Engr Help

Relative Velocity and Acceleration with Translating and Rotating Frames (cont)

So d( ω x r_AB ) /dt = α x r_AB + [ω x (ω x r_AB + v_B|₁ )]

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

here dω / dt = α where α is the angular acceleration of body 1 in frame F

Now collect terms for d v_B /dt = d v_A /dt + d v_B|₁ /dt + d( ω x r_AB) /dt

a_B = a_A + ( a_B|₁ + ω x v_B|₁ ) + α x r_AB + ω x (ω x r_AB ) + ω x v_B|₁

a_B = a_A + a_B|₁ + 2 ω x v_B|₁ ) + α x r_AC + ω x (ω x r_ac )

and for motion in a plane:

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

Meaning of terms:

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

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

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

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

r_AB is the position vector from A to B

a_B is the acceleration of point B with respect to the fixed frame F

a_A is the acceleration of point A with respect to the fixed frame F

a_B|₁ is the acceleration of B relative to A in the x₁y₁ frame

2 ω x v_B|₁ is the Coriolis acceleration of B in fixed frame F

α x r_AB is tangential acceleration of B relative to A in fixed frame F

α is the angular acceleration of body 1 in fixed frame F

- ω² r_AB is the normal acceleration of B relative to A in fixed frame F