Relative Velocity and Acceleration with Translating and Rotating Frames

Example: A bug B crawls outward at a speed v_o and an acceleration a_o relative to the rotating

arm, AC. In the position shown: θ = 30^o, ω = ω_o k = 2 k rad/sec, α = αo k = 4 k rad/sec²,

AB = r_o = 5 ft, v_o = 3 fps, a_o = 10 fps²

Find the acceleration of the bug with respect to the fixed frame of reference F.

Let xy be a fixed frame of reference, F, with origin A. Let x₁y₁ be a rotating frame of reference

attached to the rotating arm with origin C. Let i and j be unit vectors for xy and i₁, j₁ be

unit vectors for x₁y₁. Designate x₁y₁ as frame of reference 1. See the figure below.

Strategy: Apply the general relative velocity equation followed by the relative acceleration equation

for a translating/rotating frame of reference. Note: The term v_B|₁ in the relative velocity equation appears in the relative acceleration equation in the Coriolis term. So calculation of accelerations frequently is a two-step process. First apply the relative velocity equation to the problem and then

the relative acceleration equation.

v_B = v_A + v_B|₁ + ω x r_AB

Relative Velocity Equation

a_B = a_A + a_B|₁ + α x r_AB - ω² r_AB + 2 ω x v_B|₁

Relative Acceleration Equation

For the given data: v_A = 0, v_B|₁ = v_o i₁, ω = ω_o k, r_AB = r_o i₁ As in the prior example for

relative velocity:

So v_B |_F = v_o i₁ + ω_o k x r_o i₁ = v_o i₁ + r_o ω_o j₁ and for the given data

v_B |_F = 3 i₁ + 10 j₁ or in terms of i and j v_B |_F = ˗ 2.4 i + 10.2 j ft/sec (result)

Next apply the relative acceleration equation. Click here to continue with this problem.

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