• Aug 29th 2011, 10:46 PM
ILoveMathematics27
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 :)
• Aug 30th 2011, 02:43 AM
ILoveMathematics27
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
• Aug 30th 2011, 05:38 AM
Jester
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.
• Aug 30th 2011, 10:07 AM
ILoveMathematics27