Joining Coupled Modes across Operating Points

The Campbell diagram displays the frequencies of different coupled displacement modes with respect to the rotor speed together with the most important excitation frequencies given in terms of multiples of the rotor frequency (P). In addition to the frequencies of the coupled modes, the Campbell diagram displays the corresponding damping ratios, which include the effect of structural damping as well as aerodynamic damping. Both characteristics are calculated as described in the article Calculating Coupled Modes and are useful for identifying the critical operating points that need further analysis.

The Joining Process

Given a set of coupled modes at each operating point, a fundamental step in creating the resulting Campbell diagram is to identify similar modes at adjacent operating points, which allows for joining similar modes across the operating points with line segments, giving the user the impression of continuous change in frequency against rotor speed (or wind speed). This joining process is generally challenging because the coupled modes evolve and change in their contributions between the operating points.

Similar coupled modes at two adjacent operating points are identified by comparing their complex eigenvectors and frequency in term of the extended modal assurance criterion (MACX) (Vacher, Jacquier, and Buchales, 2010) with frequency weighting. More specifically, the frequency weighted MACX numbers are calculated for all combinations of coupled modes at the two operating points to form a score matrix, which are then used for joining the modes by the Gale-Shapley algorithm (Gale & Shapley, 1962). A sequence of similar modes at all operating points forms a coupled mode series, which represents a line in the resulting Campbell diagram.

To ensure that the coupled mode series in the resulting Campbell diagram primarily involves structural dynamics, an initial calculation of coupled modes with only structural states is performed at the first operating point. These structure-only modes are then joined with the coupled modes at the first operating point as described above, which effectively excludes coupled modes that mainly have contributions from aerodynamic states.

Figure 1: Illustration of the joining process showing how the coupled modes are connected at the operating points. Dots represent coupled modes. Coloured lines represent coupled mode series.

The resulting joining process is then:

1. Calculate a set of structure-only modes at the first operating point. These modes will also form the basis of the mode series, representing the lines in the resulting Campbell diagram.
2. Join the structure-only modes with the coupled modes at the first operating point. It is noted that the number of coupled modes is generally larger than the number of mode series, which means that not all coupled modes are included in a mode series.
3. Join the coupled modes that were included in a mode series at the current point (first operating point) with the coupled modes at the next point (second operating point). Repeat until the last operating point is reached.

Naming and Exclusion of Coupled Mode Series

A coupled mode series is named according to the contributions of the structure-only coupled modes at the first operating point only (more details on coupled mode naming can be found here). The contributions and therefore the shape of a coupled mode can change significantly between the range of operating points, and therefore the characteristic of a mode cannot be determined from the name alone.

A coupled mode series, which includes coupled modes with real eigenvalues (and therefore no oscillatory behaviour) at all operating points, is excluded from the resulting Campbell diagram. This is done because such modes cannot cause resonant behaviour, which is the primary purpose of the Campbell diagram to detect.

A coupled mode series is also excluded if the coupled mode frequencies at all operating points exceed a user-defined maximum value.

Last updated 26-11-2024