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Velocity Analysis
Chapter 6
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Definition
• Rate of change of position with
respect to time
– Angular
– Linear
– Position Vector
– Velocity
dt
d
dt
dRV
jPA peR
jjPA
pa epdt
dpje
dt
RV
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Definition
– Velocity (absolute)
• The velocity is always in a
direction perpendicular to
the radius of rotation and is
tangent to the path of
motion
jjPA
PA epdt
dpje
dt
RV
cossinsincos jpjjpPA V
" Absolute" PPA VV
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Definition
– Velocity (difference)
– Relative Velocity
APPA VVV
"body same on the"
PAAP VVV
APPA VVV
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Graphical Analysis
• Graphical Velocity Analysis
– Solve for
PAAP VVV
rv V
CBA ,, ocities;linear vel
, ; locitiesangular ve 43
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Graphical Analysis
• Graphical Velocity Analysis
PAAP VVV rv V
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Graphical Analysis
• Example 6-1– Given θ2, θ3, θ4, ω2 find ω3,
ω4, VA, VB and VC
– Position analysis already
performed
– 1. Start at the end of the
linkage about which you
have the most information.
Calculate the magnitude of
the velocity of point A,
22 AOvA
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Graphical Analysis
• Example 6-1– 2. Draw the velocity VA
– 3. Move next to a point
which you have some
information, point B. Draw
the construction line pp
through B perpendicular to
BO4
– 4. Write the velocity
difference equation for point
B vs. ABAAB VVV
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Graphical Analysis
• Example 6-1– 5. Draw construction line qq
through point B and
perpendicular to BA to
represent the direction of VBA
– 6. The vector equation can
be solve graphically by
drawing the following vector
diagram
BAAB VVV
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Graphical Analysis
• Example 6-1– 7. The angular velocities of
link 3 and 4 can be
calculated,
– 8. Solve for VC
4
4BO
vBBA
vBA3
CAAC VVV
3cvCA
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Instant Center of
Velocity
• An instant center of velocity is a point, common
to two bodies in plane motion, which point has
the same instantaneous velocity in each body
• The numbers of IC is calculated with;
• Linear graph is a useful way to keep track of
which IC have been found
2
1
nnC
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Instant Center of
Velocity
• Kennedy’s Rule
– Any three bodies in plane motion will have
exactly three instant centers, and they will lie
an the same straight line
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Instant Center of
Velocity
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Instant Center of
Velocity
• Slider-Crank Linkage
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Instant Center of
Velocity
• Slider-Crank Linkage
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Instant Center of
Velocity
• Slider-Crank Linkage
• Check Example 6-4:
IC for a Cam-
Follower Mechanism
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Velocity Analysis
with IC
• Once the ICs have
been found, they can
be used to do a very
rapid graphical velocity
analysis of the linkage
22 AOvA
3,1
3AI
vA 4
4BO
vB
33,1 BIvB 33,1 CIvC
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Velocity Analysis
with IC
• A rapid graphical
solution for the
magnitudes at B and C
are found from vectors
drawn perpendicular to
that line at the
intersection of the arcs
and line AI1,3 (VB’, VC’)
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• Angular Velocity Ratio
– Output angular velocity
divided by the input
angular velocity
– Can be derived by
constructing a pair of
effective links
2
4
Vm
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– Effective link pairs is
two lines, mutually
parallel, drawn through
the fixed pivot and
intersecting the coupler
extended
sin22 AOAO
sin24 BOBO
22 AOVA BA VV
sin
sin
4
2
4
2
2
4
BO
AO
BO
AOmV
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– Now the effective links
are colinear and
intersect the coupler at
the same point, I2,4
4,24
4,22
2
4
IO
IOmV
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• Mechanical Advantage
– Power in a mechanical system,
– For rotating system,
– Mechanical efficiency,
– Mechanical Advantage,
yyxx VFVFVFP
TP
out
in
out
in
out
in
out
in
in
out
in
outA
r
r
AO
BO
r
r
r
r
T
T
F
Fm
sin
sin
2
4
in
out
P
P
out
in
in
out
T
T
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Centrodes
• The path, or locus, created by a IC at
successive positions
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Centrodes
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Velocity of Slip
• Used when there is a sliding joint between
two links and neither one is the ground
– Example 6-5, 6-6
slipslip AAslip VVV2442
4
344
AO
VA
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Velocity of Slip
3
343
AO
VA
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Analysis Solution
• Position Analysis (revisited)01432 RRRR
01432 jjjj
decebeae
A
ACBB
2
4arctan2
2
4 2,1
D
DFEE
2
4arctan2
2
3 2,1
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Analysis Solution
• Velocity Analysis
01432 jjjj
decebeae
01432 jjjj
decebeaedt
d
0432 432 dt
djce
dt
djbe
dt
djae
jjj
0432
432 jjj
ejcejbeja
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Analysis Solution
• Velocity Analysis
0 BBAA VVV
4
3
2
4
3
2
j
B
j
BA
j
A
ejc
ejb
eja
V
V
V
identityEuler
part real
partimaginary
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Analysis Solution
• Velocity Analysis
43
2423
sin
sin
b
a
34
3224
sin
sin
c
a
444444
333333
222222
cossinsincos
cossinsincos
cossinsincos
jcjjc
jbjjb
jajja
B
BA
A
V
V
V
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Analysis Solution
– Slider-Crank
01432 jjjj
decebeae
032
32 dejbejajj
01432 RRRR
BAAB VVV
0 BABA VVV BAAB VV
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Analysis Solution
– Slider - Crank
2
3
23
sin
sin
b
a
3322 sinsin bad
ABBA
AB
A
jb
ja
VV
V
V
333
222
cossin
cossin
BAAB VVV
Crank-Slider Inverted - Review
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Geared Fivebar
015432 jjjjj
fedecebeae
05432
5432 jjjj
ejdejcejbeja
015432 RRRRR
25 25
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Geared Fivebar
343
54542243
cos2cos
sinsinsin2
b
da
4
5533224
sin
sinsinsin
c
dba
BAAB
C
BA
A
jd
jb
ja
VVV
V
V
V
555
333
222
cossin
cossin
cossin
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Velocity of Any Point
• Once the angular velocities of all the links are
found it is easy to define and calculate the
velocity of any point on any link for any input
position of the linkage
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Velocity of Any Point
• To find the velocity of points S & U
22220 sincos22
2
jssej
SS RR
222222 cossin22
jsjse
j
SV
44440 sincos44
4
juuej
UR
444444 cossin44
jujse
j
UV
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Velocity of Any Point
• To find the velocity of point P
PAAP
j
PA jppe
RRR
R
3333 sincos33
PAAP
j
PA jpjpe
VVV
V
333333 cossin33