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I seek to model the motion of two coupled oscillating point masses as shown below:Oscillator System

Note that x1(t) models the leftmost point mass and x2(t) is the motion of the rightmost point mass. I would like to include sliding friction and eventually static friction in my model. I have developed the two second order ODEs:

x1''[t] == -(C1/M1)*
   x1[t] + (K/M1)*(x2[t] - x1[t]) + (B/M1)*(x2'[t] - x1'[t])

x2''[t] == -(C2/M2)*
   x2[t] - (K/M2)*(x2[t] - x1[t]) - (B/M2)*(x2'[t] - x1'[t])

where B is the magnitude of sliding friction ([N]) and K = C in the diagram.

I would expect this system to oscillate and come to rest somewhere around the points x1(0) and x2(0). When I numerically solve the 2nd order DE using Wolfram Mathematica NDSolve I obtain the following plot: Simulation output

I am pleased that both masses settle at to one point as t --> infinity but I do not think both should rest at x = 0.

I was suspicious of the spring force acting on the rightmost point mass because I believe it is acting in the positive x direction. When I change x2''[t] to +C2/M2*x2[t] the system becomes unstable and diverges to infinity.

What am I missing where my oscillating masses come to rest at x = 0? Are my forces pointing in the correct direction?

(yes in my system it is possible for both M1 and M2 to pass the "walls" without repercussion.)

I am also wary of my dampening (sliding friction) force. I have experimented with dividing by the magnitude of x2-x1 but this does not yield the results I look for.

Thank you for any help!


Edit 1:

I have anchored the second point mass to the leftmost point L as suggested. Therefore my equations looks like such:

$$x_1''(t) = \frac{-C_1}{M_1}*x_1(t) + \frac{C}{M_1}*(x_2(t)-x_1(t)) - \frac{B}{M_1}*\frac{x_1'(t)}{|x_1'(t)|}$$ $$x_2''(t) = \frac{-C_2}{M_2}*(x_2(t)-L) - \frac{C}{M_2}*(x_2(t)-x_1(t)) - \frac{B}{M_2}*\frac{x_2'(t)}{|x_2'(t)|}$$

Under the influence of sliding friction alone, should my masses come to rest eventually? When I solve the system for B>>1 the masses oscillate as if they were undamped -- why might this be?

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  • $\begingroup$ Hi C. Fuhrman! This site has MathJax enabled so you can properly typeset equations so they are easier for everyone to read. $\endgroup$ – jacob1729 Oct 11 '18 at 21:40
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You need to relate the $C_2$ spring force to the position of mass 2 relative to the anchoring point on the right wall. Currently it is anchored to the origin. So, try $$ (C_2/M_2) (L-x_2(t)) $$ where $L$ is the coordinate of the right wall.

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  • $\begingroup$ i agree with this statement. Can you provide insight on my masses coming to rest under the influence of sliding friction only and not static? See "Edit 1" in original question. $\endgroup$ – C. Fuhrman Oct 11 '18 at 22:15
  • $\begingroup$ I don't think that static friction can be introduced into the equations of motion in such a manner. I would not be surprised if the numerical scheme for solving the equations were unstable if the force changes discontinuously from $+B$ to $-B$ when the velocity changes sign. And your scheme contains no "threshold" force, which is characteristic of static friction. Simply, if the mass is moving, you don't have static friction. I think you have to treat the change between static and moving regimes by hand. If you want to pursue this idea, I suggest that you ask a completely new question. $\endgroup$ – user197851 Oct 11 '18 at 23:56

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