Beam Equation - problem with boundary condition

Hello Everyone,

my PDE has a following form (E-B beam):

for using PDETOOL I had using two PDE in form:

Left end of beam is clamped and the right end is free. As I know the boundary conditions are:

for free-end

My question is how can I implement these boundary conditions in PDETOOL

[Bill Greene]

Since your equation has only one spatial dimension and time, I suggest you use the pdepe

function to solve it rather than PDE Toolbox.

[Torsten]

By introducing a third variable "u" and solving

dw/dt = u
rho*du/dt +EI*d^2v/dx^2 = f(x)
d^2w/dx^2 - v = 0

?

Looks quite challenging for pdepe.

[Bill Greene]

I have a short note describing how to use pdepe for this problem: beam equation with pdepe.

(Should have included that link in my previous comment-- sorry.)

[Abazar Rahmani]

Thanks Bill for your answer. Can you please explain the calculation of the p and q arrays (Boundary conditions) for this system of equations more? I have read your answer a few times and I get lost everytime I get to this point.

I am trying to modify your answer for a cantilever beam, and the answer I get is not intutive as the curvature of the beam is wrong when I simply use the boundary conditions that you mentioned in your files.

Thanks.

[Bill Greene]

Thank you for finding the error in my note on beam dynamics.

When I added the section on alternate boundary conditions, I assumed that

they *should* work but, unfortunately, did not actually try them. As you observed,

the boundary conditions for the cantilever beam do not produce a correct

solution in pdepe.

After considerable investigation, I believe this is due to a subtle bug in the pdepe

function. If you replace this line in pdepe (line 256 in the R2019a version of pdepe.m):

D( c == 0, 2:nx-1) = 0; 

with this line:

D( c == 0, :) = 0;

you can obtain results for the cantilever BC case that are in close agreement with

the analytical solution.

Note, this simple fix corrects the problem with this example but I have not done

enough testing to be sure it is correct for all cases.

My code for this example is here: beamFreeVibrationCantileverBC.m

[Abhigna Bhatt]

your link: : beam equation with pdepe.

is not working. please help

[Bill Greene]

I discovered that pdepe cannot solve the beam equation for general boundary conditions so removed my example.

However, I have developed a PDE solver, pde1dm, which has similar syntax to pdepe, but does correctly solve the beam equation for any of the standard boundary conditions. The pde1dm manual, which is included, contains an example that demonstrates this.

[Abhigna Bhatt]

Thank you, sir. I will definately try it. I hope it works for my dynamic nonlinear problem.

Answer 1

[Torsten]

v = vx = 0 at x=L
w = wx = 0 at x=0

?