Normal modes

A normal mode of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed-phase relationship. The free motion described by the normal modes takes place at fixed frequencies. These fixed frequencies of the normal modes of a system are known as its natural frequencies or resonant frequencies.

In linear systems each mode is entirely independent of all other modes. In general all modes have different frequencies (with lower modes having lower frequencies) and different mode shapes.

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

The top contributing states table lists the top 5 state variables of the model that govern the currently-selected eigenvalue and gives information about the mode shapes.

Eigenvalue properties

The table lists the eigenvalues of your system.

Figure 30: Eigenvalues

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 31: 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 with 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 are 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].

Contribution of state variables

The table lists the components that are contributing the most to each eigenfrequency of the system at a linearization time.

Figure 32: Top contributing states table

For each state variable, information about the variable title and the associated submodel is retrieved. Double-click on a line to highlight the component on the sketch.

The content of the table can be customized in order to display the properties that are useful in identifying the contributing states. The configuration button provides the list of attributes that can be included in the table. For instance if one submodel contains state variables with the same title, it is helpful to add the variable names since they are unique to each submodel.

Figure 33: Top contributing states properties list

The contributions of the state variables are determined by the modal participation factors that are calculated from the eigenvectors of the linear transformation of the A matrix of the state space model and that measure the interaction between the eigenvalues and the state variables. For more information, see the appendix on linear algebra.

You can set a contribution value for which only state variables with a participation factor greater than or equal to this value are displayed in the table. If the option is unchecked, the value is ignored and all state variables are displayed regardless of their contribution.