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

• eddie_spaghetti
In summary: Your Name]In summary, the conversation discusses the analysis of a 2 D.O.F. spring-mass-damper system with two time dependent forces acting on the masses. The equilibrium point is found at x1 and x2 = 0 when the input forces are 0, but there may be other equilibrium points as functions of the spring constants if the input forces are not 0. More information is needed to determine these additional equilibrium points.
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?

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

Last edited:

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!

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!

## What is the concept of Spring Mass Damper equilibrium?

Spring Mass Damper equilibrium refers to the state at which the forces acting on a spring mass damper system are balanced, resulting in no net force or acceleration. This is also known as the steady state or static equilibrium.

## How is the equilibrium position of a spring mass damper system determined?

The equilibrium position of a spring mass damper system is determined by the equilibrium condition, which states that the sum of all forces acting on the mass at any given point must be equal to zero. This position can also be calculated using Hooke's Law, which relates the displacement of the mass from its natural length to the force exerted by the spring.

## What factors affect the equilibrium position of a spring mass damper system?

The equilibrium position of a spring mass damper system is affected by the mass of the object, the stiffness of the spring, and the damping coefficient. A higher mass or stiffer spring will result in a lower equilibrium position, while a higher damping coefficient will decrease the amplitude of oscillation around the equilibrium position.

## What happens if the equilibrium position of a spring mass damper system is disturbed?

If the equilibrium position of a spring mass damper system is disturbed, the system will experience an unbalanced force and will oscillate around the new equilibrium position. This is due to the spring's tendency to return to its natural length and the damping effect of the damper resisting the motion.

## Is the equilibrium position of a spring mass damper system affected by external forces?

Yes, external forces such as applied forces, friction, and air resistance can also affect the equilibrium position of a spring mass damper system. These forces can alter the net force acting on the system and cause it to deviate from its original equilibrium position.

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