count as distinct.

Two modes closer together than the search resolution can be reported as one. This matters for a wind turbine, whose fore-aft and side-to-side tower modes are often only a few thousandths of a Hz apart: if only one tower mode appears in the results, lower the search resolution.

4 Limitations

  • The modes are undamped and real. A structure whose damping is strong, or very unevenly distributed, has complex modes that this formulation cannot represent.
  • The system is linearised about the initial position. The modes of a heavily deflected structure, and the stiffening that a static load produces, are not captured unless that state is built into the model.
  • Rotational effects are not included. A rotor at standstill and the same rotor turning do not have the same modes; see Campbell diagram.
  • The shifted system is factorised with the same Gauss elimination used elsewhere in the FEM code, and the shift assumes lumped masses.

Ashes' eigenfrequencies are checked against published reference values for several reference turbines in the Validation & benchmarks section.