Is 0,0 the Only Equilibrium Point in a 2 D.O.F. Spring-Mass-Damper System?

[eddie_spaghetti]

Hi there, thanks for taking the time to read my post, any help would be appreciated.

I am analysing a 2 D.O.F. spring-mass-damper system, which is laid out in the following way (apologies for the lack of a proper diagram, my scanner is dead):wall---spring1----mass1----spring2-----mass2----spring3----wall
wall---damper1---mass1----damper2----mass2----damper3--wall
| |
|--->x1 |--->x2There are 2 time dependent forces (F1 and F2) acting on the masses and the coordinates x1 and x2 are chosen such that the springs exert zero forces when x1 and x2 are equal to zero.

I have drawn the free body diagrams for each mass, and written the equations of motion for each as;

I missed out a couple of details but I hope my maths is correct there.

I have then been asked to find the equilibrium points( ie. the points at which xdot and xdoubledot=0.) of the system when the input forces F1 and F2 are 0. There is clearly one at x1 and x2 = 0.

Am I wrong in thinking that 0,0 is the only equilibrium point?

[edit]Think I've got it...I think I should be looking for the other equilibrium points as functions of the spring constants.[/edit]
Thanks,
Ed

 

[KataKoniK]

Dear Ed,

Thank you for sharing your post and your analysis of the 2 D.O.F. spring-mass-damper system. It seems like you have a good understanding of the system and have correctly identified the equilibrium point at x1 and x2 = 0.

To answer your question, yes, 0,0 is the only equilibrium point for this system when the input forces F1 and F2 are 0. This is because the springs are exerting zero forces at this point, and the masses are not moving (xdot = 0) and not accelerating (xdoubledot = 0).

However, if the input forces are not 0, then there may be other equilibrium points that can be found as functions of the spring constants. It would be helpful to have more information about the specific values of the spring constants and the input forces in order to determine these equilibrium points.

I hope this helps and wish you the best of luck with your analysis!

 

[piercas]

Hi Ed,

I can confirm that your understanding of equilibrium points is correct. In a spring-mass-damper system, the equilibrium point is where the forces acting on the masses are balanced and the system is at rest (xdot and xdoubledot = 0). In this case, the equilibrium point is indeed at x1 and x2 = 0, where the forces from the springs and dampers are equal and opposite.

However, depending on the specific values of the spring constants and masses, there may be other equilibrium points in the system. These points can be found by solving the equations of motion for xdot and xdoubledot = 0, as you mentioned. These additional equilibrium points may not necessarily be at x1 and x2 = 0, but they will still represent a balanced state of the system.

I would recommend exploring the system further and solving for the equilibrium points as functions of the spring constants. This will give you a better understanding of the behavior of the system and how different parameters can affect its equilibrium. Good luck with your analysis!