Simulation and analysis of the results

	Run Parameters	Final time = 20 seconds Print interval = 0.0001 seconds Discontinuities printout = active

Simulate the model, it will be solved in a few seconds. When you plot the displacement of the left-hand mass you will see that there are two instants when it oscillates. First, around 1 second with a large amplitude and then after 12 seconds with a lower amplitude. Details are given in the following plots.

Figure 9: Displacement of the left-hand mass

We will now describe the type of information that can be retrieved from each pane of the Performance Analyzer.

In Simulation mode, launch the Performance Analyzer.

Figure 10: Run statistics pane of the Performance Analyzer

The model is quick to solve, around 3 seconds of CPU time on a 2.7 GHz computer using GCC compiler.

Looking at the data on the left-hand side and at plots on the right-hand side of the pane, we see that:

• After 12 seconds, CPU time is slowing down, this can be seen because the slope of the curve is increasing.

• Around the same time the current integration step of the solver (red curve) is decreasing.

• The minimal integration time step steps down at the same moment (blue curve).

• There is a high number of discontinuities.

Conclusion:

• Something is happening in the system after around 12 seconds.

With regard to the discontinuities, it is useful to plot the time evolution of the number of discontinuities. To do this, drag and drop the line current total number of discontinuities processed onto the sketch. You can also see that this number increases continuously just after 12 seconds.

Figure 11: Discontinuities evolution

You need to make sure that the Discontinuities printout option is activated in the solver options.

Discontinuities printout option

Figure 12: Discontinuities printout option

In order to know where these discontinuities are coming from, take a closer look at the Run details window available from the Run monitor window. Make sure the Discontinuities button is toggled on and look for discontinuities happening after 12 seconds. Finally select one of these lines to highlight the triggering component on the sketch.

Figure 13: Discontinuities details

The discontinuities happen in the contact model. Every time the contact is lost or reestablished it will trigger a discontinuity for the solver. Too many discontinuities happening in a short time will slow down the solver.

Here the mass on the right-hand side is rebounding against the left-hand mass repeatedly and very rapidly.

In order to further understand the numerical side of things, move to the second pane named State contributions.

Figure 14: State contributions pane of the Performance Analyzer

This pane allows you to see which state variable of the model is controlling the integration time step of the solver.

Looking at the Cumulative contribution plot of the figure above you can see that, once again, after 12 seconds, the contribution of one variable increases notably. The details of this variable are given in the table on the left-hand side and if you double-click the line, the component is highlighted on the sketch.

This tells you that its the velocity of the mass on the right-hand side which is forcing the smaller integration time step that you were seeing in the previous pane.

With the large number of discontinuities and the influence of the mass velocity you should now be convinced that this mass, in combination with its end-stop, is the reason why the solver is slowing down.

Plot the displacement, velocity and acceleration of that mass on separate y axes.

Figure 15: Displacement velocity and acceleration of one mass

It is clear that some serious resonance is occurring after 12 seconds. Displacement, velocity and acceleration are oscillating with very large amplitude and the damping is weak.

Finally, move to the Frequencies pane. The objective here will be to associate the resonance with natural frequencies of the system. Perform a simulation run with automatic linearization thanks to the dedicated button .

Figure 16: Frequencies pane of the Performance Analyzer

The table shows the results of the automatic linearization of the model performed at the start time of the simulation equal to 0 seconds in our example.

This model is not linear (even though the stiffnesses of the springs is constant, contact can be lost), hence the results of this linearization are not valid for any time of the simulation. But in our case it will give a good insight into the system because at time 0 the gap is null. If further analysis is needed you should manually define multiple linearization times using the Simcenter Amesim Linear Analysis tool.

The table shows two oscillating modes, one at 96 Hz (with a reasonable 2.5 % damping ratio) and one at 1233 Hz (with a very low damping ratio at 0.0 %). The 1233 Hz mode is automatically colored with an yellow tag tha indicates its low damping value .

The mass on the right-hand side is excited with a variable frequency velocity excitation which varies between 1 Hz at the start of the simulation and 2001 Hz at the end. The two modes that we have identified above will then be excited. In the figure below, we plot the accelerations of the left-hand mass in the upper plot and of the right-hand side mass in the lower plot. The plots are a function of the excitation frequency.

Figure 17: Mass accelerations as a function of excitation frequency

Top: left-hand mass Bottom: right-hand mass

You can see that when the two oscillating modes are crossed, the acceleration amplitudes increase.

The right-hand side mass (bottom plot) is mainly excited at 1233 Hz and it also makes the other mass oscillate. The left-hand side mass (top plot) is excited on its own at 96 Hz and also sees the 1233 Hz mode. If you zoom in on the bottom plot at lower frequencies you will also see the impact of the 96 Hz mode but the amplitude is three magnitudes lower than for the 1233 Hz mode.

Figure 18: Zoom in on left-hand mass acceleration function of excitation frequency

The 1233 Hz oscillation is undamped and in this example, combined with the high number of discontinuities, it explains why the solver is slowing down.