Dynamics of a Rolling Cylinder on an Inclined Plane

We are sometimes intrigued by the dynamics of a rolling cylinder/ball on an inclined plane with or without slipping. This article enlightens this perspective using classical mechanics. Furthermore, it enhances our understanding of the Newton's famous third law of motion in light of bond graph. 

In figure 1 [1], a cylinder of radius r and mass m connected to a spring of stiffness k is rolling on a plane inclined at an angle α. It has a moment of inertia J with respect to an axis perpendicular to the paper plane through its center of gravity. The center of gravity of the rigid body moves at velocity v(t) parallel to the inclined plane. At the same time, the cylinder is rotating with the angular velocity ω(t). At the contact point between the cylinder and the inclined plane, a viscous friction force is acting that is proportional to the relative velocity v_r. The contact point as part of the rolling cylinder has the velocity v − v_t = v − r × ω, whereas the velocity of the contact point as part of the plane is zero. Hence, the relative velocity v_r, effecting the viscous friction force, reads v_r = v − r × ω.

Figure 1: A rolling cylinder on an inclined plane [1].

Figure 2 represents the bond graph model of the system described in figure 1. The cylinder has both translational and rotational velocity; and in the bond graph, these are highlighted by mass (I : m) and moment of inertia (I : J). The angular velocity ω is simply obtained by dividing tangential velocity with its radius r. And, the friction force plays the key role to make the relationship between translational and rotational velocities avoiding rigid mass coupling. The friction force is modelled by resistive element R which is dependent on the difference between C.G. translational velocity and tangential velocity at the contact point. Now, I would like to show the similar situation, but without considering the presence of frictional resistance or R.

Figure 2: Bond graph model of a rolling cylinder on an inclined plane [1].

Figure 3 shows a cylinder rolling on an inclined plane without slipping [2]. Here, the spring which was added in previous example is just omitted and there is no friction force at contact. The horizontal component of gravitational weight is the driving force for this rolling cylinder.

Figure 3: A cylinder rolling on an incline plane without slipping [2].

Here, I want to show little bit of math to show the dynamics of the system in figure 3. How the ball is rolling without slipping and which force contributes to that motion will be reflected by the following calculations.

The kinetic energy of the system is,

This problem presents itself with two generalized coordinates (y and θ) and one equation of constraints, which leaves us with one degree of freedom. We now apply the Lagrange equations with undetermined multipliers,

Here, Q_y and Q_θ are a force and a torque, respectively. Now, the bond graph of the system is depicted in figure 4. If we compare this figure with figure 2, we will see that there is an absence of R element which represented earlier friction and also one of I elements is in derivative causality. The above mathematical treatment is so far correct as it follows Newton’s third law where the gravitational force is the action and reaction we have found by Q_y. But, bond graph says here something more which can immediately be noticed looking into figure 4, where we have one degree of freedom, although the configuration seems to have 2 DOF. And, this poses eventually derivative causality which is certainly not physical although theoretically looks perfect. We can derive equations separately using free body diagram by following classical mechanics, but bond graph represents a total integrated system dynamics, where relevant linkages are necessary between the steps. There is obvious presence of friction in this scenario, and it could be modelled as an R element like before to present actual physical dynamics. Again, this friction force is not the driving force; the actual driving force is the gravity. If the friction would become positive and contribute to the gravity, then this would certainly violate the conservation of energy law. Conservation of energy law is universal both in Newtonian’s and Einstein’s physics, although in Einstein’s physics, mass is not constant.

Figure 4: Bond graph of a cylinder rolling on an incline plane without slipping.

At the end of the discussion, my point of view is, there has to be something at the contact between two rigid bodies, or a rigid body and solid surface, for the transmission of the forces/energies to each of the contact pairs. 

References

[1] Wolfgang Borutzky, Bond Graph Methodology: Development and Analysis of Multi-disciplinary Dynamic System Models, Springer-Verlag London Limited 2010.

[2] Classical Dynamics of Particles and Systems 5th ed - S. Thornton, J. Marion (Thomson, 2004).

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