1. ENGR 214 Chapter 15 Kinematics of Rigid Bodies
All figures taken from Vector Mechanics for Engineers: Dynamics, Beer and Johnston, 2004

2. Introduction
Kinematics of rigid bodies: relations between time and the positions, velocities, and accelerations of the particles forming a rigid body.
rectilinear translation
translation:
curvilinear translation
Classification of rigid body motions:
rotation about a fixed axis
general plane motion
motion about a fixed point
general motion

3. Motion of the plate: is it translation or rotation?
Curvilinear translation
Rotation

4. Translation Consider rigid body in translation:
direction of any straight line inside the body is constant,
all particles forming the body move in parallel lines.
For any two particles in the body,
Differentiating with respect to time,
All particles have the same velocity.
Differentiating with respect to time again,
All particles have the same acceleration.

5. Rotation About a Fixed Axis
When a body rotates about a fixed axis, any point P in the body travels along a circular path. The angular position of P is defined by q.
The change in angular position, d, is called the angular displacement, with units of either radians or revolutions. They are related by
1 revolution = 2 radians
Angular velocity, , is obtained by taking the time derivative of angular displacement:
 = d/dt (rad/s) +
Similarly, angular acceleration is
 = d2/dt2 = d/dt or  = (d/d) rad/s2

6. Rotation About a Fixed Axis: Velocity
Consider rotation of rigid body about a fixed axis AA’
Velocity vector of the particle P is tangent to the path with magnitude
The same result is obtained from

7. Rotation About a Fixed Axis: Acceleration
Differentiating to determine the acceleration,
Acceleration of P is combination of two vectors,

8. Rotation About a Fixed Axis: Representative Slab
Consider the motion of a representative slab in a plane perpendicular to the axis of rotation.
Velocity of any point P of the slab,
Acceleration of any point P of the slab,
Resolving the acceleration into tangential and normal components,

9. Examples

10. Equations Defining the Rotation of a Rigid Body About a Fixed Axis
Motion of a rigid body rotating around a fixed axis is often specified by the type of angular acceleration.
Recall
Uniform Rotation, a = 0:
Uniformly Accelerated Rotation, a = constant:

11. Sample Problem 5.1
Cable C has a constant acceleration of 9 in/s2 and an initial velocity of 12 in/s, both directed to the right.
Determine (a) the number of revolutions of the pulley in 2 s, (b) the velocity and change in position of the load B after 2 s, and (c) the acceleration of the point D on the rim of the inner pulley at t = 0.

12. Sample Problem 5.1
The tangential velocity and acceleration of D are equal to the velocity and acceleration of C.
Apply the relations for uniformly accelerated rotation to determine velocity and angular position of pulley after 2 s.

13. Sample Problem 5.1
Evaluate the initial tangential and normal acceleration components of D.
Magnitude and direction of the total acceleration,

14. General Plane Motion
A combination of translation & rotation

15. General Plane Motion
Pure translation, followed by rotation about A2 (to move B'1 to B' 2)
Motion of B w.r.t. A is pure rotation, i.e. B draws a circle centered at A
Any plane motion can be represented as a translation of an arbitrary reference point A and a rotation about A.

16. Absolute and Relative Velocity
For any two points lying on the same rigid body:
Note:
= distance from A to B
Equation can be represented graphically by a velocity diagram

17. Absolute and Relative Velocity
Assuming that the velocity vA of end A is known, determine the velocity vB of end B and the angular velocity w.
Locus for vB
Locus for vB/A vA vB q
vB/A
The direction of vB and vB/A are known. Complete the velocity diagram.

18. Absolute and Relative Velocity in Plane Motion
Selecting point B as the reference point and solving for the velocity vA of end A and the angular velocity w leads to an equivalent velocity triangle.
vA/B has the same magnitude but opposite sense of vB/A. The sense of the relative velocity is dependent on the choice of reference point.
Angular velocity w of the rod in its rotation about B is the same as its rotation about A. Angular velocity is not dependent on the choice of reference point.

19. s = displacement of center
Rolling Motion
Consider a circular disc that rolls without slipping on a flat surface O1 O2 r s A2 q
From geometry: A1
s = displacement of center

20. Sample Problem 15.2
The double gear rolls on the stationary lower rack; the velocity of its center is 1.2 m/s.
Determine (a) the angular velocity of the gear, and (b) the velocities of the upper rack R and point D of the gear.

21. Sample Problem 15.2 vD/A vB/A For any point P on the gear:
For point B:
For point D:

22. Sample Problem 15.3
The crank AB has a constant clockwise angular velocity of 2000 rpm. For the crank position indicated, determine (a) the angular velocity of the connecting rod BD, and (b) the velocity of the piston P.

23. Sample Problem 15.3
The velocity is obtained from the crank rotation data.
Locus for vD/B
The direction of the absolute velocity is horizontal. The direction of the relative velocity is perpendicular to BD.
Locus for vD

24. Sample Problem 15.3
Determine the velocity magnitudes from the vector triangle drawn to scale.

25. Instantaneous Center of Rotation
For any body undergoing planar motion, there always exists a point in the plane of motion at which the velocity is instantaneously zero (if it were rigidly connected to the body).
This point is called the instantaneous center of rotation, or C.
It may or may not lie on the body!
If the location of this point can be determined, the velocity analysis can be simplified because the body appears to rotate about this point at that instant.

26. Instantaneous Center of Rotation
To locate the C, we use the fact that the velocity of a point on a body is always perpendicular to the position vector from C to that point.
If the velocity at two points A and B are known, C lies at the intersection of the perpendiculars to the velocity vectors through A and B .
If the velocity vectors at A and B are perpendicular to the line AB, C lies at the intersection of the line AB with the line joining the extremities of the velocity vectors at A and B.
If the velocity vectors are equal & parallel, C is at infinity and the angular velocity is zero (pure translation)

27. Instantaneous Center of Rotation
If the velocity vA of a point A on the body and the angular velocity w of the body are known, C is located along the line drawn perpendicular to vA at A, at a distance
r = vA/w from A. Note that the C lies up and to the right of A since vA must cause a clockwise angular velocity w about C.

28. Velocity Analysis using Instantaneous Center
The velocity of any point on a body undergoing general plane motion can be determined easily if the instantaneous center is located.
Since the body seems to rotate about the IC at any instant, the magnitude of velocity of any arbitrary point is v = w r, where r is the radial distance from the IC to that point. The velocity’s line of action is perpendicular to its associated radial line. Note the velocity has a direction which tends to move the point in a manner consistent with the angular rotation direction.

29. Velocity Analysis using Instantaneous Center

30. Instantaneous Center of Rotation
C lies at the intersection of the perpendiculars to the velocity vectors through A and B .
The velocity of any point on the rod can be obtained.
Accelerations cannot be determined using C.

31. Sample Problem 15.4, using instantaneous center
The double gear rolls on the stationary lower rack; the velocity of its center is 1.2 m/s.
Determine (a) the angular velocity of the gear, and (b) the velocities of the upper rack R and point D of the gear.

32. Sample Problem 15.4
Point C is in contact with the stationary lower rack and, instantaneously, has zero velocity. It must be the location of the instantaneous center of rotation.

33. Sample Problem 15.5 using instantaneous center
Crank-slider mechanism
The crank AB has a constant clockwise angular velocity of 2000 rpm. For the crank position indicated, determine (a) the angular velocity of the connecting rod BD, and (b) the velocity of the piston P.

34. Sample Problem 15.5
C is at the intersection of the perpendiculars to the velocities through B and D.

35. Absolute and Relative Acceleration
Absolute acceleration of point B:
Relative acceleration includes tangential and normal components:

36. Absolute and Relative Acceleration
Given determine
Vector result depends on sense of and the relative magnitudes of
Must also know angular velocity w.

37. Absolute and Relative Acceleration
Draw acceleration diagram to scale:

38. Sample Problem 15.6
The center of the double gear has a velocity and acceleration to the right of 1.2 m/s and 3 m/s2, respectively. The lower rack is stationary.
Determine (a) the angular acceleration of the gear, and (b) the acceleration of points B, C, and D.

39. Sample Problem 15.6

40. Sample Problem 15.7
Crank AG of the engine system has a constant clockwise angular velocity of 2000 rpm.
For the crank position shown, determine the angular acceleration of the connecting rod BD and the acceleration of point D.

41. Sample Problem 15.7
From Sample Problem 15.3, wBD = 62.0 rad/s, b = 13.95o.

42. Sample Problem 15.7
Draw acceleration diagram:
Drawn to scale

43. Sample Problem 15.8 Four-bar mechanism
In the position shown, crank AB has a constant angular velocity w1 = 20 rad/s counterclockwise. Determine the angular velocities and angular accelerations of the connecting rod BD and crank DE.

44. Sample Problem 15.8 vB vD Velocities Velocity diagram vD/B vB vD/B vD
Shown here not to scale

45. Sample Problem 15.8 aD/B t aD/E t Accelerations aD/B n aD/E n
Acceleration diagram aB
aD/E n aB
aD/B n
aD/E t
Shown here not to scale