How Can I Solve the Beam Equation using PDEs and the Frequency Domain?

[athanatos]

need to solve the following beam equation:

p(x)\frac{d^2\w}{dx^2}-a\frac{d^4\w}{dx^4}-b\frac{d^2\w}{dt^2}=0

don't have experience with pde's, thanks in advance for any hints...

 

[HallsofIvy]

You might want to put [ tex ] and [ \tex ] around that!

$$p(x)\frac{d^2\w}{dx^2}-a\frac{d^4\w}{dx^4}-b\frac{d^2\w}{dt^2}=0$$

Isn't there supposed to be a function, like "y(x)" in that?

 

[saltydog]

Well, may I first suggest we put it into the form:

$$p(x)\frac{\partial^2f}{\partial x^2}-a\frac{\partial^4 f}{\partial x^4}-b\frac{\partial^2 f}{\partial t^2}=0$$

With f=f(x,t) and x being displacement and t=time. In fact, I'd say just drop the p(x) for now and just solve the simpler case for starters. Get that one working then add the p(x) and see what happens.

Well, it's linear and homogeneous so we can use "separation of variables" right? You know, let f(x,y)=g(x)h(t), substitute into the PDE and just turn the crank. You need some initial conditions and boundary conditions but if you like, you can just make up some reasonable ones just to move forward with the solution.

In fact just drop the p(x), a, and b for starters.

 

[dextercioby]

"a" & "b" are constants,parameters if u like,so there's no reason to drop them.Indeed the $p(x)$ turns it into a nasty one.I think that for arbitrary $p(x)$ is not exactly solvable...

Daniel.

 

[athanatos]

You might want to put [ tex ] and [ \tex ] around that!

$$p(x)\frac{d^2\w}{dx^2}-a\frac{d^4\w}{dx^4}-b\frac{d^2\w}{dt^2}=0$$

Isn't there supposed to be a function, like "y(x)" in that?

yep, let me try it now;

$$p(x)\frac{\partial^2w(x,t)}{\partial x^2}-a\frac{\partial^4 w(x,t)}{\partial x^4}-b\frac{\partial^2 w(x,t)}{\partial t^2}=0$$

It's basically the beam equation for transverse vibration, with the additional p(x) term corresponding to the centripudal force due to rotating the beam at constant angular velocity $$\omega$$, which will be a parameter to be varied. Know next to nothing about pde's, but need to solve, so I guess looking for a method that will work on this type of problem. Thanks in advance for any help you may be able to render.

 

[dextercioby]

You may want to separate the "t" dependence,as it has been suggested above.Then u can try a series solution to the resulting ODE,but you need to know the series expansion of $p(x)$.Perturbative/approximate solutions is all u can search for...That function really spoils everything.

Daniel.

 

[Crosson]

Substitue:

$$\omega (x,t) = X(x)*T(t)$$

i.e. function of two variables becomes a product of two one-variable functions. Then the partial derivatives become ordinary derivatives.

If you algebraicly simplify after the substitution you should get:

$$\frac{p(x)}{X(x)}\frac{d^2X(x)}{dx^2}-a\frac{1}{X(x)}\frac{d^4X(x)}{dx^4}=\frac{b}{T(t)}\frac{d^2T(t)}{dt^2}$$

The variables are separated; one side depends on t and one side on x. But t and x are independent, so the only way the can be is if both sides equal a constant. And that's the ODE that you have to solve.

 

[athanatos]

Well, may I first suggest we put it into the form:

$$p(x)\frac{\partial^2f}{\partial x^2}-a\frac{\partial^4 f}{\partial x^4}-b\frac{\partial^2 f}{\partial t^2}=0$$

With f=f(x,t) and x being displacement and t=time. In fact, I'd say just drop the p(x) for now and just solve the simpler case for starters. Get that one working then add the p(x) and see what happens.

Well, it's linear and homogeneous so we can use "separation of variables" right? You know, let f(x,y)=g(x)h(t), substitute into the PDE and just turn the crank. You need some initial conditions and boundary conditions but if you like, you can just make up some reasonable ones just to move forward with the solution.

In fact just drop the p(x), a, and b for starters.

Thanks, well, it's a beam equation, so know the boundary conditions, it's clamped at one end, free at the other.


$$p(x)\frac{\partial^2w(x,t)}{\partial x^2}-a\frac{\partial^4 w(x,t)}{\partial x^4}-b\frac{\partial^2 w(x,t)}{\partial t^2}=0$$

...yes, already have the solution for the problem w/o p(x) term, or at least I can get the separated equations easily enough, and the solution to the space equation is readily available, though am not sure how it is arrived at(undetermied coefficients?) but of course at that point am stumped, know next to nothing about pde's. The p(x) term represents the centripedal force due to rotating the beam at constant angular velocity $$\omega$$, and omega will be a parameter to be varied. Actually need to compare first few natural frequencies and modes with the unrotated beam, checking effect of $$\omega$$.
Now advice so far is to solve the equation, but can't one just formulate an eignevalue problem, then use Rayliegh-Ritz or some such scheme? ie it seems harder to get a solution to the equation, then having to get the frequency equation, etc. etc.?? instead of pluggin into a RR scheme. This problem would be self adjoint, no?? Thanks in advance for any comments...

 

[saltydog]

Thanks, well, it's a beam equation, so know the boundary conditions. The p(x) term represents the centripedal force due to rotating the beam at constant angular velocity $$\omega$$, and omega will be a parameter to be varied.

Would you kindly specify the initial and boundary values. You mentioned p(x) is the centripedal force. What is the precise form of p(x) and how is centripedal force given in terms of a function of the distance along the beam (the x-variable I'm assuming)? Please pardon my lack of familiarity with the physics of the problem.

I assume the boundary values are:

$$w(0,t)=0$$
$$w_x(0,t)=0$$

$$w_{xx}(L,t)=0$$
$$w_{xxx}(L,t)=0$$

with initial condition:

$$w(x,0)=?$$

 

[athanatos]

thanks for your reply,

addressing your notes;

[1] Yes, those are the boundary conditions

[2] Initial conditions, well, not sure I can say anything too coherent on those,
but will try...Basically, need to find the first few natural frequencies and mode shapes, so would want to solve the homogeneous problem?
Up to now have followed this procedure...get a solution with some constants, then use the bc's to whittle down some of the constants, then use solution with bc's and algebra to arrive at a frequency equation, initial conditions have not been mentioned so far in the course(of course do remember them from ode course many years ago), then from freq eq get natural freq's. Hope this helps somewhat to clarify what the task is here, am far from confident on what's relevant, basically learning modelling, and the pde solving is ancillary, unfortunately don't have any experience with this stuff.

[3] Re the p(x): here is the expression for it;

$$p(x)=\int_{x}^{L}m\omega^{2}\varphi d\varphi$$

where L=length of the beam, and $$\varphi$$ is the difference
between x and L, in doing the integration I get;

$$p(x)=\frac{1}{2}m\omega^{2}(L^{2}-x^{2})$$

though am not sure I have done this correctly, I think it should be something like this. Physically, this is an axial force along the beam, and it effects the transverse vibration of the beam as it it rotated at some constant angular velocity $$\omega$$.

 

[saltydog]

Alright, so we have:

$$\frac{1}{2}m\omega^2(L^2-x^2)\frac{\partial^2w}{\partial x^2}-a\frac{\partial^4 w}{\partial x^4}-b\frac{\partial^2 w}{\partial t^2}=0$$

$$w(0,t)=0$$
$$w_x(0,t)=0$$
$$w_{xx}(L,t)=0$$
$$w_{xxx}(L,t)=0$$

w(x,0)=some initial deflection

Anyway, using separation of variables with $w(x,t)=f(x)g(t)$, I get for the space variable:

$$a\frac{d^4f}{dx^4}-\frac{1}{2}m\omega^2(L^2-x^2)\frac{d^2f}{dx^2}+\lambda^4f=0$$

With $\lambda^4$ being the (positive) separation constant which seems to be the case but in general need to test the other two cases as well (that is, negative and zero for the separation constant).

Well, I'll go on a limb here and claim that the solution to this ODE is:

$$f(x)=\sum_{n=0}^\infty a_n x^n$$

with the first four coefficients arbitrary. Really, taken in isolation, this 4'th order ODE is doable via the Frobenius method however I'm not sure about determining the 4 arbitrary coefficients in terms of the boundary conditions but will look into it. So a possible solution to the PDE is:

$$w(x,t)=g(t)\sum_{n=0}^\infty a_n x^n$$

Well, $\lambda$ isn't really known so that might add problems to solving the ODE. I think I'll try to solve it anyway (for a constant lambda first).

 

[athanatos]

OK, thanks.

Yes regarding $$\lambda^{4}$$, that is a question! Typically, I do know one gets the so called "frequency" equation when pde allows closed form solution, and solves it numerically, and gets the natural frequencies, which can then be substituted back into the solution to the space ode, to get the modes(I think this is correct). I was thinking about just solving both time and space equations numerically, then transorming the solution to the pde into the freuqncy domain to get the natural frequencies , but one has to first estimate $$\lambda$$,
and it looks like some kind of iterative process? would be needed. Also, this is obviously not a discrete but a continuous system, so not at all clear how to get the freq domain in such a case?

Also, you mentioned Frobenius method. In poking around, the FM is always linked with solving second order ode's, and only to be used when the variable coefficients do not provide a regular point, but at least provides a regular singular point. Am guessing one must reduce the fourth order ode to two second order equations to use FM?