Physics Quiz 3

[John]

Would omega_w increase until the point on the motor curve where
tau_k/r = mu_k * W * sin(theta)
So the wheel would start spinning faster along its axis, but the rod would rotate at the same speed as before?

[Ether]

Originally Posted by Ninja_Bait

I think I have been told that speed and torque on a motor aren't necessarily linked in voltage control... but I don't understand the science behind that.

Say you have a small DC brush motor mounted on a workbench and you connect it to a fixed voltage. The motor will spin up to a certain speed and then stay there. (That's called the "free speed" or the "no load speed").

The reason the motor reaches a certain speed and then stops increasing is because as the motor speeds up it generates more and more "back emf". This back emf is a voltage that opposes the applied voltage. At free speed, the back emf is almost equal to the applied voltage. The small difference in voltage creates just enough current in the motor to keep it spinning at the free speed (some current is required at free speed to supply the torque necessary to balance the friction and other losses in the motor). This current is called the "free current" or the "no load current".


DON'T DO THE FOLLOWING, YOU MIGHT HURT YOUR HAND, IT IS A THOUGHT EXPERIMENT: Let's say Superman walks in and holds the motor shaft and gently squeezes it. What happens? The motor slows down. It slows down because it has a load on it. It also draws more current in order to generate the torque necessary to balance that load. If Superman squeezes harder, the motor slows down more. If he squeezes hard enough, the motor will stop turning. This is called "stall". The torque at stall is called the "stall torque". The current is called the "stall current".

The important take-away from all this is that, for a fixed applied voltage, the torque output and the speed of the motor are inversely related. Put more load on the motor and it slows down. Put less load on the motor and it speeds up.

Now consider what happens if you have a fixed load on the motor. If you increase the voltage applied to the motor, the motor will speed up and reach a new equilibrium speed. If you decrease the voltage applied to the motor, the motor will slow down and reach a new equilibrium speed. At the equilibrium speed, the applied voltage minus the back emf provides just enough voltage to push the current through the motor required to balance the applied load.

If you want to run some "virtual" experiments to explore these concepts, I have written a small motor calculator which is available for download here. Grab the most recent revision, at the bottom. It has all the FRC 2011 motors in it. You can change voltage, load, speed and see how the motor reacts.

[Ether]

Originally Posted by John

Would omega_w increase until the point on the motor curve where tau_k/r = mu_k * W * sin(theta)

Yes.

So the wheel would start spinning faster along its axis, but the rod would rotate at the same speed as before?

It gets kind of messy. mu_k is not really constant in real life.

Typically, omega_p would increase as omega_w increases.

Under certain conditions, I think the system tries to minimize the relative speed between the floor and the wheel. I believe this occurs when omega_p = (r/L)*cosθ*omega_w

[James Critchley]

It gets kind of messy. mu_k is not really constant in real life.

Typically, omega_p would increase as omega_w increases.

Actually the model contains the answer to this and it is a simple matter of kinematics.

All of the analyses performed are in equilibrium and part of that is the wheel slipping on the floor. In order for the force of friction to be pointed in the radial direction, the velocity of the point on the bottom of the wheel must be in that direction (for this model). Once you accelerate the wheel you will introduce a wheel slip velocity relative to the ground which is not aligned and thus results in an angular acceleration changing omega_p. You will reach a new equilibrium point if you specify a new constant wheel speed.

Quiz 4: Given omega_w determine the steady state omega_p.



Okay... got you between edits


Quiz 5: Qualitatively, what happens to the torque requirements, friction force, and reaction force at the pivot, if the motor and wheel are instead a thin disk of mass m (i.e. with rotational inertia) AND the pivot connection only resists moments about the radial axis (e.g. is a Hooke's joint)? The equations get tough, so we need only discuss the character and origin of the effects.

Answer the following:
Does the answer to quiz 4 still apply? Why or why not?
Is the applied torque requirement constant? Why or why not?
Is the friction force constant magnitude and radially oriented? Why or why not?
Is the reaction force at the pivot different in any meaningful way? How so?

[BRAVESaj25bd8]

Originally Posted by Ether

Yes.

But according to the Coulomb friction model, the total friction force must be mu_k*W.

So that means tau_k must equal mu_k*W*r*sinθ.

So, what happens if the voltage to the motor is increased?

This doesn't sound right to me. The torque applied by the motor should not be changing as theta changes. I think the problem is that in order to find the friction equal to tau_k/(r*sin(theta)), you must assume the wheel is gripping. This does make sense because it is able to rotate and drive around the pivot but is inconsistent. I feel like one of the boundary conditions is actually overconstraining the model.

[Ether]

Originally Posted by BRAVESaj25bd8

This doesn't sound right to me. The torque applied by the motor should not be changing as theta changes

tau_k is the motor torque necessary to be in equilibrium at a given angle theta, given the assumptions in the model (mu_k constant, frictionless pivot, frictionless wheel bearing, no wind resistance, no rolling friction, no friction due to rotation of the wheel about the Z axis).

What the equation is saying is that if you reduce the angle theta, it takes less tau_k to be in equilibrium. If you increase theta, it takes more.

Sustaining equilibrium when theta is very small takes very little torque. In fact, when theta is zero it takes zero torque to sustain equilibrium since the pivot is frictionless and we are ignoring wind resistance, rolling friction, friction in the wheel bearings, and friction due to rotation of the wheel about the Z-axis.

So what happens if you start with the system in equilibrium, and you then increase tau_k to tau_k' ?

[BRAVESaj25bd8]

After thinking about it more, the error I made was getting wrapped up in angular motion, thinking that theta was the wheel's angular position about the pivot as the system rotates. What it really represents, though is a constant in this problem, the angle between the wheel's axis of rotation and the pivot arm.

I think that if you increase the torque, nothing will change because the wheel is already slipping. The friction cannot be higher than W*mu so no more torque is able to be useful.

[Ether]

Originally Posted by BRAVESaj25bd8

I think that if you increase the torque, nothing will change because the wheel is already slipping. The friction cannot be higher than W*mu so no more torque is able to be useful

If you increase tau_k, the system will no longer be in equilibrium. That means something must change.

[Ninja_Bait]

Originally Posted by Ether

If you increase tau_k, the system will no longer be in equilibrium. That means something must change.

Can normal force increase to balance the increase in tau_k? I don't think that actually makes sense but it seems to be the only variable that can change in this case.

[Ether]

Originally Posted by Ninja_Bait

Can normal force increase to balance the increase in tau_k?

No.

I don't think that actually makes sense but it seems to be the only variable that can change in this case.

If the system is in equilibrium, and then you increase the motor torque that's driving the wheel, what happens?

Think torque=momentOfInertia*angularAcceleration. Then follow the causes and effects as they ripple through the system.

[James Critchley]

Ninja_Bait: Ether has already answered this question in an earlier post... I think the issue is the notion of supplying a larger torque indefinitely. Perhaps it will help to say it another way?

The system will accelerate to a speed where you are no longer able to supply torque greater than tau_k. Physical systems are power limited and power = torque * angular rate. The system will accelerate to the limits of your power supply or motor torque vs speed curve (which ever comes first). At that point the torque supplied by the motor will again be tau_k and you will have constant speeds.

If you could somehow supply a constant torque independent of speed, the system as modeled would properly accelerate forever. Other effects would eventually dominate and the model would become invalid so this wouldn't actually happen, but you would still want to patent that motor and power supply!

[Ninja_Bait]

Ah, I see. Thank you.