Recently we have been preparing to conduct an experiment with a rod sliding on a table. You might think that this sort of thing is fairly well understood by mathematicians and engineers and usually that is the case. However it turns out that even this fairly simple problem, combining rotation with dry friction, can break the normal rules of mechanics. If the rod is moved in the right way it will go into a particular configuration known as the Génot-Brogliato point. This is a particular angle and spin-rate where the standard physical equations for predicting the motion of the rod break down. As a result, if the rod reaches the Génot-Brogliato point, no one knows what will happen afterwards!

We will attempt to explain this below but first take a look at the video we made using a high-speed camera:

What you can see is a “rod” made of acrylic sliding and rotating along the surface of a table. We filmed this at 70000\,fps which seems unnecessary in retrospect so we’ve had to downsample that for the video. The Génot-Brogliato point and other interesting behaviour are not happening yet but that’s to be expected because the rod isn’t made of the right material.

So why are we investigating a rod sliding on a table? This is the simplest example of a system that demonstrates the Painleve paradox (explained below). The same paradox occurs in more complex systems and is of interest in engineering, for instance in the control of robotic manipulators. Applications affected by the paradox can be quite complex but right now we’re not even sure what happens in simple cases. The sliding rod is just a starting point: we need to fully understand this before we can confidently tackle more complex problems.

Although this looks like a fairly simple physics problem it can actually get fairly complicated! So let’s take a step back and try to understand some simpler situations first.

Consider this mass on a table that is pushed by a spring:

The two forces (shown as arrows) acting on the mass are the force *F _{s }*exerted by the spring and the reaction force

*F*from the table. Let’s assume that these forces are much larger than the force of gravity so that we can ignore gravity while thinking about this.

_{t}Now Newton’s laws can be used to understand this situation in two different ways. Newton’s first law says that if the mass is not moving then the forces acting on it must all cancel out. This tells us that *F _{s}=F_{t}*. The reaction force in the picture will take whatever value it needs to in order to cancel the spring force. However we have assumed that the spring is pushing*downwards*. Actually the spring could also pull *upwards*. In that situation for the forces to balance the reaction force would need to be negative so that it is pulling the mass downwards. We normally don’t think that a negative reaction force is possible unless the contact is “sticky” e.g. if the mass is glued to the table. So we say that it is only possible for the reaction force to be positive (upwards).

So if the spring is pulling the mass upwards, it isn’t possible for the reaction force to cancel the spring force. In this situation we assume that there won’t be any reaction force (*F _{t}=0*). Since the forces don’t cancel out, Newton’s second law comes into play. This law says that the sum of the forces acting on the mass determines its acceleration through

*F=ma*. So since the total force is the spring force the mass accelerates upwards, lifting off the table.

This mass on a table does not do the interesting things that we hoped to see in our experiment. For those we need two extra ingredients: rotation and dry friction. However this first example does illustrate one basic idea. Either a) the mass remains in contact with the table because the reaction force balances any other forces to keep it there or b) the mass lifts off and there is no reaction force.

Now let’s look at friction. Let’s suppose that our mass is now pushed by the spring downwards and to the side:

There are two types of friction force. One is static friction which is the force that stops our mass from slipping on the table when it is not moving.The other type of friction force is one that acts while the mass is slipping on the table. We are interested in the *slipping* case.

Our mass is moving to the right with speed *v*. In this situation the reaction force *F _{t}* will cancel the *vertical* part of the spring force

*F*. The friction force

_{s}*F*is in the opposite direction to the movement

_{f}*v*and is proportional to the reaction force:

*F*=

_{f}*μF*where

_{t}*μ*is the *friction coefficient*. This model of friction is known as Coulomb friction and is standard as a basic friction model in engineering.

There is another way of thinking about Coulomb friction for a slipping mass. We can combine the reaction and friction forces into a single contact force *F _{c}* that must act in a particular direction.

The direction of the contact force is given by the angle *θ _{μ}* which is related to the friction coefficient by

*tan(θ*. A larger friction coefficient gives a larger friction angle. A very smooth surface would have a small friction angle so that the contact force would be almost vertical. This way of thinking about Coulomb friction helps in understanding our sliding rod.

_{μ})=μ

We are now ready to understand the final piece of our experiment: rigid-body rotation. In our experiment we have a rod sliding on a table at the same time as rotating:

The standard laws of rigid-body mechanics tell us that the forces acting on the rod will generate *moments* which cause the rod to rotate. If the rod is at an angle below the friction angle the contact force will make it rotate anti-clockwise towards the table. If it is above the friction angle it will rotate clockwise.

Now in the clockwise case it is possible for interesting things to happen. The important point to understand here is that it is the *end* of the rod that is sliding on the table. Newton’s second law *F=ma* applies to the centre of the rod, and not necessarily to the end. The contact force causes the centre of the rod to accelerate diagonally upwards. However it also causes the rod to rotate clockwise which has the effect of accelerating the end of the rod downwards. With high enough friction it is possible that even though the contact force is acting upwards it can cause the end of the rod to accelerate downwards: this situation is the classic Painlevé paradox.

The Painleve paradox leads to the following unresolvable situation:

1) The rod is sliding but the equations for sliding require a negative contact force (not possible without glue!)

2) The rod cannot lift off because if there was no contact force the end of the rod would accelerate downwards back into contact with the table.

We call this the *inconsistent case*: the rod can’t continue sliding and can’t lift off either!

There is another scenario that arises from the Painleve paradox which is a situation in which the rod could lift off but the contact force can also *pull* it down keeping it in contact. We call this the *indeterminate case*: we don’t know whether it would continue sliding or lift off.

We normally try to analyse this situation using what we call a phase plane. The phase space for the sliding rod can be represented in two dimensions and the interesting part of it looks like this:

The two axes in this space represent the angle of the rod and the *angular velocity* of the rod (how fast it is rotating). At any time while the rod is sliding we can think of its “state” as a point on this plane. Newton’s laws, the Coulomb friction, and the rigid-body equations together tell us how the state of the rod evolves over time. We think of them as defining trajectories in the phase space.

In our experiment our rod slides down a little ski-slope and then onto the table. Then the rod has some particular angle and angular velocity so it starts at an initial point in the phase space. It should then follow the trajectory that goes through that initial point. If the friction coefficient is high enough then many of these trajectories will go through the Génot-Brogliato point. Here all the normal equations break down. Physically this means that the rod would be sliding along rotating upwards until it reaches a critical angle. At that point the rod will have reached the Génot-Brogliato point and we don’t know what will happen afterwards.

Many mathematicians have theorised about the possibility of reaching the Génot-Brogliato point over the past 100 years (including some here in Eng Maths [1,2]). However being mathematicians, none of them has attempted to physically verify their predictions.

We are in the process of getting a new rod machined to our precise specifications.

[1] Nordmark, A & Varkonyi, P. & Champneys, A., 2017, Dynamics beyond dynamic jam; unfolding the Painlev$\backslash$’e paradox singularity, ArXiv e-prints

[2] Kristiansen, KU & Hogan, SJ, 2018, ‘Le canard de Painlevé’. SIAM Journal on Applied Dynamical Systems, vol 17., pp. 859-908

Written by Oscar Benjamin and Yani Berdeni – Department of Engineering Mathematics, University of Bristol UK