KINEMATICS OF RIGID BODIES
MOTION RELATIVE TO ROTATING
AXES
In the previous article, we have used nonrotating reference axes to
describe relative velocity and relative acceleration.
Rotating as well as translating reference axes are used if the body lies
in the rotating system or is observed from a rotating system.
Let’s consider the plane motion of two particles A and B in the fixed X –
Y plane.
Y
Translating+Rotating reference axes
A(x,y)
 
r  rA / B
x
y
y

rA
 B(x,y)
rB

J
O

j
w,a

I
x

i
Fixed reference axes
X
We will consider A and B to be moving independently of one another.
We observe the motion of A from a moving reference frame x-y
which has its origin attached to B and which rotates with an angular





velocity w   ;the vector notation will be w  wk  k .
Y
Translating+Rotating reference axes
A(x,y)
 
r  rA / B
x
y
y

rA
 B(x,y)
rB

J
O

j
w,a

I
x

i
Fixed reference axes
X
The absolute position vector of A is

 
rA  rB  rA / B
 


rA / B  r  xi  yj
Time Derivatives of Unit Vectors
Since the unit vectors


i and j are now rotating with the x-y axes,
their time derivatives will not be zero.
  
di  i dj




di  dj


di d 

j
dt
dt



i  wj
1
 

dj  j d (i )




dj  di


dj
d 

i
dt
dt



j  wi
1


i  wj


j  wi
By using the cross product,
 

 
w  i  wk  i  wj
 

 
w  j  wk  j  wi
  
i  wi
  
j w j
Relative Velocity
We are now going to take the time derivative of the position vector

 
 
rA  rB  rA / B  rB  r

 
 
dr
 x i  y j  xi  yj
dt
  
here x i  w  xi

 i  w  i ,

vrel

dr d  

xi  yj



dt dt 



r
   
j w j

  
yj  w  yj

     
dr
 xi  yj  w  xi  w  yj
dt

   
dr
 xi  yj  w  r

dt 




dr
v A  vB 
dt


  
 
dr
 xi  yj  w  xi  yj




dt

  


v A  vB  w  r  vrel


r

vB
is the absolute velocity of B due to the translation of axes x-y.

v
If the x-y axes are not translating but only rotating , B = 0.
 
The term w  r appears due to the rotation of the x-y axes.

Its magnitude is r or rw and a direction normal to r .

vrel will be tangent to the path fixed in the x-y plane and its
magnitude will be equal to
s
where s is measured along the path.
Its sense will be in the direction of increasing s.
Relative Acceleration
The relative acceleration equation may be obtained by differentiating the
relative velocity equation,
  


v A  v B  w  r  v rel

   
dr
 xi  yj  w  r

dt 

    d 


a A  aB  w
 r  w  r  vrel 

 
dt **
*
a r
*
**
vrel
     
    
w  r  w  w  r  vrel   w  w  r   w  vrel



 

d 
d      
vrel   vrel  xi  yj   xi  yj   xi  yj


dt
dt






 w  x i  w  y j   xi  yj
 
 

 w  x i  y j   xi  yj




 

 w  vrel  arel


a A  aB 


arel
 
  
w  r  w  w  r  
 
2w  vrel 





aco rio lis

a rel

arel t ,  arel n


a A  aB 
 
  
w
 r  w  w  r  

 
a r
 
2w  vrel 





aco rio lis
vrel

a rel

arel t ,  arel n
  is r and its direction is tangent to the
wr
  
circle. The magnitude of w  w  r  is rw2 and its direction is from P
The magnitude of
to B along the normal to the circle.

arel may be expressed in rectangular, normal and tangential or polar
coordinates in the rotating system. Frequently, n and t components
are used as shown in the figure.
The tangential component has the
magnitude arel  vrel  s where s is the
t
distance measured along the path to A.
The
normal
magnitude a reln 
component
v 2 rel
has
the
. The sense of this

vector is always toward the center of
curvature.
Coriolis Acceleration
The term


2w  vrel
seen in the figure is called the Coriolis
acceleration. It represents the difference between the acceleration
of A relative to P as measured from nonrotating axes and from
rotating axes.
The direction is always normal to
the vector

v rel

or arel t and the
sense is established by the right
hand rule for the cross product.
The Coriolis acceleration appears when a particle or body
translates in addition to its rotation relative to a system which
itself is rotating. This translation can be rectilinear or curvilinear.
PROBLEMS
1. The disk rolls without slipping on the horizontal surface, and at the instant
represented, the center O has the velocity and acceleration shown in the figure.
For this instant, the particle A has the indicated speed
change of speed
u
u and the time rate-of-
, both relative to the disk. Determine the absolute velocity
and acceleration of particle A.
PROBLEMS
2.
For
the
represented,
link
instant
CB
is
rotating ccw at a constant
rate N = 4 rad/s and its pin A
causes a cw rotation of the
slotted
member
ODE.
Determine the angular velocity
w and angular acceleration a
of ODE for this instant.
PROBLEMS
3. Link OA has a constant clockwise angular velocity of 3 rad/s for a
brief interval of its rotation. Determine the angular acceleration aBC
of BC for the instant when =60o.
PROBLEMS
4. Determine the angular acceleration of link EC in the position shown,
2
where w  b  2 rad / s and b  6 rad / s when b=60o. Pin A is fixed
to link EC. The circular slot in link DO has a radius of curvature of 150
mm. In the position shown, the tangent to the slot at the point of
contact is parallel to AO.
AC  150 mm
150 mm
PROBLEMS
5. For the instant shown, particle A has a velocity of 12.5 m/s towards point C
relative to the disk and this velocity is decreasing at the rate of 7.5 m/s each
second. The disk rotates about B with angular velocity w=9 rad/s and angular
acceleration a=60 rad/s2 in the directions shown in the figure. The angle b
remains constant during the motion. Telescopic link has a velocity of 5 m/s and an
acceleration of 2.5 m/s. Determine the absolute velocity and acceleration of
point A for the position shown.
PROBLEMS
6. The gear has the angular motion shown. Determine the angular velocity and
angular acceleration of the slotted link BC at this instant. The pin at A is fixed to
the gear.
C
A
w=2 rad/s
2m
0.5 m
0.7 m
B
O
a=4 rad/s2
PROBLEMS
7. The pin A in the bell crank AOD is guided by the flanges of the
collar B, which slides with a constant velocity vB of 0.9 m/s along the
fixed shaft for an interval of motion. For the position =30o determine
the acceleration of the plunger CE, whose upper end is positioned by
the radial slot in the bell crank. .
PROBLEMS
8. Link 1, of the plane mechanism shown, rotates about the fixed point
O with a constant angular speed of 5 rad/s in the cw direction while
slider A, at the end of link 2, moves in the circular slot of link 1.
Determine the angular velocity and the angular acceleration of link 2 at
the instant represented where BO is perpendicular to OA. The radius
of the slot is 10 cm.
Take sin 37=06, cos 37=0.8
1
A
10 cm
2
20 cm
37o
C
37o
w1=5 rad/s
B
O
16 cm
BO  OA