Eigenvalue analysis

performing Eigenvalue Analysis

The eigenvalues of a state space model represents the natural frequencies, also know as eigenfrequency, of a physical system. The analysis of these specific frequencies allows to predict the resonance phenomenon of the system occurring at the natural frequency, as well as the amplitude of the oscillation thanks to the damping ratio.

To get the list of the eigenvalues of the equivalent linear system which is obtained at a linearization time, select the Eigenvalues item in the Linear Analysis menu.

Figure 24: Eigenvalues item in the toolbar

It opens a dedicated window that lists the eigenvalues of your system.

Figure 25: Eigenvalues tool

The eigenvalues are displayed according to the linearization time and the result set selected from the drop-down lists.

Eigenvalue properties

The content of the table can be customized in order to display the properties that are valuable for the analysis. The configuration button provides the list of properties that can be included in the table.

You can also select the unit of the frequency (Hz or Rad/s), and the value display of the damping (% or ratio).

Figure 26: Configuring eigenvalue properties

The eigenvalues are computed from the linear transformation of the A matrix of the state space model. For more information, see the appendix on linear algebra.

We consider 3 different types of eigenvalues:

• Oscillating mode

An oscillating mode is an eigenvalue of the system for which, if the system is excited at the associated frequency, the response of the system is oscillatory. The amplitude of the oscillations depends on the associated damping. One property of an oscillating mode is that the imaginary part of the eigenvalue is non-zero. There are therefore two eigenvalues associated to an oscillation mode and they are complex conjugates. By default, the oscillating modes are displayed on one line; activate the Expand oscillating modes Expand oscillating modes option to display the positive and negative values of the imaginary part of the complex conjugates on two lines.

• Time constant

A time constant is an eigenvalue of the system for which, if the system is excited at the associated frequency, the response of the system is totally damped. There are no oscillations. The value is related to the delay between the moment when there is a change in the input condition and the moment when the output has completely responded to this change. The two main properties of a time constant is that the imaginary part is equal to 0 and that the damping ratio is 100%.

• Zero

A zero is an eigenvalue that is equal to 0. By default, this type of eigenvalues is not visible in the table, so disable the Hide zero frequency modes option to display them.

Note

By clicking on a column header, you can sort the values in the column in ascending or descending order.

When < 0.01 is displayed in the table, it means that the associated value is in the range [0 0.01]. When >-0.01 is displayed, the associated value is in the range [-0.01 0].

Associated tools

When you select an eigenvalue in the table, you can open the Modal shape tool directly by clicking on the dedicated button. The tool is opened with the corresponding eigenvalue already selected. For more information, see the Modal shape topic.

You can create a pole map of the dynamic system model by clicking on the Plot button. It displays all the eigenvalues on the complex s-plane. The system is stable if all the eigenvalues are in the left-half plane, that is to say the value of the real part is negative.

Figure 27: Plot of the position of the eigenvalues