Torque on a hub as a result from a flexible arm

aerowenn

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

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.)
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

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 wont 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:

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.