Naming Mode Shapes

A structural mode shape describes an allowable pattern of translational and rotational displacements of any point of a flexible body. Using FEM, a mode shape is conveniently represented by a column vector of nodal displacements and rotations. In most cases, a mode shape represents a characteristic displacement pattern, such as the flapwise displacement of a blade or the side-to-side displacement of a tower. Naming the mode shapes with industry-standard terms can therefore facilitate easier reference.

Lower-order mode shapes are often well-defined, with displacements primarily occurring in a single direction. These cases are typically easy to categorise. However, for prebend blades with beam-level cross couplings, the mode shapes typically contain large displacements in various directions. Similarly, for multi-member towers, such as jacket support structures, where the structural members are connected in different directions, determining the primary displacement direction is not straightforward.

The problem of identifying mode shape names cannot be solved theoretically, as the problem is not rigorously defined. Hence, the presented method is a practical approach, implementing certain conditions that result in meaningful mode shape names in most cases.

The Naming Procedure

This method uses a set of predefined conditions to identify and label mode shapes based on their primary characteristics. The mode shape vector is defined as a column of the mode shape matrix, shown here, where each entry corresponds to a degree of freedom of a node in the proximal frame of the component.

Example of a normalised mode shape vector containing 2 nodes with 6 degrees of freedom each:

\[ \begin{equation} \bvector{\psi_{prox}} = \left[\begin{matrix} \underbrace{\begin{aligned} u_{x_1} \quad u_{y_1} \quad u_{z_1} \quad \theta_{x_1} \quad \theta_{y_1} \quad \theta_{z_1} \end{aligned}}_{\text{Node 1}} \quad \underbrace{\begin{aligned} u_{x_2} \quad u_{y_2} \quad u_{z_2} \quad \theta_{x_2} \quad \theta_{y_2} \quad \theta_{z_2} \end{aligned}}_{\text{Node 2}} \end{matrix}\right]^T \label{eq:modeshapevector} \end{equation} \]

where the first 6 entries correspond to the translational and rotational degrees of freedom for the first node, while the next 6 entries correspond to those of the second node. By examining the values in the vector, one can observe that the first entry is the largest. This would indicate a fore-aft or flapwise mode, depending on whether it pertains to a tower or a blade. However, this approach is not suitable, as the values of the translational and rotational degrees of freedom use differing units. Therefore, a more sophisticated method is necessary.

The procedure for naming mode shapes in Bladed involves a series of checks to determine the primary characteristic of the mode shape.

1. Determine if the Mode is Longitudinal ($u_z$)

Evaluate the following condition:

\[ \begin{equation} \bscalar{{u_z}_{max}} > \bscalar{K} \cdot \bscalar{{u_{xy}}_{max}} \label{eq:axialverticalmode} \end{equation} \]

where
${u_z}_{\text{max}} = \text{max} \left( \begin{matrix} {u_{z_1}} & {u_{z_2}} & \cdots & {u_{z_n}} \end{matrix} \right)$,
${u_{xy}}_{max}=\max{\left(\begin{matrix}\sqrt{u_{x_1}^2+u_{y_1}^2}&\sqrt{u_{x_2}^2+u_{y_2}^2}&\cdots&\sqrt{u_{x_n}^2+u_{y_n}^2}\\\end{matrix}\right)}$,
$\bscalar{K}$ is a constant calibration factor.

The left-hand side of the condition is the maximum longitudinal displacement in the mode shape vector, and the right-hand side is the maximum displacement magnitude in the x and y directions compared across all nodes.

2. Determine if the Mode is Torsional ($\theta_{z}$)

If the mode is not axial, evaluate if it is a torsional mode:

\[ \begin{equation} \bscalar{{\theta_z}_{max}} > \bscalar{K} \cdot \bscalar{{\theta_{xy}}_{max}} \label{eq:torsionalmode} \end{equation} \]

where
${\theta_z}_{\text{max}} = \text{max} \left( \begin{matrix} {\theta_{z_1}} & {\theta_{z_2}} & \cdots & {\theta_{z_n}} \end{matrix} \right)$,
${\theta_{xy}}_{max}=\max{\left(\begin{matrix}\sqrt{\theta_{x_1}^2+\theta_{y_1}^2}&\sqrt{\theta_{x_2}^2+\theta_{y_2}^2}&\cdots&\sqrt{\theta_{x_n}^2+\theta_{y_n}^2}\\\end{matrix}\right)}$

The left-hand side of the condition is the maximum torsional displacement in the mode shape vector, and the right-hand side is the maximum rotation (bending) magnitude in the x and y directions compared across all nodes.

Additionally, the following condition must also be true:

\[ \begin{equation} \quad \frac{1}{2}\frac{\bscalar{L}}{\bscalar{G_{ratio}}}\bscalar{{\theta_z}_{max}} > \bscalar{K} \cdot \bscalar{{u_{xy}}_{max}} \label{eq:tangdispl} \end{equation} \]

The left-hand side of the condition is the approximate tangential displacement resulting from the torsional rotation. This assumes a small angle approximation and uses the constant $\bscalar{G_{ratio}} = 15$, which is the typical ratio between the length and the cross-section of a wind turbine structure (both tower and blades). $L$ is a characteristic length: for whole-blade modes, it is the length of the blade; for blade part modes, it is the part length; and for towers, it is the tower height.

3. Determine if the Mode is Transverse ($u_x, u_y$)

If the mode is neither axial nor torsional, it must be a mode in either the $u_x$ or $u_y$ direction (transverse). This is determined by evaluating the largest x or y displacement at the maximum displacement magnitude, $\bscalar{{u_{xy}}_{max}}$.

List of Mode Shape Names

Mode Type	Blade	Tower
Longitudinal ($u_z$)	Axial	Vertical
Torsional ($\theta_{z}$)	Torsional	Torsional
Transverse ($u_x, u_y$)	Flapwise, Edgewise	Fore-aft, Side-side

Table of Contents

Naming Mode Shapes

The Naming Procedure

1. Determine if the Mode is Longitudinal (\(u_z\))

2. Determine if the Mode is Torsional ($\theta_{z}$)

3. Determine if the Mode is Transverse (\(u_x, u_y\))

List of Mode Shape Names

Mode Type	Blade	Tower
Longitudinal (\(u_z\))	Axial	Vertical
Torsional (\(\theta_{z}\))	Torsional	Torsional
Transverse (\(u_x, u_y\))	Flapwise, Edgewise	Fore-aft, Side-side