Systems with dissipative elements and inputs
Up to this point we have only considered lossless systems, or systems with no energy dissipating elements like (mechanical) friction or (electrical) resistance. Obviously such systems cannot actually exist, so it is natural to ask why such systems are important to study. The answer is that these systems give important insights into systems that do dissipate energy. Recall that the system had two modes:
- The lower frequency mode has the masses moving in phase with each other at a frequency of 1 rad/sec.
- The higher frequency mode has the masses moving in opposite direction (180° out of phase) at a frequency of 1.73 (√3) rad/sec.
Adding Energy to the System as a Force on the Leftmost Mass
As an example, the demonstration below reprises the two mass problem from the previous page (it might be useful to refamiliarize yourself with the problem), but it also includes light friction under the masses. It is possible to run this simulation with an input force on the leftmost mass. This force can either be sinusoidal, or a periodic pulse train that is on roughly 10% of the time. This force is included on the output graph as well as with an arrow drawn above the mass. There are a list of things to try at the bottom of the page. Go through the list to get a feel for how the system behaves to different inputs. Any energy stored in the system at the frequency of the lower frequency mode will behave as the lower frequency mode. Likewise for the higher frequency mode.
Things to try:
To increase your understanding, try the following things:
- No input (set input to "none", note that this is a zero-input response):
- Set initial conditions to x1(0)=1, x2(0)=1. As you run the simulation you will see that that masses behave much like the lower frequency (in-phase) mode of the two mass problem from the previous page, but that the oscillations decrease in size as the energy dissipates in the friction elements.
- Set initial conditions to x1(0)=1, x2(0)=-1. As you run the simulation you will see that that masses behave much like the higher frequency (out-of-phase) mode the two mass problem.
- No initial conditions (set initial conditions to x1(0)=0, x2(0)=0, note that this is a zero-state response):
- Set the input to "Sine".
- Set frequency to 1 rad/sec. The force will add energy to the system at 1 rad/sec which is the frequency of the in-phase mode of the undamped system. After transients die out, the low frequency mode dominates, and the masses move almost perfectly in phase. Note that the units of the input aren't shown. It is only there so you can see how it affects the output.
- Set frequency to 1.73 rad/sec. The force will add energy to the system at 1.73 rad/sec which is the frequency of the out-of-phase mode of the undamped system. After transients die out, the high frequency mode dominates, and the masses move almost perfectly out of phase with each other.
- Set frequency to 1.35 rad/sec. The force will add energy to the system at 1.35 rad/sec which is between the two modes. Note that after transients die out that the sinusoidal steady state doesn't closely resemble either mode and is smaller than in the two previous cases, and that after transients die out, the system oscillates at 1.35 rad/sec. It is a hallmark of linear systems that if you drive them at a single frequency, that only that frequency will be present in the system output after startup transients die out.
- Set the input to "Pulse".
- Set the frequency to 1 rad/sec. Note that after transient dies out, the lower frequency mode dominates. If you have studied Fourier theory, this is not surprising, because this is the fundamental frequency of the input wave.
- Set the input to "Pulse" and frequency to 1.73 rad/sec. You should be able to predict what happens.
- The behavior of the system tells us that if we have a lightly damped system and we excite it near the frequency of the modes of the system, that those modes will develop on the output of the system. For this system if we add energy near that of the low frequency mode, the system oscillates in that mode. If we add energy near that of the high frequency mode, the system oscillates in that mode.
- Include both initial conditions and input:
- Try including both initial conditions and an input. The effect of the initial conditions (the zero-input part of the response) dies out early on, leaving just the output due to the input (the zero-state response).
The most important insight to be seen here is that the system responds to the sinusoidal input with a sinusoidal output at the same frequency as the input (after starting transients die out). If we use a frequency near on of the modes, the system output will show only that mode (i.e., ω=1 (the frequency of the in-phase mode) yields the two masses moving in synchrony, and ω=1.73 (the frequency of the out-of-phase mode) makes the two masses oscillate out of phase with each other).
Adding Energy Another Way
Here we have a similar system but we introduce energy into the system by moving the end of the rightmost spring. In this case the graph shows the input as well as the output on the same scale, and you can see that the output actually gets larger thant the input.Try the same things as you did previously, and you'll see similar behavior. The take home lesson is that if you introduce energy to the system at a frequency near one of the mode frequencies, you'll get behavior that looks like that mode. To introduce energy to the system, the previous example uses a force on the leftmost mass; the example below moves the end of the rightmost spring.