kinema tics of mechanisms

8/12/2019 Kinema Tics of Mechanisms

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Chair of Mechanics

Kinematics of multi-loop mechanisms


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Chair of Mechanicsmultibody system

Multibody system:

lZ

jZ

iZ

:Zj rigid body coordinates

open

1

j

i

:i auxiliary parameters(naturaljoint coordinates)

closed


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Chair of Mechanicsconstraint equations

Constraint equations:

3z

2z

1z

z

zd

z 0=);(y q

i

j

)(

open:

q

0

closed:

);(g q

0=);(g q

.


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Chair of Mechanicsblock diagrams I

Example:

transmission element four-bar mechanism

nonlinear transmission function

)(=

r

s

d

kinematical transformer


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Chair of Mechanicsblock diagrams II

Coupling

connected leavers A B

BA

2

11 2

Block diagram

2 1 +=

linear coupling relation

nonlinear linear nonlinear

1

1

2

1

2

2

21


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Chair of Mechanicsblock diagrams III

Branching:

+=

+=

+=

4

3

2

1

11

3

2

1

Block diagram:

11

2

2

2

2

33

3

4

4

1 1

3

44


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Chair of Mechanicsblock diagrams IV

General spatial transmission chain with branching points


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Chair of Mechanicsblock diagrams V

Block diagram

The oriented block diagram represents: - solution flow (numerical)

- kinematical structure (construction)

Advantages: - global relationships are visualized independently of construction details

- clear graphical representation for quick overview


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Chair of Mechanicstopological analysis of mechanisms I

Example:

2L

1L 1L

2L

preferable bad choice

257n7n

5n

L

G

B

===

indeed independent!

choice of independent loops as small as possible!


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Chair of Mechanicstopological analysis of mechanisms II

Example:

3L1L1L

2L 2L

3L

3710n

10n7n

L

G

B

===

wrong correct

1L3L 2L""""

Examples of independent loops:


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Chair of Mechanicstopological analysis of mechanisms III

Examples of independent loops:

all loops are planar four-bar mechanisms

a) number of bodies

number of joints (joint C has to be counted twice)

number of loops 4913n

13n

9n

L

G

B

===

4L3L

2L

1L

C

B

A

degrees of freedom: per loop 1f iL=

) f 11


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Chair of Mechanicstopological analysis of mechanisms IV

b) number of bodies

number of joints (all joints are to be counted twice!)

number of loops 51116n

16n

11n

L

G

B

===

degrees of freedom: per loop 1f iL=

all loops are planar four-bar mechanisms

D

C

BA

5L

4L

3L2L

1L


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Chair of Mechanicstoplogical analysis of mechanisms VI

number of bodies:

number of joints:

(spherical joints are not decomposed!)

number of loops: 4610n

10n

6n

L

G

B

===

4L3L

2L

1L

degrees of freedom:

per loop:

with isolated DOF:without isolated DOF: 4610f

6612f

iL

iL ==

d) five-link wheel supsension


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Chair of Mechanicstopological analysis of mechanisms VII

Linear relationships for couplings:

1. spatial revolute joint:

321 const

1L1L

2L

2L

3L

3L

3

2

1

2 spatial prismatic joint:


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Chair of Mechanicstopological analysis of mechanisms VIII

2. spatial prismatic joint:

321 const

2

1

1L

2L

3L

3

G l l ti hi b t j i t di t f th ti l i ti j i t


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Chair of Mechanicstopological analysis of mechanisms IX

General relationship between joint coordinates for the spatial prismatic joint:

3

2

1

3

2

1

1L

2L

3L

ji

23

12

1

.

.

.

k

2

1

.

.

.

3

2

1

.

.

.

= 0

= +

= +

= +

P = +matrix P: )G(n1)G(n BL rank r = nB ( G ) )G(n1)G(np BLi

Application of the coupling relationships:


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Chair of Mechanicstopological analysis of mechanisms X

Application of the coupling relationships:

1) single revolute joint:

2L

2K

1K

1L 1 2

number of bodies:number of loops:

number of couplings:

coupling equation: ==

==11nnp

2n2n

BL

L

B

1 2 const 360

2) multiple joint with single coupling:

3K

2K

1K1L

2L

3L

1

2

31 2

number of bodies:

number of loops:


coupling equation: ==

==11nnp

3n

3n

BL

L

B

3 const 360

3) general case:


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Chair of Mechanicstopological analysis of mechanisms XI

3K

2K

1K

3L

4L4

3

2L

2

11L 1

4

32

number of bodies:number of loops:


coupling equations:

==

===

21nnp

4n3n

BL

L

B

3 const

const 360

0

typical coupling:


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Chair of Mechanicstopological analysis of mechanisms XII

typical coupling:

13

13 21

21

1321 const

Basic building blocks for structuring constraint equations


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Chair of Mechanics

Basic building blocks for structuring constraint equations

I Loops nonlinearrelationships (locally)

four-bar mechanism in general case:

planartransformer

spatial transformer

2i

4i

4i

3i

3i

2i

1i

1i

6 outputs

3 outputs

inputs

inputsLif

Lif

topological analysis ofmechanisms XIII

II knots between loops linear relationships (global)


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Chair of Mechanics

II knots between loops linear relationships (global)

binary:

360

general case:

Lif inputs 1 outputs

.

.

.

counting rule:

per joint Gi

1)(d)(dp BL ii bodiesloops

independent, linear equations

topological analysis ofmechanisms XIV

Examples of kinematical networks


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Chair of Mechanicstopological analysis ofmechanisms XV

C

B

A

L4

L1

L3

L2

a) 4 independent loops

3 couplings (joints A, B, C)

4.,.,1i;1f iL =3,2,1i;1p i =

overall DOF:

in general:

or GRBLER - formula

1314f = iiL pff

Examples of kinematical networks

CA

L4

L3

L2

1fl=

L1

B

1fl=1fl= 1fl=

block diagram:


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Chair of Mechanicstopological analysis ofmechanisms XVI

b) 5 independent loops

4 couplings (joints A, B, C, D)L5

L4L3L2

L1

overall DOF:

1415f =B

L4

L1

1fl=

1fl=1fl=

1fl=

AL2

1fl=

D

CL3

block diagram L5


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d) mechanism for unloading a tipping waggon


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Chair of Mechanicstopological analysis of mechanisms XIX

3 21

3

1

2

AA B

B

CC D

D

EE

FF

GGHH

M

M

N N

KK

LL

3

2

1

B

D

C

E

F

G H

K

L

MN

A

L4

L3

L2

L1

L5

Block diagram:L


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Chair of Mechanicstopological analysis of mechanisms XX

5 independent loops

4 couplings

(joint A: pi = 3

joints F, K: pi = 1)

overall DOF:

1415f =

A3

A2

A1

F

K

L4L3

L1

L2

L5

1fl=

1fl=1fl=

1fl=

2fl=

e) spatial mechanism (swashplate of a helicopter)

2 independent loops (degrees of freedom f = 2)


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Chair of Mechanicstopological analysis ofmechanisms XXI

2 independent loops (degrees of freedom fLi = 2)

3 couplings

H

SS

H

R P

R

s

2

1

1L

2L

swashplate

L2

s

L1

2fL=2fL=

2

1

block diagram:

overall DOF:

1322f =


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Structure of kinematic solution flow

a) decomposable mechanisms


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Chair of Mechanicstopological analysis ofmechanisms XXIII

) p

C

B

A

L4

L2L1

L3

f = 1

q

orientend block diagram (variant 2):

L1 L2 L4

1fl= 1fl= 1fl=

1fl=

A

B

C

L3

q

Structure of kinematic solution flow

a) decomposable mechanisms


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Chair of Mechanicstopological analysis ofmechanisms XXIV

) p

C

B

A

L4

L2L1

L3

f = 1

q

q~kinematic flow can not be found,solution: pseudo input

(e.g. at L1 )

q~

iterative determination of q~

oriented block diagram (variant 3):

L1 L2 L4

1fl= 1fl= 1fl=

1fl=

A

B

C

L3

qq~

Structure of the system of equations (recursive) for input at L1 (variant 1):

1


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Chair of Mechanicstopological analysis ofmechanisms XXV

=

xxxx

xxx

xx

111

xxxx

xxx

xx

11

xxxx

xxx

xx

11

xxxx

xxx

xx

1

gJ

L1 L2 L3 L4

L4

C

L3

B

L2

A

L1

q

Jacobian

b) non-decomposable mechanisms


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Chair of Mechanicstopological analysis ofmechanisms XXVI

L1

L5

L4L3L2

oriented block diagram: L5

L2 L3 L4

L1

A

B

C

D

1fl=

1fl= 1fl= 1fl=

1fl=

q

q

solution flow can not be determined.solution: pseudo input(e.g. at L2 )

q~

q~

q~

iterative determination of q~

Solution withpseudo input:


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Chair of Mechanicstopological analysis ofmechanisms XXVII

L1

L4L3L2

L5

oriented blockdiagram:

q~pseudo input

L2

L1

L3 L4

L5

A

B

C

D

1fl=

1fl=

1fl=

1fl=

1fl=

1L1 L

2 L3 L4 L5

q

Structure of the systemof equations:


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Chair of Mechanicstopological analysis ofmechanisms XXVIII

=

xxxx

xxxxx

111

xxxx

xxx

xx

111

xxxx

xxx

xx

111

xxxx

xxx

xx

1xxxx

xxx

xx

J

q

L1

L2

L3

L4

L5

D

C

B

A

q~

iterativelysolvedequations

)1...,0,1...,0,1...,0(hT

circular solution flow:

L


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Chair of Mechanicstopological analysis ofmechanisms XXIX

L2 L3 L4

L5

L1

oriented block diagram:

L2

L1

L4L3

L5

q

B

A C

Dcircle !

1fl=

1fl= 1fl= 1fl=

1fl=

Structure of the system of equations:

1 q


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Chair of Mechanicstopological analysis ofmechanisms XXX

=

xxxx

xxx

xx

111

xxxx

xxx

xx

111

xxxx

xxx

xx

111

xxxx

xxx

xx

111xxxx

xxx

xx

gJ

L1 L3 L4 L5 L2

L2

D

L5

C

L4

B

L3

A

L1

q

domain ofiteration

Jacobian


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Possible structures of equations for different choices of pseudo inputs


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Chair of Mechanicstopological analysis ofmechanisms XXXII

large iteration domain small iteration domain

q~

preferable

1q~

2q~

nested iteration domains local iteration domains

2q~

1q~

q~

kinema tics of mechanisms

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