1. Plane Kinematics of Rigid Bodies
study of body motion without reference to force.
Rigid Body
It has dimensions. (particle doesn’t have it).
distance between 2 points in the body remains unchanged.
assumption validity? i.e. there is no real rigid body.
Plane Motion
Definition: All parts of the body move in parallel planes.

2. Treat the body as thin slab object.
Plane Motion
Definition:
All parts (points) of the body move in parallel planes.
Movement of one cutting face (corresponding to its motion plane) describes movement of the whole body.
Motion Plane
Treat the body as thin slab object.
Any slab object (cutting face) is okay,
But we usually use the plane where the object’s C.G.is in.
C.G.
All corresponding points in other motion plane
have the same velocity and acceleration

4. Rigid-Body Plane Motion
(Pure) Translation II
(Pure) Rotation
III
General Motion

5. Type of Plane motion (Pure) Translation
C.G.
(Pure) Translation
Definition: A line between two points in the body remains parallel through out the motion.
Motion of one point can be used to describe motion of the whole body.
Rectilinear
Translation
Curvilinear
Translation
C.G.
Treat it as a particle

6. Types of Plane Motions (cont.)
Fixed-axis (Pure) Rotation:
All , move along circular path with center at
Not having the same velocity and acceleration (depends on circle radius r)
Use techniques for particle (circular) motion (n-t, r-)
Important Property of Fixed-Axis Rotation
Rotation axis
motion plane
Rotation Axis
Motion Plane

7. Types of Plane Motions (cont.)
General Plane motion:
need new techniques in this chapter

8. 5/2 Rigid Body’s Rotation
Angle between any two lines on “rigid” object does not change during the time
“Rotating” concept is basically
a concept on rotation of “line”.
same w,a?
a concept on (whole) rigid body.
a line b
any
Define  angular position (+/-) +
angular velocity
angular acceleration
any Reference
Any lines on a rigid body in its plane of motion have the same angular displacement, velocity and acceleration
for rigid body

9. (a) Angular Motion Relations
Define  angular position (+/-)
any line
angular velocity
angular acceleration +
any Reference
Similar to rectilinear motion
Sign convention of all variables must be consistent!

10. Angular Motion Relations
Observed the similarity with the linear motion
2) Graphical meanings
1) Integrals Calculation

11. CCW:+
a=4t
a=4t
a=4t
w = - 60p
w = 0 w=
A flywheel rotating freely at 1800 rev/min clockwise is subjected to a variable
counterclockwise torque which is first applied at time t = 0. The torque
produces a counterclockwise angular acceleration  = 4t rad/ss, Determine the
total number of revolutions, clockwise plus counterclockwise, turned by the
flywheel during the first 14 seconds of torque application. +

13. Fixed-Axis (Pure) Rotation (scalar notation)
Can we find? rw P
Rigid
body
points
rww
whole body (rigid body) ra O
Point P w a
n-t coord:
Fixed-Axis
(Pure) Rotation Only !

14. (a) the acceleration of point C on the cable in contact with the drum
The pinion A of the hoist motor drives gear B, which is attached to the the hoisting drum. The load L is lifted from its rest position and acquires an upward velocity of 2 m/s in a vertical rise of 0.8 m with constant acceleration. As the load passes this position, compute
(a) the acceleration of point C on the cable in contact with the drum
(b) the angular velocity and angular acceleration of the pinion A.
w,a
CCW
CCW

15. n n t t = 0 = 0 share the same share the same Non-slipping
floor- fixed
Non-slipping
Non-slipping
share
the
same
= 0
share
the
same
Why?
= 0
Without gear teeth, the ball is not guaranteed to roll without slipping.
Two possible motion:
- roll without slipping
- roll with slipping.
If it do really roll without slipping,
It should have some “motion constraint”.
[ fixed relation between w and v ]
(Next session: we will find out this)
In case of No gear-teeth

18. Fixed-Axis (Pure) Rotation (vector notation)
The body rotating about the O axis
Represent “angular velocity” w as Vector
Direction: “Right-hand rule”
From the last section, the magnitude of the velocity is
Think in 3D Vector
We consider only Plane motion of rigid body
The cross product can be used to establish the direction:

19. Fixed Axis Rotation (vector notation)
Differentiating the velocity with respect to time:
Direction depends on
We consider only “Plane motion” of rigid body
Direction:

20. Fixed-Axis (Pure) Rotation (vector notation)

21. The rectangular plate rotates clockwise.
If edge BC has a constant angular velocity of 6 rad/s. Determine the vector expression of the velocity and acceleration of point A, using coordinate as given.
Rigid body

23. Equations Review: (Pure) Translation
Movement of one point describes movement of the whole body.
(Pure) Rotation
rotation of one line describes rotation of the whole body.
Whole body shares the same “angular” quantities.
Reference +
any line
Absolute motion
Relative motion
General Plane Motion

24. 5/3 Absolute Motion (of General Plane Motion)
Use geometric constraint (which define the configuration of the body) to obtain the velocity and acceleration (in general motion)
Idea: write a (position) constraint equation which always applies regardless of the system’s configuration, then differentiate the equation to get velocity and acceleration. w?
for complex constraint method of relative motion may be easier.

25. Define the displacement and its positive direction.w?
Define the displacement and its positive direction.
The variable must be measured from the fixed reference point or line.
Find the equation of constraint motion.
Equation must be true all during the motion.
Differentiate it to find (angular) velocity and acceleration.

26. 5/56. Express the angular velocity and angular acceleration of the connecting rod AB in terms of the crank angle for a given constant b

27. w SP5/4 A wheel of radius r rolls on a flat surface without slipping.
motion constraint
SP5/4 A wheel of radius r rolls on a flat surface without slipping.
Determine angular motion of the wheel in terms of the linear motion of its center O. Also determine the acceleration of a point on the rim of the wheel as the points comes into contact with the surface on which the wheel rolls w
Point O:
rectilinear motion
Point C’s trajectory

28. SP5/4 A wheel of radius r rolls on a flat surface without slipping.
Determine angular motion of the wheel in terms of the linear motion of its center O. Also determine the acceleration of a point on the rim of the wheel as the points comes into contact with the surface on which the wheel rolls
Not depend on a(t), w(t) are!
velocity=0
Acceleration in the
direction of axis x = 0 O D
When Point D comes to contact the surface,
It also has a velocity (=0), and
acc. as above.
C’’

29. velocity=0 No slipping Acceleration in the direction of axis x = 0 O O
Analogy
No slipping
Rel. vel. =0
Rel acc. = 0

30. Non-slipping Condition
floor- fixed

31. “no slipping” implies:
1) Contact point { C , C’ } on two body has no relative velocity.
2) Contact point { C, C’ } on two body has same tangential component of acceleration

32. O and L has same vertical velocity & acceleartion
Each cables do not slip. Load-supporting pulleys are rigid body. A B O
O and L has same vertical velocity & acceleartion

35. General Plane Motion of Rigid Body
Introduction
นิยามการเคลื่อนที่ของวัตถุเกร็ง
Rotation
วิธีอธิบายการเคลื่อนที่แบบหมุน
- Calculation methods
usually for any t
using constraint equation
Absolute motion
ใช้คำนวณ V A
Translating-only observer
usually for
some t
(instant)
using its geometric shape at that instant
Relative velocity
ใช้คำนวณ V
Observer is at the point of rigid body where its velocity = 0
Instantaneous Center of
Zero Velocity (ICZV)
ใช้คำนวณ V
Relative acceleration
ใช้คำนวณ A V
Motion relative to
rotating axes
Translating and rotating
Observer
ใช้คำนวณ V A

36. 5/4 Relative Velocity relative velocity concept
Different
viewpoint
General Plane Motion = Translation + Rotation
“simultaneous”
Motion of point (observer) B, detected by O
= Motion of plate moving “translationally”
Since the distant between the two points on a rigid body is constant,
an observer at one point will see the other point move in a circular motion around it!
Wait!
B really sees A moving circularly?
B sees A has no movement !?!?!? O

37. Applying the relative concept
Which one?
Only this case
Observer B is sitting on the magic carpet.
Observer B is on the plate
Rotating observer (attached to B)
non-rotating observer (attached to B)
Rotating frame
non-rotating frame A A A B B
B see A
having a velocity perpendicular with its distance.
B see A no moving at all

38. Relative Velocity (non-rotating observer)
We use: non-rotating observer (frame) attached to B
Absolute world:
Relative world:
same?
Only when
non-rotating observer
(see next page)
Observer at B see A moving in a circle around it
Observer O detects:
Observer B detects:

39. Translating observer see same w,a as absolute Observer
The rotating of rigid body
= The rotating of line compared with “fixed” reference axis
same w
same a B
B’s reference
line B O
O’s reference line

40. Translating-Rotating observer sees different w,a with absolute Observer
The rotating of rigid body
= The rotating of line compared with “fixed” reference axis
different w
different a B
B’s reference
line B
The rotating B’s “reference line”,
observed by absolute observer. O
O’s reference line

41. Understanding the equation
absolute
absolute
Non-rotating,
Moving with B
absolute
Non-rotating,
Moving with B
always perpendicular to line AB. Its direction can be deduced from 
Important key:
Hint on solving problems
Identify the known and unknown
Any 2 points on the same rigid body A
Above equation (2D: “3D-fake”) can be solved when there are at most 2 unknown scalar quantities B
Above equations usually contains 5 scalar quantities (not including position vector r)
Also works with A as the observer

42. Relative Vector Analysis on General Plane Motion
Fixed-Axis (Pure)
Rotation w P w P G w P
w of observer at G
= w (of rigid body)
of observer at O G

44. A, B on the same rigid body (bar AB)
Velocity at A
is key point to find w
A, B on the same rigid body (bar AB)
Solved by
Vector Analysis

45. +   + y x From i j k A B C O 3D vector calculation (i,j,k):
Sign indicates angular direction
(right hand rule) x y

46. A, B on the same rigid body (bar AB)? M ?
Solved by Graphical Method CW CW
need to find the angular direction from the figure

47. middle link

49. Note: relative velocity technique
Direction is simply found:
graphical solution:
simplest, be careful about sign /
3D (i, j, k) vector:
complicated, automatic sign indication
: same in absolute and relative world
Non-rotating
To know of the rigid body A B A B

51. D ? M ?
wABC
CCW
Graphical solution

52. vB D ? M ? vA
Vector solution
Direction? CW

54. Relative Velocity (Part 2)
Another usage of the relative velocity equation B
A on OD D
2 points need not be in the same rigid body
B on
screw
parerell
For constrained sliding contact between two links in a mechanism.
Pick points A and B as coincident points, one on each link
(the points may be imaginary).
some reason later!
The observer on B no longer see A moving around it in a circle.
c.c.w.

56. q f B O P
0.175
0.175cosq
cosq M ? ? D CW
Ans

58. y C x vQ vP 200 vQ/P 100 vO O CCW Plus Non-slipping condition b ? ? q
Q on slot C
V=1.5
q=30
Non-slipping
condition D C y O
100 x b vQ vP
200 ? ?
vQ/P
Plus q vO
CCW

61. 5/6 Relative Acceleration
concept
Different
viewpoint
General Plane Motion = Translation + Rotation
“simultaneous”
Motion of point (observer) B, detected by O
= Motion of plate moving “translationally”
Since the distant between the two points on a rigid body is constant,
an observer at one point will see the other point move in a circular motion around it!
Wait!
B really sees A moving circularly?
B sees A has no movement !?!?!? O

62. Relative Acceleration (non-rotating observer)
We use: non-rotating observer (frame) attached to B
Absolute world:
Relative world:
same?
Only when
non-rotating observer
(see the proof at relative velocity part)
Observer at B see A moving in a circle around it
Observer O detects:
Observer B detects:

63. Understanding Equations
Non-rotating
: the same both in absolute and relative (translation-only) world
Hint on solving problems
Identify the known and unknown
Above equation (2D: “3D-fake”) can be solved when there are at most 2 unknown scalar quantities
Above equations usually contains 6 scalar quantities
(not including position vector r)

64. = +
Given :
(constant)
Find :

65. Given :
(constant)
Find :
(at this instant)
ANS
ANS

66. Find :
(at this instant)
Find velocity first,
Before acceleration

67. Given :
(constant)
Find :
(at this instant)
ANS
ANS

68. Practice before I.C.Z.V

69. Non-slipping condition
5/124 The center O of the disk has the velocity and acceleration shown in the figure. If the disk rolls without slipping on the horizontal surface, determine the velocity of A and the acceleration of B for the instant represented.
Non-slipping condition
You can calculate using point O and D. a w
You can calculate using point O and and D. D
I.C.Z.V

71. vA vE
wOA O vA B vD
wBD

72. aOA
wOA=20 O

75. I.C.Z.V General Plane Motion (non-rotating observer)
Cross-Vector Approach
Graphical Approach
any point: A, B (on same rigid body moving in GPM)
New technique
I.C.Z.V
B is special point : I.C.Z.V

76. Note: relative velocity technique
Direction is simply found:
graphical solution:
simplest, be careful about sign /
3D (i, j, k) vector:
complicated, automatic sign indication
: same in absolute and relative world
Non-rotating
To know of the rigid body A B A B

78. Checkpoint: circular motion
Don’t know it is
fixed-point rotation
or not
(General Motion) ?
Fixed-point rotation
(Rotation)
from rotation point
valid method?
Yes! but show your reason!
I.C.Z.V concept

79. 5/5 Instantaneous Center of Zero Velocity
General
Plane
motion
5/5 Instantaneous Center of Zero Velocity
Extension theory using relative velocity. B A P
w of observer at C
= w of rigid body
(in Absolute Observer’s perception) Z w
Z is called I.C.Z.V (the point where its velocity at that instant is zero)
each point on the body can be though of as rotating around point Z.
- can find w easily by geometry
- can find velocity and its direction of any points easily by geometry
For calculating v and w only

80. 5/5 Instantaneous Center of Zero Velocity
For a moving body at each instant of time, there is (always) a point on the rigid body (or on the extended body) with zero velocity.
This point is call the “Instantaneous center of zero velocity” (I.C.Z.V.)
How to find that point I.C.Z.V ?
must be
perpendicular to CA
General
Plane motion B A
w of observer at C
= w of rigid body
(in Absolute Observer’s perception)
Absolute velocity = relative velocity observing from an I.C.Z.V.
For calculating velocity only, each point on the body can be though of as rotating around point C.
I.C.Z.V

81. instantaneous velocity only
Finding an I.C.Z.V. A B
not a
rigid body
“instantaneous”
Translational motion z
General Plane motion
I.C.Z.V at Inf. A B A B A B z z z B A
I.C.Z.V for calculating
instantaneous velocity only
w of observer at C
= w of rigid body
(in your perception) A B z
az usually  0
(Even vz = 0) D

82. Direction of (absolute) velocity of two point in the same rigid body
Arm OB of the linkage has a clockwise angular velocity of 10 rad/s in the position shown where  = 45°. Determine the velocity of A, the velocity of D, and the angular velocity of link AB for the instant shown.
w of what? OA, AB, BO
solved by
relative
velocity
I.C.Z.V D
350 M ?
381 ?
Direction of (absolute) velocity
of two point in the same rigid body w
Thus, we can locate the
instantaneous center of velocity,
which is point C
You have to find
Direction yourself

83. Kinematics of Rigid Bodies
ICZV

84. a x y
Vector Diagram vC rC q
ICZV vB q rB q vA
CCW

85. Non-slipping motion ICZV of P1 CW ICZV of sun ICZV of A CW ICZV of P2
Sun Gear: Fixed-Axis (Pure) rotation
CCW
Planet Gear: general Plane motion

86. ICZV
ICZV
100 mm

87. Non-slipping condition
I.C.Z.V
Absolute motion
rectilinear
I.C.Z.V n
circle t
5/140
I.C.Z.V

89. vC1 I vC1 =8r II d III vH2 = 8r vH2 vH3 vC2 vH3 vC2 4 rad/s VD1 ICZVCW
VD2 II d
4 rad/s
vC2
vH2
= 8r
ICZV
VD3
III
vC2
vH3
VD0
vH2
vH3
CCW

91. Practice Before Rotating Observer

92. 5/134 The sliding collar moves up and down the shaft, causing an oscillation of crank OB. If the velocity of A is not changing as it passes the null position where AB is horizontal and OB is vertical, determine the angular acceleration of OB in that position.
CCW
ICZV
of AB CW

93. 5/134 The sliding collar moves up and down the shaft, causing an oscillation of crank OB. If the velocity of A is not changing as it passes the null position where AB is horizontal and OB is vertical, determine the angular acceleration of OB in that position.
CCW
ICZV
of AB =0 CW

95. ICZV of DAB
is at infinite CW
3D vector solution CW
Graphical solution

97. middle link

100. 5/134 The sliding collar moves up and down the shaft, causing an oscillation of crank OB. If the velocity of A is not changing as it passes the null position where AB is horizontal and OB is vertical, determine the angular acceleration of OB in that position.
CCW
ICZV
of AB =0 CW

102. ICZV at inf.
AB translational. CW CW

103. CCW
CCW