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# Mine Hoist Problem, #54

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Mechanics Corner
A Journal of Applied Mechanics and Mathematics by DrD, # 54

Mine Hoist Problem

Introduction

As we saw in the previous problem (A Hoisting Problem for Engineers, #53), hoisting problems can be fun. This has gotten me to thinking about some other hoist related problems, and here I present three variations on a new problem. The problem is much the same in all three cases, with only small changes in the geometry. These changes do appear to significantly modify the system response.
The problem involves a mass M to be lifted straight up (against gravity) from a deep mine. The lift is accomplished by allowing another weight to descend under gravity, all with no friction involved. The second weight (the one that falls) is a large wheel set (much like the two wheels and rigid axel commonly used on railroad freight cars) that runs on a track down the hillside. The track geometry is such that the center of the wheel (with radius R) traces the curve x=cy² (see figures for clarification) as it rolls without slipping on the track. For each case, the well axis begins at x=0, y=0 with respect to the coordinate system shown. A cable from the load to be lifted is connected to the rolling wheel assembly. The connection details vary from case to case. For each case, you are asked to formulate and solve the system kinematics, then formulate the system dynamics, and finally make a simulation (using Octave, Scilab, Matlab, or your own computer integration code). In each case, the system has only one degree of freedom, and you are free to choose any suitable variable to describe the system state (the generalized coordinate). It appears likely that the axel vertical position, y, would be a good choice for this purpose. Please send me your proposed solutions (neatly handwritten is acceptable, provided the contrast is high; ink on paper is better than pencil on paper). When I receive a solution that appears to be correct, I will post it for all to see and recognize the person who made the solution.
I want all readers to know up front that I do not have solutions for any part of this problem. I've posted too many solutions without giving all of you the opportunity to enjoy working them out for yourselves. At this point, I've not assigned any numeric values but rather have given symbolic names to the various parameters. As time passes, if some of you get to the point of needing numerical values to proceed with calculations, please send me a message and I'll come up with some numbers. With that caveat, please consider the following cases.

Well, here we are on 31 August with 151 views (presumable this first page only), and only 4 downloads (necessary to really see what the problem is). That is just slightly more than 2.5% of people bothering to actually look at the problem. How could this group be more lacking in curiosity?

DrD

If a picture is worth a thousand words, a video could be worth a thousand pictures. This should be presented in video format

And just how would I do that, Timothy?

1)Do you want a video of the system (which only exists on paper at this time) and is therefore impossible to do?

or

2) Do you want a video of me talking about the problem? What would be the point of that? the written words "hold still" and can be read and re-read as many times as necessary.

What do you have in mind, Timothy?

DrD

My apologies.  I understand that would be impossible.

I down loaded the problem and it is a classic mechanics view point.  But as I have done many systems for mines not actually applicable to be considered viably functional.

If you are attempting a large mass lift then this would be completed by a large gear box system with controls for braking and over running loads.  The key here is speed to mass and functional gear box and proportional speed control either with a VFD or a proportional hydraulic systems.  Normally a triple redundant brake design is incorporated to ensure a safety level.

So I think it would be more applicable to address mass loads in a real world analysis either using electro proportional hydraulic lift system or a variable frequency  drive with feedback for motion control.  All integrating a active controlled redundant brake design for safety.

Reference active designs from Siemag and ABB controls

If we complicated the problem as you recommend, it would be fa beyond the likelihood of anyone working it. As it is, I am unaware of anyone having work the problem.

One of the big points that you evidently missed is the kinematic aspect. I am well aware that slopes that follow exactly parabolic form are exceedingly rare. It was never intended to represent any actual mine, but rather a hypothetical situation intended as an exercise for folks to develop their skills.

DrD

Dear Mark,

After thinking further about your response, I finally decided that I should tell you more about the situation.

The load to be lifted out of the shaft is a massive blob of pure gold, studded with 50 carat diamonds. Your team has been searching for it for months, and now it has been found. Now, do you use the primitive lift described in the problem, or do you place an order for the modern equipment you recommended?

Oh, I should also add a couple of other factors that may influence your choice.

The action all occurs at a very high altitude (22,000 ft above sea level) in a remote part of the world. There is no electric power available. You could order a diesel generator set along with the rest of your gear, but the diesel might have difficulty producing full power due to the thin atmosphere.

As a final consideration, you know that there are at least two "bad guy" teams searching for this treasure, and you've found their tracks in the area. Your gear is not available with Amazon Prime, so delivery may take months. Which lifting rig would you use?

DrD

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