angular motion

[steveB]

Let's do another case. This time the pulley is already rotating with angular velocity, ω, of 2π rad/s. There is no angular acceleration which means torque is zero and if we further assume that there is no friction involved then the rocket isn't really doing any work; it's basically in off mode.

The setup looks like this. But this time as I mentioned above the pulley was already rotating at a constant angular velocity of 2π rad/s which translates to linear velocity of 2π m/s when the rope which has 2 kg mass suspended to it and resting on ground gets somehow attached to it. The vertical height between the pulley and ground could be approximated to be 20 meters.

Our purpose is to lift the mass to some height above the ground and we also want to keep things as calm as they could be; I mean no jerky motion.

We can assume that the rocket has some kind of a sensor which regulates the angular velocity at 2π rad/s. It turns the rocket on if the velocity is under 2π rad/s and off if it goes above. Further, we can also assume that the sensor makes rocket to exert different amount of force as required.

As soon as the rope gets attached to the pulley, the angular velocity of pulley will go down for an infinitesimal amount of time and during that time both pulley and mass would have same speed.

Then, instantly, the rocket turns on and starts exerting force. The rocket aims to get the system to 2π rad/s within 1 second. Therefore the system needs the acceleration of (2π-5. 9924)/1s=0.291 rad/s/s for one second. This time we need to know rocket force and I believe that the formula derived by you can be used, α=( R Ft - R m g )/ ( I + m R^2). After rearranging Ft={α/(R(I+mR^2))}+mg={0.291/(20+2)}+2(10)=20.013 N. Once the system has reached 2π rad/s angular velocity then the rocket just has to exert 20 N force to counterbalance the clockwise torque to 2 kg mass. Do you agree with everything I have said?

Very interesting problem you stated here. This is actually a somewhat complicated control system problem.

I think I agree with just about everything you said. I'll need to read through it again with a fresh mind later to be sure. There are a few nit-picky things I can say.

First, I would say that your statement "... As soon as the rope gets attached to the pulley, the angular velocity of pulley will go down for an infinitesimal amount of time and during that time both pulley and mass would have same speed ..." is not quite correct. I might say that "once the pulley is connected to the mass by the rope, the pulley will decelerate for a short amount of time and during this time, the mass will accelerate. Eventually, both speeds will converge and stabilize to one value".

Second, the value of the acceleration you calculated to get to the correct speed in one second, while perhaps correct in value (note that I didn't check to see if you got it right), represents a very simplified answer that could be considered an average target value for the control system. A real control system would not be able to achieve this exact acceleration profile (basically a step profile you specified).

A real problem like this would need to consider the stretch, spring force and damping of the rope, the dynamic response of the rocket force and the details of the type of control system used and the gain values used in the feedback design. Design would have to consider stability, robustness and meeting the specifications as you have provided them.

Like I said, this is a very interesting problem, and not trivial. Hence, your answer, though basically correct ( I think), and demonstrative of good thinking on your part, should be viewed as an approximate first pass answer to a problem that is not trivial and one that might have various types of solutions depending on the system parameters and final chosen control system design.

 

[steveB]

This question came to my mind while writing about the case when the pulley rotating at constant angular velocity gets connected to the rope. The pulley possessed angular moment and momentum is always conserved; linear and angular momentums work independent of each. When the pulley gets joined/connected to rope and mass starts get lifted, some of angular momentum gets transformed into linear momentum. I don't see how angular momentum is conserved in this case.

Let's try a simple case. Suppose a bicycle is being driven at a constant speed which means constant angular velocity and hence angular momentum. Once the brakes are applied, where does angular momentum disappear? Thank you.

For the simple case remember that application of breaks is a dissipative process. In dissipative processes such as friction, which generates heat, we are simplifying the problem by ignoring billions and billions of degrees of freedom in the problem. We do this to make the problem tractable, but in the process we will sometimes mask some of the fundamental laws of nature such as conservation laws.

Also, consider that force can be defined as time rate of change of linear momentum and torque can be defines as time rate of change of angular momentum. Hence, in any situation where an external force or torque is acting, momentum will not be conserved in the system under analysis. If you were to use a Lagrangian approach to studying some of these problems, some of this might be more obvious, but even with simpler methods it is clear that we often isolate our system from the universe and allow momentums, forces and energies to come and go through the walls of our laboratory and they interact with the outside universe. Hence, if the entire universe were considered, then energy and momentum would indeed be conserved, but the problem will then usually be completely intractable.

Another case occurs in thermodynamic analysis where the aggregate average effects are calculated and the many intractable degrees of freedom are ignored. We end up with a law of entropy which says that order and information is gradually lost and physics is not time reversible in these analyses. However, this is a false prediction caused by the ignoring of degrees of freedom and in fundamental laws of physics we find that there is no entropy change, information is never truly lost (in principle) and physics is time reversible.

So, in your original question "When the pulley gets joined/connected to rope and mass starts get lifted, some of angular momentum gets transformed into linear momentum. I don't see how angular momentum is conserved in this case." we can apply these ideas and conclude that if you find linear and angular momentum not independently conserved, there must be an interaction with the outside universe that will balance the lost momentums. You then need to look for either external forces/torques (e.g. chemical energy creating rocket force) or dissipative processes (e.g. rope damping) that might carry energy and momentum to and fro between our system and the universe at large.

 

[PG1995]

Thanks a lot, Steve.

Very interesting problem you stated here. This is actually a somewhat complicated control system problem.

I believe that once you told me that your work was related to lifting bales using induction motor. Whatever we are discussing might relate to that. I think that we should try to isolate this discussion from control theory so that things remains tractable but I will still go on to ask some simple questions which are not directly related to physics.

Second, the value of the acceleration you calculated to get to the correct speed in one second, while perhaps correct in value (note that I didn't check to see if you got it right), represents a very simplified answer that could be considered an average target value for the control system. A real control system would not be able to achieve this exact acceleration profile (basically a step profile you specified).

Agreed. In reality the acceleration profile would look more like this and it will need some time before the acceleration reaches a constant value.

A real problem like this would need to consider the stretch, spring force and damping of the rope, the dynamic response of the rocket force and the details of the type of control system used and the gain values used in the feedback design. Design would have to consider stability, robustness and meeting the specifications as you have provided them.

Note to self:
Initially I was thinking that stretch, spring force and damping of the rope are all the same but I believe that all the terms are important and convey separate meanings; there could still be some overlap. I would still say that 'stretch' is little redundant but semantically it's common practice in English to put two words of similar meanings next to each other. 'spring force' tells force required for each unit extension of rope and 'damping of the rope' means how the rope behaves once the rope tries to get back to its normal shape and length after stretch - how its oscillations die. Have a **broken link removed**.

This is not part of the note to self. I can't resist myself from asking you this. How do you describe 'the gain values used in the feedback design'? It might be little difficult to explain this in the context of current discussion so you might like to use this op-amp example (source) with negative feedback.

For the simple case remember that application of breaks is a dissipative process. In dissipative processes such as friction, which generates heat, we are simplifying the problem by ignoring billions and billions of degrees of freedom in the problem. We do this to make the problem tractable, but in the process we will sometimes mask some of the fundamental laws of nature such as conservation laws.

I agree with you. So, in short, we opt for something more tractable to mask/avoid many complications. For example during the application of breaks angular momentum should be conserved but proving or resolving such a problem will require extensive labor and time therefore we rather talk in terms of heat dissipation or conservation of energy.

Another case occurs in thermodynamic analysis where the aggregate average effects are calculated and the many intractable degrees of freedom are ignored. We end up with a law of entropy which says that order and information is gradually lost and physics is not time reversible in these analyses. However, this is a false prediction caused by the ignoring of degrees of freedom and in fundamental laws of physics we find that there is no entropy change, information is never truly lost (in principle) and physics is time reversible.

It's interesting what you say about entropy. On one hand entropy seems a very simple concept but on the other hand it's rather a difficult concept. I read about entropy in high school and the definition there was very simple - entropy of a system always increases as time goes by; in other words the system get disordered gradually. I think that we can discuss it some other time.



So far, I have intentionally used a rocket instead of some motor although in one of my previous posts I provided the following two links:
1: http://lancet.mit.edu/motors/colorTS1.jpg (DC motor)
2: http://raise.spd.louisville.edu/ECE252/images/L19-18.gif (induction motor)

Let's talk about DC motor first. We can see that both torque and speed are tied to each other and dependent on each other. In the discussion above the rocket was just applying a certain amount of torque to rotate the pulley at a certain speed and its speed was basically tied to that of the pulley and the pulley's speed was dependent upon the applied torque. Sometimes, we might need a particular speed and a certain amount of torque which aren't really possible to get from the given motor curve. For example, if you need more torque then you need to sacrifice speed but what if you need enough torque without sacrificing the speed too much. How do we resolve such issue? The same goes for induction motor. Thank you.

Best wishes
PG

Helpful links:
1: **broken link removed** (induction motor)
2: http://www.nrcan.gc.ca/energy/products/reference/15433 (induction motor)
3: http://www.science20.com/hammock_physicist/what_entropy-89730 (entropy)

 

[steveB]

I can't resist myself from asking you this. How do you describe 'the gain values used in the feedback design'? It might be little difficult to explain this in the context of current discussion so you might like to use this op-amp example (source) with negative feedback.

As you say, it is difficult to explain gain without the proper context and without an actual feedback topology specified, but still we can do it.

Your example of an OPAMP has only 1 gain value to worry about. It so happens that systems that are well approximated as a simple first order system can use a very high gain and a single gain too. Various opamp circuits are of this type because many opamps are "compensated" with a low frequency pole that forces the system to behave as a simple first order system.

Many systems are not simple first order systems, and typically mechanical systems fall into this class. Hence, the example you presented would most likely need a more sophisticated feedback system. For example, a simple PID feedback could be tried, and this would have 3 different gain values to be optimized. We could talk about more sophisticated feedback systems, but this is sufficient to get the point across, I think.

Let's talk about DC motor first. We can see that both torque and speed are tied to each other and dependent on each other. In the discussion above the rocket was just applying a certain amount of torque to rotate the pulley at a certain speed and its speed was basically tied to that of the pulley and the pulley's speed was dependent upon the applied torque. Sometimes, we might need a particular speed and a certain amount of torque which aren't really possible to get from the given motor curve. For example, if you need more torque then you need to sacrifice speed but what if you need enough torque without sacrificing the speed too much. How do we resolve such issue? The same goes for induction motor. Thank you.

This is the issue of "actuator limits" that every control system designer must deal with. Feedback gains are limited by actuator limits. The more limited to actuator (in this case the torque provider), the lower the gains typically need to be. Lower gains imply lower bandwidth or slower response for the feedback system. Hence, as much as you might like to leave feedback concerns out of your question, it creeps into the discussion automatically. Feedback is a critical aspect of the problem you presented.

 

[steveB]

Just to finish the thought about actuator limits, it should be obvious that the actuator must have some minimum capability to allow the system to meet established specifications. If the actuator is "undersized" then no feedback system will help it work. It would then simply be incapable of providing the required drive to meet specifications.