4-bars

8/20/2019 4-bars

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Link

A component forming a part of a chain; generally rigid with provision at each end for connection to

two other links

Mechanism

A combination of rigid bodies (links) connected by kinematic pairs.

Kinematic pair

A joint which is formed by the contact between two bodies and allows relative motion between them.

Machine A collection of mechanisms which transmit force from the source of power to the resistance to be

overcome

Kinematics

A branch of dynamics dealing with motion in time and space but disregarding mass and forces

Kinetics A branch of physics that deals with the relation of force and changes of motion

Dynamics

A branch of mechanics that deals with matter (mass) in motion and the forces that produce or

change such motion. Mechanics deals with force and energy in their relation to the material bodies.

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A mechanism is defined as a combination of rigid bodies connected

by kinematic pairs.

A kinematic pair is a joint which is formed by the contact between two

bodies and allows relative motion between them.

he contact element on a body* which joins to form a kinematic pair*

is called pairing element.

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!ach link in the slider/crank mechanism shown here has two pairing elements.

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urface contact pairs are lower pairs.

he commonly used lower pairs include

(1) "evolute -air

() -rismatic -air

() crew -air

(0) ,ylindrical -air

(2) pherical -air

(3) -lanar -air

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2

Degrees of freedom: 1

Symbol: R

Relative motion: Circular

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revolute.&8AM

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3


Symbol: P

Relative motion: linear

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prismatic.&8AM

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Symbol: H

Relative motion: Helical

,"!4 -A#" (:!&#,A& -A#")

screw.&8AM

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Symbol: C

Relative motion: Cylindrical

,&#$8"#,A& -A#"

cylidrical.&8AM

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Symbol: S Relative motion: Spherical

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spherical.&8AM

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1>


Symbol:

Relative motion: Planar

-&A$A" -A#" (?&A -A#")

planar.&8AM

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11

:igher pairs (joints) have either a line contact or a point contact.

:igher pairs e@ist in cam mechanisms* gear trains* ball and roller bearings and roll/slide

joints* etc.

?or planar motion* both line contact higher pairs and point contact higher pairs have two

degrees/of/freedom.

he only constraint at the contact point is along the common normal.

A pin/in/slot joint (rolling contact with sliding) is also a higher pair with a line contact

between the pin and the slot.

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1

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higher.&8AM

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1

A kinematic chain is an assemblage of links by pairs. 4hen one link of a

kinematic chain is held fi@ed* the chain is said to form a mechanism. he

fi@ed link is called the ground link or frame.

A closed chain is a consecutive set of links in which the last link is connected

to the first. All links have at least two pair elements. here are single loop

closed chains and multi/loop closed chains.

An open chain is the one in which the last link is not connected to the first

link. At least one link has a single pair element.

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A closed chain mechanism. An open chain mechanism.

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10

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2 bar linkage.&8AM

'round

lider/crank

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+#$!MA#, ,:A#$ %-!$

fanuc robot.&8AM

'round

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A four/bar mechanism is composed of four links (including the ground link) and

four kinematic pairs.

-lanar four bar mechanisms are the simplest closed/chains such as crank-rocker

and slider-crank mechanisms.

A dyad is a combination of two links connected by a joint. A four/bar mechanism is

composed of two dyads.

Many planar mechanisms can be viewed as a combination of a four/bar

mechanism with one or more dyads.

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,rank/rocker

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A spatial mechanism is one in which one or more links do not move in planar motion.

#n the ",," mechanism shown here* the input (blue disk) and the output (yellow

disk) move in different planes that are not parallel to each other.

he coupler link has three dimensional spatial motion and does not move parallel to

a single plane herefore* the mechanism is defined as a spatial mechanism.

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"

revolute

"

revolute

,

cylindrical

,

cylindrical

spatial.&8AM

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1

he degrees of freedom of a mechanical system is the number of independent inputs

reuired to determine the position of all links of the mechanism.

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1 2

1

2

Gruebler's equation for planar mechanisms

# DOF = 3(n -1) - 2f - f

n number of lins

f number of lo!er pairs (1DOF)

f number of hi"her pairs !ith (2DOF)

A planar mechanism containing n links (including the ground link) has (n/1) degrees of freedom

before they are connected by pairs.

A lower pair has the effect of providing two constraints between the connected links. herefore* f 1 lower pairs will remove f

1 degrees of freedom from the system.

A higher pair provides one constraint. o* f 2 higher pairs will remove f

2 degrees of freedom from the

system.

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>

0 bar linkage.&8AM

8!'"!! %? ?"!!8%MB 0 =A" &#$+A'!

1 2

1

2

# DOF = 3(n -1) - 2f - f

n =

f =

f = $

# DOF = 3( -1) - 2 - $ = 1×

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1

8!'"!! %? ?"!!8%MB 2 =A" &#$+A'!

2 bar linkage.&8AM

1 2

1

2

# DOF = 3(n -1) - 2f - f

n = %

f = %

f = $

# DOF = 3(% -1) - 2 % - $ = 2×

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8!'"!! %? ?"!!8%MB ,"A$+ A$8 &#8!"

crank mechanism.&8AM

1 2

1

2

# DOF = 3(n -1) - 2f - f

n =

f =

f = $

# DOF = 3( -1) - 2 - $ = 1×

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&n

& i

i=1

&

i

Gruebler's equation for spatial mechanism

# DOF (n - n - 1) f

n number of lins

n number of *oints

f number of +e"rees of free+om of *oint

≥ ∑

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0

ball joint.&8AM

&n

& i

i=1

&

i

,umber of +e"rees of free+om for spatial mechanism

# DOF (n - n - 1) f

n number of lins

n number of *oints


≥ ∑

&

i

# DOF ( - - 1) 1 1 1 3 = $

n number of lins

n number of *oints


≥

ball joint >1.&8AM

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2

8!'"!! %? ?"!!8%MB 2 =A" &#$+A'!

4% ,#",6# A"! -%#=&!

2 bar linkage.&8AM

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3

4% ,#",6# A"! -%#=&!

1 2

1

2

Gruebler's equation

# DOF = 3(n -1) - 2f - f

n = %

f = %

f = $

# DOF = 3(% -1) - 2 % - $ = 2×

,#",6# 1 ,#",6# 8#A!M=&

#f after specifying two independent variables defining the

linkage position (here the angular position of two links

connected to ground) the number of possible positions of

remaining links are finite* the number of degrees of freedom

is eual

2 bar linkage.&8AM

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4% ,#",6# A"! -%#=&! =6 8#A!M=& # "!C6#"!8 %

M%5! ?"%M %$! ,#",6# % :! %:!"

1 2

1

2

Gruebler's equation

# DOF = 3(n -1) - 2f - f

n =

f =

f = $

# DOF = 3( -1) - 2 - $ = 1×

,#",6# 1 ,#",6# 8#A!M=&

#f after specifying one independent variable to define thelinkage position (here the angular position of the red link)

the number of possible positions of remaining links are

finite* the number of degrees of freedom is eual 1

crank rocker.&8AM

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4hen three links are joined by a single pin* two pairs must be counted.

4hen n links are joined by a single pin* (n/1) pairs must be counted.

1 2

1

2

# DOF = 3(n -1) - 2f - f

n =

f = .

f = $

# DOF = 3 ( -1) - 2 . - $ = 1×

3 bar linkage.&8AM

wo pin joints here

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here are instances when 'rueblerDs formula predicts a seemingly e@cessive number of

degrees of freedom. his may involve a passive or redundant degree of freedom.

he redundant degrees of freedom does not influence the overall motion of themechanism.

he rotation of the roller about its own a@is is a redundant degree of freedom and it

does not affect the motion of the output follower.

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"edundant degree

of freedom between

arm and roller

1 2

1

2

1

2

# DOF = 3(n -1) - 2f - f

n =

f = 3

f = 1

# DOF = 3 ( -1) - 2 3 - 1 = 2

n = 3

f = 2

f = 1

# DOF = 3 (3

inc

orrect

correct

-1) - 2 2 - 1

1 =

×

×cam and follower.&8AM

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>

here are mechanisms whose computed degrees of freedom may be Eero or

negative. hey can* nevertheless* move due to special proportion* for e@ample* the

five/bar linkage.

=ecause of the parallel configuration* the linkage can move. his is called

overconstrained linkage* in which one of the two couplers provides a redundant

constraint.

"emove the link which provides redundant constraint in calculating the degrees of

freedom.

2 bar linkage overconstrained.&8AM

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1 2

1

2

1

2

# DOF = 3(n -1) - 2f - f

n = %

f =

f = $

# DOF = 3 (% -1) - 2 - $ = $

n =

f =

f = $

# DOF = 3 (

inc

orrect

correct

-1) - 2 - 1

$ =

×

×

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1

he spring in a mechanism can be replaced by a dyad.

he punch mechanism shown has one degree of freedom.

he input is the slider. he motion of the green link is controlled not only by the

red link but also by the spring force and the contact force between the pawl and

the part being punched.

1 2

1

2

# DOF = 3(n -1) - 2f - f

n = /

f = 1$

f = $

# DOF = 3 (/ -1) - 2 1$ - $ = 1×

-"#$' ,%$$!,#%$

lider

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?or the purpose of kinematic analysis* a planar higher/pair mechanism can be replaced

by an euivalent lower/pair mechanism based on instantaneous velocity euivalence.

!ach higher pair is replaced by two lower pairs and a link.

he degrees of freedom of the euivalent mechanism is the same as the original

mechanism.

he instantaneous velocity and acceleration relationships between links and of the

original and the lower/pair euivalent mechanism are the same.

he euivalence is instantaneous. =ecause the positions the center of curvature changes

as the mechnism moves* different mechanism position will generate a different euivalent

linkage.

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he higher mechanism (left) and its euivalent

linkages (right)* in which , and , are centers

of curvature of contact curves on part and

part at point , respectively

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two cams concept.&8AM

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two cams.&8AM

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&1

&

&

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#f one link can perform full rotation relative to another link of four bar linkage (we may also

say Gif there is to be continuous motionH) the sum of the length of the shortest and the

longest link must not be larger than the sum of the lengths of the two other links.

#f the above condition is satisfied the four bar link is called Grashof mechanism.

ma0 min a b L L L L+ ≤ +

1 3 2 $ L L L L+ ≤ +

:ereB

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1 & / &1 I &> J &

2 & I & / &1 J &>3 &> I & K &1 J&

4 &1 J & I &> J&

5 & I &1 J & J &> &> I &1 J & J &

2 ! 3 LLL &1 I &

2 ! 4 LLL &1 I &>

3 ! 4 LLL &1 I &

&1 F & min

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,rank is the shortest link

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'"A:%? M!,:A$#MB "#$%K #&"K'#

ground

8riven

link

coupler

crank

,rank is the shortest link

crank rocker.&8AM

#nput (crank) rotates* output crank (driven link) oscillates

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'"A:%? M!,:A$#MB D#$G L(%K

crank

ground

8rivenlink

coupler

?i@ed link is the shortest link

drag link.&8AM

#nput (crank) rotates* output crank also rotates

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'"A:%? M!,:A$#MB D&)*L' #&"K'#

crank

ground

8riven

link

coupler

,oupler is the shortest link

double rocker.&8AM

#nput (crank) and output crank both oscillate

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0>

he process of choosing different links of a kinematic chain for the frame is known

as kinematic inversions.

he relative motions between the various links are not altered but their absolute

motions may be change drastically.

=y fi@ing different links three different types of four/bar mechanisms are derived

from the original four/bar mechanism. hese are crank/rocker* double/crank (or

drag/link) and double/rocker mechanisms. he crank is the link which can rotatecomplete 3> degrees.

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01

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'"%6$8

,rank rocker 8ouble rocker 8rag link

=y fi@ing different links three different types of four/bar mechanisms are derived from the original four/bar

mechanism. hese are crank/rocker* double/crank (or drag/link) and double/rocker mechanisms.

he crank is the link which can rotate complete 3> degrees.

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,"A$+

,"A$+

,"A$+

'"%6$8

'"%6$8

'"%6$8

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0

'"A:%? M!,:A$#MB ".$%G' /&(%+

crank

ground

8rivenlink

coupler

ma0 min a b L L L L+ = +

change point.&8AM

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0

'"A:%? M!,:A$#MB ".$%G' /&(%+

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00

"A$M##%$ A$'&! K 0 bar linkage

transmission angle.&8AM

crank

ground

8riven

link

coupler

Ma@imum transmission angle

Minimum transmission angle

"ecommended transmission angle

(angle between coupler centerline and the driven crank

centerline)

0>>

I A I 10>>

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02

Ma@imum transmission angle

"ecommended transmission angle

/0>> I A I 0>>

"A$M##%$ A$'&! K crank slider

crank mechanism.&8AM

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03

hese type of kinematic chains have four binary links and two ternary links.

A single degree of freedom chain has all lower/pair (pins or sliders) singledegree of fredom joints.

#n 4att/type si@ link chain* the two ternary links are directly connected to each

other. ?igure shows the two distinct ways in which two ternaries and four binary

can be arranged.

4A #/=A" &#$+A'!

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#n tephenson chains* the two ternary links are separated by a binary link.

&ike the 4att/chain* all tephenson chains have single degree of freedom.

=oth 4att and tephenson chains have two loops.

!-:!$%$ #/=A" &#$+A'!

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crank mechanism with offset.&8AM

C6#,+ "!6"$ M!,:A$#M

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0<

uick return.&8AM

C6#,+ "!6"$ M!,:A$#M

4-bars

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