Free body diagram for an inverted pendulum in the rolling sphere

[GottfriedLenz]

Homework Statement

To get equations of motion of the system, which consists of the pendulum in the spherical enclosure

Relevant Equations

NA

So, to obtain the motion equations I initially plotted the free-body diagram (see picture). Then I’ve tried to get equations, but I’m not sure, do I have done it rightl. I will be gratefull if someone could help me.

 

[TSny]

Welcome to PF.

The problem statement is not clear. Were you given the problem in written form? If so, please quote the entire problem statement exactly as given to you.

Also, define the symbols that you use in the your equations and diagram.

 

[GottfriedLenz]

Thank you! Unfortunately, the task is formulated not in English, so I can’t simply quote it. It's kind of an inverted pendulum on a cart problem. But only instead of a cart, the pendulum is attached to the motor shaft that is placed in the centre of the spherical enclosure. Consequently, by changing the inclination of the pendulum one can make the sphere move and vice versa. I shall determine the position of the sphere in a two-dimensional plane in dependence on the inclination of the pendulum.

In the diagram, I’ve used the next notations:
CG - centre of gravity;
theta - inclination of the pendulum;
l_x - distance between the centre of the sphere and the centre of gravity;
N_pend - reaction force of the pendulum and N_sph is corresponding reaction of the sphere
P - force of gravity.

And the symbols in equations are:
m_pend - mass of the pendulum;
M_sph - mass of the spherical enclosure;
x - displacement of the spherical body;
r - radius of the sphere;
b - friction coefficient.

 

[TSny]

Thanks for providing more information. But I'm still not clear on the setup.

So, there is a motor located inside the sphere?

Is the pendulum rod locked to the motor's shaft so that the pendulum always rotates at the same angular speed as the motor's shaft?

Does the motor's shaft rotate at a constant speed relative to the body of the motor? Or is it more complicated?

Does the body of the motor rotate with the sphere?

You said that $P$ represents the force of gravity. In your diagram you have $P_{sp}$ and $P_{pend}$. But $P_{pend}$ points upward. So, it's not a force of gravity. Rather, it looks like $N_{pend}$ and $P_{pend}$ might be the horizontal and vertical components of the force that the motor shaft exerts on the pendulum rod. $N_{sp}$ and $P_{sp}$ are the corresponding reaction forces that the pendulum exerts on the motor shaft. Is this correct?

What does the force $F$ represent that acts at the point of contact between the sphere and the surface?

The friction force at the point of contact between the sphere and the surface has the form $b \dot x$, which seems a little strange. What type of friction is this?

Does the sphere roll without slipping on the surface?

 

[GottfriedLenz]

Thank you and sorry for the delay in responding.

So, there is a motor located inside the sphere?

That’s right.

Is the pendulum rod locked to the motor's shaft so that the pendulum always rotates at the same angular speed as the motor's shaft?

Yes.

Does the motor's shaft rotate at a constant speed relative to the body of the motor? Or is it more complicated?

I’m not sure that I’ve understood your question correctly. Taking back to the analogy with an inverted pendulum on a cart, the motor is needed to hold the pendulum in the vertical position. The equations of motions, which I tried to obtain, are aimed at building a controller.

Does the body of the motor rotate with the sphere?

Yes.

You said that P represents the force of gravity. In your diagram you have Psp and Ppend. But Ppend points upward. So, it's not a force of gravity. Rather, it looks like Npend and Ppend might be the horizontal and vertical components of the force that the motor shaft exerts on the pendulum rod. Nsp and Psp are the corresponding reaction forces that the pendulum exerts on the motor shaft. Is this correct?

Yes, you are right.

What does the force F represent that acts at the point of contact between the sphere and the surface?

I have to say that it is my total misunderstanding of acting forces. I’ve meant this force makes the sphere move when the pendulum biasing. But now I see that at list direction is wrong.

The friction force at the point of contact between the sphere and the surface has the form bx˙, which seems a little strange. What type of friction is this?

One more time, my fault. Here should be the force of rolling friction. It’s really not bx.

Does the sphere roll without slipping on the surface?

Yes.

 

[haruspex]

the motor is needed to hold the pendulum in the vertical position

So the diagram represents a small perturbation from the vertical, but does the motor accelerate to correct it, or merely to hold it at the perturbed position? If the former, it is a control theory problem. You would need to specify the motor's logic, no?

 

[GottfriedLenz]

The motor is supposed to correct the perturbation and make the pendulum vertical. However, I’m a little bit confused with your words:

If the former, it is a control theory problem.

Did you mean that in the latter it is not a control theory problem?

 

[thegoldbering]

Thank you! Unfortunately, the task is formulated not in English, so I can’t simply quote it. It's kind of an inverted pendulum on a cart problem. But only instead of a cart, the pendulum is attached to the motor shaft that is placed in the centre of the spherical enclosure. Consequently, by changing the inclination of the pendulum one can make the sphere move and vice versa. I shall determine the position of the sphere in a two-dimensional plane in dependence on the inclination of the pendulum.

In the diagram, I’ve used the next notations:
CG - centre of gravity;
theta - inclination of the pendulum;
l_x - distance between the centre of the sphere and the centre of gravity;
N_pend - reaction force of the pendulum and N_sph is corresponding reaction of the sphere
P - force of gravity.

And the symbols in equations are:
m_pend - mass of the pendulum;
M_sph - mass of the spherical enclosure;
x - displacement of the spherical body;
r - radius of the sphere;
b - friction coefficient.

What is the original language the question is originally formulated? Is it Spanish or German?

 

[haruspex]

The motor is supposed to correct the perturbation and make the pendulum vertical. However, I’m a little bit confused with your words:Did you mean that in the latter it is not a control theory problem?

If it were merely required to hold it in the perturbed position then it would be fairly straightforward, and would not involve control theory.
Since the aim is to restore it to the vertical, there are options in how to do that. Essentially, you need to pick a gain factor.
I'm no expert on the topic, but the ideal gain would be something analogous to critical damping of a forced oscillation. I.e., you want to get there as fast as possible without overshooting.

 

[GottfriedLenz]

What is the original language the question is originally formulated? Is it Spanish or German?

The original language is Russian.

Since the aim is to restore it to the vertical, there are options in how to do that. Essentially, you need to pick a gain factor.

Thank you!