Schematic diagram of cycloid motions. (a): epicycloid; (b): hypocycloid. Dotted line represents the trajectory of cycloid motions of grains. Solid line and dashed line indicate the primary circle and the secondary circle, respectively. r₂ and r₁ represent the radii of the primary and the secondary circles, respectively. ω₂ and ω₁ represent the angular velocities of the primary and the secondary circles, respectively. Panels (a) and (b) occur when ω₁ω₂ > 0 and ω₁ω₂ < 0, respectively. Panels (c)–(f) reproduce the trajectories as shown in Figs. 2(a)–2(d) in term of Eqs. (1)–(4). The parameters are ω₁ = 0.14, 0.15, 0.42, 0.22 rad/s, ω₂ = 44.52, 27.45, −45.13, −32.63 rad/s, r₁ = 20.39, 20.27, 12.31, 13.17 mm, r₂ = 0.21, 0.14, 0.25, 0.12 mm in Figs. 2(a)–2(d) and Figs. 3(c)–3(f), respectively.