Linearisation & Campbell diagram

Articles on this page:

• Coupled wind turbine frequencies in parked condition
• Determining coupled mode types
• Plot coupled mode shapes from Campbell diagram
• Damped vs undamped frequencies in Campbell
• Detailed guidance (with example) on how to plot coupled mode shapes
• Energy contributions reported by Campbell diagram calculation
• The best practice Bladed model settings for model linearisation calculation when designing a control system
• What affects the damped frequency result for floating wind turbine
• Dynamic non-linear p-y curve from Bladed 4.9 and later versions
• Linear analysis of two-blade turbines
• How to debug if Campbell diagram modes give correct tower contributions?
• Memory error when performing Campbell analysis
• Why do coupled mode damping values of +/-1 sometimes appear in Campbell results?
• A damping value of -1 (negative damping) for rigid body modes

Problem

I would like to know if it is possible to obtain the coupled wind turbine frequencies with the rotor blocked (parked condition), and if so, how.

Solution

This is a feature available in Bladed from version 4.3 onwards. In the model linearisation screen you will find a tick box for parked as per below:

Keywords

Parked; Coupled frequencies

Problem

How do I work out which coupled modes are which in the Campbell diagram such as in-plane, out-of-plane etc...?

Solution

The coupled modes in the Campbell diagram are named after their most significant uncoupled mode contribution. This naming is not always reliable and does not give a full description of the mode. Therefore, it is recommended that users always work out what a coupled mode is based on the contributions from uncoupled modes. You can see the uncoupled contributions by clicking on a point in the diagram and looking in the separate 'GHView' window. In the example below, you can see that this is a mode that couples 'mode 2' in the three blades to form a coupled rotor mode. Mode 2 directly relates to the corresponding mode in the modal analysis results in the relevant project. So in this case, it would be the 2nd blade mode in the list.

You can also view the coupled modes by clicking the 'View' button in the GHView window. This can help you work out the phase of the contributing uncoupled modes, which tells you whether the modes are symmetric or anti-symmetric.

In 4.7 and later, the names of the uncoupled modes have more complete names (e.g. 1st flapwise mode), and the phase of each contribution is also output to the GHView window.

Keywords

Campbell; Coupled modes; Mode; Modal

Problem

The Campbell diagram calculates coupled mode shapes based on contributions from the uncoupled mode shapes.

How can the coupled mode shapes be plotted?

Solution

Bladed 4.17 onwards:

A coupled mode animator was introduced in Bladed 4.17 for visualising coupled mode shapes - find out more in the Bladed documentation.

Bladed 4.0 to 4.16 (from 4.8 to 4.16, this only applies to tower modes):

NOTE: the workarounds listed below do not work for Bladed 4.17 and later.

This is not currently directly supported in Bladed versions 4.0 to 4.16, but can be achieved as follows.

Information about the coupled mode shapes are output in the .$01 file from a Campbell diagram calculation. This file describes the contributions of each uncoupled mode to each coupled mode.

The coupled mode shapes are based on a contribution from each uncoupled mode. This is the information in the “MAG” (magnitude) values. So, you can re-create the coupled mode shapes by multiplying each magnitude by the corresponding uncoupled mode shape. You can find the mode shapes in the $PJ file for older versions (Bladed 4.7 and earlier); look in the RMODE module. In Bladed 4.8 and later this information is in the XML section under .

You should also include the phase, which takes a value of 0 or +/-PI/2 without damping, but can take other values once damping is included. The way to do this is to take sin(phase) then for example 0 will give a weighting of 0 and PI/2 will give a weighting of 1. If desired, you could then increment the phase for all components and plot the coupled mode shapes at different points in the harmonic cycle. Eigen modes from other software (e.g. Sesam) are usually calculated without damping. So it may be convenient to set structural damping to zero for the purpose of comparison.

This process is fairly straightforward if you want to look at the coupled tower modes. However, it gets more tricky if you want to include the rigid body blade motion that results from tower deflection. You’d need to calculate this coupling yourself as it’s not directly stored in the mode shape result (e.g. you need to take account of tilt, azimuth, blade root length etc). Also note that since the introduction of multi-part blade modelling in v4.8, you can no longer apply the methods in this article to blade / rotor modes, but only to tower modes.

* Note that in Bladed 4.8 and later, the uncoupled mode shapes are also given in the $VE file. However, the modes shapes in the $VE are scaled differently compared to the $PJ file mode shapes in Bladed 4.8 to 4.15. For these Bladed versions it's important to use the values from the $PJ file as these are consistent with the mode shapes that are used in the linearisation calculation. In Bladed versions 4.16.1 and above, scaling is no longer an issue as the mode shapes match between the two files.

There is now a downloadable package, available in the Detailed guidance (with example) on how to plot coupled mode shapes section below, containing a worked example (simplified) that helps guide you through the process.

Keywords

Campbell diagram; Magnitude phase; Coupled modes

Problem

When you look at the Campbell diagram in Data View, which of the frequency values is displayed? Is it damped or undamped?

Solution

It is the undamped frequency that is displayed in the Campbell viewer. Damped frequencies are also available in the output groups of the Campbell diagram calculation.

For details of how the damped and undamped frequencies are calculated from the linearised system matrices, see Calculating Coupled Modes in the Bladed documentation (or Section 11.2 in PDF versions of the Theory Manual).

Keywords

Campbell; Frequency; Damped; Undamped

Problem

Visualising coupled mode shapes is a common requirement. In Bladed 4.17, a coupled mode animator was introduced, but there is no coupled mode animator in versions below 4.17.

Solution

Bladed 4.17 onwards:

A coupled mode animator was introduced in Bladed 4.17 for visualising coupled mode shapes. Read more about it in the Coupled Mode Animator article in the Bladed Documentation.

Bladed 4.0 to 4.16 (from 4.8 to 4.16, this only applies to tower modes):

NOTE: the workarounds listed below do not work for Bladed 4.17 and later.

The information needed to reconstruct coupled modes is in the .$01 file and the .$PJ file generated during a Campbell diagram run.

Please refer to the Plot coupled mode shapes from Campbell Diagram section above for background.

There is a downloadable zip package which contains worked example files for a simple demo case that allows the user to construct coupled modes for the tower. It also helps users to understand how the uncoupled modes are combined together. Note that it does not provide a solution to showing coupled modes for the entire turbine.

Keywords

Coupled modes; Campbell; Mode shapes

Problem

What is the difference between the contribution of uncoupled mode shapes to coupled shapes reported in the $01 and $CM files when running a Campbell diagram?

Solution

The uncoupled contributions to coupled modes in the $01 file are obtained from normalising entries in the rows of the calculated A matrix. These contributions represent displacement magnitudes of each uncoupled mode that contributes to the coupled mode shape. Contributions below a specified tolerance will be set to zero. In version 4.15, the tolerance is set to 1e-6 and the values are presented in scientific notation, while in all previous versions, it is set to 1e-3 using standard notation. The change makes it easier to evaluate whether specific modes remain influential when dynamic stall or wake models are active.

The energy contribution is derived by normalising the magnitudes of uncoupled mode shapes to coupled modes. The meaning of the energy contributions are ill-defined for a system which includes servo systems anyway (as explained below) and so we would not recommend using it to compare against other tools. These results are saved to the $CM file.

The stiffness and mass (and damping) matrices make sense for a system which just represents a physical structure, whose degrees of freedom are all second-order modes representing movement of parts of the structure. Then everything is straightforward, and the percentages should be expressed as energy in the sense of ½kx² terms where k is stiffness and x is displacement. The problem is that the system dynamics include non-structural modes, which affect the dynamic response but can’t be represented in the same way. This could include pitch actuator dynamics, generator/power converter dynamics, perhaps with internal control loops, measurement sensors, etc. The dynamics of these components partly come from electronic circuits (including digital controls) rather than masses and stiffnesses, so it’s not possible to define the energy in those modes, and indeed the concept of mass and stiffness matrices is no longer strictly useful. If any of these additional dynamics can be represented as second-order modes then one can ‘pretend’ that they represent masses and stiffnesses with modes containing a certain energy, although this is quite artificial; and in practice there is no need for them to be second-order, and often they are not. Because of this, the concept of % contributions becomes ill-defined. Bladed uses an algorithm to come up with these contributions in order to give the user some idea of what a given coupled mode represents physically, but the algorithm is not theoretically rigorous – it can’t be, because the problem is not rigorously posed. It’s just trying to be a bit helpful, and the algorithm seems to produce helpful results at least most of the time. It would be possible to generate a Campbell diagram including only the structural dynamics, and then everything would be easy, and mass and stiffness matrices could be calculated. However, ignoring the non-structural dynamics is not correct, because they do have an influence on the frequencies and damping of the coupled modes.

Keywords

Coupled modes; Campbell diagram; Mode shapes

Problem

What are the best practice Bladed model settings for model linearisation calculation when designing a control system?

Solution

Although use of some of the detailed modelling features like multi-part blade and dynamic stall are important in time domain simulations, it is often a reasonable simplification to remove them in the linearisation calculation. They typically do not significantly affect the tuning in the frequency range of the controller. Including these features significantly increases the number of modes and therefore states in the state space model. This can lead to issues in the linearisation post processing and can make linear design in SISO tools more difficult.

The user can often get sufficiently accurate results with a simpler linear model by:

1. Simplifying the blade to a single part blade. To simplify, click 'delete selected blade parts' in Flexibility Modeller screen - Bladed will reassign stations automatically. The 'axial loads only' Blade geometric stiffness model is recommended for single part blades. Enough blade modes should be used to capture the blade 1st torsional mode.
2. Disabling dynamic stall. This significantly reduces the number of states and in most cases does not influence the dynamics relevant to the controller. To disable dynamic stall, go to Specify -> Aerodynamic control.

Keywords

Linearisation; Controller; Flexibility; Multi-parts blades

Problem

What affects the damped frequency result

Solution

The Wamit result, used for the BEM hydrodynamics definition, is a frequency domain dependent definition. The hydrodynamics modelled from the Wamit results are only included in time domain simulations in Bladed where a time history is used to derive the frequency dependent hydrodynamic contributions at each point in time. The model linearisation is a “timeless” calculation so there is no contribution from any of the frequency dependent Wamit results in the linear model. Therefore, changing the Wamit result will not change the damped frequency as the damping is not effected. In general, the quadratic damping and the radiation damping is not captured by the linear model. Only linear damping is present. As a workaround it is possible to tune a linear damping coefficient based on free decay simulations in Bladed (parked time domain simulations where initial platform position is defined not in equilibrium, the platform then moves to the equilibrium position during the simulation). From this, you are able to estimate the amount of damping that should exist. This can then be added to the model pre-linearisation as a global linear damping term to compensate for the lost damping in the linearisation process. This makes the linear model more representative of the actual system which is helpful for control design. After control design, for the time domain simulations, you would remove this added linear damping otherwise you will be double accounting for the frequency dependent damping.

Keywords

Floating; Linearisation

Problem

Does it mean the ’finite element loads’ correction will alter the natural frequency of offshore wind turbines in Bladed flexibility calculation?

Solution

The Campbell diagram uses the defined uncoupled tower modes as input. The point of the ‘finite element deflections’ feature is to recreate the dynamics that would be seen with a far higher number of modes defined. Therefore the Campbell diagram output and time series output with ‘finite element deflections’ enabled would converge in terms of frequencies for the case where the Campbell diagram is run with a model that has a very high number of uncoupled tower modes defined. For a Campbell diagram run with a ‘standard’ model definition (relatively small set of defined uncoupled tower modes), the coupled tower frequencies would not identically match ‘finite element deflections’ time series frequencies.

Keywords

Finite element deflections; Campbell diagram

Problem

Is Bladed’s linearisation appropriate for the analysis of coupled modes, and control design for two-bladed rotors? Does the theory behind the linearisation take the periodicity of the two-bladed turbine into account?

Solution

Bladed has been widely used for the analysis of two-blade turbines (see public examples below)

The MBC transformation is applied for two-blade turbines, resulting in the correct coupling between rotor and tower modes for a given azimuth. But the analysis should be repeated at different azimuth angles.

For a 2-blade turbine, it’s not possible to remove all of the periodicity to produce a linear time invariant system, so technically the coupled modes should be determined by Floquet analysis rather than a conventional eigen analysis. However, Bladed uses eigen analysis for this process, giving reasonable results that have been applied successfully to perform control design of two-blade turbines.

Example applications in public domain:

• Control design for 20MW reference 2-blade turbine
• Implementation of individual pitch control on NREL CART2 turbine
• PhD thesis on control design for 2-blade turbines

Floquet recommendation example:

• Paper showing advantage of Floquet analysis over Eigen analysis

Keywords

Two-blade; Control modes; Teeter

Problem

For instance, a client simulates a Campbell diagram for their turbine model and recognizes that the Tower 2nd modes are not present when dynamic stall is included. How to evaluate that?

Solution

To check if this makes sense or not, the user is suggested to have a look at “$01” file provided in the simulation results (human readable if ASCII data is selected as the output). Here one will see a list of contributions from the uncoupled modes:

It can be depicted that there are 1330 uncoupled modes that are used to compute 87 coupled modes. As an example, the “Tower modes” labelled in above figure map directly with the uncoupled modes in the flexibility screen in Bladed User Interface (sorted by frequency) as shown in above figure, e.g.,

• Tower mode 1 = 1st side-side mode
• Tower mode 2 = 1st fore-aft mode
• etc.

Note that for blade uncoupled modes these can be more complex depending on the use of multi-blade coordinate transformation MBC (transformation to non-rotating frame) as:

• Without MBC - displacement contributions to coupled modes from whole blade modes. But whole blade modes are different at each operating point.
• With MBC - displacement contributions to coupled modes from rotor modes calculated in MBC.

See below a screenshot of the “$01” file where the contributions of the uncoupled modes are listed. These are sorted by the frequency of the coupled modes (in vertical direction, see figure). The magnitudes of the contributions are indicated by the keyword “MAG” and are sorted according to the uncoupled modes listed on top of the “$01” file. Contributions below a specified tolerance will be set to zero. In version 4.15, the tolerance is set to 1e-6 and the values are presented in scientific notation, while in all previous versions, it is set to 1e-3 using standard notation. The change makes it easier to evaluate whether specific modes remain influential when dynamic stall or wake models are active.

The fifth value of “MAG” corresponds directly with the “Tower mode 5” or 2nd fore-aft mode“ of the tower. It can be observed here that the value of this mode remains zero for all frequencies, which seems questionable (see below figure for more detail).

If the dynamic stall model is deactivated (as already recommended in practice for Campbell diagram), the contribution of the “2nd fore-aft mode” of the tower is now present for some frequency points, and is seemingly more correct.

The following two articles provide some other examples on how to interpret the results of the “$01” file and possibilities to plot the modes:

• Plot coupled mode shapes from Campbell Diagram
• Detailed guidance (with example) on how to plot coupled mode shapes

Keywords

Linearisation; Dynamic stall; Modes; Campbell

Problem

"Unable to allocate aligned memory..." error.

Solution

Introduction:

Memory allocation errors such as *** ERROR: Unable to allocate aligned memory... Campbell analysis can be attributed to models that have a high number of degrees of freedom, leading to excessive memory demand.

The dynamic aero terms like wake and stall introduce many more degrees of freedom. A practical solution is to minimize the complexity of the model to reduce memory usage.

Solution:

1. The most effective method to avoid these errors is to disable dynamic wake and dynamic stall terms. Dynamic terms introduce additional states, increasing memory requirements. By disabling them, you can lower the number of states and decrease memory usage, potentially resolving the allocation issues.
2. Once the Campbell analysis is running smoothly, you can appropriately add options for dynamic wake and dynamic stall to achieve more accurate results.

Keywords

Campbell analysis; Memory

Problem

When I run a Campbell diagram and view the Damping values in the Coupled Modes output group, sometimes a value of +1 or -1 is seen for the damping. What does this mean?

Solution

If you get either +1 or -1 for damping values, this corresponds to non-oscillatory modes of the system, i.e. not corresponding to actual physical modes. For this reason it should be OK to ignore these, as they will not influence the resonances of the system which is the main point of the Campbell calculation.

The reasons for this are mentioned in the Theory Manual. The damping ratio is the cosine of the angle "theta" in the Argand diagram. A damping value of +/-1 means theta is zero, i.e. a value on the real axis which means no oscillation. As the above section of the Theory Manual explains, these coupled modes should be excluded from the Campbell diagram itself if their damping values are +/1 at all operating points:

Coupled mode series that have a real eigenvalue, and therefore no oscillatory behaviour, at all operating points are excluded from the results. This is because they cannot cause resonant behaviour which is primarily what the Campbell diagram is used to check for. This generally excludes modes that are mainly contributed by aerodynamic states.

Keywords

Campbell; Damping

Problem

There can be a damping value of -1 (negative damping) for rigid body modes, e.g. Rotor rigid body mode = -1, in old versions of Bladed (e.g. 4.9).

Solution

A damping value of -1 indicates that the coupled mode is first-order and associated with a positive real eigenvalue. This result only appears for rigid body modes, which are associated with eigenvalues of small magnitude and therefore correspond to low frequencies. And rigid body modes are irrelevant for stability analysis and can be ignored.

If you find such cases please try to do the simulations again using Bladed 4.18 or later versions. The Campbell diagram analysis has been improved in Bladed 4.18 – both in terms of accuracy and the presentation of the results. For example, the resulting diagram is now more streamlined and no longer includes irrelevant aerodynamic modes, which were often distracting in Bladed 4.16 and earlier.

Keywords

Campbell; Damping; Linearisation