Relative Velocity and Acceleration in a Plane (2 points in the same Link)

Example: The figure below shows a model of the shoulder socket at A (fixed point), upper

arm, AB, lower arm BC, and a spinning disk of radius r with center at C. Point P of the

disk is directly to the right of C. The elbow at B is bent at 90^o. The following data:

AB = 0.45 m, BC = 0.675 m, r = 0.1 m, ω₁ = 5 k rad/sec, ω₂ = 3 k rad/sec,

ω₃ = - 5 k rad/sec Find the velocity of P at this instant. Find the velocity of P relative to B.

Recall for any two points, R and S, in the same rigid link the relative velocity equation is:

v_S = v_R + ω x r_RS

Strategy: Apply the relative velocity equation for each link, moving from the upper arm,

to the lower arm, and finally to the disk.

For the upper arm v_B = v_A + ω₁ k x (˗ AB i ) v_A = 0 so v_B = - AB ω₁ j = - 2.25 j m/s

For the lower arm v_C = v_B + ω₂ k x (˗ BC j ) so v_C = - 2.25 j + BC ω₂ i

v_C = - 2.25 j + 2.025 i = 2.025 i - 2.25 j m/s

For the disk v_P = v_C - ω₃ k x ( r i ) so v_P = 2.025 i - 2.25 j ˗ r ω₃ j

v_P = 2.025 i - 2.25 j ˗ 0.5 j = 2.025 i - 2.75 j m/s (result)

Now the velocity of P relative to B is v_P ˗ v_B = 2.025 i - 2.75 j ˗ (- 2.25 j )

v_P/B = 2.025 i ˗ 0.50 j m/sec (result)

Click here for another example.

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