Example 4: Root locus analysis

Root locus plot

For this we consider the same example as shown in figure Figure 90, the MassSpringDamper system.

For a root locus the roots are those of the characteristic equation, in other words the eigenvalues of the A matrix. The B, C and D matrices are of no interest in this analysis. It follows that it is not necessary to have any control or observer variables. If you have any defined, it will not cause any problems, they will simply be ignored.

The analysis is based on a batch run with only one varying parameter. For each batch run a linearization is done. To be meaningful this must be done at an equilibrium point.

Procedure

1. Leave the parameters at their default values.

2. Set up a batch run with the damper rating:

Switch to Parameter mode and then select Configure > Study parameters, or use Ctrl + B to open the Study Manager.

Select the spring damper component to open the contextual Parameters window and drag and drop the damper rating to the into the Input parameters list of the Parameters tab.

Go to the Studies tab and set up the batch parameters as follows: varying from 0 with a step size of 250 N/(m/s) with 40 points above.

3. Ensure the system is in equilibrium as before.

4. Start the batch run to generate the .jac files for each value of damper rating. Set Simulation type to Batch in the Run Parameters dialog box.

5. When the run process is finished, select the Root Locus button in the Linear analysis toolbar to access the root locus data.

You obtain a dialog box shown in figure Figure 97 which allows you to select the linearization time (and, if necessary, the custom scale). Figure 97: Root Locus

6. Click on Plot to obtain the graph shown in figure Figure 98.

Figure 98: Root locus plot

Note there are 3 types of symbols:

: first value for each eigenvalue (corresponding to 0 damping here),

: last value for each eigenvalue (corresponding to 10000 N/(m/s)),

: all other eigenvalues.

The grid contains 2 types of curve:

circles display the iso-frequency (constant frequency)

lines correspond to the iso-damping rate (constant damping)

Normally, as in this example, all eigenvalues have zero or negative real parts. It is possible to have eigenvalues with positive real parts but this indicates a serious stability problem. To remind you of the interpretation of a root locus plot, look at the custom root locus plot below. Note that on the real axis there are no oscillations. The frequency of oscillation increases as the imaginary part increases. There is no damping for eigenvalues on the imaginary axis. Damping increases as the eigenvalue moves to the left of the imaginary axis. The amplitude grows to the right of the imaginary axis. If your plot contains extreme eigenvalues, you can use the Custom scale option to limit the range of frequencies displayed. To illustrate this, we will create another root locus plot, and use the custom scale to zoom in on a part of it.

7. Select the Root Locus button in the Linear analysis toolbar to access the root locus data.

8. Select the Custom scale check box.

The Maximum frequency [Hz] field is enabled.

9. Enter the value 5 and click Plot.

A new graph is generated, with the frequencies limited to exclude the extreme value: Figure 99: Custom root locus plot

Note that typical answers depending on the eigenvalue location in a Root Locus plot are the following: Figure 100: Typical answers in a Root Locus plot