AE352_F23_HW1

Frame Origin Basis Vectors Rotation or Translation Basis conversion F O ( ⃗ f 1 , ⃗ f 2 , ⃗ f 3 ) N/A N/A G O ′ ( ⃗ f ′ 1 , ⃗ f ′ 2 , ⃗ f ′ 3 ) Translation by −− → OO ′ ( ⃗ f ′ 1 , ⃗ f ′ 2 , ⃗ f ′ 3 ) = ( ⃗ f 1 , ⃗ f 2 , ⃗ f 3 ) H O ′ ( ⃗ f ′′ 1 , ⃗ f ′′ 2 , ⃗ f ′′ 3 ) Rotation ϕ about ⃗ f ′ 3 ⃗ f ′′ 1 = cos ϕ ⃗ f ′ 1 + sin ϕ ⃗ f ′ 2 , ⃗ f ′′ 2 = − sin ϕ ⃗ f ′ 1 + cos ϕ ⃗ f ′ 2 , ⃗ f ′′ 3 = ⃗ f ′ 3 Q ... ... ... ... 2. Express the following vectors in the specified coordinates: (a) { − → OP } F = (b) { − → CH } Q = (c) { −− → OH } F = (d) Write out a simple criterion (an equation) for when the hook H coincides with the payload attachment point at P . 3. In the process of trying to affix itself to the payload, we want the hook to be at zero velocity relative to the payload (otherwise we’re possibly swinging and crashing the hook into the payload at high speed..!). First, find the following velocities, and note that θ ( t ) , h ( t ) , a ( t ) and l ( t ) are all time varying quantities in our problem. (a) ⃗ v Q ( H ) (b) ω Q / F (c) ⃗ v F ( H ) (d) Write a criterion for the velocity of the hook to be zero relative to the attachment point P . 2