Coupled Oscillations: Understanding the Equations

[Lagrange53110]

Homework Statement

[URL]http://imgur.com/2KIhk[/URL]

Hi!

If you please look at the image. I have already determined the equations of motion.
They are as follows:

Ma1 = 2kx1+kx2
Ma2 = -2kx2+kx1+kx3
Mx3 = -2x3 +kx2

Now... what I don't understand is how does this physically happen?

If you look at say mass m1. Then when the spring is oscillating it pulls the mass to the left
on the left side of m1 which is I suppose: -kx1 and the spring on the right is also doing that at x1 so then also -kx1 and then when it oscillates further at x2 the right spring goes +kx2 which gives us: ma1 = -2kx1+kx2.

I just want to is this the correct analysis for determining the equations?
I'm just stomped and wish I knew a perfect 100% method on finding these equations.
I just "know" they're right... I just don't know why... sadly

 

[mathfeel]

Homework Statement

[URL]http://imgur.com/2KIhk[/URL]

Hi!

If you please look at the image. I have already determined the equations of motion.
They are as follows:

Ma1 = 2kx1+kx2
Ma2 = -2kx2+kx1+kx3
Mx3 = -2x3 +kx2

Now... what I don't understand is how does this physically happen?

If you look at say mass m1. Then when the spring is oscillating it pulls the mass to the left
on the left side of m1 which is I suppose: -kx1 and the spring on the right is also doing that at x1 so then also -kx1 and then when it oscillates further at x2 the right spring goes +kx2 which gives us: ma1 = -2kx1+kx2.

I just want to is this the correct analysis for determining the equations?
I'm just stomped and wish I knew a perfect 100% method on finding these equations.
I just "know" they're right... I just don't know why... sadly

Can't see your picture, but I think you have a sign error. It probably should be:
$M \ddot{x}_1 = - 2 k x_1 + k x_2 = k(x_2 - x_1) - k x_1$
$M \ddot{x}_2 = - 2 k x_2 + k x_1 + k x_3 = k(x_3 - x2) + k(x_1 - x_2)$
$M \ddot{x}_3 = - 2 k x_3 + k x_2 = k (x_2 - x_3) - k x_3$

In general, if you have N mass, the equation of motion on the j-th one (if j is not the first or last one) that's connected to two neighbors is:

$M \ddot{x}_j = k x_{j+1} + k x_{j-1} - 2 k x_{j}$

 

[Lagrange53110]

Can't see your picture, but I think you have a sign error. It probably should be:
$M \ddot{x}_1 = - 2 k x_1 + k x_2 = k(x_2 - x_1) - k x_1$
$M \ddot{x}_2 = - 2 k x_2 + k x_1 + k x_3 = k(x_3 - x2) + k(x_1 - x_2)$
$M \ddot{x}_3 = - 2 k x_3 + k x_2 = k (x_2 - x_3) - k x_3$

In general, if you have N mass, the equation of motion on the j-th one (if j is not the first or last one) that's connected to two neighbors is:

$M \ddot{x}_j = k x_{j+1} + k x_{j-1} - 2 k x_{j}$

Okay that makes sense! That is the same conclusion I came up with the 3 masses scenario. Because the two xvalues to the left and the right cause positive "tugs" if you will.
What is the equation in general for the endpoint masses.

Would be in your case -2x(j) +2x(j+1)? for the left? and -2x(j)+2x(j-1)?

 

[mathfeel]

Okay that makes sense! That is the same conclusion I came up with the 3 masses scenario. Because the two xvalues to the left and the right cause positive "tugs" if you will.
What is the equation in general for the endpoint masses.

Would be in your case -2x(j) +2x(j+1)? for the left? and -2x(j)+2x(j-1)?

For the end point, it depends on boundary condition. Since I can't see your picture, I am going to assume the j=1 and j=N mass is connected to fix wall by a spring, then

$m \ddot{x}_1 = k (x_2 - x_1) - k x_1 = k x_2 - 2 k x_1$
$m \ddot{x}_N = k (x_{N-1} - x_{N}) - k x_{N} = k x_{N-1} - 2 k x_{N}$

 

[Lagrange53110]

For the end point, it depends on boundary condition. Since I can't see your picture, I am going to assume the j=1 and j=N mass is connected to fix wall by a spring, then

$m \ddot{x}_1 = k (x_2 - x_1) - k x_1 = k x_2 - 2 k x_1$
$m \ddot{x}_N = k (x_{N-1} - x_{N}) - k x_{N} = k x_{N-1} - 2 k x_{N}$

but why?

I am confused how these equations are derived.

I have derived the middle mass equation.

But I am unsure about some conventions I have used...

please can you show me... what happens physically?