Non-linearities

As soon as some non-linearities are introduced, the A matrix and its eigenvalues will not be constant. To experiment with this:

Procedure

1. Change the system to that shown in figure Figure 65 and use Premier submodel.

Note Simcenter Amesim Run users can get the modified system through Help > Get demo from the AMERunTutorials directory. The file is called SimpleMassSpringNL.ame.

The 2-port mass icon has changed. The new mass has the submodel MECMAS21 and differs from MAS002 in that it contains an option for friction. With the default parameter values there is no friction and hence, if you do a series of linearizations with the defaults you will get exactly the same results as with MAS002. Figure 65: New submodel MECMAS21

2. To make life more interesting, change the Coulomb friction force and stiction force levels to 50 N,

3. Increase the amplitude in the sine wave source to 1000, and change the frequency to 0.3 Hz and start a new run.

The new parameters give a force between –1000 N and +1000 N and the mass will be alternating between being in motion and being stationary held in friction.

4. Do a run and plot the mass displacement to get the graph below:

Figure 66: Mass displacement

At some linearizations you get precisely the same eigenvalues as previous with MAS002. These correspond to the system being in motion. However, when it is held in stiction, the eigenvalues are very different as in figure Figure 67. Figure 67: Eigenvalues with a stiction

Note You must click the gear icon and deselect the Hide zero frequency modes checkbox to display the values:

Technically this is a non-linear system or more accurately a non-linear model. We could also describe it as a model with precisely two distinct linear modes. The two modes naturally are separated by a discontinuity.

5. Try changing the viscous friction and/or windage in MECMAS21 (currently both zero) to non-zero values.

Results

You will now have the linear (constant A matrix and eigenvalues) results when in stiction and you will also have varying A matrix and varying eigenvalues when in motion.

So far we have linearized at times at which the system is not necessarily in equilibrium and looked at the A matrix and its eigenvalues at these times. This can be useful in analyzing why a simulation run is fast or slow. One obvious factor is the size of the system. Big systems tend to have a large number of state variables, which can make the system very slow. However, there are other factors involved and the eigenvalues give information on these. In general eigenvalues are complex numbers. Roughly speaking the solution is made of components, one component corresponding to each eigenvalue. The real part gives an indication of the damping of the component of the solution. The imaginary part, if it is non-zero, indicates oscillations and gives an estimate of the frequency.

Both the real and imaginary parts have units 1/s. Normally the real part is negative or zero. If it is negative then

is a time constant in seconds. Very small time constants correspond to very large negative eigenvalues and very rapid transient behavior. This in turn leads to very small integration steps. If this rapidly dies out, there are no problems and the integration step increases.

We can illustrate this with the first order lag system shown below.

Figure 68: First order lag system

Note that this system contains a run stats component and its submodel RSTAT. This component is in the Simulation category and enables you to monitor information about the integration processed such as steps size, method used, number of discontinuities processed, etc.

Set the following parameter:

Submodel	Title	Value	Unit
LAG1	time constant [s]	1.0e-8	s

The run will be accomplished very rapidly so that the step size starts at about

10 -2.5 (about 0.003) and rises to 10+2.5 (about 300). This is shown in figure Figure 65.

Figure 69: Step size evolution

However, if some input(s) to the system never allows this component of the solution to die out, then you have a slow simulation. To summarize:

An eigenvalue with a negative real part, which has a very large value, corresponds to a very small time constant. When the corresponding component of the solution is excited, the run will be slow. If this component is continually excited, the whole simulation will be slow.

If the imaginary part of an eigenvalue is non-zero, the corresponding component of the solution will be oscillatory. When this component of the solution is excited, the integrator step size is unlikely to get any bigger than a tenth of the corresponding period (or time for a complete oscillation).

Perhaps the worst situation is a very high frequency component which is undamped or very lightly damped.

	This can be illustrated with a very simple system comprising a second order lag with a natural frequency of 1000 Hz and a damping ratio of 1.0e-3. The corresponding period is 10e-3 seconds and so the step size cannot exceed 10e-4. This is seen in figure Figure 70. However, if the damping ratio is 1, the component is damped out and hence the step size rises.

Figure 70: Step size