@acer In the usual eigenvalue problem in linear algebra, finding the eigenvalues of a matrix M amounts to finding the values of λ so that the matrix M − λ I is singular.  His is a nonlinear eigenvalue problem where we have a matrix M(λ) which is a nonlinear function of the parameter λ, and we want to find λ so that M(λ) is singular.  That's beyond the scope of LinearAlgebra:-Eigenvalues.

@PhD_Wallyson Look at the equation just before the implicitplot command.  Call that equation Eq. It involves only rho and lambda.  Given rho=4.3, you want to calculate the corresponding lambda values.  So we do:

eval(Eq, rho=4.3):
fsolve(%, lambda=0.001..5, maxsols=5);

and that produces

0.4074182340, 1.033338698, 1.741806657, 2.444688425, 3.111634975

The reason for the 0.001 in the call to fsolve is to avoid the lambda=0 solution.

@PhD_Wallyson In the worksheet that you have posted, note paragraph just above where the plot_mode( ) proc is defined.  It says that in principle the determinant of M is zero, but in practice is may be not quite zero due to floating point roundoffs.  It also says that you need to keep an eye on the value of the determinant and take corrective action is if it too far from zero.

Also note the message, printed in blue, just before each of your three plots.  We have things like this:

Determinant is -2.99856e+025.  Increase Digits if not near zero!

Certainly 10^25 is not close to zero, therefore the result is pretty much nonsense.

As the message suggests, one way around this is to increase the number of digits that Maple carries in its floating point calculations.  The default is 10 digits.  If you increase the digits of accuracy to 30, then the determinants will be almost zero.

The digits of accuracy is specified by the Digits variable.  Therefore near the beginning of your worksheet you need to insert:

Digits := 30;

Although that will work, I strongly advise against doing that  because that's the wrong approach to solving the problem.  Consider, for instance, that you want to carry out the equivalent calculations in Matlab instead of Maple.  Matlab's precision is something like Digits = 16, and that's not adjustable.  Therefore your program won't work at all in Matlab. As you have seen, if you use centimeters instead of meters, Maple works just fine with the default of 10 digits.  In engineering calculations it is quite common (and often absolutely necessary) to use sensible units in order to scale the computational ranges of variables into what the machine can handle.

@vv I had not expected the maxsols option to work in this case since according to the documentation, "this option is valid only for a single univariate polynomial equation" [my emphasis].  Perhaps the documentation needs to be updated.

@Neel Your hand-written notes on solving the forced wave equation are on the right track.  I have written my take on your notes and have completed the calculations in the following worksheet.  Perhaps this can be of some help.

I must point out that very little of Maple's computational power is used in this worksheet.  These calculations could have been easily done by hand.  Don't get distracted by some of the obscure Maple commands that you may encounter in the worksheet..  Instead, focus on what they are meant to do.

Download wave-equation-forced-vibration.mw

After deriving equation (22), the authors identify three special cases for which solutions are available in the literature.  Let's look at their Case 1, that is, a/L=0 and mass_ratio=0.  The equation then should reduce, and they assert that it does, to the classical clamped-free cantilever model.  But I don't see that.

Setting a/L=0 and mass_ratio=0 in equation (22), I get 0=0.  (I did that by hand; no need for Maple for that.)  But that's not correct.  The equation should reduce to cos(βL) cosh(βL) = −1.

Can you verify my calculation?  If I am correct, then there must be a typo in equation (22).  That needs to be fixed before going ahead with the rest of the calculations.

@Neel What you have here is so far removed from what would be the right approach to the solution of the problem, that it leaves little to salvage.  In my earlier message I provided a general outline for solving the forced vibration problem.  What you have in your worksheet does not follow that outline at all.  The entire worksheet needs to be discarded and started over again.

But you should not feel bad about that.  My outline assumes some familiarity with orthogonal basis functions generated by solutions of boundary value problems, and how to expand an arbitrary function in such a basis.  It appears that you have not seen that material, so I wouldn't blame you for that.

I can provide the somewhat long and messy solution to the problem which you are attempting to solve, but without the proper mathematical background it won't be understandable.

Why are you interested in this problem?  If it is a student project, you should really begin with learning the basic mathematical ideas behind what is called eigenfunction expansions. Furthermore, the beam problem is one or two steps more advanced than a beginner problem in this subject, which would be:

First: The heat equation:
w_t= w_xx + f(x,t),
w(0,t)=0,   w(L,t)=0,
w(x,0) = g(x),

Second: The wave equation:
w_tt = w_xx + f(x,t),
w(0,t)=0,   w(L,t)=0,
w(x,0) = g(x),   w_t(x,0) = h(x).

You need to be able to confortably handle these before attempting the beam equation.  Do you have a textbook that shows how to solve these problems?  Learning that will be the first in your progression.  By the way, notice the initial conditions g(x), and h(x).  You will need to specify initial conditions to solve your beam problem as well.  I didn't see initial conditions in your worksheet.
 

@panke Specify the version of your Maple when you ask a question.  There is a pull-down menu just before you submit your question in which you enter the version.  Please don't ignore it.

The phase space of a 5th order ODE is five-dimensional.  The current version of Maple does three-dimensional pictures quite well.  You will have to wait for a future release of Maple to do plots in five dimensions.

@vercammen If you know about Maple's procs, then a better way would be through a proc, as in

make_prob := proc({n:=10, alpha:=-60,  beta:=20, P_val:=4})
    local Qs := alpha + beta*P;
    printf("There are %a identical firms.  "
           "Supply is given by Qs = %a.  "
           "When P = %a then quantity supplied is equal to %a.  "
           "Calculate the ...\n", n, Qs, P_val, eval(Qs, P=P_val));
end proc:

Then you may call the proc any number of times with the desired arguments.  Missing arguments take on default values as indicated in the first line of the proc's definition.  For example:

make_prob();   # all default args
There are 10 identical firms.  Supply is given by Qs = -60+20*P.  When P = 4 then quantity supplied is equal to 20.  Calculate the ...

make_prob(n=40, P_val=5);   # some optional args
There are 40 identical firms.  Supply is given by Qs = -60+20*P.  When P = 5 then quantity supplied is equal to 40.  Calculate the ...

@Neel There is no general way for that.  Each case needs to be treated on its own, but the steps are pretty much always the same..  So if you see what I did for the clamped-clamped case, you may do the same to analyze the clamped-free case (a cantilever beam).

Here is a pointer to simplify your calculations just a bit.

In what I presented earlier, I went along with general solution of the ODE produced by Maple as a linear combination of exponential and trigonometric functions with coefficients _C1 through _C4.  For our purposes that's not the best form of the general solution.  The first two terms may be replaced with hyperbolic functions, as in _C1*sinh(mu*x/L) + C_2*cosh(mu*x/L).  Since _C1 and _C2 are arbitrary, the changed expression is equivalent to the original.  I suggest that you make that change in the worksheet (manually) and then continue the calculations.  You will see that things work out more smoothly that way.

@Aminaple The idea suggested by Dr. Venkat Subramanian is quite good and I like it better than the solution that I presented earlier.  Here are the details of this alternative approach.

Download mw2.mw

@Neel 

@Neel What are the boundary condtions?  I cannot tell by reading through your worksheet.

@janhardo Your interpretation of my calculations is quite correct.  Well done!