Worksheet: mw.mw

Animation extracted fron worksheet mw.mw. The triangle marks the osculating plane. The red dot marks the center of curvature.

I don't have MapleSim but the animation can be produced quite easily in Maple; see the attached worksheet. Here is a sample demonstration where the left end is clamped and the right end is hinged, but you may change the boundary conditions in the worksheet as you wish.

Worksheet: animate-beam.mw

This should be self-explanatory:

Download mw.mw

I find your question quite vague, so here is my attempt to figure out what it is asking.  The calculation that follows is inconclusive.  You need to state more clearly what you have in mind.

Here it is:

Worksheet: mw.mw

You did not provide your equations in the Maple input form, so I retyped them. I don't guarantee that I have made no mistakes. Check!

Download mw.mw

As to your question "Or solve by hand?": I don't think you would want to do that.

General advice: It is important to distinguish between the assignment operator ":=" and the equality operator "=".  The primary purpose of the assignment operator ":=",, is to define labels for expressions to make it easier to refer to them later.  Clearly you don't intend ω as a mere label, rather, you intend ω to enter as a variable in your equations.

So this is what you need to do:

rels1 := diff(phi(t),t) + diff(beta(t),t) = omega(t);
rels2 := diff(rels1, t);
simplify(myvector, {rels1, rels2});

Note how rels1 and rels2 serve as labels for their respective equations.

Download mw.mw

restart;
f:=n->piecewise(n=0,1,n>=1,sum('f(k)',k=0..n-1));
seq(f(i), i=0..10);
        1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512

You have

plot(l, ....);

What is l? It is not defined as far as I can tell.  Perhaps you wanted to say

plot(x[1], ....);

I may not be understanding your question correctly, but perhaps the following can help.

Download FourierSeries.mw

Solving this PDE is a textbook example an application of the Fourier series.  Tomleslie has already sketched the first half of the process.  In the attached worksheet I have carried it out all the way to the end.  There is one correction, however.  Tomleslie's calculations end up with X(x) expressed as an exponential function and Y(y) as a trigonometric function.  That won't work.  It should be the other way around in order to satisfy the boundary conditions bc[3] and bc[4].  The solution turns out to be

which agrees with what Maple 2017's pdsolve() produces, as shown by Kitonum.

Here's what the solution looks like:

Worksheet:   mw.mw

Update: The previous worksheet had a couple of careless typos, although the final solution was correct.  In this new version I have fixed the typos.

Worksheet ver 2: mw-ver2.mw

@Bendesarts After fixing the brackets issue we can make some progress but the results are not very encouraging.  Let's assume all variables other than g are positive:

indets(TableauRouth) minus {g}:
(x -> x>0)~(%):
rels := %[];
                  rels := 0 < ca, 0 < ka, 0 < keqc, 0 < ma, 0 < meqc, 0 < p

and then try the easier task of solving the equality TableauRouth[4,1]=0 for g:

solve(TableauRouth[4,1]=0, g) assuming rels;

If you try that, you will see that Maple does succeed in solving the equation but the result is so huge as to being useless. This tells us that there is no simple necessary and sufficient condition in terms of g for the stability of your system.

You should refer to matrix entries through square brackets, not parentheses.

Change TableauRouth(2,1)  to TableauRouth[2,1] to fix the Maple coding problem.  After that you will have to deal with the mathematics of your question, and that may not be very easy.