Modal Analysis Output

Generalised mass and stiffness properties

During modal analysis, Bladed calculates the “generalised mass matrix” and “generalised stiffness matrix” by transforming the flexible component finite element mass/stiffness matrices using the normalised mode shape matrix \(\bmatrix{\Psi}\) as shown below. The purpose of this is to transform the finite element mass and stiffness matrices into modal space (often referred to as the generalised coordinates).

\[ \begin{align} \bmatrix{M}_{ge}&= \bmatrix{\Psi}^T\bmatrix{M}_{fe}\bmatrix{\Psi} \label{eq:modalelementmass} \\ \bmatrix{K}_{ge}&= \bmatrix{\Psi}^T\bmatrix{K}_{fe}\bmatrix{\Psi} \label{eq:modalstiffness} \end{align} \]

where

• \(\bmatrix{M}_{fe}\) and \(\bmatrix{K}_{fe}\) are the finite element mass and stiffness matrices for the complete flexible component. They are described in terms of \(6\bscalar{N_{n}} \text{-by-} 6\bscalar{N_{n}}\) square matrices, where \(\bscalar{N_{n}}\) is the total number of nodes in the flexible component.

• \(\bmatrix{M}_{ge}\) and \(\bmatrix{K}_{ge}\) are the modal mass and stiffness matrices. They are described in terms of \(\bscalar{N_{m}} \text{-by-} \bscalar{N_{m}}\) square matrices, where \(\bscalar{N_{m}}\) is the total number of modes specified for the flexible component

• \(\bmatrix{\Psi}\) is the mode shape matrix, which is used to transform between the finite element and modal domain. Consequently, the mode shape matrix has the dimensions below, with one mode shape defined on each column.

The diagonal elements of this generalised mass and stiffness matrices are then reported by Bladed for each mode.

\[ \begin{equation} \bmatrix{\Psi} = \ \begin{bmatrix} . & . & . & . & . & . & & & & \\ . & . & . & & & & & & & \\ . & . & . & & & & & & & \\ . & & & . & & & & & & \\ . & & & & . & & & & & \\ . & & & & & & & & & \\ & & & & & & & & & \\ & & & & & & & & & \\ < & - & - & - & \bscalar{N_{n}} & - & - & - & - & > \end{bmatrix}\begin{matrix} /\backslash \\ | \\ | \\ | \\ \ \ 6\bscalar{N_{m}} \\ | \\ | \\ | \\ \backslash\text{/} \end{matrix} \label{#eq:modeshapematrix} \end{equation} \]

Natural frequency calculation

Using the Rayleigh principle (Clough and Penzien, 1993), the mode natural frequencies can be calculated as

\[ \begin{equation} \bscalar{\omega_{i}}^{2} = \ \frac{\bscalar{K_{ge,i}}}{\bscalar{M_{ge,i}}}, \label{eq:naturalfrequencies} \end{equation} \]

where \(\bscalar{K_{ge,i}}\) and \(\bscalar{M_{ge,i}}\) are the terms on the leading diagonal of the generalised stiffness and mass matrices.

Note on the “effective modal mass”

There is also a quantity called the “effective modal mass” associated with each mode shape, which is not reported by Bladed. This quantity can represent (for example) the part of the total mass in each mode shape that responds to a specified unit displacement (for example a unit translation of the whole component in a certain direction). The “effective modal mass” for a particular mode, \(i\), can be expressed as:

\[ \begin{equation} \bscalar{M}_{\text{eff},i} = \frac{\bscalar{l}_{i}^{2}}{\bscalar{M}_{ge,i}}, \label{eq:effectivemodalmass} \end{equation} \]

where

\[ \begin{equation} \bvector{l} = \ \begin{bmatrix} \bscalar{l}_{1} \\ \bscalar{l}_{2} \\ . \\ . \\ {\ \bscalar{l}}_{\bvector{N_{m}}} \end{bmatrix} = \ \bvector{\Psi}^{T} \bmatrix{M}_{fe} \bvector{r} \label{eq:eachmodeshape} \end{equation} \]

and \(\bvector{r}\) is the displacement vector of each degree of freedom when the component is subject to a unit displacement.

It is possible to add up the “effective modal mass” for each mode and compare that to the total component mass to evaluate the contribution of each mode to a certain displacement field. This summation of “effective modal mass” is sometimes used to help choose the number of modes required for a simulation.

However, it is not clear in general which set of displacements should be used to calculate the “effective modal mass”. For example, a unit displacement of the whole component is appropriate for (say) simulating an earthquake, but does not give information in general as to the significance of a mode when subject to arbitrary forcing and displacement. For this reason, the “effective modal mass” is not reported by Bladed.