Modal shapes in energies

To show the influence of this mass, we can do the analysis using energies, and Simcenter Amesim allows us to do this. We click on the Energies tab, and the display changes as follows:

Figure 80: Energy and modal shapes

If we click on Plot now, we get velocity squared, which is not what we want. For kinetic energy we want which means for each observer we must put in a factor corresponding to . In fact when we do energy modal shapes we usually get involved and since we are only interested in relative proportions we can omit it.

We enter the masses in kg under the I,R,C factor column. Note how the last column changes!

Figure 81: The energies corresponding to the two velocities

From an energy point of view, the first velocity contribution is negligible. This is confirmed by selecting Plot:

Figure 82: Energy Plot

Interestingly, for the other mode, the situation is reversed:

Figure 83: Energy Plot

Clearly, the large mass is not responsive to this high frequency mode. The very small mass is.

The question now arises: under what circumstances should you use magnitude analysis, and under what circumstances should you use energy analysis?

1. Often you are only interested in finding out which state variables are involved in a particular eigenvalue/mode of the model. You select this mode because it is interesting or perhaps is causing you trouble. You make all state variables State Observers and then look at the modal shapes for this mode. Often this analysis can be done when not in an equilibrium position. For a big system, the vast majority of state variables will not respond to this mode, and they will have 0.0 in the magnitude (or energy) column. It is only a very small number of states that do respond. Having identified these, the objective is achieved, and no further modal shape analysis is necessary. Here, you classify the observers as Yes/No. The choice between Magnitude and Energy is irrelevant.

2. You want to know why your simulation run is very slow. The answer lies in identifying the states that are associated with the highest modes of the model. Almost certainly the analysis will be done when the system is not in an equilibrium position. This is a special case of 1: the choice is irrelevant.

3. For those observer variables that do respond to the mode, you need to know the relative amounts by which they respond. Perhaps in an effort to eliminate problems caused by this mode, you want to concentrate on the observer that responds most. Normally Energy is a much better choice than Magnitude. This is particularly true when the observers are in different units and are from completely different domains. It is difficult to compare an observer that is a velocity in m/s with another that is a flow rate in L/min. However, as we will see, the Energy choice involves much more work than Magnitude.

Modal shape with mixed domain

The following system involves the mechanical translation and rotation domain. Build the model as follows, using Premier submodel and default values except where indicated in the screenshot.

Figure 84: Mixed domain example

Note

Simcenter Amesim Run users can get the system through Help > Get demo from the Tutorials directory. The file is called ModalShape2.ame.

Select the mass (velocity at port 1) and the rotary load (shaft speed port 2) as state observers, set an LA time and start a run. There are 4 state variables and the eigenvalues are as follows:

Figure 85: Eigenvalues of mixed domain system

Selecting the last eigenvalue and clicking Modal Shapes gives the following dialog box:

Figure 86: Modal shapes for mixed domain system

We must now try to compare the response of the linear velocity with that of the rotary velocity in some meaningful way. Energies provide this way, so we click on Energies.

Figure 87: Energy modal shapes

Note that the conversion of the observer variable to SI units is done for us but it is our responsibility to provide the I,R,C factor. For the linear velocity the appropriate factor is m (the mass). I, R, C factors must use SI units so the mass must be expressed in kg and the moment of inertia in kg.m². For the rotary velocity we have I, the moment of inertia (see the table below).

Figure 88: Factors for energy modal shapes

Looking at the values in the Energy column, we have a meaningful comparison of the responses. Here is the plot:

Figure 89: Plot of energy modal shapes

Note that you have a free choice of the variables you will select as observers. Not all of them can be converted to an energy, for example a Reynolds number of flow in a pipe.

Below is a table of the factors for converting common quantities to energy. In all cases there is a factor of which is omitted.

Observer	Factor	Nomenclature
linear velocity	m	m = mass in kg
rotary velocity	I	I = moment of inertia in kgm 2
hydraulic pressure		V = volume in m 3 B = bulk modulus in Pa
hydraulic flow rate		= density in kg/m 3 L = length in m A = cross section in m 2
spring force		k = spring stiffness in N/m
rotary torque		k = spring stiffness in Nm/rad
current	L	L = self inductance in H
voltage	C	C = capacitance in F
magnetic voltage		µ = magnetic permeability H/m A = cross section in m 2 L = length in m
temperature	mCp	m = mass in kg Cp = specific heat capacity at constant pressure

The ModalShapes menu

In the ModalShapes menu, we have the following items to operate on the chart:

With....	You can....
Start/pause animation	Animate the modal shape through time. If the eigenvalue is complex, you see oscillations and damping. For real eigenvalues you see damping. For a zero eigenvalue the shape is constant and does not evolve with time. For an eigenvalue with a positive real part the shape grows exponentially with time and so a warning message is displayed and no animation is shown.
Start/pause all animations	Animate all the modal shapes if you have several on the same plots.
Stop animation	Stop the animation and reinitialize the display.
Add observer titles	Display the titles of the observer variables, the frequency and the damping values of the selected mode.
Temporal view	Create a temporal curve of the magnitude for each observer.