@wlferguson19  Change the line
pdsol:-animate(t=0..0.3, frames=40);
to
pdsol:-animate(t=0..0.3, frames=40, title="time = %f");

Does that do what yiou want?

@wlferguson19 Oh, I see;  I didn't notice that you had written Ux to indicate derivatives.  All's well.

The line
t:= textplot3d([-1.1,1.1,1,typeset(`#mover(mi(` || v ||  `),mo("→"))`)]):
can be entered more succinctly as
t:= textplot3d([-1.1,1.1,1,`#mover(mi("v"),mo("→"))`]);
and of course this is independent of the Physics package.

@Jjjones98 I can't think of a way for doing that. Perhaps someone else can come up with an idea.

restart;
with(plots):
S:=cat("Happy New Year 2020!   "$3):
N:=length(S): h:=2*Pi/N:
frame := proc(t)
  seq(textplot([cos(-k*h+t), sin(-k*h+t),S[k+1]],
                rotation=-Pi/2-k*h+t,
                'font'=["times","roman",20]),
       k=0..N-4);
  display([%]);
end:
nframes := 60:
frames := seq(frame(2/3*Pi*i/nframes), i=0..nframes-1):
display([frames], insequence, axes=none);

@Carl Love The rope is assumed to have no flexural resistance.  Think of it as a chain.

It is possible to put in flexural resistance.  That model is called a beam.  (An Euler beam, to be precise.)  The PDE will be different.  Each of the two x derivatives will be replaced by second order derivatives with respect to x, so altogether the left-hand side of the equation will turn into a fourth order differential operator.  The right-hand side will retain the second order derivative with respect to t.

You are correct in observing that the y coordinate of the end is constant.  This is because the equation is limited to small displacements and as such the points move laterally but not up and down.  (The up and down movement of the points is of the second order in displacements and therefore it is ignored.)  It is possible to derive the equation of motion without the smallness assumption but that is an unpleasant mess.

The term f'''(η₀) appears in equation (16) but neither f nor η₀ shows up in the subsequent equations.  I see that you did not include that term in your worksheet.  Perhaps the paper that you are reading explains what needs to be done with that term. Let us know.

Moreover, it would be good to see equation (15) and the problem's boundary conditions which are referenced in what you have shown.

@Carl Love There is viscous damping between the ball and the container, meaning a resistive force which is proportional to their relative velocities. There is dry friction between the container and the floor, and that's responsible for the jerky motion.

The viscous damping enters the equations of motion through the term c*(v₂−v₁). Taking c=0 is possible but in that case the ball will forever oscillate after the container has come to rest.

As to marketability, I don't know; that's above my pay grade :-)

@Preben Alsholm I had completely missed the event_doublecross option in the doucmentation, and that's exactly what is needed here.  Thank you very much for pointing it out.  Now I am happy :-)

@Carl Love Motivated by your comment, I made a model of something similar to sloshing water in a bucket.  The "water" in the model is just a point of mass which bounces back and forth in a container which in turn can slide on the floor.   I think that calculation deserves a post of its own, so I will post it in a new thread.

@mmcdara I have read your suggestion regarding an alternative approach to treating dry friction and I like it quite a bit.  The result produced through it are quite compatible to those obtained through dsolve/events and have the advantage that they involve straightforward numerical solution of a system of differential equations and therefore are more robust than the delicate approach through events.

I will post a solution of a different problem which I solved by applying the friction model suggested by you.

Thanks for the very helpful input.

@mmcdara I am out on travel now. I will return in two days and carefully read your comments and then reply.  I thank you very much for your extensive feedback.

@mmcdara I have tried smoothing out the friction term but have found the results to be unsatisfactory in the sense that even with quite small eps (in the notation of your code fragment), the solution remains noticeably different from that obtained without smoothing. The discrepancy is most noticeable in the plot of v(t).  The original v(t) has flat parts, indicating a motion with constant velocity (sticking to the belt).  In the smoothed out version, v(t) has nothing resembling flat parts.  This is with eps=1e-3.  Changing eps to 1e-4 leads to a premature halt in the solver as in the unsmoothed case.

@acer Thanks for the feedback.  I rarely think of Explore as an animation tool but I can see that it can be very helpful, especially during code development. As to DocumentTools:-Tabulate, I will have to read the docs because I know nothing about it.  

@Carl Love The automatic root finder is a good idea but we should be aware that it can skip over some roots.  For instance, the roots 245.3396184, 481.5561050, 530.0902992, 578.7535346 are missing in what you have shown.