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Presentation03: Kinematics analysis of mechanisms

Outline

• Four-bar linkage: introduction; velocity and acceleration analyses (graphical approach).approach).

• Crank-slider mechanism: position, velocity, and acceleration analyses (graphical and analytical approaches)(graphical and analytical approaches).

• General analytical approach: the matrix formulation.

• Elements for the analytical study of Relative Motions.

FOUR-BAR LINKAGE

Four-bar linkageFour bar linkage

GRASHOF’s rulea: longest bar, b: shortest barc, d: intermediate length bars.

– a + b < c + d Grashof mechanism– a + b > c + d non-Grashofian mechanism – a + b = c + d Change-point mechanism

FOUR-BAR LINKAGE

Grashof-type four-barypCRANK – ROCKER

Dead-point configurations

Grashof-type four-barTWO CRANKS

Grashof-type four-barTWO ROCKERS

FOUR-BAR LINKAGE

Change-point mechanism

Isosceleslinkage

Table lampParallelogram linkage Antiparallel

linkagelinkage

Locomotive

FOUR-BAR LINKAGE

Position analysis

Known: geometry, θ1O1O3

O1AAB

2 BA

ABBO3

1

2

3A

2 BA

4

θ1

O1 O3

13

2’

A

θ1 3

4O1 O33’B’

θ1

B

FOUR-BAR LINKAGE

Position analysis

A NO SOLUTION

1

Aθ1

4O1 O3

AA

B≡B’

4

1

O O

θ1SINGULARITY4O1 O3

FOUR-BAR LINKAGE

Velocity analysis

C24Known: geometry, position, 1

B

2

vA ≡ C

1

2

3

BA

A

vB

C12 ≡≡ C23 1 vA, 2 vB, 3

C ≡4

11

O

3

B

C ≡ OC34 ≡4O1C14 ≡ O3

FOUR-BAR LINKAGE

Acceleration analysis

1 aA

a (a ? a )Known: geometry, position, 1 (assumed as constant), 2, 3

2

3

BA

2 aBAn (aBAt? aBAn)

3aBn (aBt? aBn)

1 3

aAaBAnaBn

aBt

= +4O1 O3

aA

aBaBn

aB = aBn + aBt

aA aBAn

aBt aBAtaA aBAn

aBAt

aBn aB aB = aA + aBAt + aBAt

RRRP (or 3R-P) KINEMATIC CHAIN

Crank-Slider mechanism

Crank-Slotted mechanism

CRANK-SLIDER MECHANISM

Velocity analysis

C24

1 vA, 2 vB24

C342

A

1

≡ C12

C34C31

2vA

B1 2

3C23 ≡

1 vBO

4C14 ≡

CRANK-SLIDER MECHANISM

Kinematic analysis: analytical methodA

lr

O B

s

cos( ) cos( ); sin( ) sin( )Bs r l Position

2 2

sin(2 ) cos( )(sin( ) );2 cos( )1 sin ( )

Bs r

Velocity

2 2

2 2

cos( ) cos( ) sin( ) sin( );

sin( ) cos( ) sin( ) cos( )s r r l l

Acceleration

sin( ) cos( ) sin( ) cos( )Bs r r l l

KINEMATIC ANALYSIS: ANALYTICAL METHOD

Matrix formulation

Position: q s1 DOF systems:q := independent variables := dependent variables( , ) 0f q s

Velocity:

Closure equations

q s q s

s : dependent variables

Velocity:

1( , )0 ( ) ( )

d f q s f fq s s B h q q k q q

, ,q s q s

1det( ) 0B

( ) ( )q q q q qdt q s

Bh

1det( ) 0B

Acceleration:

2 2'( )( )d s k qk k k

, , , ,q s q s q s

SINGULARITY

2 2( )( ) qs k q q q k q k qdt q

KINEMATIC ANALYSIS: ANALYTICAL METHOD

Matrix formulation: example (Crank-slider)

( , ) 0f q s Position: Closure equations:

:

q

s

cos( ) cos( ) 0

sin( ) sin( ) 0Br l s

r l

:B

ss

Velocity: 1( , )0 ( ) ( )

d f q s f fq s s B h q q k q q

dt q s

det( ) 0B

Bh

sin( ) sin( ) 10

r l

( ) ( )0

cos( ) cos( ) 0B

r l s

det( ) cos( ) 02

B l

KINEMATIC ANALYSIS: ANALYTICAL METHOD

Matrix formulation: example (Crank-slider)

Velocity: 1( , )0 ( ) ( )

d f q s f fq s s B h q q k q q

dt q s

det( ) 0B

Bh

1 cos( )0 sin( )r

2

sin( )cos( ) cos( )

cos( )1 tan( ) sin( ) tan( ) cos( )B

rl

rs r r

Acceleration: 2 2'( )( )d s k qs k q q q k q k qdt q

( ) i ( )

2

cos( ) sin( )cos( ) cos( )

sin( ) tan( ) cos( ) cos( ) tan( )sin( )Bs r r r r

sin( ) tan( ) cos( ) cos( ) tan( )sin( )B r r r r

RELATIVE MOTION

Position

0 1 1 0 1 1 1 1 0 0 0 0( - ) ( - ) ( - ) x y x yP O P O O O i j i j

1 Velocity

P

y1

20( - )

Pd P O

dt v

y0

jO1

x1j1 i1 1 1

0 0 0 0 1 1 1 1 1 1x y x y x yd ddt dt

i ji j i j

j0

i0 x0O01 1( )O r T rP O v v v v

RELATIVE MOTION

0 1 1 0 1 1 1 1 0 0 0 0( - ) ( - ) ( - ) x y x yP O P O O O i j i j

0 0 0 0 1 1 1 1 1x y x y ( )P P O

v i j i j

0 1 1 0 1 1 1 1 0 0 0 0( ) ( ) ( ) y yj j

Acceleration

1 1 1

0 0 0 0 1 1 1 1 1 1 1( )x y x y x y ( )P

d d d P OP Odt dt dt

i ja i j i j

2( ) ( ) 2

dt dt dt

P O P O

a a v1 1( ) ( ) 2O r rP O P O

a a v

a a aT r C a a a

KINEMATICS: SUMMARY

T i /P bl M h dTopic/Problem

• Kinematics of a particle

Methods

Cartesian planar vectors; p

• Kinematics of a rigid body

pComplex Numbers

Cartesian planar vectors;Kinematics of a rigid body(Rivals theorem, Instant Centre ofRotation, Kennedy-Aronhold theorem, Rotational/Translational/Rolling motions)

Cartesian planar vectors; Complex Numbers

g )

• Kinematic analysis of mechanisms(Position, Velocity, and Acceleration analyses,

Graphical approach;Analytical approaches:(Position, Velocity, and Acceleration analyses,

Relative motion)Analytical approaches:

• explicit formulation

• matrix formulation

• Complex Numbers