Modeling a mass-spring-damper system

[VitaX]

Homework Statement

Homework Equations

The Attempt at a Solution

This is one of our past homework assignments with the solution given to us. I'm trying to work my way through each part of this assignment but am getting stuck on a few aspects of the model.

My approach:

I first set up a FBD modeling each mass in the system and analyzing the forces acting on that mass. So for mass 1 in part a) I would say there is the spring and damping force acting to the left of the mass and another spring force acting to the right of the mass. What I'm having a bit of difficulty understanding is the the displacement of each mass and how its oriented in the diagram. Do we always take the origin at the center of mass for the object? Is it wrong to place the origin at the wall where spring 1 is connected? I'm getting a little confused on such small matters like this but its effecting my ability to move forward. For instance in part b) I'm confused about the 2nd differential equation modeling mass 2. Why are both forces negative?

 

[rude man]

Think of moving mass 2 a bit to the right. Don't both springs force it to the left?

 

[VitaX]

Hmmm but how I understand the forces on springs in part b) is that the force from spring 1 acting on the wall is towards the right. Thus equal and opposite on mass 1 makes that spring force to the left. That's what I thought you do for spring 3 in part b) as well. You look at the force spring 3 is exerting on the wall and that should be to the left I thought? Then I would say equal and opposite force on mass 2 would make that spring force 3 acting on mass 2 towards the right. Am I looking at this completely wrong?

 

[ehild]

You have to relate the spring forces to the displacements. If m2 displaces to the positive direction (to the right) it makes spring 3 shortened. The spring pushes both the wall and m2 away from its centre. The wall experiences a force to the right, and m2 experiences a force to the left from spring 3, opposite to its displacement.

ehild

 

[rude man]

Hmmm but how I understand the forces on springs in part b) is that the force from spring 1 acting on the wall is towards the right. Thus equal and opposite on mass 1 makes that spring force to the left. That's what I thought you do for spring 3 in part b) as well. You look at the force spring 3 is exerting on the wall and that should be to the left I thought? Then I would say equal and opposite force on mass 2 would make that spring force 3 acting on mass 2 towards the right. Am I looking at this completely wrong?

Never mind what the forces on the walls are. Look at the motion of the two masses. For each mass, if you push the mass to the right, both springs apply force to the left. One pushes, one pulls, but they both apply force to the direction opposite to motion.

 

[VitaX]

Ah now that makes sense. I should have thought of it like that. By the way, when dealing with the spring k2 in these problems, is the reason why you have to take the position x2 into account when looking at mass 1 because the masses are connected via that spring? I guess what I'm saying is at first I was a little confused why its written as k2(x2-x1) for the differential equation for mass 1.

 

[CAF123]

Ah now that makes sense. I should have thought of it like that. By the way, when dealing with the spring k2 in these problems, is the reason why you have to take the position x2 into account when looking at mass 1 because the masses are connected via that spring?

Yes, when m2 is displaced to the right, this lengthens the spring but also acts to shift m1 (in effect to contract the spring) so we compensate for that with the minus.

I guess what I'm saying is at first I was a little confused why its written as k2(x2-x1) for the differential equation for mass 1.

The spring force on m2 due to k2 acts to the left. Since m1 and m2 are connected via k2, there is a force on m1 to the right due to this spring. That is why there is a plus '+k2(x2-x1)'

 

[ehild]

Ah now that makes sense. I should have thought of it like that. By the way, when dealing with the spring k2 in these problems, is the reason why you have to take the position x2 into account when looking at mass 1 because the masses are connected via that spring? I guess what I'm saying is at first I was a little confused why its written as k2(x2-x1) for the differential equation for mass 1.

Assume that the positive direction is to the right and x2>x1.
The increase of length of spring 2 is x2-x1. It exerts the force k(x2-x1) inward on the masses attached to the ends: So m1 experiences k(x2-x1) positive force (to the right) and m2 experiences -k(x2-x1) force (that is, force to the left).

Note that the spring forces on mass m_i are of opposite sign as the "own" displacement x_i and of the dame signs as the displacements of the other masses.

ehild