# Coulomb damping time calculation

I'm designing a movable cover on rails under a spring and with added friction to dampen the movement. The model that describes it's movement the best is Coulomb damping model. The cover is moved from it's equillibrium position to one side and realeased. I want to calculate the time it's needs to get back to equillibrium position.

The paper on the link bellow defines the equations. https://www.studocu.com/en-us/document/university-of-toledo/mechanical-vibration/lecture-notes/ln-6-coulomb-friction/3247643/view

The initial preload distance $$x_0$$ needs to be large enough so that the spring force is larger than the static friction.

My problem is that when I put the values in the equation and try to calculate the time mentioned above I get the error from the acos() part of the function, as the value in the acos() brackets is smaller than -1 even though that the force in the spring is larger than the friction. If I make the spring stiffness larger, then the equation works, when the part in the acos() brackets is larger than -1. The equation I'm using is on the picture in the link bellow.

$$x(t) = \left(x_0- \frac{\mu m g }{k}\right) \cos\left(\omega_n t \right) + \frac{\mu m g}{k}$$

I really do not understand what this limitation means?

## 2 Answers

Although its very difficult to follow your question and the paper, the way you are presenting them, I think that what is happening is that if the value of $$x_0$$ is low and the spring is not stiff enough the the force of friction manages to stop the lid in its tracks. So the time you require is infinite.

That is why when you increase the stiffness of the spring for the same $$x_0$$, you get a larger force. The larger force means that the mass gets enough kinetic energy to move up to zero deflection.

• I'm sorry if my formation of the question is not ok. Do you perhaps have any pointers on how to present it better? Nov 2 '20 at 7:13
• @user2882635 I would replicate the relevant equations in your post. And/or paste clear images of the equations which are displayed in the post. Pointing to a paper which is 6 pages long with no pages number, and multiple exercises, made it hard for me to guess which equations you refer to. Nov 2 '20 at 7:35
• thanks for the comment! I will most definitely keep these pointers in mind Nov 9 '20 at 8:23

The answer is given near the top of the page:

The body will only move if the spring force is greater than the static friction force: $$kx_0 > u_smg$$.

If the spring constant is too small and the body doesn't move at all, you will get impossible values for the $$\text{acos}()$$ function when you assume it did move.