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?

  • $\begingroup$ 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

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.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.