The app is "Paper" by WeTransfer. Running on an iPad Pro 13 inch and using an Apple Pencil.

It's a good question. I'm not sure if I can explain it thoroughly using only text and without the use of a video, but let me try...The full solution for the displacement field is: w(x,t) = W(x)·T(t) = Σ d_n(...)·(A cos ω_n·t + B sin ω_n·t) = (...)·(A _n cos ω_n·t + B _n sin ω_n·t) where the "..." is meant to represent the remaining terms in equation 18 and the Σ implies that this is summed for n = 1 to ∞. So what I have done is I have combined the constants, d_n's with the A and B to form A_n's and B_n's - so we have eliminated the d_n's. Now we can solve for the A_n's and B_n's by applying the initial conditions. This yields two Fourier series which can be solved for A_n and B_n. It's a little hard to explain without showing an example, so perhaps I will need to make a video for this. Hope I haven't confused you.

@@Freeball99 I'm with you! Instead of directly calculating d_n you combine it with the other constants A and B (since this will yield the same result, just another set of constants). If I understood you correctly? Thanks again! Really nice of you to take time and answer my question.

I think it's easiest to solve it graphically by dividing both side by cosh βnL and the plotting each side of the equation for different values of βnL and seeing where they intersect. Also, you could do this standard numerical methods by subtracting 1 from each side and finding the roots of the resulting equation on the left-hand side (i.e. the values of βnL that make this equation zero).

Not sure I follow exactly what you mean by "different force acting at every end of the beam". Perhaps you could draw a figure and send it to me at: apf999@gmail.com

Thanks for your suggestion, I'll add this to my list. I have some other videos to make first.

@@Freeball99 Thank you! I was trying to model the shaft whirling problem with a continuous beam. So, the equation that I'm trying to solve is basically the beam vibration equation, but with an harmonic force excitation. Something like, d^2 v(x,t)/dt^2 + c^2 d^4 v(x,t)/dx^4 = F_0 e^(i w t)

[A] is the coefficient matrix, so if the determinant is non-zero then this means that the two equations are linearly independent of one another and so only one solution exists. Clearly, the solution d1=0 and d2=0 satisfies this equation, but this is a trivial solution (it means that the response is zero at all times). In order to remove this constraint, we require the determinant of the [A] matrix to be zero. This means that the two equations are NOT linearly independent (eg. this is like having equations such as d1 + d2 = 1, and 2d1 + 2d2 = 2 - these two equations are really just the same equation multiplied by a constant - and so the determinant of the coefficient matrix would be zero). As a result, we don't have enough information to uniquely determine d1 and d2. This is why when determining the eigenvector, we arbitrarily set either d1 or d2 equal to 1. This, effectively, introduces a second independent equation which allows us to solve for the other value. This reduces the problem to figuring out under what conditions the determinant of [A] is zero, which gives rise to the so-called "frequency equation" from which the natural frequencies are determined. In other words, at what frequencies does the system need to vibrate in order to keep the determinant equal to zero thereby allowing non-trivial solutions? There are the natural frequencies and give rise to the mode shapes.

I'm not sure what you mean by "global buckling". Are you asking how to find the point when the assembled truss structure buckles?

@@Freeball99 Yes. Like the truss buckling not the individual bars

I believe a way to do it would be to examine the fundamental frequency for the assembled structure as you load it. If it's true buckling that you're analyzing, then at the onset of buckling, the fundamental frequency should go to zero - i.e. the compressive forces in the truss ought to produce a de-stiffening effect.