FRF Partial decomposition

Here we will consider a linear system whose response function is G(s), this function can be written as a rational fraction G(s)=N(s)/D(s) where N(s) and D(s) are the numerator and denominator polynomials, respectively.

The partial fraction decomposition, also known as partial fraction expansion is an operation that consists in expressing the fraction as a sum of several fractions with a simpler denominator such as:

where λi and λi* are the eigenvalues, ai and ai* are called the residues; and assuming that the degree of the polynomial N(s) is lower than the degree of the polynomial D(s).

For more information, see the appendix on linear algebra.

The response function is then a sum of pairs, each pair containing the contribution of a single mode. A residue represents the strength of a mode; the stronger a resonance is, the larger the residue. You can therefore analyze the contribution of each mode to the global response function by plotting the Bode diagram of the equivalent response function of each pair.

To see the partial fraction contributions, click the check box in the Decomposition settings group to enable the display.

Figure 37: Displaying of partial fraction contributions

Note

Only the contributions of the oscillating modes are displayed

Decomposition settings

In addition to the display option for partial fractions, the decomposition settings group allows you to focus your analysis on a restricted number of contributing modes and a limited frequency range.

Figure 38: Decomposition settings group

The number of contributing modes is the number of partial fraction plots that are displayed on the Bode diagram. The modes to be displayed are determined by calculating the global maximum of all partial fractions over the specified frequency range and by keeping the highest ones defined by this number.

The frequency range is defined by the Min. frequency and Max. frequency parameters. A light blue zone is displayed on the graph to identify the boundaries of the frequency range.

Figure 39: Frequency range graph zone

Note

The maximum number of contributing modes is defined by the total number of oscillating modes for the selected linearization time and result set pair.

Minimum and maximum frequency values are not allowed beyond the values of the start and final frequency parameters of the Bode diagram. If you change these Bode diagram parameter values, the minimum and maximum frequency values are automatically adjusted. Click the Update decomposition button to refresh the plot.

Graph interaction

To get information about the mode of the partial fraction curves, you can activate the Data inspector mode. This is very convenient when you need to select a specific curve from many.

The data inspector switches the graphs into a mode that allows you to get information about the curves. To activate this mode, click the button on the plot toolbar.

When the mode is activated, all curves on the graph are shaded. If you move the pointer over a curve, it is highlighted and its color turns orange. In addition to the highlighting, a label is displayed near to the pointer. The label content depends on the curve element that is highlighted; it contains the type of the curve (global response function or partial fraction) and the value of the eigenfrequency is shown for the partial fraction.

Figure 40: Data inspector

If you click on a curve, the orange color remains. All groups in the Modal analysis window are synchronized when the data inspector mode is activated. When you select a curve on the Bode diagram, the corresponding mode is selected in the eigenvalues list and thus the top contributing states are updated. The same synchronization is performed when you select an eigenvalue in the list, the corresponding curve of the partial fraction is highlighted.

Note

Only one mode can be selected at a time.