Tutorial Sheet 5: 3D Kinematics

Topics covered are:

• 3D velocity and acceleration
• Reference frames

Tips

• Cross products will get much larger - the order of calculation matters, and you will need to use the full version of the acceleration equations.
• Mechanisms are now in 3D, so practice visualisiing the actual movements to avoid mistakes.
• Some will like to use column vectors, write it how you feel works best for you!

Question 1

The aeroplane’s angular velocity vector relative to an earth-fixed reference frame, expressed in terms of the body-fixed coordinate system shown, is $\omega$ = 0.62i+0.45j−0.23k rad/s. The coordinates of point A of the airplane are (3.6, 0.8, −1.2) m. What is the velocity of point A relative to the velocity of the aeroplane’s center of mass?

Answer

Question 2

The angular velocity of the cube relative to the primary reference frame, expressed in terms of the body-fixed coordinate system shown is $\omega$=−6.4i+8.2j+12k rad/s.The velocity of the center of mass G of the cube relative to the primary reference frame at the instant shown is $v_G$=26i+14j+32k m/s. What is the velocity of point A of the cube relative to the primary reference frame at the instant shown?

Answer

v_A = 22.2i+32.4j+17.4k \text{ m/s}$$

Question 3

Using the cube in Q2, the coordinate system shown is fixed with respect to the cube. The angular velocity of the cube relative to the primary reference frame, $\omega$=−6.4i+8.2j+12k rad/s, is constant.The acceleration of the center of mass G of the cube relative to the primary reference frame at the instant shown is $a_G$=136i+76j−48k m/s $^2$. What is the acceleration of point A of the cube relative to the primary reference frame at the instant shown?

Answer

Question 4

The origin of the secondary coordinate system shown is fixed to the center of mass G of the cube. The velocity of the center of mass G of the cube relative to the primary reference frame at the instant shown is $v_G$=26i+14j+32 m/s. The cube is rotating relative to the secondary coordinate system with angular velocity $\omega_{rel}$ =6.2i−5j+8.8k rad/s. The secondary coordinate system is rotating relative to the primary reference frame with angular velocity $\omega$ = 2.2i+4j−3.6k rad/s.

(a) What is the velocity of point A of the cube relative to the primary reference frame at the instant shown?
(b) If the components of the vectors $\omega_{rel}$ and $\omega$ are constant, what is the cube’s angular acceleration relative to the primary reference frame?

Answer

(a)

(b)

Question 5

Relative to an earth-fixed reference frame, points A and B of the rigid parallelepiped are fixed and it rotates about the axis AB with an angular velocity of 30 rad/s. Determine the velocities of points C and D relative to the earth-fixed reference frame.

Answer

Question 6

Using the parallelepiped in Q5, relative to the xyz coordinate system shown, points A and B of the rigid parallelepiped are fixed and the parallelepiped rotates about the axis AB with an angular velocity of 30 rad/s. Relative to an earth fixed reference frame, point A is fixed and the xyz coordinate system rotates with angular velocity −5i+8j+6k rad/s. Determine the velocities of points C and D relative to the earth-fixed reference frame.

Answer

Question 7

Relative to an earth-fixed reference frame, the vertical shaft rotates about its axis with angular velocity $\omega_O$=4 rad/s. The secondary xyz coordinate system is fixed with respect to the shaft and its origin is stationary. Relative to the secondary coordinate system, the disk (radius 8 cm) rotates with constant angular velocity $\omega_d$=6 rad/s. At the instant shown, determine the velocity of point A

(a) Relative to the secondary reference frame.
(b) Relative to the earth-fixed reference frame.

Answer

(a)

(b)

Question 8

The object in figure (a) is supported by bearings at A and B in figure (b). The horizontal circular disk is supported by a vertical shaft that rotates with angular velocity $\omega_O$=6 rad/s. The horizontal bar rotates with angular velocity $\omega$=10 rad/s. At the instant shown,

(a) What is the velocity relative to an earth-fixed reference frame of the end C of the vertical bar?
(b) What is the angular acceleration vector of the object relative to an earth-fixed reference frame?
(c) What is the acceleration relative to an earth-fixed reference frame of the end C of the vertical bar?

Answer

(a)

(b)

(c)

Question 9

The point of the spinning top remains at a fixed point on the floor, which is the origin O of the secondary reference frame shown. The top’s angular velocity vector relative to the secondary reference frame, $\omega_{rel}$=50k rad/s, is constant. The angular velocity vector of the secondary reference frame relative to an earth-fixed primary reference frame is $\omega$=2j+5.6k rad/s. The components of this vector are constant. (Notice that it is expressed in terms of the secondary reference frame.)

(a) Determine the velocity relative to the earth-fixed reference frame of the point of the top with coordinates (0, 20, 30) mm.
(b) What is the top’s angular acceleration vector relative to the earth-fixed reference frame
(c) Determine the acceleration relative to the earth fixed reference frame of the point of the top with coordinates (0, 20, 30) mm

Answer

(a)

(b)

(c)

Question 10

The radius of the circular disk is R=0.2 m, and b=0.3 m. The disk rotates with angular velocity $\omega_d$=6 rad/s relative to the horizontal bar. The horizontal bar rotates with angular velocity $\omega_b$=4 rad/s relative to the vertical shaft, and the vertical shaft rotates with angular velocity $\omega_O$=2 rad/s relative to an earth-fixed reference frame. Assume that the secondary reference frame shown is fixed with respect to the horizontal bar.

(a) What is the angular velocity vector $\omega_{rel}$ of the disk relative to the secondary reference frame?
(b) Determine the velocity relative to the earth-fixed reference frame of point P, which is the uppermost point of the disk.

Answer

(a)

(b)