Engr Help

Relative Velocity and Acceleration with Translating and Rotating Frames

Strategy: Obtain the relative acceleration equation by taking the time derivative of each term in

the relative velocity equation in the fixed frame, F i.e. v_B = v_A + v_B|₁ + ω x r_AB

where v_B is the velocity of point B with respect to the fixed frame F

v_A is the velocity of point A with respect to the fixed frame F

v_B|₁ is the velocity of B with respect to A in x₁y₁ frame

ω x r_AB is the velocity of point B with respect to point A in the fixed frame F

ω is the angular velocity of body 1 in the fixed frame F

See the figure below.

The time derivative of each term in the fixed frame F is

d v_B /dt = d v_A /dt + d v_B|₁ /dt + d( ω x r_AB ) /dt

The last two terms on the right hand side become:

d(v_B|₁₎ /dt = d(v_B|₁) /dt |₁ + ω x v_B|₁ = a_B|₁ + ω x v_B|₁

d( ω x r_AB ) /dt = dω / dt x r_AB + ω x d (r_AB )/dt

Now recall d(r_AB )/dt = ω x r_AB + v_B|₁ and ω x d (r_AB )/dt = ω x (ω x r_AB + v_B|₁ )