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Could you please help me solve a system of 3 ODEs?

Hello. I'm working on a problem involving 3 masses connected by springs. One of the masses has a driving force Focos(wt). These masses move along a line.

Using the Euler-Lagrange equations (I know some simple classical mechanics) I have derived the equations of motion, shown below:

[IMG]http://img849.imageshack.us/img849/9243/mathshelp.png[\IMG]

k is the spring constant, m is the mass of each of the three objects. The displacements from equilibrium for each object are given by x, y and z. At the end I just convert from equilibrium displacement to actual position by adding on the equilibrium positions :)

I have spent days trying to figure out how to solve them analytically but I have had no luck. I have been going through a method in Goldstein's Classical Mechanics but cannot get things to work :( I have put the equations into matrix form, hoping that will help. I would love any help :)

I have a numerical solution of this, and I am confident it is accurate, so I have a comparison ready :)

Re: Could you please help me solve a system of 3 ODEs?

Okay, I have a solution to the homogenous system of equations (so without the driving force) but I am struggling to find a particular solution that will allow me to solve the non-homogenous system of equations.

I have tried many solutions, as shown below.

Note that for simplicity I have taken k,m,Fo and omega all to be 1.

http://img717.imageshack.us/img717/2430/mathshelp2.png

Re: Could you please help me solve a system of 3 ODEs?

Here's a suggestion. Your equations are

$\displaystyle \ddot{x} =\frac{k}{m}( y - x) + \frac{F_0}{m} \cos \omega t$

$\displaystyle \ddot{y} = \frac{k}{m}(x+z-2y)$

$\displaystyle \ddot{z} = \frac{k}{m}(y-z)$

If you add the three equations together you get

$\displaystyle (x+y+z)'' = \frac{F_0}{m} \cos \omega t$.

This you can integrate twice. You get $\displaystyle x + y + z = stuff$

Then eliminate $\displaystyle x + z$ in your second equation to get a single ODE for $\displaystyle y$. Solve. Then return to the first equation and solve for $\displaystyle x$.

On a side note, particular choices for $\displaystyle \omega $ **will make** a difference so be careful.

Re: Could you please help me solve a system of 3 ODEs?

Thank you for your reply.

Before reading it though, I had already solved the system of equations. I didn't use your method, I used matrices and such, finding the eigenvalues and eigenvectors and went from there. The results match my numerical solution.

I get what you are saying about particular choices for omega. My analytic solution won't work for omega = 1, 3^0.5 because those are the natural frequencies. In the numerical solution you do see resonance, as you would hope!

For all other frequencies i used a particular solution of bfcos(omega*t), where b is a vector which i calculated and f = (Fo/m, 0, 0) is the force vector. That worked fine. For resonant frequencies i tried tbfcos(omega*t) but the maths got out of hand pretty quickly.

I'll look at your suggestions in the morning, as I am dead tired now.