[Feb 27] Pendulum

Today's class was mostly a review and an application of what we did last week. It gave me a bigger picture of what is going on with the two parts of the circuit how they are connected.

First, this is the whole set-up:

The big picture is, on the left side, we feed in a voltage proportional to torque and the voltage coming out of the left side is proportional to the negative of speed. Then this voltage proportional to the speed comes in to the right side, gets integrated and comes out as a voltage proportional to the negative of position. Then this voltage was fed back into the left side, which generates a proportional torque.

This process can be described in the following diagram:

If there is no integrator, in other words, if torque is proportional to speed, then we have a damper. With the integrator, ideally, we should see a relationship between speed and position of a virtual spring as following:

The pendulum will slow down and eventually stop.

However, since we have the integrator, which eventually makes the torque proportional to the negative of position, the system will not slow down and stop.

Then Oscar demonstrated the analogy with the spring. When spring operates a force, it does three things: measure the displacement, compute how much force it should exert, and exert it. In the set-up, different parts do the same three things (as shown in the first graph). As the motor spins, voltage is proportional to negative of speed, which is "measuring" the speed; the integrator on the right is "computing" the voltage; then the part where the output of the right part is inserted as the input of the left side is "exerting" the voltage back into the system.

Then we move on to talk about the case where in the left part of the circuit, the combined resistors don't exactly cancel out the effect of the inner resistance of the motor. The voltage of the motor is -Vemf - Rm * I (because of the virtual ground, it's all negative). Since V is proportional to speed, I is proportional to torque, the output of the system on the left is voltage proportional to -w + αT. The output of the part on the right will then be voltage proportional to -θ + α ∫ T dt, which is proportional to torque. This is unstable because as long as you have a little torque, it is going to generate more torque in the same direction through the integrator, which will generate more torque through the cycle.


Here is the abstraction:

Then we started building the circuit. 


I asked Oscar the function of the two potentiometers in the system. The one on the left side is to determine whether the system is stable or not. When we reach extremes (please refer to the first graph), the system is going to be either extremely unstable or stable. When the potentiometer is in position (1), it is in a very stable stage (the motor will stop spinning); when the potentiometer is in position (2) in the graph, it is in a very unstable stage (the motor will spin in one direction very fast);. The analogy is as following:

In (1), it's as if a ball in a sink: very stable. In (2), however, it's as if a ball on top of a hill: a force in either side is going to bring the ball into motion in only one direction. It is only in the middle can the system truly function as a "spring" as the ball can oscillate back and forth.


The function of the potentiometer on the right is to control the frequency of the spinning of the motor. As we have deducted last week, the integration generates Vout = Vin + 1/RC *  ∫ Vin dt. The potentiometor controls the R in this equation. If R is very big,  Vout becomes very small: a small torque will be generated, which means low frequency; if R is very small,  Vout becomes very big: a big torque will be generated, which means high frequency.


Here is our circuit:

Here is our graph on the oscilloscope:

The blue curve indicates the speed; its probe is connected to the left side of the circuit in the first graph in this blog. The yellow curve indicates position; its probe is connected to the right side of the circuit.

This is another version, setting speed and position as the x and y axis:

The reason that it is not like the hand-drew graph shown earlier in this blog is because it is not a perfect circuit.

This is our spinning pendulum!