What are the key steps for drawing a mass-spring free body diagram?

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To draw a mass-spring free body diagram for two equal masses connected by three springs on a frictionless table, each mass experiences two spring forces: one from the wall and one from the adjacent mass. The equilibrium condition indicates that all three springs will have the same extension due to identical spring constants and masses. The horizontal displacements Y1 and Y2 represent the positions of the first and second masses, respectively, starting from rest. The equations of motion for each mass should exclude unnecessary displacement terms, focusing instead on the relative displacements between the masses. Clarifying these relationships helps in accurately setting up the problem for further analysis.
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I am trying to get started on a problem. I have two masses M1 = M2 and three springs all with the same constant. The weights and springs are on a frictionless table and the outer springs are attached to walls

Wall ~~~Mass~~~Mass~~~Wall

How do I draw the free-body diagram? I am confused about the forces. I know each mass has a spring force pulling it towards the wall and the middle spring should be a force on each mass in the opposite direction that the wall spring force is. Are these the only forces? Or do I need to show both wall spring forces on each mass?
 
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Each mass connects to two springs and thus has two spring forces acting on it.
 
Yeah for each mass two spring forces are acting on it...thats all...
so for equilibrium and same spring constants and masses ... alll 3 springs will show same extension/deflection...
 
R Power said:
Yeah for each mass two spring forces are acting on it...thats all...
so for equilibrium and same spring constants and masses ... alll 3 springs will show same extension/deflection...

So I need to find the horizontal displacements Y1 and Y2 with Y1 the displacement on the first mass in the right direction and Y2 the displacement on the 2nd mass in the right direction.

The motion is started from rest with Y(0) = [1 1] and Y'(0) = [0 0]

Y(0) and Y'(0) should be vertical vectors, i.e one column but I did not know how to represent them that way.


I am also given that M=3kg and K=9N/m

I just want help in setting it up. I am coming up with these two equations but I don't know if it is right:

For the first mass -k(y1 +d1) + k(y2-y1+d2) = my''
For the 2nd mass -k(y2-y1+d1) + k(y2+d2) = my''

I'm not sure whether I should have the d terms in there or not (d is displacement) since they are starting at rest.
 
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the d terms need not be there i think...then the equations seem correct.
Yeah! no d terms I'm pretty sure!

Reason: (y2 - y1) itself is the deflection of the second spring (starting from left) and y1 is the displacement of the first spring. So there is no way these extra d1 and d2 come from.

Actually I mean- d2= y2 - y1
d1= y1
 
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R Power said:
the d terms need not be there i think...then the equations seem correct.
Yeah! no d terms I'm pretty sure!

Reason: (y2 - y1) itself is the deflection of the second spring (starting from left) and y1 is the displacement of the first spring. So there is no way these extra d1 and d2 come from.

Actually I mean- d2= y2 - y1
d1= y1

Thanks. I'm still working on it but that helps. I'm a 50 something engineering masters student and this stuff is not coming back to me very easy as it's been a long time since I got my BS. Of course back then, resources like this weren't here either!

Thanks!
 

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