Hi Valery, Hi Alvaro,I find useful some additional comments on the subject targeted in yours interesting posts.
A. The catenary cable problem, you are dealing with, seems as being physically speaking this:
• A cable defined by its 2 endpoints (the 4 coordinates can be reduced to 2 information
(H; V) if the origin point of the cable is taken in the origin of the axis system, and by the developed length
L of the cable.
• A load (mass) is applied on the cable between its 2 ends.
Note: When the 3-input data of a cable
(H; V; L) are given, its geometrical equation (the catenary) is completely determined by using
3 nonlinear equations. The nonlinear 3 equation system can be transformed in
3 independent equations, as it is shown here:
https://en.smath.com/for...nary-Cable-Analysis.aspx #14 Posted
B. If I’m not wrong, your objective is to find the new position of the cable (the cable being axially rigid) that assures the equilibrium.
Globally, this problem, IMHO, has 3 distinct technical instances:
i) The intermediary load is fixed on the cable length on a given distance over the developed cable. And your target is to find the final equilibrium position; that means
2 unknowns.
ii) This load can roll (slide) over the cable and you look for the position where the load finishes in equilibrium. This means
3 unknowns (
the position of the rolling load on the cable length and the 2 coordinates of this point).
iii) This load can move only in one direction (vertical, for instance); the second DOF (horizontal, for instance) is blocked. That’s only
1 unknown - case technically pertinent also.
I find nice the graphical representation as the intersection of 2 distinct cables; the real solution also includes a third information - the position on the cable's length.
If I understood well your development, it seems that it’s belonging to the ii) case?
What do you think about some other solutions?
For instance, a method useful for a general case (more than a force between the cable’s ends) can be based on the concept of FEM (?).
Best regards,
Ioan 
