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Rolling

Key Concepts: Rolling implies the "no slip" condition between the point of contact of a

wheel or gear and its mating surface such as pavement or track for a gearing system. Although

the velocity of the point of contact of the wheel or gear is zero, its acceleration is not zero. The direction of the acceleration of the point of contact is toward the center of curvature of the

curved surface. For curved surfaces, the acceleration of the center of the wheel may have both

normal and tangential components. See the figures below.

In a Nut Shell: Three common cases occur for rolling on a surface. See the table below. Let C

be the center of the wheel of radius r and P be a point on the wheel in contact with the surface.

Case 1: Rolling of a wheel on a flat surface

Case 2: Rolling of a wheel on a concave upward surface.

Case 3: Rolling of a wheel on a concave downward surface.

where Q is the center of curvature, ρ is the radial distance from Q to C, and both dθ/dt and

d²θ/dt² are measured in the fixed frame of reference.

Case 1: v_P = 0 a_p = - r (dθ/dt)² j a_C = r (d²θ/dt²) i

Case 2: v_P = 0 a_p = [(1 + r/ρ) r(dθ/dt)²] e_n a_C = r (d²θ/dt²) e_t + (rdθ/dt)²/ρ e_n

Case 3: v_P = 0 a_p = [(- 1 + r/ρ) r(dθ/dt)²] e_n a_C = r (d²θ/dt²) e_t + (rdθ/dt)²/ρ e_n

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