Wrote

My students solve this problem at the intersection of mathematics, mechanics and computer science (STEM education - see
https://www.routledge.com/25-Problems-for-STEM-Education/Ochkov/p/book/9781032174624).
We take an inextensible cable of length L and with a known specific gravity (the mass of one meter of cable). We attach a load of known mass to this cable at a distance l from one end of the cable. Next, the cable with the load is attached at two points. It is necessary to determine the geometry of the cable sag and the force at the attachment points. Then you need to change the value of l from zero to L and create an animation.
I can do this in Mathcad, but I cannot do it in SMath. I turned to you for help.

Wrote

The dotted lines are a phantom continuation of the catenary.
If G=0 we have one line.

Wrote

Here is my force-balance approach to this problem.

I divide the cable weight into a number of nodes equally distributed along the cable, then add the cable car weight to one of the nodes (the one nearest to the distance that it was added in the original catenary file here, though this can be changed as desired). The nodes are then assumed to be connected by weightless, inextensible links.
It took me some time to get the syntax right! I’ve no doubt an SMath expert could significantly improve the implementation.

CableCar.sm (44 КиБ) скачан 23 раз(а).

Wrote

Hi StvMath,

I enjoy your solution, based on the discretization of the continuum.
As always with such an approach, the results are enough dependent on the input data values and decisions.

I played a little with your tool by introducing (“provoking it”) some limit situations, namely:
• a small or a null cable weight;
• a small or a null “cable car weight”.

Please try such cases and tell us your evaluations. Normally, in this way, some analytical results could be checked (?).

Once again, thanks for your interesting work,

Best regards,
Ioan

Wrote

Of course, if SMath had its own Minimize (or Maximize) function

(%i1) load("lbfgs";
(%o1)           /usr/share/maxima/5.47.0/share/lbfgs/lbfgs.mac
(%i2) f(x,y):=(x-y)/((x^2+5)*(y^2+5))+(x^2+y^2)/20000;
                                                    2    2
                                     x - y         x  + y
(%o2)               f(x, y) := ----------------- + -------
                                 2        2         20000
                               (x  + 5) (y  + 5)
(%i3) lbfgs(f(x,y),[x,y],[1,1],1e-10,[-1,0]);
(%o3)          [x = - 1.2860935388262762, y = 1.286093516249451]