Velocity Analysis

Chapter 6

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

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

Definition

– Velocity (difference)

– Relative Velocity

APPA VVV

"body same on the"

PAAP VVV

APPA VVV

Graphical Analysis

• Graphical Velocity Analysis

– Solve for

PAAP VVV

rv V

CBA ,, ocities;linear vel

, ; locitiesangular ve 43

Graphical Analysis

• Graphical Velocity Analysis

PAAP VVV rv V

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

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

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

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

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

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

Instant Center of

Velocity

Instant Center of

Velocity

• Slider-Crank Linkage

Instant Center of

Velocity

• Slider-Crank Linkage

Instant Center of

Velocity

• Slider-Crank Linkage

• Check Example 6-4:

IC for a Cam-

Follower Mechanism

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

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’)

• 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

– 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

– Now the effective links

are colinear and

intersect the coupler at

the same point, I2,4

4,24

4,22

2

4

IO

IOmV

• 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

Centrodes

• The path, or locus, created by a IC at

successive positions

Centrodes

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

Velocity of Slip

3

343

AO

VA

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

Analysis Solution

• Velocity Analysis

01432 jjjj

decebeae

01432 jjjj

decebeaedt

d

0432 432 dt

djce

dt

djbe

dt

djae

jjj

0432

432 jjj

ejcejbeja

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

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

Analysis Solution

– Slider-Crank

01432 jjjj

decebeae

032

32 dejbejajj

01432 RRRR

BAAB VVV

0 BABA VVV BAAB VV

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

Geared Fivebar

015432 jjjjj

fedecebeae

05432

5432 jjjj

ejdejcejbeja

015432 RRRRR

25 25

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

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

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

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