Chapter 5
Plane Kinematics of Rigid Bodies
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Dynamics
Undergraduate course, 2nd year
92
Plane Kinematics of Rigid Bodies
Introduction
Introduction to Plane Kinematics of Rigid Bodies I
(5/1)
Rigid-Body Assumption
1. Rigid body is a system of particles for which the distances between the particles remain
unchanged.
2. In rigid-body kinematics, we must also account for the rotational motion of the body.
(Motion of a rigid body)
 (Motion of a point on the body)  (Relative rotational motion of other points)
Plane Motion of a Rigid-Body
1. Translation is defined as any motion in which every line in the body remains parallel to its
original position at all times.
 Rectilinear translation
 Curvilinear translation
2. Rotation about a fixed axis is the angular motion about the axis. All particles in a rigid body
move in circular paths about the axis of rotation. All lines in the body rotate through the same
angle in the same time.
3. General plane motion is a combination of translation and rotation.
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93
Plane Kinematics of Rigid Bodies
Introduction
Introduction to Plane Kinematics of Rigid Bodies II
(5/1)
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Dynamics
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94
Plane Kinematics of Rigid Bodies
Rotation
Rotation I
(5/2)
Rotation (or Angular Motion)
1. All lines in a rigid body in its plane of motion have
the same angular displacement, angular velocity, and angular acceleration.
 &  2 : angles of the lines 1 & 2
  1
  : difference between two angles
  2  1  
(Relation between 1 &  2 )
  2  1  
(Angular displacements for time interval t )
  2  1
   
(Same angular displacements, owing to   0 for rigid bodies)
(Same angular velocities)
 2  1
(Same angular accelerations)
2
1
Differential Equations for Angular Motion
Rectilinear motion of a particle ( s , v , a )
Angular motion of a line on a body (  ,
d
 
dt
d
ii)  
 
dt
ds
 s
dt
dv
ii) a 
 v
dt
i)  
i) v 
iii) vdv  ads or a  v
P Kim (Chonbuk National University)
,  )
dv
ds
iii)  d    d  or   
Dynamics
d
ds
Undergraduate course, 2nd year
95
Plane Kinematics of Rigid Bodies
Rotation
Rotation II
(5/2)
Analytical Integration for Angular Motion
Acceleration
Constant
aa
 
Function of Time
a  f (t )
  g (t )
Function of Velocity
a  f (v )
  g ( )
Rectilinear motion of a particle [Ch.2]
 v  v0  at
   0   t
1 2
at
2
 v 2  v02  2 a  s  s 0 
1
    0  0t   t 2
2
2
2
    0  2    0 
 s  s 0  v0 t 
t
a  f (s)
  g ( )
P Kim (Chonbuk National University)
t
 v  v0   f (t ) dt
    0   g (t ) dt
 s  s 0   vdt
    0    dt
1
v0 f (v) dv
v
v
 s  s0  
dv
v0 f ( v )
t
0
t
0
v
t
 v 2  v02  2  f ( s ) ds
s0
t

s
s0
0
t
0
s
Function of Displacement
Angular motion of a line on a body
1
ds
v(s)

0
1
d
g ( )
   0  

0

d
g ( )

  2   02  2  f ( ) d 
0
t
Dynamics



1
d
0  ( )
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96
Plane Kinematics of Rigid Bodies
Rotation
Rotation III
(5/2)
Rotation about a Fixed Axis
1. When a rigid body rotates about a fixed axis, all points (other than those on the axis) move
in concentric circles about the fixed axis.
2. Any point (such as the point A) on a rigid body moves in a circle of radius r. Thus, its
position vector is perpendicular to its velocity vector (i.e., v  r ).
[Circular motion w.r.t. the (n  t ) coordinate system]
 v  ve t  re t  r e t


v2
2
2
a  e n  ve t  r e n  re t  r e n  r e t

r
or
 v  r

v2

 r 2  v
an 
r

 at  v  r
[Expressions using the vector cross product]
 v  r  ω  r
 r
 a  v  ω  r  ω
 r
 ω  (ω  r)  ω
 ω v  αr
Normal
v  ωr
a n  ω  (ω  r)   2 r
at  α  r
Tangential
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Plane Kinematics of Rigid Bodies
Absolute Motion of Rigid Bodies
Absolute Motion of Rigid Bodies
(5/3)
Absolute-Motion Analysis of Rigid Bodies’ Motion
1. Absolute motion of a rigid body is the motion observed directly from the origin of an
absolute coordinate system.
2. The position of any point on a body is determined by using the geometric relations, which
define the configuration of the body, and then the velocity and acceleration are obtained by its
time derivatives.
3. Geometric relations are constraint conditions to relate the motions of all points (or particles)
on a body.
(Ex. 5.4)
[Position vector of a point on a body]
r  x i  yj
[Geometric relations]
s  r
 x  s  r0 sin   r  r0 sin 

 y  r  r0 cos 
For the point C (r  r0 ),
 x  r   sin  

 y  r 1  cos  
[Velocity & Acceleration]
v  r & a  
r
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98
Plane Kinematics of Rigid Bodies
Relative Motion due to Rotation
Relative Motion due to Rotation of Rigid-Body I
(5/4)
Relative Velocity due to Rotation
1. The distance b/w two points A and B on the same rigid body is always constant. Thus, the
motion of one point A, as seen by an observer translating with the other point, must be circular
since the radial distance from the reference point B to the observed point A does not change.
[Position vector]
rA  rB  rA / B
at time t (state 1)
rA  rB   rA / B  at time t  t (state 2)
[Displacement vector]
 rA  rA    rB   rB    rA / B   rA / B 
 rA  rB  rA / B
for time interval t
[Note] rB : translation, rA / B : rotation
[Velocity vector]
r
rA
r
 lim B  lim A / B at time t
t  0  t
t  0  t
t  0
t
drA drB drA / B


dt
dt
dt
 v A  v B  v A / B  instantaneous velocity at time t
lim
[Relative velocity vector]
 v A / B  ω  rA / B
 similar to the rotation about the "fixed" point B
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99
Plane Kinematics of Rigid Bodies
Relative Motion due to Rotation
Relative Motion due to Rotation of Rigid-Body II
(5/4)
Relative Velocity due to Rotation (cont.)
1. All lines in a rigid body have the same angular displacement, velocity, and acceleration.
v A / B  ω  rA / B
v B / A  ω  rB / A  ω   rA / B   ω  rA / B   v A / B
 v A/ B   v B / A
Interpretation of the Relative-Velocity Equation
(General motion) = (Translation) + (Relative rotation)
 v A  v B  ω  rA / B
where
ω  θ (absolute angular velocity of the body)
v A / B  rA / B (i.e., v A / B is always perpendicular to rA / B )
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100
Plane Kinematics of Rigid Bodies
Relative Motion due to Rotation
Relative Motion due to Rotation of Rigid-Body III
(5/4)
Solution of the Relative-Velocity Equation
 [Absolute-motion analysis]
(Ex. 5.7)
rA  xi  yj   r  r0 sin  2     i   r  r0 cos  2     j

 v A  xi  y j   r  r0 cos  2     i   r0 sin  2     j
 v A   0.3(10)  0.2(10)(  1 2)  i   0.2(10)( 3 2)  j  4i  3 j
v A  v A  4.36 m/s
 [Principle of relative motion] v A  v O  v A / O
i) Scalar-geometric method
 v A / O  rA / O
 2
2
2
v A  vO  v A / O  2vO v A / O cos  2 3 
vA
120
 v A  3 2  2 2  2  3  2  cos  2 3   4.36 m/s
v A/O
vO
ii) Vector method
v A  v O  v A / O  v O  ω  r0
 3i   10k    0.2 cos  6  i  0.2 sin  6  j
 v O  r
 
vO
3

 10 rad/s
0.3
r
 3i  2 cos  6  k  i   2 sin  6  k  j
 3i  3  j  1 i 
 v A  4i  3 j &
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v A  v A  4.36 m/s
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101
Plane Kinematics of Rigid Bodies
Relative Motion due to Rotation
Relative Motion due to Rotation of Rigid-Body IV
(5/5)
Instantaneous Center of Zero Velocity
1. Rigid body may be in pure rotation about an axis (normal to the plane of motion), which is
called the instantaneous axis of zero velocity (or instantaneous axis of rotation).
2. The intersection of the instantaneous axis of zero velocity with the plane of motion is known as
the instantaneous center of zero velocity (or instantaneous center of rotation).
 v A  v A / C  rA / C  & v B  v B / C  rB / C 
 rA / C  rB / C (or rA / C  rB / C )  v A  v B (or v A  v B )
vA
: instantaneous angular velocity
rA
v  r
 v B  rB / C   rB / C  A   B / C v A
 rA  rA / C

3. Instantaneous center (point C) of zero velocity may lie on or off the body.
4. v A (  v A / C )  rA / C , v B (  v B / C )  rB / C : The absolute velocity vector of any point on a rigid body are
perpendicular to the relative position vector observed from the point C.
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102
Plane Kinematics of Rigid Bodies
Relative Motion due to Rotation
Relative Motion due to Rotation of Rigid-Body V
(5/5)
Instantaneous Center of Zero Velocity (cont.)
(Ex. 5.11)
 Point C: instantaneous center of zero velocity  v C  0
 [Principle of relative motion] v A  v C  v A / C
i) Scalar-geometric method
0
 v A  v A / C  rA / C   v A  rA / C

v A  v A / C  rA / C 
 2
2
2
 rA / C  rO / C  rA / O  2 rO / C rA / O cos  2 3 
 rA / C  (0.3) 2  (0.2) 2  2(0.3)(0.2) cos  2 3   0.436
 v A  rA / C   0.436(10)  4.36 m/s
 For any point A on the body,
rA / C  rj  r0  cos  i  sin  j  r0 cos  i   r  r0 sin   j
ω   k
v A  ω  rA / C   k   r0 cos  i   r  r0 sin   j
 vO  r
 
vO
3

 10 rad/s
r
0.3
  r0 cos   k  i     r  r0 sin   k  j
  r0 cos   j    r  r0 sin    i 
 v A   r  r0 sin   i   r0 cos   j
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103
Plane Kinematics of Rigid Bodies
Relative Motion due to Rotation
Relative Motion due to Rotation of Rigid-Body VI
(5/6)
Relative Acceleration due to Rotation
 v A  v B  v A / B   a A  a B  a A / B , where a A / B is the relative acceleration
1. Remember that the relative motion of a point A on a rigid body w.r.t. any other point B (i.e., the
origin of a moving coordinate system) must be circular motion about the point B.
(Relative motion on a rigid body) = (Rotation about a "fixed" axis) on page 97
So, we rewrite the relative acceleration vector, a A / B , as follows:
 a A/ B
 (a A / B ) n  ω  (ω  rA / B )   2 rA / B
 (a A / B ) n  (a A / B ) t where 
 (a A / B ) t  α  rA / B
( a A / B ) n  rA / B  2
or 
( a A / B ) t  rA / B

 a  a B  (a A / B ) n  (a A / B ) t
2. Relative acceleration ( a A / B ) depends on the absolute angular acceleration ( α ) as well as the
absolute angular velocity ( ω ).
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Plane Kinematics of Rigid Bodies
Motion Relative to Rotating Axes
Motion Relative to Rotating Axes I
(5/7)
- Use of rotating reference axes greatly facilitates the solution of many problems in kinematics
where motion is generated within a system or observed from a system which itself is rotating.
 X  Y : a fixed (absolute) coordinate system
[constant unit vectors] I and J
 x  y : a rotating (relative) coordinate system
[rotating unit vectors] i and j
 Position vector
rA  rB  r  rB   xi  yj , *[Note] r  rA / B w.r.t. a non-rotating (x -y ) system
Time Derivatives of Rotating Unit Vectors i and j
d   dt : infinitesmal change in angle during time interval dt
di  d  j & dj   d  i
 di d 
i  ω  i
 dt  dt j   j   (k  i )  ω  i


j  ω  j
 dj   d i   i   ( k  j)  ω  j
 dt
dt
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Plane Kinematics of Rigid Bodies
Motion Relative to Rotating Axes
Motion Relative to Rotating Axes II
(5/7)
Relative Velocity
 Position vector: rA  rB  rA / B , where rA / B  r  xi  yj
 Velocity vector: v A  v B  v A / B
 v A / B  r A / B  r , w.r.t. a non-rotating (x -y ) system
 Relative velocity vectors 
w.r.t. a rotating (x -y ) system
 v rel  xi  y j,
d
v A / B  r A / B  r   xi  yj
dt
 xi  yj   xi  y j  i  ω  i, j  ω  j, v rel  xi  y j


 x  ω  i   y  ω  j  v rel
 ω  xi  ω  yj  v rel  ω   xi  yj  v rel
 rA / B  xi  yj
 ω  rA / B  v rel
 Relative velocity vector:
v A / B  ω  rA / B  v rel
 Velocity vector:
v A  v B  ω  rA / B  v rel
*[Note] ω  rA / B  v A / B  v rel
 the term ω  rA / B is the difference b/w relative velocities w.r.t. rotating and non-rotating (x-y) systems.
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106
Plane Kinematics of Rigid Bodies
Motion Relative to Rotating Axes
Motion Relative to Rotating Axes III
(5/7)
Relative Velocity (cont.)
 ω  rA / B  v A / B  v rel  v P / B
 v rel  v A / P

Transformation of a Time Derivative b/w Non-Rotating and Rotating Systems
 V  Vx i  V y j : an arbitrary vector
 dV 





 i  ω  i, j  ω  j
  Vx i  V y j  Vx i  V y j
 dt  XY
 Vx i  Vy j  Vx  ω  i   V y  ω  j  Vx i  Vy j  ω  Vx i  ω  V y j


 Vx i  Vy j  ω  Vx i  V y j
 dV 

  ωV
 dt  xy
 )  (V
 )  ωV
 (V
XY
xy
i)
ii)
P Kim (Chonbuk National University)
 dV 



  Vx i  V y j, V  Vx i  V y j
dt

 xy
i) part of the total derivative of V measured relative to the x-y system
ii) part of the total derivative due to the rotation of the x-y system
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Plane Kinematics of Rigid Bodies
Motion Relative to Rotating Axes
Motion Relative to Rotating Axes IV
(5/7)
Relative Acceleration
 Velocity vector: v A  v B  v A / B  v B  (ω  rA / B  v rel )
 Acceleration vector: a A  a B  a A / B
a A / B  v A / B , w.r.t. a non-rotating (x -y ) system
 Relative acceleration vectors 
xi  
yj, w.r.t. a rotating (x -y ) system
a rel  
d
a A / B  v A / B   ω  rA / B  v rel 
dt
  rA / B  ω  r A / B  v rel  α  ω
 , r A / B (  v A / B )  ω  rA / B  v rel , v rel  ω  v rel  a rel
ω
 α  rA / B  ω   ω  rA / B  v rel   ω  v rel  a rel
 α  rA / B  ω   ω  rA / B   ω  v rel  ω  v rel  a rel
 α  rA / B  ω   ω  rA / B   2ω  v rel  a rel
v rel  ( xi  yj)  ( 
xi  
yj)
 x (ω  i )  y (ω  j)  a rel
 ω  xi  ω  y j  a rel  ω   xi  y j  a rel
 ω  v rel  a rel
 Relative acceleration vector: a A / B  α  rA / B  ω   ω  rA / B   2ω  v rel  a rel
 Acceleration vector:
P Kim (Chonbuk National University)
a A  a B  α  rA / B  ω   ω  rA / B   2ω  v rel  a rel
Dynamics
Undergraduate course, 2nd year
108
Plane Kinematics of Rigid Bodies
Motion Relative to Rotating Axes
Motion Relative to Rotating Axes V
(5/7)
Relative Acceleration (cont.)
 a A  a B  α  rA / B  ω   ω  rA / B     2ω  v rel   a rel 
i)
iii)
ii)
i) the acceleration of the circular motion of the position vector, r (  rP / B  rA / B ) , w.r.t. non-rotating axes
 ω   k , α   k , rA / B  r ( e N ), w.r.t. the (N -T -z ) coordinate system
ω  (ω  rA / B )   k   k  r ( e N )    r 2 k  (k  e N )
 a P / B  α  r A / B  ω   ω  rA / B 


  r 2 k  ( eT )  r 2 k  eT  r 2 e N
 where (a P / B ) N  ω  (ω  rA / B )  r 2 e N

(a P / B )T  α  rA / B  r eT
α  rA / B   k  r ( e N )   r k  e N   r ( eT )  r eT
ii) the acceleration of the relative motion of the point A w.r.t. rotating axes
 a rel  (a rel ) n +(a rel ) t w.r.t. the (n-t -z ) coordinate system
 (a rel ) n 
vrel

e n , (a rel ) t  vrel e t
iii) Coriolis acceleration
 a A  a B  α  rA / B  ω   ω  rA / B     2ω  v rel   a rel 
 2ω  v rel  a A  a B  α  rA / B  ω   ω  rA / B    a rel
 a A  a B  a P / B  a rel  a A  a P  a rel  a A / P  a rel
 2ω  v rel  a A / P  a rel
P Kim (Chonbuk National University)
Dynamics
Undergraduate course, 2nd year
109
Plane Kinematics of Rigid Bodies
Motion Relative to Rotating Axes
Motion Relative to Rotating Axes VI
(5/7)
Coriolis acceleration
 a Cor  2ω  v rel  a A / P  a rel  the difference b/w the acceleration of A relative to P as measured
from nonrotating axes and from rotating axes
a Cor  2ω  v rel   k  vrel e t  vrel k  e t  (vrel )e n
 ω   k , v rel  vrel e t  
 a Cor  (vrel )e n
 Coriolis acceleration is in the n-direction normal to the vector, v rel .
(Simple case)
 Purely rotating (x -y ) coordinate system  a O  0

  v rel  const.  a rel  0
ω  const.  α  0


a A  a O  α  rA / O  ω   ω  rA / O   2ω  v rel  a rel
 a A  ω   ω  rA / O   2ω  v rel
P Kim (Chonbuk National University)
Dynamics
Undergraduate course, 2nd year
110
Plane Kinematics of Rigid Bodies
Motion Relative to Rotating Axes
Motion Relative to Rotating Axes VII
(5/7)
Rotating vs. Nonrotating Systems
 a P / B  α  rA / B  ω   ω  rA / B 
 a A / P  a Cor  a rel  2ω  v rel  a rel

cf) a A  a P  2ω  v rel  a rel
where a P  a B  α  rA / B  ω   ω  rA / B 
(Summary: notations of physical quantities)
P Kim (Chonbuk National University)
Dynamics
Undergraduate course, 2nd year
111