A Journal of Applied Mechanics and Mathematics by DrD, # 54
© 25 August, 2020
Mine Hoist Problem
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.