Rolling friction and energy dissipation in a spinning disc

Abstract

This paper presents the results of both experimental and theoretical investigations for the dynamics of a steel disc spinning on a horizontal rough surface. With a pair of high-speed cameras, a stereoscopic vision method is adopted to perform omnidirectional measurements for the temporal evolution of the disc's motion. The experiment data allow us to detail the dynamics of the disc, and consequently to quantify its energy. From our experimental observations, it is confirmed that rolling friction is a primary factor responsible for the dissipation of the energy. Furthermore, a mathematical model, in which the rolling friction is characterized by a resistance torque proportional to the square of precession rate, is also proposed. By employing the model, we perform qualitative analysis and numerical simulations. Both of them provide results that precisely agree with our experimental findings.

Terminal retrograde turn of rolling rings

“…Increasing the thickness of the disk changes the dynamics because of the existence of an unstable, invertedpendulum-like, static equilibrium [4,12,13]. Nevertheless, the center of mass of the disk with the global position vector r C always moves on a spiral trajectory [5,6] for low inclination angles, while the orbital angular momentum vector L = r G ×ṙ C per unit mass is almost aligned with the angular velocity ω of the disk and we have L · ω > 0. Here r G is the position vector of the center of mass with respect to the contact point of the body with the surface.…”

Retrograde motion of a rolling disk

“…The most thorough results of exploring the mechanisms of spinning disk energy dissipation, including experimental ones, are presented in Refs [10,23]. According to their authors, after a short stage of slip, the main factor affecting the dynamics of Euler's disk is the rolling friction, which is modelled in Refs [12,23] as a viscous contact with a quadratic dependence on the motion velocity. This is confirmed by the rather good agreement between the results of numerical simulations for the nutation angle and the precession speed, and the experimental results.…”

On rolling friction

“…Считая удар неупругим, определим компоненты ударного импульса I n и I t из условия мгновенной остановки новой точки контакта. Как следует из формулы (14), момент, пропорциональный w, создаётся не только нормальной, но и касательной составляющими ударного импульса: ∈( , ), где значение w 1 определяется равенством w 0 = 0 в формуле (9), а значение w 2 -формулой (11). Иными словами, колесо должно перекатиться через верхнюю точку и не потерять контакт с опорой.…”

“…Природа сопротивления качению круглых тел (шара или цилиндра) существенно зависит от упругих свойств тела и поверхности. В частности, при движении автомобиля по шоссе это сопротивление главным образом обусловлено гистерезисом при деформации шин [9]. Для описания зависимости коэффициента трения от угловой скорости предлагается использовать эмпирическую формулу, идентичную (3).…”