Two Springs and Two Masses. Equation of Motion

[harrisiqbal]

Homework Statement

Derive the equation of motion for the following Mechanical System. You need to introduce an additional variable to represent position of the bottom mass.

I drew a picture ;D
http://www.mediafire.com/imageview.php?quickkey=mzkx0mimmwd&thumb=4

Homework Equations

EOM:
Mx(double Dot) = reactionary forces

Bx(dot) is the damping effect: B = damping coefficient times the velocity

The Attempt at a Solution

Now, this is apparently a vibration type problem. But I'm assuming too many things when trying to write the Eq. of Motion.. so I don't feel confident about the question. When the K1 spring is pulled towards the right, M1 stars moving towards the right... Obviously.. Now, a few questions come up, the figure assumes that M2 has a direction to right ( Atleast I think.. that's the Xout in the picture). So what does that mean? The friction between M1 and M2 is static friction so M2 does not slide and just moves to the right until spring K2 pulls it back and then Kinetic friction comes in.

But I was told by a few people, to not think of this as having friction but actually like the masses are damping each others velocities. So the Eq. Of Motion would have Bx(dot) instead of a couple of Kx.

I just need a little bit of direction on this problem.

Heres what I have come up with.

Assuming M2 slides the opposite way as M1 moves to the right. So M2 basically oscillates back and forth just a tiny bit as the bigger mass M2 moves right and left.

So that means that the two masses provide a damping effect on each other.

So if i was to write the EOM: [ displacements have to be defined in the figure obviously so just assume that I defined them right]

(M1 + M2)x(double dot) = K1(X3 - X2) - Bx(dot) - Bx(dot) - K2(Delta S)

But I'm not confident about this because I'm not sure I get the whole scenario conceptually.

Thanks For The Help!

 

[KataKoniK]

Thank you for your post. It seems like you are on the right track with your attempt at the solution. However, there are a few things that can be clarified to give you a better understanding of the problem and how to approach it.

Firstly, it is important to understand the concept of damping and how it affects the motion of the system. Damping is a force that opposes the motion of the system and is proportional to the velocity of the system. This means that the faster the system moves, the stronger the damping force will be. In your system, the masses M1 and M2 provide a damping effect on each other as they move back and forth. This means that the equation of motion for each mass will have a term that includes the velocity of the other mass.

Secondly, it is important to understand that the figure you have drawn is a simplified representation of the system. In reality, M2 would not slide to the right as M1 moves to the right. This is because of the static friction between the two masses. Instead, M2 will remain stationary until the spring K2 is stretched enough to overcome the static friction and start moving M2 to the left. This is where the damping effect comes into play. As M2 moves to the left, it will also provide a damping effect on M1, slowing down its motion.

With these concepts in mind, let's look at the equation of motion for each mass. For M1, the equation of motion would be:

M1x(double dot) = K1(X3 - X2) - Bx(dot) - B(x(dot) - y(dot))

Here, the first term on the right-hand side represents the force exerted by the spring K1, the second term represents the damping force caused by M1's own velocity, and the third term represents the damping force caused by the velocity of M2. Note that the velocity of M2 is represented by (x(dot) - y(dot)) because M2 is moving in the opposite direction to M1.

Similarly, for M2, the equation of motion would be:

M2y(double dot) = -K1(X3 - X2) - K2(Delta S) - B(y(dot) - x(dot))

Here, the first term represents the force exerted by the spring K1, the second term represents the force exerted by the spring K2, and the third term represents the