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Rolling

Example: The figure below shows a fixed ring gear of radius, R, a planetary gear of radius, r ,

and a carrier link OC pinned at O. The gear teeth are not shown. At the instant shown the planet has an angular velocity ω = 4 k rad/sec and an angular acceleration α = ˗ 3 k rad/sec².

B is a point on the planetary gear directly above of its center at this instant of time.

Find the velocity and acceleration of the center of the planetary gear, the velocity and acceleration

of point A on the planetary gear, the angular velocity and acceleration of the carrier link, and the radius of curvature of point B. Data include: R = 10 in. and r = 2 in.

Start by applying the relative velocity equation between points P and C.

v_C = v_P + ω x r_PC = v_P + ωk x rj = 0 i + 0 j - r ω i =

- (2)(4) i = - 8 i in/sec (result)

The velocity of the contact point P of the planetary gear v_P= 0 i + 0 j in/sec. (no slip)

Also, the acceleration of the contact point P, (for a concave surface up) is

a_P= ( 1 + r/ρ) r ω² j = [1 + r/(R ˗ r)] r ω² j = ( 1 + 2/8) (2)(4²) j = 40 j in/sec²

Continue by applying the relative acceleration equation between points P and C.

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