Relative-Motion Analysis: Velocity
• Translation only
Kinematics
– Position



rB  rA  rB / A
– Velocity


vB  vA
– Acceleration


aB  aA
y
A
B
rA
rB
x
Relative-Motion Analysis: Velocity
• Transl. & Rotation
y
(General Plane Motion)
– Position



rB  rA  rB / A
– Velocity (time deriv)



vB  v A  vB / A
 

where v B / A    rB / A
and (ω is rotation of
member about A)
For our problems, we will just need to plug in for
each of these variables to get vB.and often ω
drA
A
rB/A
B
drA
drB
rA
rB
x
dθ
rB/A (new)
drB/A
Review of Cross Products
• See Section 4.2 for full details
 
A  B  Ax
ˆj
kˆ
Ay
Az
Bx
By
Bz
iˆ
or
To use, must define right-hand
x, y, z coordinate system
Example Problem
If rod AB slides along the horizontal slot with a velocity of 60 ft/s,
determine the angular velocity of link BC at the instant shown.
(F16-11, 48 rad/s)
What about the velocity of the pin at C, and the
angular velocity of wheel OC at that instant?
(104 ft/s up)
Special Case for Velocity Solution
Rolling without slip
Can also have slip, in that instance
direction of vA is at least known but
magnitude unknown
Example Problem
A bowling ball is cast on the “alley” with a backspin of ω = 10
rad/s while its center O has a forward velocity of vO = 8 m/s.
Determine the velocity of the contact point A in contact with the
alley. (16-58, 9.20 m/s to the right)
Instantaneous Center of Zero Velocity
• Relate velocity of two
points on right body
Rolling without slip (not
always)
 


v B  v A    rB / A
• What if choose a point A
which is instantaneously
stationary (i.e. vA = 0)
 

v B    rB / A ;
vB  r
• Can we find an instant
point with this property
to relate to?
What if we want velocity at each point on rim?
(each point will instantaneously rotate about axis
fixed to that point)
Instantaneous Center of Zero Velocity
• Does an I.C. always exist?
–
–
–
–
At some instant, yes
Consider curvilinear motion in particle mech.
For rigid body?
I.C. need not be ON the body?
• To find I.C.
– Identify instantaneous direction
of velocity for each point
– Draw perpendicular lines from each
– Intersection is I.C. at that instant
• To solve
vPoint = ωrPoint/IC
Graphic Examples
To find I.C.
Identify instantaneous direction
of velocity for each point
Draw perpendicular lines from each
Intersection is I.C. at that instant
To solve
vPoint = ωrPoint/IC
Example Problem
If link CD has an angular velocity of 6 rad/s, determine the
velocity of point E on link BC and the angular velocity of link AB
at the instant shown.
(16-89, 6 rad/s CCW, 4.76 m/s, 40.9° above – x)