How Do You Model a Damped Harmonic Oscillator with External Forces?

[gjfelix2001]

Homework Statement

Give the equations of motion of the following system:

http://www.jelp.org/imagenes/mech.jpg

Homework Equations

So, i assume the following cases (the diagram i so deficient).

1) Black Point (x1) is fixed
2) There's a force applied at x1 (black dot)
3) The position of the black dot is a function of time f(t).

The Attempt at a Solution

For 1) I think the equation of motion is:

http://www.texify.com/img/%5CLARGE%5C%21m%5Cddot%7Bx%7D%2Bkx%2BG%5Cdot%7Bx%7D%3D0.gif

It's a dampeda harmonic oscillator, Right?

2) If a force f(t) is applied to the system, then we have :

http://www.texify.com/img/%5CLARGE%5C%21m%5Cddot%7Bx%7D%2Bkx%2BG%5Cdot%7Bx%7D%3DF%28t%29.gif

like the Damped and driven harmonic oscillator, am i right?

3) If by example, the full system (the damper too) is moving to the left (i.e. negative X).

For this, i tried the following:

The position of the mass is X=f(t)+L+lo , where L is the natural length of the spring, lo is the elongation or strecht of the spring. Then, taking the time derivatives of X, and substituing in the damped harmonic oscillator equation (Case 1), i get to:

http://www.texify.com/img/%5CLARGE%5C%21m%5Cddot%7Bf%7D%2Bk%28f%28t%29%2BL%2Bl_0%29%2BG%5Cdot%7Bf%7D%3D0.gif

Please, tell me if i am right in the whole problem. Thanks

 

[Ma77h3w]

I've not done this for a while.. but.. resolve the forces around the mass M
I get something like (cant get latex thing workings so decript!)
m(x''₂) - G(x'₂) + k(x₁ - x₂) = 0

then you know x₁ is a function of time f(t)
so this can be substituted in for x1 i believe
then rearrange so f(t) is subject.
I may be slightly wrong.. but I had a go :)

 

[Ma77h3w]

P.S. force from spring = k( x1- x2)
which is the bit that may have confused you