Question about Free-Body-Diagrams (MSc Thesis)

[Laura85]

TL;DR Summary

I'd like some feedback for solving the Equations of Motion after setting up some Free-Body-diagrams. I have 6 unknown variables and 5 equations and am not sure about the solution.

Hello,

I'm doing an FBD exercise and I'm hoping you guys and girls can give me some feedback. I've already done a lot, but I'm not completely sure about it and since it's part of my Msc Thesis, it's important that I get it right. After setting up the FBDs, I get 5 equations and 6 unknown variables, so I set the resulting moment around the center, Mc, to zero. This allows me to solve the equations and I get the answer that I want, but I'm not sure if it's right since the bodies are rotating. I can't really find a reason to justify that. On the other hand, I also can't see any other equations or ways to solve it.

So, my objective is to show that a C-leg is better in climbing an obstacle then a wheel and that it is due to its geometry decomposing torque more effectively into a positive momentum. To do that, I’ve made an FBD of both bodies when hitting an obstacle. Using the Equations of Motion, I then want to express the resulting moment around point A, Ma, in terms of a set of variables. A positive Ma means the body successfully climbs the obstacle, so plotting Ma against the obstacle height allows me to compare the two. The bodies are assumed to be massless, but there is a load Fg on the center point C. Fg , r and T are constant variables in the calculations.

I've put all my work in the attachment. I'm sorry it's so long, but the deductions take up a lot of space.

Thank you very much in advance,
Laura

 

[Kyouran]

I only looked at the beginning, but it looks a bit odd to me. For starters, why would there be a tangential force Ft at A? If you assume no slip, then the tangential force there should be zero. If you do assume slip, then you'd need an equation involving the friction coefficient somewhere.

And then there's also the fact that this is a dynamic problem so you'll need to take into account the kinematics of the problem.

 

[Laura85]

Hi Kyouran, thanks for taking the time.

Yes, I am assuming no friction or slippage. But I was also assuming there would be a tangent reaction force in A? I figured the wheel would be pushing against the surface, causing a counter moment against the torque. Isn't that the same as if the wheel was on the ground?

 

[hutchphd]

As far as I can tell you are really overcomplicating this. There are two reasons why the C- leg will be better:

1. The torque required at axle C will be smaller.
2. The required frictional force at the interface to the step will be less

Condition 1 follows immediately from the fact that d can be shorter for the C-leg than the wheel for any finite step.
Condition 2 follows because $\theta$ can always be chosen larger for the C-leg and the necessary friction coefficient goes like cot $\theta$
Also the C-leg can step higher than the axle !
The rest of the analysis is secondary to these points and you need to be clear as to the question you are asking. Good luck.

 

[Laura85]

Hi hutchphd,

Thanks for your reply. I'm mostly looking for someone checking my deductions and some feedback and this is very helpful. I didn't think about the effect of the length of d yet, I thought it was more about the tangent force having a smaller angle wrt the interface. I should look into that.

Ofcourse the C-leg will do better, but if I know what factors are important it could help with optimizing the shape in the next part of the research. Torque is important due to limitations of the motor.

 

[Dr.D]

@Laura85 You say in #3 that you are assuming no friction or slip. Looking at the first problem (the rolling disk striking the step), these two assumptions are incompatible. If there is no friction any where, then the impact at A must involve slip. If there is no slip, this is only possible if there is friction. So, just which way do you want to have it?

For the second problem (the C-leg), what is happening here? is the C-shaped arm rotating before impacting a stop, or something else? Is there a driving torque applied to the body? I think this problem needs some clarification as to what is going on.

 

[Laura85]

Hey,
Thank you for your help.

I looked at it again and I think I get what you guys are saying now. Since there is a counter force, there must be friction between wheel/leg and the obstacle. Would it be plausible saying Fw=Ft=Fn*mu ? That would give me an extra equation and might make it easier to solve as well.

With the C-leg/wheel, I was assuming both would be starting from rest and then a torque is applied on the center axis.

 

[Dr.D]

Writing Fn*mu for the friction force might be valid if you assume that there is slipping at the point of contact. It is usually not true if there is no slip.

 

[Laura85]

Writing Fn*mu for the friction force might be valid if you assume that there is slipping at the point of contact. It is usually not true if there is no slip.

Ok, thanks. I'll try work it out and see where that leads me.