Lecture 2a : Kinematics Examples
Rotating Reference Frame
d ( . ) /dt
=
( . )’ +  x ( . )
or using Coriolis’ theorem
Coriolis’ theorem : Differentiating a vector expressed in terms of a
rotating frame, given:
0) basis vectors of rotating frame i, j, k
1) position vector of a particle P, r(t) = x(t) i + y(t) j + z(t) k
2) angular velocity vector of the rotating frame
(t) =
x(t) i + y(t) j + z(t) k
The absolute velocity of P is
dr/dt =
x ' (t) i + y ' (t) j + z ' (t) k
+
 x r
Gerard Leng, ME Dept, NUS
Coriolis’ theorem is actually a differentiation rule:
d ( . ) / dt
=
( . )’ +
Gerard Leng, ME Dept, NUS
x( . )
Apply the theorem twice to find absolute acceleration
Denote
r ' (t) =
x ' (t) i + y ' (t) j + z ' (t) k
r '' (t) =
x '' (t) i + y '' (t) j + z '' (t) k
where i, j, k are the basis vectors of a rotating frame
The absolute acceleration is
d2r/dt2
=
r ''
+ 'xr
+  x ( x r )
+ 2xr'
Gerard Leng, ME Dept, NUS
Example 1 : Linear motion on a rotating base
j

1. An ant starts at the edge of a
disc of radius R and crawls
towards the centre at a constant
speed v.
v
O
P
i
2. The disc is rotating at a constant
speed  about the k axis.
3. What is the absolute velocity
and acceleration of the ant?
Gerard Leng, ME Dept, NUS
j
1. The reference frame ijk is a rotating
frame with angular velocity vector
 = k.
i
P
O
r
2. The position vector of the ant is
r(t) = (R – v t) i
0  t  2R/v
3. Apply Coriolis’ theorem:
dr/dt
=
r’
+
x r
=
=
Gerard Leng, ME Dept, NUS
4. The absolute velocity is v =
-vi +
 (R – v t) j
Applying Coriolis’ theorem again:
dv/dt
=
v’
+
x v
=
=
The absolute acceleration is a = - (2v) j - 2 (R – v t) i
Gerard Leng, ME Dept, NUS
Example 2 : Circular motion on a rotating base
1. Disc 1 rotates about point
O at a constant speed .


R
O
r
o
disc 2
disc 1
P
2. Disc 2 is attached to disc 1
at point o and rotates at a
constant speed of  with
respect to Oo.
3. Find the absolute velocity
and acceleration of point P
on the rim of disc 2.
4. The initial position of P is
as shown in the diagram.
Gerard Leng, ME Dept, NUS
1. Question : Where to put the rotating reference frame ?
Hint : Two possibilities - point O or point o.
P
j
i
2. What is the angular velocity vector
of the rotating frame ?
o
O

=
k
3. Now write the position vector of
point P, r = x i + y j
x
y
=
=
Gerard Leng, ME Dept, NUS
4. Apply Coriolis’ theorem to get absolute velocity
v
v
=
r’ +
x r
r ( -sin (t) i + cos(t) j )
+
=
=
 k x [ ( R + r cos (t) )i + r sin(t) j ]
[ - r sin (t) - r  sin(t) ] i
+ [ r cos (t) +  ( R + r cos(t) ) ] j
=
- r (  +  ) sin (t) i + [ R + r (  + ) cos(t) ] j
Gerard Leng, ME Dept, NUS
4. Apply Coriolis’ theorem again to get absolute acceleration
a
a
=
v’ +  x v
= { -r (  +  ) sin (t) i + [ R + r (  + ) cos(t) ] j }’
+  k x{- r (  +  ) sin (t) i + [ R + r (  + ) cos(t) ] j }
= - r (  +  ) cos(t) i - r (  + ) sin(t) j
-  [ R + r (  + ) cos(t) ] i - r  (  -  ) sin (t) j
= - [ R2 + r (  + )2 cos(t) ] i Gerard Leng, ME Dept, NUS
r (  + )2 sin (t) j
So what happens when  = -  ?
v =
- r (  +  ) sin (t) i + [ R + r (  + ) cos(t) ] j
=
a = - [ R2 + r (  + )2 cos(t) ] i -
r (  + )2 sin (t) j
=
This suggest P is
Animate
Gerard Leng, ME Dept, NUS
Example 3 : Quick return mechanism
B

P
O
R
1. A crank OP of length R rotates at a
constant angular speed . The crank is
connected to a slider at P.
2. The rod AB passes through the slider
and rotates about point A where OA = L.
3. Find the angular velocity of the rod AB
and the speed of the slider relative to the
rod AB at the extreme position of the
mechanism.
L
A
Animate
4. Find the acceleration of the slider
relative to the rod AB and the angular
acceleration of the rod AB.
Gerard Leng, ME Dept, NUS
Solution
1. The extreme position occurs when
angle OPA is 90o.
j
B
O
2. The absolute velocity of P is R
along PB
P
3. Where should we put the reference
frame?

A
4. At the extreme position, the absolute velocity
of P written in terms of ij is
v
i
=
5. The frame ij rotates with an angular velocity
k. Let AP = r j with  and r unknown.
Gerard Leng, ME Dept, NUS
6. Using Coriolis theorem
v
=
or
Equate components

This means that at the extreme position
i) the rod AB has 0 angular velocity
ii) The slider is moving at a speed of R relative to the rod.
Gerard Leng, ME Dept, NUS
7. Using Coriolis’ theorem again,
a
=
=
=
v’ + k x v
( r’ + k x r)’ + k x v
r’’ + ’ k x r + k x r’ + k x v
At the extreme position :
a =
,
and
r =
,
 =
r’’ =
,
Hence
 The acceleration of the slider relative to the rod AB, r’’ =
The angular acceleration of the rod AB, ’ =
Gerard Leng, ME Dept, NUS
Example 4 : Geneva mechanism


1. Disc A , radius R, rotates at a
constant speed  rad/s.
P
2. Disc B, radius rotates
intermittently at (t) rad/s.
B
A
3. Find the distance AB for the
mechanism to work.
4. Hence determine  and ’
for the time the peg P enters
and leaves the slot.
Gerard Leng, ME Dept, NUS
1. Observation For the peg to enter
the slot, its velocity must be parallel
to the slot at the point of entry.
j
P
R
R
R
 Angle BPA is 90o.
Hence BA =
B
A
2. Place a rotating frame at B
aligned with the slot. The angular
velocity vector of the frame is k.
i
3. Let the position vector of P be r = r(t) j . Apply Coriolis’ theorem
dr/dt
=
Gerard Leng, ME Dept, NUS
At the point of entry, r = R j
Hence

4. Apply Coriolis’ theorem again
a
=
r’’ + ’ k x r + k x r’ + k x v
At the point of entry,

Gerard Leng, ME Dept, NUS
Question
What is  and ’ when the peg is in the slot ?
Hint : Angle BPA is not 90o and BP is less than R.
j
P
r
R
B
2 R
A
R
i
Gerard Leng, ME Dept, NUS