How Can a Non-Homogeneous 4th Order PDE for Cantilever Beam Vibration Be Solved?

[harsha@iitm]

Hi everyone... I'm new to this fourm...Can anyone tell me how a non homogenous 4th order pde can be solved. its a cantilever beam transverse vibration problem. can i use variables seperable? i got solution for homogenous eq... but how abt non homogenous part.

the eq is
$$\frac{\partial^4 y(x,t)}{\partial x^4} -A*\frac{\partial^2y(x,t)}{\partial t^2}=f(x,t)$$

 

[Chris Hillman]

Two suggestions for homogeneous vibration equation

That's a version of the Euler beam equation... oh, I see you already know that! OK, first try setting f=0 and consider the initial value problem
$$y_{tt} = A \, y_{xxxx}, \, y(0,x) = p(x), \; y_t(0,x) = q(x)$$
..oh, I see, you already solved that!

Well for the benefit of others: for an elegant solution using integral transforms, try using Laplace transform (wrt t) then Fourier transform (wrt x), then solve the resulting algebraic equation, then inverse Laplace and inverse Fourier (in that order--- you can think about why this is OK!) This also works in higher dimensions and is a lovely illustration of the power of integral transforms, as is a similar analysis for the usual wave equation. (Unfortunately, in many other examples it is not so easy to find the requisite inverse transforms of the functions which arise in the course of trying to follow this method!) If you are familiar with Lie's method of solving PDEs (even nonlinear PDEs) using internal symmetries, these also provide useful information here and for analogous equations formulated in noneuclidean backgrounds. It can be helpful to observe that by analogy with constructing "conjugate harmonic" solutions (up to additive constant) to the Laplace equation, we can construct "conjugate" solutions to the beam equation (up to a more complicated additive term).

For the inhomogeneous vibration equation, for some f the integral transform approach might still bring joy. And "group analysis" as per Lie will pick out those terms f which yield simpler "point symmetry groups" of the inhomogeneous beam equation; in these cases symmetry approaches (among others) will yield more solutions. What happened when you tried separation of variables? If all else fails, you can look through eqworld.ipmnet.ru/ (seems to be down at time of posting)

 

[harsha@iitm]

Thank u very much chris. that was really helpful...i'll try solving it. with variables seperable i dnt know how to solve for particular integral for a pde . f(x,t) in my case is

$$f(x,t)= a\sqrt{x}*(cos(\omega_{1}t)+cos(\omega_{2}t))$$

i'll try with laplace and Fourier transforms and let u know if i have any further problem.

 

[Chris Hillman]

Green with envy

Oooh, harsha, you lucky fellow, doesn't that (the integral transform method) just work out beautifully?!

I should credit this method of solution to Boussinesq of Boussinesq equation fame from somewhat before (I think) the turn of the last century. I learned about it from the textbook of Graff on nonlinear waves in elastic solids, who I guess learned about it from the famous old textbook of Morse and Feshback, whom I guess learned it from the original paper of Boussinesq. I think the original paper might be cited in the CRC book by Polyanin, who is also the creator of EqWorld.

Harsha, this is such a nice example that now I'm itching to write about it, but I'll restrain myself until you've had a chance to work it out for yourself. Slightly revised hint: first try the homogeneous initial value problem (IVP)
$$u_{tt} + a^2 \, u_{xxxx} = 0, \;<br /> u(x,0) = p(x), \; u_t(x,0) = q''(x)$$
(where the reason for the double derivative wrt x in the last equation will become clear when we consider conjugate solutions of the beam equation!) and then start playing with some inhomogeneous terms. The key is to recall the relation between convolution and ordinary product when we evaluate the Fourier transform! Also, to pay attention to domains (we probably want to restrict to $x>0$ for your function, or else cut off to the left). The answer is (if I haven't goofed) a sum of convolutions, the first two terms agreeing with the Boussinesq solution of the homogeneous IVP and the others coming from your f(x,t).

If there are any lurkers out there with a yen to learn about PDEs generally (good for you!), try Ronald W. Guenther and John W. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations, Dover reprint, which has a nice (but more elementary) discussion of the beam equation! An excellent book for integral transforms is Dean G. Duffy, Transform Methods for Solving Partial Differential Equations, CRC, 1994.