![rolling disk graph rolling disk graph](http://www.aimsciences.org/fileAIMS/journal/article/jgm/2020/1/PIC/1941-4889_2020_1_53-6.jpg)
** Run this code first to obtain the symbolic equations of motion called by the main script. Solve the rolling disk’s equations of motionĭerive the rolling disk’s equations of motion ** The animations in Figures 2, 3, and 4 were generated using the following MATLAB code: Main script: Animation of a thin disk rolling under the action of a force that causes the disk to turn right. It is important to note that the force is applied throughout the duration of the simulation, causing the disk to accelerate and travel in an outward spiraling path.įigure 4. An animation of this simulated behavior is provided in Figure 4. We can cause the disk to initiate a right turn by exerting a forward force on the right side of the axle at a distance from the disk’s center. Suppose the disk now includes a very light axle that passes through the disk’s center along its spin axis. However, their physical interpretation is still an open question.įinally, we may extend our analysis of the rolling disk to very simply demonstrate the counterintuitive nature of making a turn on a bicycle or motorcycle: to turn right, for example, one must push forward on the right handlebar. (4) Surprisingly, the rolling disk has associated with it two other conserved quantities that were discovered in the late 19th century by Appell, Chaplygin, and Korteweg (see ). Animation of a thin rolling disk undergoing a steady motion in which the instantaneous point of contact with the horizontal surface traces out a circle.īecause the instantaneous point of contact with the horizontal surface has zero velocity, the static friction force acting on the disk does no work, and thus the disk’s total mechanical energy is conserved: Taking the state vector, we haveĪssuming the horizontal surface is sufficiently rough to prevent slipping, a typical simulated rolling motion of the disk is animated in Figure 2, where we trace out the path of the instantaneous point of contact with the surface to further highlight the disk’s behavior.įigure 3. We may conveniently express this system of equations in the first-order form suitable for numerical integration in MATLAB. The resulting equations of motion from applying the balance laws coupled with the non-integrable constraints form a set of differential equations that describe the disk’s orientation and the lateral translation of its mass center over time. The disk’s weight, the vertical reaction force exerted by the surface on the bottom of the disk, and the static friction force acting at the instantaneous contact point comprise the net force and contribute to the net moment about the disk’s center of mass. We may obtain the disk’s equations of motion by applying a balance of linear momentum and a balance of angular momentum with respect to the disk’s mass center: and, respectively. One of these constraints, (that is, the disk maintains contact with the surface), is integrable (i.e., holonomic), while the remaining two constraints, and, are non-integrable (or non-holonomic). (1) The disk is subject to three constraints arising from the fact that the instantaneous point of contact with the fixed horizontal surface must have zero velocity for the disk to roll without slipping. Using a 3-1-3 set of Euler angles, are related to, , and and their rates of change as follows: The angular momentum of the disk about its mass center is then, where denote the corotational components of the disk’s angular velocity. The disk’s instantaneous point of contact with the surface is located relative to the center of mass according to, where the basis vector is attached to the disk but not spinning with it, i.e., the basis and the corotational basis are separated by a spin about the direction.īecause the disk is axisymmetric, its principal moments of inertia and. We assume the disk’s reference configuration is such that the disk lays flat in the horizontal plane, and therefore the disk’s spin axis corresponds to the corotational basis vector. The orientation of the disk is parameterized by set of Euler angles:, , and. The bases and are related by a spin about the direction. The basis, which is attached to the disk but not spinning with it, more conveniently describes the location of the disk’s instantaneous point of contact with the horizontal surface. Schematic of a thin rolling disk illustrating the alignment of the corotational basis. We locate the disk’s mass center using a set of Cartesian coordinates,, where is the space-fixed basis, and thus the disk’s linear momentum. Consider a thin axisymmetric disk with mass and radius that rolls without slipping over a stationary and rough horizontal plane, as illustrated in Figure 1.