Torque on a hub as a result from a flexible arm

[aerowenn]

I'm working on a research project as a graduate student in engineering. I'm modeling a set of equations in Simulink and MATLAB derived from a journal entry.

I'm a little rusty on classical mechanics as my specialization is electrical engineering. I want to model the angular torque on a hub as a result from a flexible arm that has mass (and therefore inertia). The hub is a flat (or nearly flat) solid cylinder that also has mass.

The arm is a type of flexible material that can move on its own (piezoelectric to be specific). The flexible arm will have a sinusoidal motion of some kind, this motion will translate into torque on the hub. I would like to go from the motion of the arm to an output of the torque on the hub. It doesn't have to be specific, just a general way of solving or going about such a problem.

I have included a diagram to help demonstrate what I mean. The axis of rotation (though not attached to anything, no friction) is labeled "O". Any help with this would be much appreciated!


http://sphotos-a.xx.fbcdn.net/hphotos-prn1/150695_4622504175131_1167341224_n.jpg

 

[haruspex]

Do you mean a torque arising from the reaction against the arms as they move through some fluid, or from the inertia of the arms themselves (or both)?
If the first, you need some expression for the force per unit length that arises for a given rate of movement of a section of the arm.

 

[aerowenn]

Sorry, I should have specified that. This would be in an ideal vacuum, so only from the inertia of the arms themselves.

 

[haruspex]

That's relatively easy, then.
Consider an element ds of an arm. At time t it will be (in polar relative to hub centre) at some position (r=r(t), θ=θ(t)). The tangential acceleration is $r\ddot{\theta}+2\dot{r}\dot{\theta}$. See http://en.wikipedia.org/wiki/Polar_coordinate_system#Vector_calculus
If the element has mass dm (=ρds if uniform density) then the reactive torque is $-(r\ddot{\theta}+2\dot{r}\dot{\theta})r.dm$

 

[SteamKing]

The arms flap, and the hub spins.

 

[aerowenn]

That's relatively easy, then.
Consider an element ds of an arm. At time t it will be (in polar relative to hub centre) at some position (r=r(t), θ=θ(t)). The tangential acceleration is $r\ddot{\theta}+2\dot{r}\dot{\theta}$. See http://en.wikipedia.org/wiki/Polar_coordinate_system#Vector_calculus
If the element has mass dm (=ρds if uniform density) then the reactive torque is $-(r\ddot{\theta}+2\dot{r}\dot{\theta})r.dm$

I see, this makes sense. Will this account for the flexibility of the arm as well? I suppose I would specify that motion somehow. Also, this gives torque from the arm, so I can translate this directly to where the arm is mounted on the central hub correct?

The hub will have a moment of inertia as well, and the end result I'm trying to reach is its angular motion around the axis "O" (in the center). In the end I would have a torque from the arm on one side of the equation and the torque applied to the hub on the other.

Solving for the motion on the hub would it come out to angular acceleration?

I really appreciate the help, this seems to be a problem that's not easily found by searching or flipping through textbooks.

 

[aerowenn]

While I'm here, let me pose a conceptual question:

If these arms were to have a motion profile in which they move much faster in one direction than the other, translating to a higher acceleration in one direction (but for a shorter time) in one direction, can this result in an overall angular velocity of the hub? It wouldn't be smooth I imagine, but can a rotation be created in this way?

It doesn't seem possible to me, but I believe one of the research sources is indicating this to be the case.

 

[haruspex]

If these arms were to have a motion profile in which they move much faster in one direction than the other, translating to a higher acceleration in one direction (but for a shorter time) in one direction, can this result in an overall angular velocity of the hub? It wouldn't be smooth I imagine, but can a rotation be created in this way?

you can't get that result just by moving faster one way than the other, but you can get it by moving the arms one way while extended and the other while drawn in. It's how cats land on their feet.

 

[aerowenn]

you can't get that result just by moving faster one way than the other, but you can get it by moving the arms one way while extended and the other while drawn in. It's how cats land on their feet.

I didn't think so. I believe I'm misreading or misunderstanding the source document. It discusses a satellite performing planar motions to defeat an external torque. Perhaps disturbance torque or something, the source doesn't really matter.

Neglecting the modes of the flexible arm, the equation looks like this:

$J\ddot{\theta}+ {\sum}({\rho}\ddot{\theta}qin^2 + 2{\rho}\dot{\theta}qin\dot qin + Jn\ddot qin)$


Where J is the inertia of the hub, Jn is the inertia of the arm, and qin is the motion of the arm. The summation (I'm not versed in Latex) is i=1 to k, I think this is for point mass along the arm.

 

[aerowenn]

Sorry, all that is equal to the external torque. I was trying to take this equation equal to zero and produce a rotation. But it didn't make sense.

 

[Ryoko]

The system can't gain angular momentum without an external couple force being applied.

 

[haruspex]

The system can't gain angular momentum without an external couple force being applied.

It doesn't need to gain angular momentum to change angular position.
1. Arms flattened against hub, anticlockwise, say. System stationary.
2. Swing arms outstretched clockwise until flat against hub again. System stationary once more, but hub has rotated anticlockwise.
3. Curl tip of arm back, anticlockwise, but keeping it flat against itself. Slide tip anticlockwise around hub until arm back in original position. System stationary once more.
Hub rotated clockwise during step 3, but not as much as it rotated anticlockwise at step 2.

 

[Ryoko]

My point was that if the arms are moving in opposite directions, there is a couple due to the equal but opposite forces across the body as a result of the symmetry. If the arms are waving clockwise, the central body must turn counter-clockwise to keep the total angular momentum at zero. Everything stops rotating when the arms reach the end of their travel.

 

[haruspex]

My point was that if the arms are moving in opposite directions, there is a couple due to the equal but opposite forces across the body as a result of the symmetry. If the arms are waving clockwise, the central body must turn counter-clockwise to keep the total angular momentum at zero. Everything stops rotating when the arms reach the end of their travel.

Sure, but the question asked was whether a 'rotation can be created', and whether it can 'result in an overall angular velocity of the hub'. Taking 'overall' to mean long-term average, the answer is yes.

 

[aerowenn]

Sorry, I've been away. Yes, what I am working asks the question on whether or not a long term change in angular position of the hub can be created by some motion of the arms. They would move in the same direction (both CCW or CW when they move).

The arms would consist of a piezoelectric material, so they can only deflect in the same fashion either direction. They can't replicate the way a cat would do this to change position. The arms can't be shorter in one direction than the other. If you could imagine the inverse of the diagram shown above, that's exactly the way it would look if the arms were to move the other direction.

My question was whether or not how these arms move could create a long term change in the position of the hub around the "O" axis.

I had the impression that a very fast motion, say in the CCW direction, and a very slow motion (estimate a 4th of the time it takes for the CCW motion) in the CW direction would produce a higher angular acceleration in the CCW direction than the CW direction and therefore higher torque. However, it seems that the higher torque would occur for a smaller time than the lower torque which should equal out to a zero net change in position of the hub over time. This is a flapping motion if you will.

The paper I'm referencing for this is attached, but perhaps I'm reading it wrong. It isn't my background so I'm a bit unsure of how I'm reading it. Equation (20) is the Lagrange equation of motion that I was using to attempt to prove this theory.

 

[aerowenn]

I should note that (zeta) on the right hand side is equal to zero for my purpose, as there are no outside torques present. I'm trying to create one.

 

[Ryoko]

The math in that paper is a bit over my head. But I can offer this that you need to consider the nature of angular momentum and that it can be defined as the product of torque x time. So you can move an object slowly with over a long time with a small torque or move it fast with a large torque for a short time and still get the same angular momentum. Since the arms have nothing to push against in space, the system can't gain any angular momentum by flapping the arms. The speed doesn't really matter. The center body will shift in the opposite direction of the arms to conserve the total angular momentum. You won't be able to cause the body to spin, but you'd be able to change it's rotation angle.

 

[aerowenn]

That's relatively easy, then.
Consider an element ds of an arm. At time t it will be (in polar relative to hub centre) at some position (r=r(t), θ=θ(t)). The tangential acceleration is $r\ddot{\theta}+2\dot{r}\dot{\theta}$. See http://en.wikipedia.org/wiki/Polar_coordinate_system#Vector_calculus
If the element has mass dm (=ρds if uniform density) then the reactive torque is $-(r\ddot{\theta}+2\dot{r}\dot{\theta})r.dm$

I see that we can't create an angular velocity without morphing the shape of the arms. But I need to be able to bring this before my advisor and explain it well. I might have simply misunderstood his reasoning behind using this paper to begin with.

With the equation of motion as you posted (the same form of what I posted just after now that I look at it), can the differential equation be solved in the usual way? Is there a general solution for that second order differential since the position "r" is also a function of time as is theta?

It's been a while since calc IV, but everything I can find in books and reference only point to examples where the function (say "r" in this case) is being multiplied by constants, not time-varying functions such as theta.

Wolfram doesn't like this guy either. These seem to be a bit ugly, but I do like the challenge.

Thanks for all the help from all of you up until now by the way, it's been invaluable.

 

[haruspex]

I might have simply misunderstood his reasoning behind using this paper to begin with.

The paper you cited is concerned with feedback and stability. From what I understand, you're not trying to construct a system with sensors and feedback control - you just want to operate the arms in a way to have a precalculated effect. (Of course, you could add some simple feedback logic that keeps asking where the system is in relation to where it ought to be and has another go at getting there, but you don't need to get into the complexities of that paper for that.)

With the equation of motion as you posted (the same form of what I posted just after now that I look at it), can the differential equation be solved in the usual way? Is there a general solution for that second order differential since the position "r" is also a function of time as is theta?

That wasn't a differential equation, just an expression for a component of a differential equation. To get an equation you need to supply a proposed pattern of movements of the arms as a function of time. I assume this will largely be constrained by the capabilities of the mechanism. You have the basics of what the movements must be (for hub clockwise: arms in, arms clockwise, arms out, arms anticlockwise, etc.), so the next step is to specify an exact pattern of movement following that scheme.

 

[aerowenn]

That wasn't a differential equation, just an expression for a component of a differential equation. To get an equation you need to supply a proposed pattern of movements of the arms as a function of time. I assume this will largely be constrained by the capabilities of the mechanism. You have the basics of what the movements must be (for hub clockwise: arms in, arms clockwise, arms out, arms anticlockwise, etc.), so the next step is to specify an exact pattern of movement following that scheme.

Well to illustrate a bit further, this research is for an attitude control mechanism for small satellites. There will be feedback control in the end, but you are correct in thinking it will not be very complex. There shouldn't be a need for complexity as long as the motion of the device is predictable (get attitude, make attempt at desired attitude until desired error margin is reached).

With the types of devices we are planning on using for the arms, there will be no morphing capability. It will be a uniform motion that looks the same in both directions. They will flex, but 30 degrees to the CW will look equal and opposite to 30 degrees to the CCW. So I won't be able to produce angular velocity by tucking in the arms in one direction.

Luckily, the flapping motion isn't the only option for getting a net change. But what I do need to do is prove that the flapping motion won't work, and why. I could argue that momentum is conserved in the system (which is true), but I need something to base it off of. I've not been able to find documentation specific enough to demonstrate this.

It would also be nice if I could model the system and show that no angular velocity is developed on the hub with any prescribed motion of the arms. The proposed motion profile was a sine wave that rises quickly and falls slowly (shown below). So I have that. I can take the derivative and get velocity, then acceleration. This is a simple piecewise equation set.

If you're aware of some documentation or a good way to go about modeling this in say MATLAB/Simulink or something similar I would appreciate the advice.

Again, thanks for the input so far, it's been very helpful!


Arm tip motion:
http://sphotos-a.xx.fbcdn.net/hphotos-ash4/480263_4638912265323_1686660715_n.jpg

 

[haruspex]

As discussed, the angular momentum of hub+arms will always be zero, so there'll be no build-up of angular velocity. The remaining question is whether a change of position can be achieved. Given that the flapping in one direction (as a trajectory, not paying attention to speed) is just a mirror image of that in the other direction I would say not even that can be done. You should be able to prove that by a symmetry argument.

 

[Ryoko]

Let me toss a suggestion out -- The motion of the arms is pretty messy to deal with. Replace the arms with a simple thin ring that can rotate independently from the body. The ring is driven by the central body. The ring will be allowed to change its diameter, but not it's mass which gives it the ability to change its moment of inertia much like the arms do. This model will allow you to mimic the arms without the nasty math.

 

[aerowenn]

I like that, thanks for the suggestion, it would make it easier to model I think.

First though I need to prove mathematically that it can't be done without changing the diameter (or length of arms). I would prefer to take the equations of motion and show this, because I'm going to need them anyway. I suppose I need some good material to go over and learn how to solve that thing.

 

[Ryoko]

If you want a compromise, you can break the arms up into a series of segments and model the segments as concentric rings.

 

[aerowenn]

Ok, I'm missing something critical here, it seems like I've solved a similar problem before but I just can't get it or find examples anywhere.

How would I take the equation for the beam, having it's components theta and r both with second derivatives and solve for something like angular velocity or position? Would I have to specify the motion of the beam first? If so, how would I do that? Flexural rigidity if it's flexible and all that is still above my head I suppose.

 

[Ryoko]

If you have an equation describing the motion of the beam, you can find the velocity and acceleration forces by taking the appropriate derivatives. Just remember that forces act thru the center of gravity. Also, if you rotate the body on an axis away from the CG, you need to apply a correction to the angular moment of inertia reflecting the offset.