Euler-Bernoulli's beam ODE or PDE

Hi guys,

I'm trying to develop a script of Euler-bernoulli's beam equation for the purpose of my project.

I've been researching the internet with no success.

I'm fimiliar with the theory but don't know how to translate it into matlab.

I've done multi degree of freedom system using ODE45 and it works,

but I can't find a way to use EB eqn.

I've came across a statement: Usually a PDE will have boundary conditions, while an ODE has an initial condition.

In E-B eqn we have both, so is it PDE or ODE?

and then will I be able to do it using ODE45?

If not, where to go? what to use instead?

If yes, I'd appreciate some hints where to start with coding.

Here's the link to my MDOF problem I solved with some help here.

(https://uk.mathworks.com/matlabcentral/answers/441268-ode45-2dof-unable-to-perform-assignment-because-the-left-and-right-sides-have-a-different-number-of)

I'm not expecting the code written for me, i just need some suggestions and guidance

Many thanks

Piotr

P.S.

the eqn has a form of:

E*I*d4w/dx4(x,t)+rho*A*d2w/dt2(x,t)=f(x,t)

[Amin Mohammed]

Very interesting question, I just came accross situation and would love to see the documentation suggested by @Bill Greene, the link provided is proken now, could you kindly help me please to access there?

[Lubos Smolik]

Amin, you can try to explore Bill's repository (link).

Answer 1

[Bill Greene]

Take a look at this example: beam dynamics with pdepe

[Piotr Haciuk]

Thank you Bill,

I'll have a look later on.

My main aim is to control the vibrations, will I be able to doi it once I adopt your suggestion?

Thank you

[Piotr Haciuk]

Dear Bill,

I've used your example code as a reference to mine.

I've changed few things, i.e. force as a static load, and initial conditions = zero

I got some results, but

looking on the shape of the final displacemet which is a nice curve i'm wondering

how many mode shapes are included in the results?

Best regards,

Piotr

[Bill Greene]

I selected my initial condition to excite only a single mode. For a general initial condition or load, an infinite number of modes are excited.

[Piotr Haciuk]

Dear Bill

I was wondering if the case was moving load along the beam,

will I still be able to use pdepe?

also if the case would be at least two moving loads along the beam in opposite direction,

will it work?

Kindest

Piotr

[Amin Mohammed]

I just came accross situation and would love to see the documentation suggested by @Bill Greene, the link provided is proken now, could you kindly help me please to access there?

Answer 2

[John D'Errico]

Euler-Bernoulli

https://en.wikipedia.org/wiki/Euler–Bernoulli_beam_theory

might appear in several aspects. It kind of depends on what you are doing with it.

For example, in a purely static form, thus with no time dependence, this is a 4th order ODE. However much of the time, a beam will have boundary conditions applied at each end of the beam. That makes it a boundary value problem. ODE45 and tools like that are designed to solve initial value problems. You can use IVP solvers to solve boundary value problems, using perhaps a shooting method. But it is easier to use tools like BVP4C to solve a boundary value problem.

Once you add in a time dependency however, this becomes a problem with two variables, so it is a PDE. The time dependency enters in differently however, than as just a spatial variable. For example, a classical PDE might be to predict the steady state temperature of a body using Fourier's law of heat conduction. So just a Laplacian in perhaps 2 or 3 dimensions. This becomes a simple elliptic PDE, subject to boundary constraints.

Your beam is not really any different.

What happens however, when you add in that time dependency? Typically, this looks like an initial value problem in time. That is, you know the state of the beam at time = 0. Then based on a variety of forces on the beam, on the constraints at the boundaries, you wish to see how this system will evolve over time.

A standard way to solve such a problem is the method of lines.

https://en.wikipedia.org/wiki/Method_of_lines

http://www.pdecomp.net/Scholarpedia/MOLfinal.pdf

[Piotr Haciuk]

Thank you John, I'll have a look later and see if my brain can make a use of it.