# Motion of a bead threaded onto a vertical ring with friction using a differential equation

There are similar questions posted but unfortunately all deal with the case of no gravity.

I am trying to create a general model for the motion of a small bead of mass $$m$$ which has been threaded onto a circular ring or radius $$r$$ which is fixed in a vertical plane. The coefficient of friction between the ring and the bead is $$\mu$$.

Initially (at time $$t = 0$$), the bead is located at the left-hand point on the ring which is at the same elevation as its centre, and is being pushed downwards at a speed of $$u$$ (see image):

My attempt was (where $$\omega$$ and $$\alpha$$ are angular speed and angular acceleration respectively):

$$(1)$$ Resolving forces radially: $$R - mg \sin \theta = mr\omega ^2$$

$$(2)$$ Resolving forces tangentially: $$mg \cos \theta - \mu R = mr \alpha$$

Rearranging $$(1)$$ for $$R$$ and substituting it into $$(2)$$ gives: $$r \alpha = g \cos \theta - \mu r \omega ^2 - \mu g \sin \theta$$ which translates into the differential equation

$$\theta '' + \mu (\theta ')^2 + \dfrac{g}{r}(\mu \sin \theta - \cos \theta) = 0$$

The fact that this equation has no analytical solution isn't an issue in my case, but numerically solving it (with sensible values $$r = 1$$, $$g = 9.81$$, $$\mu = 0.1$$, $$\theta (0) = 0, \theta '(0) = 1$$) and graphing the results shows that the oscillations do not die down as expected due to friction, meaning my equation must be wrong.

Any help in correcting the model is much appreciated!

(Please note I am aware this problem can be solved using conservation of energy. I specifically want to explore the differential equation approach.)

• i doubt velocity would remain the same? , it is a complex situation of gain of energy, through a change in gravitational potential energy, so it is very to from differential equation of like this system which includes energy gain through this,l May 28, 2020 at 13:34
• @Yuvraj His EoM doesn't assume velocity remains the same. $\theta'(t)$ is the instantaneous angular velocity (in time $t$). Obviously it changes all the time, hence the need for a differential equation.
– Gert
May 28, 2020 at 13:38
• I can't find anything wrong with the EoM. Can you describe your numerical method and the results a little more?
– Gert
May 28, 2020 at 13:41
• @Gert The solver I am using is Runge-Kutta 4th order. I saw somewhere that this method isn't great for problems involving friction (which can suddenly change direction, as pointed out in the answer) but seems to have worked fine in my case. May 28, 2020 at 14:18

One choice would be $$mg \cos \theta - \mu R\cdot\text{sgn}(\omega) = mr \alpha$$