Derive an expression for the velocity vector of the collar

Question

Question 1:

Derive an expression for the velocity vector of the collar D, ^NV^D. The upper arm (B) rotates about a horizontal axis ( nˆ_x) at a rate q_^·1 The lower arm (C) Rotates about the longitudinal axis of B (bˆ_z), at a rate q^·₂. The collar (D) is located a distance q₃ from the point Q. At time t = 0, q₁ = q₂ = 0 and nˆ₂ = bˆ₂ = cˆ₂.

You may leave your answer in a mixed basis.

Question 2:

The figure below shows a planetary gear train. The ring gear (R) is stationary in N. The small gear (A) rotates at a rate of 5 rad/sec in the nˆ_z direction(^Nω^A = 5nˆ_z). Both gears (A and B) are mounted on the carrier (E). While not labeled, the center of gear A is point A_O and gear B is point B_O.

Find the angular velocity of B relative to N (^Nω^B). You may leave your answer in a mixed Basis.

Question 3:

The figure below shows some kind of contraption. The rod AB moves horizontally an amount q3 and the rod CE moves vertically an amount q2. A roller at point D allows relative vertical motion between the piston (P) and the end of rod R (point D).

Determine the velocity of point D in terms of an independent combination of q_^·1, q^·₂ and q^·₃

Question 4:

The figure below shows some kind of contraption. The shaft (S) rotates relative to N about the nˆy axis at a rate of ω₁, as shown. The disk (D) rotates relative to S about the sˆx axis at a rate of ω₂ , as shown. At time t = 0, q₁=q₂=0 and n_x = s_x = dx. The point P lies on the edge of the desk (the position vector from center of the disk to P is given by r^→P/G = Rdˆx. Determine the velocity of P in N, ^Nv^→P.