presentation03 kinematic analysis of mechanisms

7/24/2019 Presentation03 Kinematic Analysis of Mechanisms

1/17

Presentation03: Kinematics analysis of mechanisms

Outline

Four-bar linkage: introduction; velocity and acceleration analyses (graphical

.

Crank-slider mechanism: position, velocity, and acceleration analyses

.

General analytical approach: the matrix formulation.

Elements for the analytical study of Relative Motions.


2/17

FOUR-BAR LINKAGE

-

GRASHOFs rule

a: longest bar, b: shortest bar

c, d: intermediate length bars.

a + b < c + d Grashof mechanism

a + b > c + d non-Grashofian mechanism

a + b = c + d Change-point mechanism


3/17

FOUR-BAR LINKAGE

Grashof-t e four-bar

CRANK ROCKER

Dead-point configurations

Grashof -type four-bar

TWO CRANKSGrashof-type four-bar

TWO ROCKERS


4/17

FOUR-BAR LINKAGE

Change-point mechanism

Isosceles

linkage

Table lampParallelogram linkage Antiparallel

Locomotive


5/17

FOUR-BAR LINKAGE

Position analysis

Known: geometry, 1

O1O3

O1A

B

BO3

13

A2

B

4

1

O O

13

2

4O1 O33

1


6/17

FOUR-BAR LINKAGE

Position analysis

A NO SOLUTION

1

1

4O1 O3

BB

1 1SINGULARITY

1 3


7/17

FOUR-BAR LINKAGE

Velocity analysis

C24Known: geometry, position,

1

2

v

2

3

A

vB

C12

23

1 vA, 2 vB, 3

1

3

34 114 3


8/17

FOUR-BAR LINKAGE

Acceleration analysis

1 aA

Known: geometry, position, 1 (assumed

as constant), 2, 3

2B

A

2 BAn BAt BAn

3aBn (aBt? aBn)

1

aBAnaBnaBt

4O1 O3

aBaBn

aB aBn aBt

aA aBAn

aBt aBAtaA aBAn

aBAt

aBn aB aB = aA +aBAt +aBAt


9/17

RRRP (or 3R-P) KINEMATIC CHAIN

Crank-Slider mechanism

Crank-Slotted mechanism


10/17

CRANK-SLIDER MECHANISM

Velocity analysis

C

1 v

A,

2 v

B

C

A C12C31 vA

B2

3

C23

1 vBO

4

C14


11/17

CRANK-SLIDER MECHANISM

Kinematic analysis: analytical method

A

lr

O B

s

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

s r l Position

2 2

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

2 cos( )1 sin ( )

Bs r

Velocity

2 2

2 2

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

Acceleration

B


12/17

KINEMATIC ANALYSIS: ANALYTICAL METHOD

Matrix formulation

Position: q s 1 DOF systems:

q:= independent variable=( , ) 0f q s Closure equations

1( , )0

d f q s f f s s B h k

, ,

1det( ) 0B

dt q s

Bh 1

det( ) 0B

Acceleration:

'd s k

, , , ,q s q s q s

SINGULARITY

s q q q q q

dt q


13/17


Matrix formulation: example (Crank-slider)

( , ) 0f q s Position:Closure equations

:q

cos( ) cos( ) 0

sin( ) sin( ) 0

Br l s

r l

Bs

Velocity: 1( , ) 0 ( ) ( )d f q s f f

q s s B h q q k q qdt q s

det( ) 0B

Bh

sin sin 1r l

0cos( ) cos( ) 0

B

r ls

det( ) cos( ) 02

B l


14/17


Matrix formulation: example (Crank-slider)

Velocity: 1( , )

0 ( ) ( )d f q s f f

q s s B h q q k q q

dt q s

det( ) 0B

Bh

1 cos( )0 sinr

2 cos( ) cos( )

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

lrs r r

Acceleration: 2 2'( )( )

d s k qs k q q q k q k q

dt q

2

cos s n

cos( ) cos( )

sin tan cos cos tan sins r r r r


15/17

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

2

0( - )P

d P O

dt v

y0O

1

x1j1 i1

1 1

0 0 0 0 1 1 1 1 1 1x y x y x yd d

dt dt

i j

i j i j

0

i0x0O0

1 1( )

O r T r P O v v v v


16/17

RELATIVE MOTION

- - - x xP O P O O O i i

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

v i j i j

Acceleration

1 1 10 0 0 0 1 1 1 1 1 1 1

( )x y x y x y ( )

P

d d d P OP O

i ja i j i j

2

1 1O r r

T r C


17/17

KINEMATICS: SUMMARY

op c ro em

Kinematics of a particle

et o s

Cartesian planar vectors;

Complex Numbers

(Rivals theorem, Instant Centre ofRotation, Kennedy-Aronhold theorem,

Rotational/Translational/Rolling motions)

Complex Numbers

Kinematic analysis of mechanisms

Position Velocit and Acceleration anal ses

Graphical approach;

Anal tical a roaches:Relative motion)

explicit formulation

matrix formulation

Complex Numbers

presentation03 kinematic analysis of mechanisms

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