Two Spring System: Steady State Motion

[mwkfun]

Homework Statement

Consider a system consisting of two springs suspended from the ceiling. The first has a spring constant k-1, the second k-2. They are connected by a mass m and the second spring also has a mass m connected at the bottom. A periodic force is applied to the upper mass. What is the steady state motion for each mass?

Homework Equations

Fcoswt
x1 = Acos wt
x2 = Bcos wt

The Attempt at a Solution

 

[tiny-tim]

Welcome to PF!

Hi mwkfun! Welcome to PF!

Show us what you've tried, and where you're stuck, and then we'll know how to help.

 

[mwkfun]

I tried two different ways...I attempted to turn the system into an electrical circuit equivalent and I tried with free body diagrams and the following equations:
mX=mg + k1x1 + k2x2 - Fcoswt
mX=mg + Fcoswt - k1x1 - k2x2

(For these two eqns. the X means X double dot indicating acceleration)

mX2 = mg + k2x2 - Fcoswt
mX2 = mg + Fcoswt - k2x2

(again, X2 indicates x double dot...I don't know how to type it the correct way)
I think I should be solving for x1 and x2, but I am not sure if I have the correct equations. The eqns. reflect the free body diagrams I have drawn. Thanks for any help you can give.

 

[mwkfun]

I am also wondering whether I should use the x1= Acoswt and x2 = Bcoswt as a substitutions or if they are specific solutions to the differential eqns.

 

[tiny-tim]

… I tried with free body diagrams and the following equations:
mX=mg + k1x1 + k2x2 - Fcoswt
mX=mg + Fcoswt - k1x1 - k2x2

(For these two eqns. the X means X double dot indicating acceleration)

mX2 = mg + k2x2 - Fcoswt
mX2 = mg + Fcoswt - k2x2

Hi mwkfun!

(on this forum, it's best to use dashes instead of dots: X₁'' )

I don't understand why you've put each of these equations in pairs

Anyway, you need to take into account that the displacement of the lower mass is not the same as the displacement of the lower spring … it's the displacement of the lower spring minus the displacement of the upper spring

 

[mwkfun]

Thanks so much for your help. I think I got it!