Engr Help

Transformation of Coordinates

Example: To illustrate coordinate transformation consider a bug B crawling outward at a speed

v_oand 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

Let xy be a fixed frame of reference, F, with origin A and unit vectors i and j . Let x₁y₁ be a

rotating frame of reference R attached to the rotating arm AC with origin C and unit vectors i₁, j₁ .

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

Strategy: Apply the general relative velocity equation for a translating/rotating frame of reference.

v_B = v_A + v_B|₁ + ω x r_AB

Relative Velocity Equation

For the given data: v_A = 0, v_B|₁ = v_o i₁, ω = ω_o k, r_AB = r_o i₁

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

v_B |_F = 3 i₁ + 10 j₁ (result in terms of components in R)

The final step is to transform from components in R to components in F.

i.e. Transform from i₁ j₁ to i j . Recall:

i₁ = i cos θ + j sin θ and j₁ = ˗ i sin θ + j cos θ

The result is: v_B |_F = ˗ 2.4 i + 10.2 j ft/sec (result in terms of components in F)