Kom Material

8/9/2019 Kom Material

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acceleration o the ,ody and sometimes in elastic deormation and other

eects0

#ery day we deal with orces o one 6ind or another0 A pressure is a orce0

+he earth e%erts a orce o attraction or all ,odies or o,7ects on its surace0

+o study the orces acting on o,7ects we must 6now how the orces areapplied the direction o the orces and their #alue0 8raphically orces are

oten represented ,y a #ector whose end represents the point o action0

A mechanism is what is responsi,le or any action or reaction0 !achines are

,ased on the idea o transmitting orces through a series o predetermined

motions0 +hese related concepts are the ,asis o dynamic mo#ement0

.0.02 +or/ue

+or/ue1 )omething that produces or tends to produce rotation and whose

eecti#eness is measured ,y the product o the orce and the perpendicular

distance rom the line o action o the orce to the a%is o rotation0

onsider the le#er shown in *igure .-.0 +he le#er is a ,ar that is ree to turn

a,out the %ed point A called the ulcrum: a weight acts on the one side o

the le#er and a ,alancing orce acts on the other side o the le#er0

*igure .-. A le#er with ,alanced orces

+o analy;e le#ers we need to nd the tor/ues o the orces acting on the

le#er0 +o get the tor/ue o orce < a,out point A multiply < ,y l. its

distance rom A0 )imilarly * % l2 is the tor/ue o * a,out ulcrum A0

.02 !otion


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!otion1 a change o position or orientation0

.020. !otion Along a )traight Path


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>.-3?

!ore generally acceleration is

>.-4?

.0202 inear !otion in )pace

+he picture ,ecomes more complicated when the motion is not merely alonga straight line ,ut rather e%tends into a plane0 &ere we can descri,e the

motion with a #ector which includes the magnitude and the direction o

mo#ement0

Position #ector and displacement #ector

+he directed segment which descri,es the position o an o,7ect relati#e to an

origin is the position #ector as d. and d2 in *igure .-2

*igure .-2 Position #ector and displacement #ector

$ we wish to descri,e a motion rom position d. to position d2 or e%ample

we can use #ector d. the #ector starts at the point descri,ed ,y d. and goes

to the point descri,ed ,y d2 which is called the displacement #ector0

>.-5?

Velocity #ector

*or a displacement d occurring in a time inter#al t the a#erage #elocity


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during the inter#al is

>.-?

learly Va#e has the direction o d0

$n the limit as delta t approaches ;ero the instantaneous #elocity is

>.-C?

+he direction o V is the direction o d or a #ery small displacement: it is

thereore along or tangent to the path0

Acceleration #ector

+he instantaneous acceleration is the limit o the ratio Vt as t ,ecomes #ery

small1

>.-E?

.0203 !otion o a Rigid Body in a Plane

+he pre#ious sections discuss the motion o particles0 *or a rigid ,ody in a

plane its motion is oten more comple% than a particle ,ecause it is

comprised o a linear motion and a rotary motion0 8enerally this 6ind omotion can ,e decomposed into two motions >*igure .-3? they are1

+he linear motion o the center o the mass o the rigid ,ody0 $n this part o

the motion the motion is the same as the motion o a particle on a plane0

+he rotary motion o the rigid ,ody relati#e to its center o mass0


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*igure .-3 !otion o a rigid ,ody in a plane

.03 FewtonGs aw o !otion

.030. FewtonGs *irst aw

.-9?

+he proportionality constant m #aries with the o,7ect0 +his constant m is

reered to as the inertial mass o the ,ody0 +he relationship a,o#e em,odies

FewtonGs law o motion >FewtonGs second law?0 As


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>.-.H?

in which a is the acceleration o the o,7ect0 .-..?

$ m @ . 6g and a @ .msec2 than * @ . newton0

*orces and accelerations are #ectors and FewtonGs law can ,e written in

#ector orm0

>.-.2?

.04 !omentum and onser#ation o !omentum

.040. $mpulse

+ry to ma6e a ,ase,all and a cannon ,all roll at the same speed0 As you can

guess it is harder to get the cannon ,all going0 $ you apply a constant orce

* or a time t the change in #elocity is gi#en ,y /uation .-90 )o to get the

same # the product *t must ,e greater the greater the mass m you are trying

to accelerate0

+o throw a cannon ,all rom rest and gi#e it the same nal #elocity as a

,ase,all >also starting rom rest? we must push either harder or longer0


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)uppose we apply the same impulse to a ,ase,all and a cannon ,all ,oth

initially at rest0 )ince the initial #alue o the /uantity m# is ;ero in each case

and since e/ual impulses are applied the nal #alues m# will ,e e/ual or the

,ase,all and the cannon ,all0 'et ,ecause the mass o the cannon ,all ismuch greater than the mass o the ,ase,all the #elocity o the cannon ,all

will ,e much less than the #elocity o the ,ase,all0 +he product m# then is

/uite a dierent measure o the motion than simply # alone0 .-.3?

Velocity and momentum are /uite dierent concepts1 #elocity is a 6inematical/uantity whereas momentum is a dynamic one connected with the causes

o changes in the motion o masses0

Because o its connection with the impulse which occurs naturally in FewtonGs

law >/uation .-9? we e%pect momentum to t naturally into Fewtonian

dynamics0 Fewton did e%press his law o motion in terms o the momentum

which he called the /uantity o motion0 .-.4?

where # and #G are the #elocities ,eore and ater the impulse0 +he right-hand

side o the last e/uation can ,e written as

>.-.5?

the change in the momentum0 +hereore


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Power is the rate at which wor6 is done0

$n the British system power is e%pressed in oot-pounds per second0 *orlarger measurements the horsepower is used0

.horsepower @ 55Ht Il,s @ 33HHHtIl,min

$n )$ units power is measured in 7oules per second also called the watt >


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+a,le o ontents

omplete +a,le o ontents

. Physical Principles

.0. *orce and +or/ue

.0.0. *orce

.0.02 +or/ue

.02 !otion

.020. !otion Along a )traight Path

.0202 inear !otion in )pace

.0203 !otion o a Rigid Body in a Plane

.03 FewtonGs aw o !otion

.030. FewtonGs *irst aw

.0302 FewtonGs )econd aw

.04 !omentum and onser#ation o !omentum

.040. $mpulse

.0402 !omentum

.0403 onser#ation o !omentum

.05


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ams

C 8ears

E Kther !echanisms

$nde%

Reerences

Arrow to Bottom

39-245

Rapid Design through Virtual and Physical Prototyping

arnegie !ellon "ni#ersity

ourse $nde%

&R

$ntroduction to !echanisms

'i (hang

with

)usan *inger

)tephannie Behrens


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+a,le o ontents

2 !echanisms and )imple !achines

!echanism1 the undamental physical or chemical processes in#ol#ed in or

responsi,le or an action reaction or other natural phenomenon0

!achine1 an assem,lage o parts that transmit orces motion and energy in a

predetermined manner0

)imple !achine1 any o #arious elementary mechanisms ha#ing the elementso which all machines are composed0 $ncluded in this category are the le#er

wheel and a%le pulley inclined plane wedge and the screw0

+he word mechanism has many meanings0 $n 6inematics a mechanism is a

means o transmitting controlling or constraining relati#e mo#ement >&unt

CE?0 !o#ements which are electrically magnetically pneumatically operated

are e%cluded rom the concept o mechanism0 +he central theme or

mechanisms is rigid ,odies connected together ,y 7oints0

A machine is a com,ination o rigid or resistant ,odies ormed and

connected do that they mo#e with denite relati#e motions and transmit

orce rom the source o power to the resistance to ,e o#ercome0 A machine

has two unctions1 transmitting denite relati#e motion and transmitting

orce0 +hese unctions re/uire strength and rigidity to transmit the orces0

+he term mechanism is applied to the com,ination o geometrical ,odies

which constitute a machine or part o a machine0 A mechanism may thereore

,e dened as a com,ination o rigid or resistant ,odies ormed and

connected so that they mo#e with denite relati#e motions with respect to

one another >&am et al0 5E?0

Although a truly rigid ,ody does not e%ist many engineering components are


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rigid ,ecause their deormations and distortions are negligi,le in comparison

with their relati#e mo#ements0

+he similarity ,etween machines and mechanisms is that

they are ,oth com,inations o rigid ,odies

the relati#e motion among the rigid ,odies are denite0

+he dierence ,etween machine and mechanism is that machines transorm

energy to do wor6 while mechanisms so not necessarily perorm this

unction0 +he term machinery generally means machines and mechanisms0

*igure 2-. shows a picture o the main part o a diesel engine0 +he

mechanism o its cylinder-lin6-cran6 parts is a slider-cran6 mechanism asshown in *igure 2-20

*igure 2-. ross section o a power cylinder in a diesel engine

*igure 2-2 )6eleton outline

20. +he $nclined Plane

*igure 2-3a shows an inclined plane AB is the ,ase B is the height and A

the inclined plane0


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*igure 2-3 $nclined plane

"sing an inclined plane re/uires a smaller orce e%erted through a greater

distance to do a certain amount o wor60

etting * represent the orce re/uired to raise a gi#en weight on the inclined

plane and < the weight to ,e raised we ha#e the proportion1

>2-.?

20.0. )crew Lac6

Kne o the most common application o the principle o the inclined plane is

in the screw 7ac6 which is used to o#ercome a hea#y pressure or raise a

hea#y weight o < ,y a much smaller orce * applied at the handle0 R

represents the length o the handle and P the pitch o the screw or the

distance ad#ances in one complete turn0

*igure 2-4 +he screw 7ac6

Feglecting the riction the ollowing rule is used1 +he orce * multiplied ,y the

distance through which it mo#es in one complete turn is e/ual to the weight

lited times the distance through which it is lited in the same time0 $n one

complete turn the end o the handle descri,es a circle o circumerence 2R0

+his is the distance through which the orce * is e%erted0

+hereore rom the rule a,o#e


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>2-2?

and

>2-3?

)uppose R e/uals .E in0 P e/uals .E in0 and the weight to ,e lited e/uals

.HHHHH l,0 then the orce re/uired at * is then ..H l,0 +his means that

neglecting riction ..H l,0 at * will raise .HHHHH l,0 at #elocity ratio? o the large to the smaller is as . to 20

*igure 2-5 8ears

+he gear that is closer to the source o power is called the dri#er and the

gear that recei#es power rom the dri#er is called the dri#en gear0

2020. 8ear +rains


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A gear train may ha#e se#eral dri#ers and se#eral dri#en gears0

*igure 2- 8ear train

%ed? to the

same shat0 +he num,er o teeth on each gear is gi#en in the gure0 8i#en

these num,ers i gear A rotates at .HH r0p0m0 cloc6wise gear B turns 4HH

r0p0m0 >rotations per minute? countercloc6wise and gear turns .2HH r0p0m0

cloc6wise0

*igure 2-C ompound gears

20202 8ear Ratios

$t is important when wor6ing with gears to 6now what num,er o teeth thegears should ha#e so that they can mesh properly in a gear train0 +he si;e o

the teeth or connecting gears must ,e match properly0

203 Belts and Pulleys


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Belts and pulleys are an important part o most machines0 Pulleys are nothing

,ut gears without teeth and instead o running together directly they are

made to dri#e one another ,y cords ropes ca,les or ,elting o some 6inds0

As with gears the #elocities o pulleys are in#ersely proportional to their

diameters0

*igure 2-E Belts and pulleys

Pulleys can also ,e arranged as a ,loc6 and tac6le0

204 e#er

205


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@ the eMciency o a machine


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Reerences

3 !ore on !achines and !echanisms

30. Planar and )patial !echanisms

!echanisms can ,e di#ided into planar mechanisms and spatial mechanisms

according to the relati#e motion o the rigid ,odies0 $n a planar mechanisms

all o the relati#e motions o the rigid ,odies are in one plane or in parallel

planes0 $ there is any relati#e motion that is not in the same plane or in

parallel planes the mechanism is called the spatial mechanism0 $n other

words planar mechanisms are essentially two dimensional while spatial

mechanisms are three dimensional0 +his tutorial only co#ers planar

mechanisms0

302 Jinematics and Dynamics o !echanisms

Jinematics o mechanisms is concerned with the motion o the parts without

considering how the inuencing actors >orce and mass? aect the motion0

+hereore 6inematics deals with the undamental concepts o space and time

and the /uantities #elocity and acceleration deri#ed there rom0

Jinetics deals with action o orces on ,odies0 +his is where the the eects o

gra#ity come into play0

Dynamics is the com,ination o 6inematics and 6inetics0


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305 Pairs &igher Pairs ower Pairs and in6ages

A pair is a 7oint ,etween the suraces o two rigid ,odies that 6eeps them in

contact and relati#ely mo#a,le0 *or e%ample in *igure 3-2 a door 7ointed to

the rame with hinges ma6es re#olute 7oint >pin 7oint? allowing the door to ,e

turned around its a%is0 *igure 3-2, and c show s6eletons o a re#olute 7oint0

*igure 3-2, is used when ,oth lin6s 7oined ,y the pair can turn0 *igure 3-2c is

used when one o the lin6 7ointed ,y the pair is the rame0

*igure 3-2 Re#olute pair

$n *igure 3-3a a sash window can ,e translated relati#e to the sash0 +his 6ind

o relati#e motion is called a prismatic pair0 $ts s6eleton outlines are shown in

, c and d0 c and d are used when one o the lin6s is the rame0

*igure 3-3 Prismatic pair

8enerally there are two 6inds o pairs in mechanisms lower pairs and higher

pairs0 2D?

mechanisms there are two su,categories o lower pairs -- re#olute pairs and

prismatic pairs as shown in *igures 3-2 and 3-3 respecti#ely0 Point- line- or

cur#e-contact pairs are called higher pairs0 *igure 3-4 shows some e%amples

o higher pairs !echanisms composed o rigid ,odies and lower pairs are

called lin6ages0

*igure 3-4 &igher pairs

30 Jinematic Analysis and )ynthesis


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$n 6inematic analysis a particular gi#en mechanism is in#estigated ,ased on

the mechanism geometry plus other 6nown characteristics >such as input

angular #elocity angular acceleration etc0?0 Jinematic synthesis on the

other hand is the process o designing a mechanism to accomplish a desiredtas60 &ere ,oth choosing the types as well as the dimensions o the new

mechanism can ,e part o 6inematic synthesis0 >)andor N rdman E4?

4 Basic Jinematics o onstrained Rigid Bodies

40. Degrees o *reedom o a Rigid Body

40.0. Degrees o *reedom o a Rigid Body in a Plane

+he degrees o reedom >DK*? o a rigid ,ody is dened as the num,er o

independent mo#ements it has0 *igure 4-. shows a rigid ,ody in a plane0 +o

determine the DK* o this ,ody we must consider how many distinct ways

the ,ar can ,e mo#ed0 $n a two dimensional plane such as this computer

screen there are 3 DK*0 +he ,ar can ,e translated along the % a%is

translated along the y a%is and rotated a,out its centroid0

*igure 4-. Degrees o reedom o a rigid ,ody in a plane

40.02 Degrees o *reedom o a Rigid Body in )pace

An unrestrained rigid ,ody in space has si% degrees o reedom1 three

translating motions along the % y and ; a%es and three rotary motions


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around the % y and ; a%es respecti#ely0

*igure 4-2 Degrees o reedom o a rigid ,ody in space

402 Jinematic onstraints

+wo or more rigid ,odies in space are collecti#ely called a rigid ,ody system0

R-pair?


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*igure 4-4 A planar prismatic pair >P-pair?

40202 ower Pairs in )patial !echanisms

+here are si% 6inds o lower pairs under the category o spatial mechanisms0

+he types are1 spherical pair plane pair cylindrical pair re#olute pair

prismatic pair and screw pair0

*igure 4-5 A spherical pair >)-pair?

A spherical pair 6eeps two spherical centers together0 +wo rigid ,odies

connected ,y this constraint will ,e a,le to rotate relati#ely around % y and ;

a%es ,ut there will ,e no relati#e translation along any o these a%es0

+hereore a spherical pair remo#es three degrees o reedom in spatial

mechanism0 DK* @ 30

*igure 4- A planar pair >-pair?

A plane pair 6eeps the suraces o two rigid ,odies together0 +o #isuali;e this

imagine a ,oo6 lying on a ta,le where is can mo#e in any direction e%cept o

the ta,le0 +wo rigid ,odies connected ,y this 6ind o pair will ha#e two

independent translational motions in the plane and a rotary motion around

the a%is that is perpendicular to the plane0 +hereore a plane pair remo#es

three degrees o reedom in spatial mechanism0 $n our e%ample the ,oo6

would not ,e a,le to raise o the ta,le or to rotate into the ta,le0 DK* @ 30

*igure 4-C A cylindrical pair >-pair?


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A cylindrical pair 6eeps two a%es o two rigid ,odies aligned0 +wo rigid ,odies

that are part o this 6ind o system will ha#e an independent translational

motion along the a%is and a relati#e rotary motion around the a%is0 +hereore

a cylindrical pair remo#es our degrees o reedom rom spatial mechanism0

DK* @ 20

*igure 4-E A re#olute pair >R-pair?

A re#olute pair 6eeps the a%es o two rigid ,odies together0 +wo rigid ,odies

constrained ,y a re#olute pair ha#e an independent rotary motion around

their common a%is0 +hereore a re#olute pair remo#es #e degrees o

reedom in spatial mechanism0 DK* @ .0

*igure 4-9 A prismatic pair >P-pair?

A prismatic pair 6eeps two a%es o two rigid ,odies align and allow no relati#e

rotation0 +wo rigid ,odies constrained ,y this 6ind o constraint will ,e a,le to

ha#e an independent translational motion along the a%is0 +hereore aprismatic pair remo#es #e degrees o reedom in spatial mechanism0 DK* @

.0

*igure 4-.H A screw pair >&-pair?

+he screw pair 6eeps two a%es o two rigid ,odies aligned and allows arelati#e screw motion0 +wo rigid ,odies constrained ,y a screw pair a motion

which is a composition o a translational motion along the a%is and a

corresponding rotary motion around the a%is0 +hereore a screw pair remo#es

#e degrees o reedom in spatial mechanism0


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403 onstrained Rigid Bodies

Rigid ,odies and 6inematic constraints are the ,asic components o

mechanisms0 A constrained rigid ,ody system can ,e a 6inematic chain a

mechanism a structure or none o these0 +he inuence o 6inematic

constraints in the motion o rigid ,odies has two intrinsic aspects which are

the geometrical and physical aspects0 $n other words we can analy;e the

motion o the constrained rigid ,odies rom their geometrical relationships or

using FewtonGs )econd aw0

A mechanism is a constrained rigid ,ody system in which one o the ,odies is

the rame0 +he degrees o reedom are important when considering a

constrained rigid ,ody system that is a mechanism0 $t is less crucial when the

system is a structure or when it does not ha#e denite motion0

alculating the degrees o reedom o a rigid ,ody system is straight orward0

Any unconstrained rigid ,ody has si% degrees o reedom in space and three

degrees o reedom in a plane0 Adding 6inematic constraints ,etween rigid

,odies will correspondingly decrease the degrees o reedom o the rigid ,ody

system0


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$n *igure 4-..a a rigid ,ody is constrained ,y a re#olute pair which allows

only rotational mo#ement around an a%is0 $t has one degree o reedom

turning around point A0 +he two lost degrees o reedom are translational

mo#ements along the % and y a%es0 +he only way the rigid ,ody can mo#e is

to rotate a,out the %ed point A0

$n *igure 4-.., a rigid ,ody is constrained ,y a prismatic pair which allows

only translational motion0 $n two dimensions it has one degree o reedom

translating along the % a%is0 $n this e%ample the ,ody has lost the a,ility to

rotate a,out any a%is and it cannot mo#e along the y a%is0

$n *igure 4-..c a rigid ,ody is constrained ,y a higher pair0 $t has two

degrees o reedom1 translating along the cur#ed surace and turning a,out

the instantaneous contact point0

$n general a rigid ,ody in a plane has three degrees o reedom0 Jinematic

pairs are constraints on rigid ,odies that reduce the degrees o reedom o a

mechanism0 *igure 4-.. shows the three 6inds o pairs in planar mechanisms0

+hese pairs reduce the num,er o the degrees o reedom0 $ we create a

lower pair >*igure 4-..a,? the degrees o reedom are reduced to 20

)imilarly i we create a higher pair >*igure 4-..c? the degrees o reedom are

reduced to .0

*igure 4-.2 Jinematic Pairs in Planar !echanisms

+hereore we can write the ollowing e/uation1

>4-.?


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* @ total degrees o reedom in the mechanism

n @ num,er o lin6s >including the rame?

l @ num,er o lower pairs >one degree o reedom?

h @ num,er o higher pairs >two degrees o reedom?

+his e/uation is also 6nown as 8rue,lerGs e/uation0

%ample .

oo6 at the transom a,o#e the door in *igure 4-.3a0 +he opening and closing

mechanism is shown in *igure 4-.3,0 etGs calculate its degree o reedom0

*igure 4-.3 +ransom mechanism

n @ 4 >lin6 .33 and rame 4? l @ 4 >at A B D? h @ H

>4-2?

Fote1 D and unction as a same prismatic pair so they only count as one

lower pair0

%ample 2

alculate the degrees o reedom o the mechanisms shown in *igure 4-.4,0

*igure 4-.4a is an application o the mechanism0


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*igure 4-.4 Dump truc6

n @ 4 l @ 4 >at A B D? h @ H

>4-3?

%ample 3

alculate the degrees o reedom o the mechanisms shown in *igure 4-.50

*igure 4-.5 Degrees o reedom calculation

*or the mechanism in *igure 4-.5a

n @ l @ C h @ H

>4-4?

*or the mechanism in *igure 4-.5,

n @ 4 l @ 3 h @ 2

>4-5?


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Fote1 +he rotation o the roller does not inuence the relationship o the input

and output motion o the mechanism0 &ence the reedom o the roller will

not ,e considered: $t is called a passi#e or redundant degree o reedom0

$magine that the roller is welded to lin6 2 when counting the degrees o

reedom or the mechanism0

40402 Jut;,ach riterion

+he num,er o degrees o reedom o a mechanism is also called the mo,ility

o the de#ice0 +he mo,ility is the num,er o input parameters >usually pair

#aria,les? that must ,e independently controlled to ,ring the de#ice into a

particular position0 +he Jut;,ach criterion which is similar to 8rue,lerGs

e/uation calculates the mo,ility0

$n order to control a mechanism the num,er o independent input motions

must e/ual the num,er o degrees o reedom o the mechanism0 *or

e%ample the transom in *igure 4-.3a has a single degree o reedom so it

needs one independent input motion to open or close the window0 +hat is

you 7ust push or pull rod 3 to operate the window0

+o see another e%ample the mechanism in *igure 4-.5a also has . degree oreedom0 $ an independent input is applied to lin6 . >e0g0 a motor is mounted

on 7oint A to dri#e lin6 .? the mechanism will ha#e the a prescri,ed motion0

405 *inite +ransormation

*inite transormation is used to descri,e the motion o a point on rigid ,ody

and the motion o the rigid ,ody itsel0

4050. *inite Planar Rotational +ransormation


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*igure 4-. Point on a planar rigid ,ody rotated through an angle

)uppose that a point P on a rigid ,ody goes through a rotation descri,ing a

circular path rom P. to P2 around the origin o a coordinate system0 4-?

where

>4-C?

40502 *inite Planar +ranslational +ransormation

*igure 4-.C Point on a planar rigid ,ody translated through a distance

)uppose that a point P on a rigid ,ody goes through a translation descri,ing a

straight path rom P. to P2 with a change o coordinates o >% y?0 4-E?

where


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>4-9?

40503 oncatenation o *inite Planar Displacements

*igure 4-.E oncatenation o nite planar displacements in space

)uppose that a point P on a rigid ,ody goes through a rotation descri,ing a

circular path rom P. to P2G around the origin o a coordinate system then a

translation descri,ing a straight path rom P2G to P20 4-.H?

and

>4-..?

4-.2?

where D.2 is the planar general displacement operator 1

>4-.3?


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40504 Planar Rigid-Body +ransormation

4-.4?

40505 )patial Rotational +ransormation

4-.5?

where


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u% uy u; are the othographical pro7ection o the unit a%is u on % y and ;

a%es respecti#ely0

s @ sin

c @ cos

# @ . - cos

4050 )patial +ranslational +ransormation

)uppose that a point P on a rigid ,ody goes through a translation descri,ing a

straight path rom P. to P2 with a change o coordinates o >% y ;? we can

descri,e this motion with a translation operator +1

>4-.?

4050C )patial +ranslation and Rotation !atri% or A%is +hrough the Krigin

)uppose a point P on a rigid ,ody rotates with an angular displacement a,out

an unit a%is u passing through the origin o the coordinate system at rst and

then ollowed ,y a translation Du along u0 +his composition o this rotational

transormation and this translational transormation is a screw motion0 $ts

corresponding matri% operator the screw operator is a concatenation o the

translation operator in /uation 4-C and the rotation operator in /uation 4-90

>4-.C?

40 +ransormation !atri% Between Rigid Bodies

400. +ransormation !atri% Between two Ar,itray Rigid Bodies

*or a system o rigid ,odies we can esta,lish a local artesian coordinate

system or each rigid ,ody0 +ransormation matrices are used to descri,e the

relati#e motion ,etween rigid ,odies0


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4004 Application o +ransormation !atrices to in6ages

A lin6age is composed o se#eral constrained rigid ,odies0 i6e a mechanisma lin6age should ha#e a rame0 +he matri% method can ,e used to deri#e the

6inematic e/uations o the lin6age0 $ all the lin6s orm a closed loop the

concatenation o all o the transormation matrices will ,e an identity matri%0

$ the mechanism has n lin6s we will ha#e1

+.2+23000+>n-.?n @ $

5 Planar in6ages

50. $ntroduction

50.0. *igure 5-.a?O +he mechanism

shown in *igure 5-., transorms the rotary motion o the motor into an

oscillating motion o the windshield wiper0

*igure 5-.


39/82

etGs ma6e a simple mechanism with similar ,eha#ior0 +a6e some card,oard

and ma6e our strips as shown in *igure 5-2a0

+a6e 4 pins and assem,le them as shown in *igure 5-2,0

Fow hold the in0 strip so it canGt mo#e and turn the 3in0 strip0 'ou will see

that the 4in0 strip oscillates0

*igure 5-2 Do-it-yoursel our ,ar lin6age mechanism

+he our ,ar lin6age is the simplest and oten times the most useul

mechanism0 As we mentioned ,eore a mechanism composed o rigid ,odies

and lower pairs is called a lin6age >&unt CE?0 $n planar mechanisms there are

only two 6inds o lower pairs --- re#olute pairs and prismatic pairs0

+he simplest closed-loop lin6age is the our ,ar lin6age which has our

mem,ers three mo#ing lin6s one %ed lin6 and our pin 7oints0 A lin6age that

has at least one %ed lin6 is a mechanism0 +he ollowing e%ample o a our,ar lin6age was created in )imDesign in simdesignour,ar0sim

*igure 5-3 *our ,ar lin6age in )imDesign

+his mechanism has three mo#ing lin6s0 +wo o the lin6s are pinned to the

rame which is not shown in this picture0 $n )imDesign lin6s can ,e nailed tothe ,ac6ground there,y ma6ing them into the rame0

&ow many DK* does this mechanism ha#eO $ we want it to ha#e 7ust one we

can impose one constraint on the lin6age and it will ha#e a denite motion0

+he our ,ar lin6age is the simplest and the most useul mechanism0


40/82

Reminder1 A mechanism is composed o rigid ,odies and lower pairs called

lin6ages >&unt CE?0 $n planar mechanisms there are only two 6inds o lower

pairs1 turning pairs and prismatic pairs0

50.02 *unctions o in6ages

+he unction o a lin6 mechanism is to produce rotating oscillating or

reciprocating motion rom the rotation o a cran6 or #ice #ersa >&am et al0

5E?0 )tated more specically lin6ages may ,e used to con#ert1

ontinuous rotation into continuous rotation with a constant or #aria,le

angular #elocity ratio0

ontinuous rotation into oscillation or reciprocation >or the re#erse? with a

constant or #aria,le #elocity ratio0

Kscillation into oscillation or reciprocation into reciprocation with a constant

or #aria,le #elocity ratio0

in6ages ha#e many dierent unctions which can ,e classied according on

the primary goal o the mechanism1

*unction generation1 the relati#e motion ,etween the lin6s connected to the

rame

Path generation1 the path o a tracer point or

!otion generation1 the motion o the coupler lin60

502 *our in6 !echanisms

Kne o the simplest e%amples o a constrained lin6age is the our-lin6

mechanism0 A #ariety o useul mechanisms can ,e ormed rom a our-lin6

mechanism through slight #ariations such as changing the character o thepairs proportions o lin6s etc0 *urthermore many comple% lin6 mechanisms

are com,inations o two or more such mechanisms0 +he ma7ority o our-lin6

mechanisms all into one o the ollowing two classes1

the our-,ar lin6age mechanism and

the slider-cran6 mechanism0


41/82

5020. %amples

Parallelogram !echanism

$n a parallelogram our-,ar lin6age the orientation o the coupler does not

change during the motion0 +he gure illustrates a loader0 K,#ioulsy the

,eha#ior o maintaining parallelism is important in a loader0 +he ,uc6et

should not rotate as it is raised and lowered0 +he corresponding )imDesign

le is simdesignloader0sim0

*igure 5-4 *ront loader mechanism

)lider-ran6 !echanism

+he our-,ar mechanism has some special congurations created ,y ma6ing

one or more lin6s innite in length0 +he slider-cran6 >or cran6 and slider?

mechanism shown ,elow is a our-,ar lin6age with the slider replacing an

innitely long output lin60 +he corresponding )imDesign le issimdesignslider0cran60sim0

*igure 5-5 ran6 and )lider !echanism

+his conguration translates a rotational motion into a translational one0

!ost mechanisms are dri#en ,y motors and slider-cran6s are oten used totransorm rotary motion into linear motion0

ran6 and Piston


42/82

'ou can also use the slider as the input lin6 and the cran6 as the output lin60

$n this case the mechanism transers translational motion into rotary motion0

+he pistons and cran6 in an internal com,ustion engine are an e%ample o

this type o mechanism0 +he corresponding )imDesign le is

simdesigncom,ustion0sim0

*igure 5- ran6 and Piston

'ou might wonder why there is another slider and a lin6 on the let0 +his

mechanism has two dead points0 +he slider and lin6 on the let help the

mechanism to o#ercome these dead points0

Bloc6 *eeder

Kne interesting application o slider-cran6 is the ,loc6 eeder0 +he )imDesign

le can ,e ound in simdesign,loc6-eeder0sim

*igure 5-C Bloc6 *eeder

50202 Denitions

$n the range o planar mechanisms the simplest group o lower pair

mechanisms are our ,ar lin6ages0 A our ,ar lin6age comprises our ,ar-

shaped lin6s and our turning pairs as shown in *igure 5-E0


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*igure 5-E *our ,ar lin6age

+he lin6 opposite the rame is called the coupler lin6 and the lin6s whic6 are

hinged to the rame are called side lin6s0 A lin6 which is ree to rotate through

3H degree with respect to a second lin6 will ,e said to re#ol#e relati#e to the

second lin6 >not necessarily a rame?0 $ it is possi,le or all our ,ars to

,ecome simultaneously aligned such a state is called a change point0

)ome important concepts in lin6 mechanisms are1

ran61 A side lin6 which re#ol#es relati#e to the rame is called a cran60

Roc6er1 Any lin6 which does not re#ol#e is called a roc6er0

ran6-roc6er mechanism1 $n a our ,ar lin6age i the shorter side lin6

re#ol#es and the other one roc6s >i0e0 oscillates? it is called a cran6-roc6er

mechanism0

Dou,le-cran6 mechanism1 $n a our ,ar lin6age i ,oth o the side lin6s

re#ol#e it is called a dou,le-cran6 mechanism0

Dou,le-roc6er mechanism1 $n a our ,ar lin6age i ,oth o the side lin6s roc6

it is called a dou,le-roc6er mechanism0

50203 lassication

Beore classiying our-,ar lin6ages we need to introduce some ,asic

nomenclature0

$n a our-,ar lin6age we reer to the line segment ,etween hinges on a gi#en

lin6 as a ,ar where1

s @ length o shortest ,ar

l @ length o longest ,ar

p / @ lengths o intermediate ,ar


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8rashoGs theorem states that a our-,ar mechanism has at least one

re#ol#ing lin6 i

s l Q@ p /

>5-.?

and all three mo,ile lin6s will roc6 i

s l p /

>5-2?

+he ine/uality 5-. is 8rashoGs criterion0

All our-,ar mechanisms all into one o the our categories listed in +a,le 5-.1

+a,le 5-. lassication o *our-Bar !echanisms

ase l s #ers0 p / )hortest Bar +ype

. Q *rameDou,le-cran6

2 Q )ide Roc6er-cran6

3 Q oupler Dou,l roc6er

4 @ Any hange point

5 Any Dou,le-roc6er

*rom +a,le 5-. we can see that or a mechanism to ha#e a cran6 the sum othe length o its shortest and longest lin6s must ,e less than or e/ual to the

sum o the length o the other two lin6s0 &owe#er this condition is necessary

,ut not suMcient0 !echanisms satisying this condition all into the ollowing

three categories1


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negligi,le ,ending action? and is directedalong B0 *or a gi#en orce in the coupler lin6 the tor/ue transmitted to the

output ,ar >a,out point D? is ma%imum when the angle ,etween coupler ,ar

B and output ,ar D is 20 +hereore angle BD is called transmission angle0

>5-3?

*igure 5-.. +ransmission angle


46/82

sometimes called a toggle point?0

*igure 5-.H Dead point

$n *igure 5-.. i AB is a cran6 it can ,ecome aligned with B in ull e%tension

along the line AB.. or in e%ion with AB2 olded o#er B220


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*igure 5-.2 K#ercoming the dead point ,y asymmetrical deployment >V

engine?

5020 )lider-ran6 !echanism

+he slider-cran6 mechanism which has a well-6nown application in engines

is a special case o the cran6-roc6er mechanism0 Fotice that i roc6er 3 in

*igure 5-.3a is #ery long it can ,e replaced ,y a ,loc6 sliding in a cur#ed slot

or guide as shown0 $ the length o the roc6er is innite the guide and ,loc6

are no longer cur#ed0 Rather they are apparently straight as shown in *igure

5-.3, and the lin6age ta6es the orm o the ordinary slider-cran6 mechanism0

*igure 5-.3 )lider-ran6 mechanism

5020C $n#ersion o the )lider-ran6 !echanism

$n#ersion is a term used in 6inematics or a re#ersal or interchange o orm or

unction as applied to 6inematic chains and mechanisms0 *or e%ample ta6inga dierent lin6 as the %ed lin6 the slider-cran6 mechanism shown in *igure

5-.4a can ,e in#erted into the mechanisms shown in *igure 5-.4, c and d0

Dierent e%amples can ,e ound in the application o these mechanisms0 *or

e%ample the mechanism o the pump de#ice in *igure 5-.5 is the same as

that in *igure 5-.4,0

*igure 5-.4 $n#ersions o the cran6-slide mechanism

*igure 5-.5 A pump de#ice


48/82

Jeep in mind that the in#ersion o a mechanism does not change the motions

o its lin6s relati#e to each other ,ut does change their a,solute motions0

ams

0. $ntroduction

0.0. A )imple %periment1 use your hand as a guide?0


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A cam may ,e dened as a machine element ha#ing a cur#ed outline or a

cur#ed groo#e which ,y its oscillation or rotation motion gi#es a

predetermined specied motion to another element called the ollower 0 +he

cam has a #ery important unction in the operation o many classes o

machines especially those o the automatic type such as printing presses

shoe machinery te%tile machinery gear-cutting machines and screwmachines0 $n any class o machinery in which automatic control and accurate

timing are paramount the cam is an indispensa,le part o mechanism0 +he

possi,le applications o cams are unlimited and their shapes occur in great

#ariety0 )ome o the most common orms will ,e considered in this chapter0

02 lassication o am !echanisms

*igure -2a,cde?

Rotating ollower >*igure -2?1

+he ollower arm swings or oscillates in a circular arc with respect to the

ollower pi#ot0

+ranslating cam-translating ollower >*igure -3?0

)tationary cam-rotating ollower1

+he ollower system re#ol#es with respect to the center line o the #ertical

shat0


50/82

*igure -3 +ranslating cam - translating ollower

020. *ollower onguration

Jnie-edge ollower >*igure -2a?

Roller ollower >*igure -2,e?

*lat-aced ollower >*igure -2c?

K,li/ue at-aced ollower

)pherical-aced ollower >*igure -2d?

0202 *ollower Arrangement

$n-line ollower1

+he center line o the ollower passes through the center line o the camshat0

Kset ollower1

+he center line o the ollower does not pass through the center line o the

cam shat0 +he amount o oset is the distance ,etween these two center

lines0 +he oset causes a reduction o the side thrust present in the roller

ollower0

0203 am )hape

Plate cam or dis6 cam1

+he ollower mo#es in a plane perpendicular to the a%is o rotation o the

camshat0 A translating or a swing arm ollower must ,e constrained tomaintain contact with the cam prole0

8roo#ed cam or closed cam >*igure -4?1

+his is a plate cam with the ollower riding in a groo#e in the ace o the cam0


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*igure -4 8roo#ed cam

ylindrical cam or ,arrel cam >*igure -5a?1

+he roller ollower operates in a groo#e cut on the periphery o a cylinder0 +he

ollower may translate or oscillate0 $ the cylindrical surace is replaced ,y a

conical one a conical cam results0

nd cam >*igure -5,?1

+his cam has a rotating portion o a cylinder0 +he ollower translates or

oscillates whereas the cam usually rotates0 +he end cam is rarely used

,ecause o the cost and the diMculty in cutting its contour0

*igure -5 ylindrical cam and end cam

0204 onstraints on the *ollower

8ra#ity constraint1

+he weight o the ollower system is suMcient to maintain contact0

)pring constraint1

+he spring must ,e properly designed to maintain contact0

Positi#e mechanical constraint1

A groo#e maintains positi#e action0 >*igure -4 and *igure -5a? *or the cam

in *igure - the ollower has two rollers separated ,y a %ed distance

which act as the constraint: the mating cam in such an arrangement is oten

called a constant-diameter cam0

*igure - onstant diameter cam


52/82

A mechanical constraint cam also ,e introduced ,y employing a dual or

con7ugate cam in arrangement similar to what shown in *igure -C0 ach cam

has its own roller ,ut the rollers are mounted on the same reciprocating or

oscillating ollower0

*igure -C Dual cam

0205 %amples in )imDesign

Rotating am +ranslating *ollower

*igure -E )imDesign translating cam

oad the )imDesign le simdesigncam0translating0sim0 $ you turn the cam

the ollower will mo#e0 +he weight o the ollower 6eeps them in contact0 +his

is called a gra#ity constraint cam0

Rotating amRotating *ollower

*igure -9 )imDesign oscillating cam

+he )imDesign le is simdesigncam0oscillating0sim0 Fotice that a roller is

used at the end o the ollower0 $n addition a spring is used to maintain thecontact o the cam and the roller0

$ you try to calculate the degrees o reedom >DK*? o the mechanism you

must imagine that the roller is welded onto the ollower ,ecause turning the

roller does not inuence the motion o the ollower0


53/82

03 am Fomenclature

*igure -.H illustrates some cam nomenclature1

*igure -.H am nomenclature

+race point1 A theoretical point on the ollower corresponding to the point o

a ctitious 6nie-edge ollower0 $t is used to generate the pitch cur#e0 $n the

case o a roller ollower the trace point is at the center o the roller0

Pitch cur#e1 +he path generated ,y the trace point at the ollower is rotated

a,out a stationary cam0

reerence circle?1 +he smallest circle rom the cam center

through the pitch cur#e0

Base circle1 +he smallest circle rom the cam center through the cam prole

cur#e0

)tro6e or throw1+he greatest distance or angle through which the ollower

mo#es or rotates0

*ollower displacement1 +he position o the ollower rom a specic ;ero or rest

position >usually its the position when the ollower contacts with the ,ase

circle o the cam? in relation to time or the rotary angle o the cam0


54/82

Pressure angle1 +he angle at any point ,etween the normal to the pitch cur#e

and the instantaneous direction o the ollower motion0 +his angle is

important in cam design ,ecause it represents the steepness o the cam

prole0

04 !otion e#ents


55/82

e%cept at the end o the stro6e would ,e ;ero as shown in c0 +he diagrams

show a,rupt changes o #elocity which result in large orces at the ,eginning

and the end o the stro6e0 +hese orces are undesira,le especially when the

cam rotates at high #elocity0 +he constant #elocity motion is thereore only o

theoretical interest0

>-.?

0402 onstant Acceleration !otion

onstant acceleration motion is shown in *igure -..d e 0 As indicated in e

the #elocity increases at a uniorm rate during the rst hal o the motion and

decreases at a uniorm rate during the second hal o the motion0 +he

acceleration is constant and positi#e throughout the rst hal o the motion

as shown in and is constant and negati#e throughout the second hal0 +his

type o motion gi#es the ollower the smallest #alue o ma%imum acceleration

along the path o motion0 $n high-speed machinery this is particularly

important ,ecause o the orces that are re/uired to produce the

accelerations0

-2?

-3?


56/82

0403 &armonic !otion

A cam mechanism with the ,asic cur#e li6e g in *igure -Cg will impart simple

harmonic motion to the ollower0 +he #elocity diagram at h indicates smooth

action0 +he acceleration as shown at i is ma%imum at the initial position

;ero at the mid-position and negati#e ma%imum at the nal position0

>-4?

05 am Design

+he translational or rotational displacement o the ollower is a unction o the

rotary angle o the cam0 A designer can dene the unction according to the

specic re/uirements in the design0 +he motion re/uirements listed ,elow

are commonly used in cam prole design0

050. Dis6 am with Jnie-dge +ranslating *ollower

*igure -.2 is a s6eleton diagram o a dis6 cam with a 6nie-edge translating

ollower0


57/82

Parameters1

ro1 +he radius o the ,ase circle:

e1 +he oset o the ollower rom the rotary center o the cam0 Fotice1 it could,e negati#e0

s1 +he displacement o the ollower which is a unction o the rotary angle o

the cam -- 0

$


58/82


59/82


60/82

,etween the ollower >AB? and the line o two pi#ots >AK? is H0 $t can ,e

calculated rom the triangle KAB0

-?

+o get the corresponding 6nie-edge o the ollower in the in#erted

mechanism simply turn the ollower around the center o the cam in the

re#erse direction o the cam rotation through an angle o 0 +he 6nie edge will

,e in#erted to point J which corresponds to the point on the cam prole in

the in#erted mechanism0 +hereore the coordinates o point J can ,e

calculated with the ollowing e/uation1

>-C?

Fote1


61/82

r1 the radius o the roller0

$!1 a parameter whose a,solute #alue is . indicating which en#elope cur#e

will ,e adopted0

R!1 inner or outer en#elope cur#e0


62/82

+he calculation o the coordinates o the point P has two steps1

alculate the slope o the tangent tt o point J on pitch cur#e aa0

alculate the slope o the normal nn o the cur#e aa at point J0

)ince we ha#e already ha#e the coordinates o point J1 >% y? we can e%press

the coordinates o point P as

>-E?

Fote1

cam prole? lies inside the pitch cur#e1 R! @ .

otherwise1 R! @ -.0C 8ears

8ears are machine elements that transmit motion ,y means o successi#ely

engaging teeth0 +he gear teeth act li6e small le#ers0

C0. 8ear lassication

8ears may ,e classied according to the relati#e position o the a%es o

re#olution0 +he a%es may ,e

parallel

intersecting

neither parallel nor intersecting0

&ere is a ,rie list o the common orms0


63/82

8ears or connecting parallel shats

8ears or connecting intersecting shats

Feither parallel nor intersecting shats

8ears or connecting parallel shats

)pur gears

+he let pair o gears ma6es e%ternal contact and the right pair o gears

ma6es internal contact

Parallel helical gears

&erring,one gears >or dou,le-helical gears?

Rac6 and pinion >+he rac6 is li6e a gear whose a%is is at innity0?

8ears or connecting intersecting shats

)traight ,e#el gears

)piral ,e#el gears

Feither parallel nor intersecting shats

rossed-helical gears

&ypoid gears


64/82

C-.?

or

>C-2?


65/82


66/82

+o o,tain the e%pected #elocity ratio o two tooth proles the normal line o

their proles must pass through the corresponding pitch point which is

decided ,y the #elocity ratio0 +he two proles which satisy this re/uirement

are called con7ugate proles0 )ometimes we simply termed the tooth proles

which satisy the undamental law o gear-tooth action the con7ugate proles0

Although many tooth shapes are possi,le or which a mating tooth could ,e

designed to satisy the undamental law only two are in general use1 the

cycloidal and in#olute proles0 +he in#olute has important ad#antages -- it is

easy to manuacture and the center distance ,etween a pair o in#olute gears

can ,e #aried without changing the #elocity ratio0 +hus close tolerances

,etween shat locations are not re/uired when using the in#olute prole0 +he

most commonly used con7ugate tooth cur#e is the in#olute cur#e >rdman N

)andor E4?0

C03 $n#olute ur#e

+he ollowing e%amples are in#olute spur gears0


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*igure C-3 $n#olute cur#e

+he cur#e most commonly used or gear-tooth proles is the in#olute o acircle0 +his in#olute cur#e is the path traced ,y a point on a line as the line

rolls without slipping on the circumerence o a circle0 $t may also ,e dened

as a path traced ,y the end o a string which is originally wrapped on a circle

when the string is unwrapped rom the circle0 +he circle rom which the

in#olute is deri#ed is called the ,ase circle0

$n *igure C-3 let line !F roll in the countercloc6wise direction on the

circumerence o a circle without slipping0


68/82

C04 +erminology or )pur 8ears

*igure C-4 shows some o the terms or gears0

*igure C-4 )pur 8ear

$n the ollowing section we dene many o the terms used in the analysis o

spur gears0 )ome o the terminology has ,een dened pre#iously ,ut we

include them here or completeness0 >)ee >&am 5E? or more details0?

Pitch surace 1 +he surace o the imaginary rolling cylinder >cone etc0? that

the toothed gear may ,e considered to replace0

Pitch circle1 A right section o the pitch surace0

Addendum circle1 A circle ,ounding the ends o the teeth in a right section o

the gear0

Root >or dedendum? circle1 +he circle ,ounding the spaces ,etween the teeth

in a right section o the gear0

Addendum1 +he radial distance ,etween the pitch circle and the addendum

circle0

Dedendum1 +he radial distance ,etween the pitch circle and the root circle0

learance1 +he dierence ,etween the dedendum o one gear and the

addendum o the mating gear0

*ace o a tooth1 +hat part o the tooth surace lying outside the pitch surace0

*lan6 o a tooth1 +he part o the tooth surace lying inside the pitch surace0

ircular thic6ness >also called the tooth thic6ness? 1 +he thic6ness o the

tooth measured on the pitch circle0 $t is the length o an arc and not the

length o a straight line0

+ooth space1 +he distance ,etween ad7acent teeth measured on the pitch

circle0


69/82

Bac6lash1 +he dierence ,etween the circle thic6ness o one gear and the

tooth space o the mating gear0

ircular pitch p1 +he width o a tooth and a space measured on the pitch

circle0

Diametral pitch P1 +he num,er o teeth o a gear per inch o its pitch

diameter0 A toothed gear must ha#e an integral num,er o teeth0 +he circular

pitch thereore e/uals the pitch circumerence di#ided ,y the num,er o

teeth0 +he diametral pitch is ,y denition the num,er o teeth di#ided ,y

the pitch diameter0 +hat is

>C-5?

and

>C-?

&ence

>C-C?

where

p @ circular pitch

P @ diametral pitch

F @ num,er o teeth

D @ pitch diameter

+hat is the product o the diametral pitch and the circular pitch e/uals 0

!odule m1 Pitch diameter di#ided ,y num,er o teeth0 +he pitch diameter is

usually specied in inches or millimeters: in the ormer case the module is


70/82

the in#erse o diametral pitch0

*illet 1 +he small radius that connects the prole o a tooth to the root circle0

Pinion1 +he smaller o any pair o mating gears0 +he larger o the pair is called

simply the gear0

Velocity ratio1 +he ratio o the num,er o re#olutions o the dri#ing >or input?

gear to the num,er o re#olutions o the dri#en >or output? gear in a unit o

time0

Pitch point1 +he point o tangency o the pitch circles o a pair o mating

gears0

ommon tangent1 +he line tangent to the pitch circle at the pitch point0

ine o action1 A line normal to a pair o mating tooth proles at their point o

contact0

Path o contact1 +he path traced ,y the contact point o a pair o tooth

proles0

Pressure angle 1 +he angle ,etween the common normal at the point o tooth

contact and the common tangent to the pitch circles0 $t is also the angle

,etween the line o action and the common tangent0

Base circle 1An imaginary circle used in in#olute gearing to generate the

in#olutes that orm the tooth proles0

+a,le C-. lists the standard tooth system or spur gears0 >)higley N "ic6er EH?

+a,le C-. )tandard tooth systems or spur gears

+a,le C-2 lists the commonly used diametral pitches0

oarse pitch 2 2025 205 3 4 E .H .2 .

*ine pitch 2H 24 32 4H 4E 4 9 .2H .5H 2HH

+a,le C-2 ommonly used diametral pitches


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>C-.4?

+hese e/uations can ,e com,ined to gi#e the #elocity ratio o the rst gear in

the train to the last gear1

>C-.5?

Fote1

+he tooth num,er in the numerator are those o the dri#en gears and the

tooth num,ers in the denominator ,elong to the dri#er gears0

8ear 2 and 3 ,oth dri#e and are in turn dri#en0 +hus they are called idlergears0 )ince their tooth num,ers cancel idler gears do not aect the

magnitude o the input-output ratio ,ut they do change the directions o

rotation0 Fote the directional arrows in the gure0 $dler gears can also

constitute a sa#ing o space and money >$ gear . and 4 meshes directly

across a long center distance their pitch circle will ,e much larger0?

+here are two ways to determine the direction o the rotary direction0 +he

rst way is to la,el arrows or each gear as in *igure C-0 +he second way is

to multiple mth power o =-.= to the general #elocity ratio0 internal contact gear pairs do not

change the rotary direction?0 &owe#er the second method cannot ,e applied

to the spatial gear trains0

+hus it is not diMcult to get the #elocity ratio o the gear train in *igure C-,1

>C-.?

C0C Planetary gear trains

Planetary gear trains also reerred to as epicyclic gear trains are those in

which one or more gears or,it a,out the central a%is o the train0 +hus they

dier rom an ordinary train ,y ha#ing a mo#ing a%is or a%es0 *igure C-E

shows a ,asic arrangement that is unctional ,y itsel or when used as a part

o a more comple% system0 8ear . is called a sun gear gear 2 is a planet


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lin6 & is an arm or planet carrier0

*igure C-E Planetary gear trains

*igure C-C Planetary gears modeled using )imDesign

+he )imDesign le is simdesigngear0planet0sim0 )ince the sun gear >the

largest gear? is %ed the DK* o the a,o#e mechanism is one0


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>C-.C?

or

>C-.E?

C0C02 %ample

+a6e the planetary gearing train in *igure C-E as an e%ample0 )uppose F. @

3 F2 @ .E . @ H 2 @ 3H0


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*igure E-. Ratchet

A is the ratchet wheel and B is an oscillating le#er carrying the dri#ing pawl

0 A supplementary pawl at D pre#ents ,ac6ward motion o the wheel0

countercloc6wise? motion as ,eore0

+he amount o ,ac6ward motion possi,le #aries with the pitch o the teeth0+his motion could ,e reduced ,y using small teeth and the e%pedient is

sometimes used ,y placing se#eral pawls side ,y side on the same a%is the

pawls ,eing o dierent lengths0

+he contact suraces o wheel and pawl should ,e inclined so that they will

not tend to disengage under pressure0 +his means that the common normal

at F should pass ,etween the pawl and the ratchet-wheel centers0 $ this

common normal should pass outside these limits the pawl would ,e orced

out o contact under load unless held ,y riction0 $n many ratchetmechanisms the pawl is held against the wheel during motion ,y the action

o a spring0

+he usual orm o the teeth o a ratchet wheel is that shown in the a,o#e

*igure ,ut in eed mechanisms such as used on many machine tools it is

necessary to modiy the tooth shape or a re#ersi,le pawl so that the dri#e

can ,e in either direction0 +he ollowing )imDesign e%ample o a ratchet also

includes a our ,ar lin6age0

$ you try this mechanism you may turn the cran6 o the lin6 mechanism0 +he

roc6er will dri#e the dri#ing pawl to dri#e the ratchet wheel0 +he

corresponding )imDesign data le is1


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asandrew0cmu0educitcerapidprotosimdesignratchet0sim

E02 K#errunning lutch

A special orm o a ratchet is the o#errunning clutch0 &a#e you e#er thought

a,out what 6ind o mechanism dri#es the rear a%le o ,icycleO $t is a ree-

wheel mechanism which is an o#errunning clutch0 *igure E-2 illustrates a

simplied model0 As the dri#er deli#ers tor/ue to the dri#en mem,er the

rollers or ,alls are wedged into the tapered recesses0 +his is what gi#es the

positi#e dri#e0 )hould the dri#en mem,er attempt to dri#e the dri#er in the

directions shown the rollers or ,alls ,ecome ree and no tor/ue is

transmitted0

*igure E-2 K#errunning clutch

E03 $ntermittent 8earing

A pair o rotating mem,ers may ,e designed so that or continuous rotation

o the dri#er the ollower will alternately roll with the dri#er and remain

stationary0 +his type o arrangement is 6now ,y the general term intermittent

gearing0 +his type o gearing occurs in some counting mechanisms motion-

picture machines eed mechanisms as well as others0

*igure E-3 $ntermittent gearing

+he simplest orm o intermittent gearing as illustrated in *igure E-3 has the

same 6ind o teeth as ordinary gears designed or continuous rotation0 +his


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e%ample is a pair o .E-tooth gears modied to meet the re/uirement that

the ollower ad#ance one-ninth o a turn or each turn o the dri#er0 +he

inter#al o action is the two-pitch angle >indicated on ,oth gears?0 +he single

tooth on the dri#er engages with each space on the ollower to produce the

re/uired motion o a one-ninth turn o the ollower0 During the remainder o a

dri#er turn the ollower is loc6ed against rotation in the manner shown in thegure0

+o #ary the relati#e mo#ements o the dri#er and ollower the meshing teeth

can ,e arranged in #arious ways to suit re/uirements0 *or e%ample the dri#er

may ha#e more than one tooth and the periods o rest o the ollower may ,e

uniorm or may #ary considera,ly0 ounting mechanisms are oten e/uipped

with gearing o this type0

E04 +he 8ene#a


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An e%ample o this mechanism has ,een made in )imDesign as in the

ollowing picture0

+he corresponding )imDesign data le is1

asandrew0cmu0educitcerapidprotosimdesigngene#a0sim

E05 +he "ni#ersal Loint

+he engine o an automo,ile is usually located in ront part0 &ow does it

connect to the rear a%le o the automo,ileO $n this case uni#ersal 7oints are

used to transmit the motion0

*igure E-5 "ni#ersal 7oint

+he uni#ersal 7oint as shown in *igure E-5 is also 6nown in the older literature

as &oo6eGs coupling0 Regardless o how it is constructed or proportioned or

practical use it has essentially the orm shown in *igure E- consisting o two

semicircular or6s 2 and 4 pin-7ointed to a right -angle cross 30

*igure E- 8eneral orm or a uni#ersal 7oint


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+he dri#er 2 and the ollower 4 ma6e the complete re#olution at the same

time ,ut the #elocity ratio is not constant throughout the re#olution0 +he

ollowing analysis will show how complete inormation as to the relati#e

motions o dri#er and ollower may ,e o,tained or any phase o the motion0

E050. Analysis o a "ni#ersal Loint

*igure E-C Analysis o a uni#ersal 7oint

$ the plane o pro7ection is ta6en perpendicular to the a%is o 2 the path o a

and , will ,e a circle AJB as shown in *igure E-C0

$ the angle ,etween the shats is the path o c and d will ,e a circle that is

pro7ected as the ellipse ABD in which

K @ KD @ KJcos @ KAcos

>E-.?

$ one o the arms o the dri#er is at A an arm o the ollower will ,e at 0 $

the dri#er arm mo#es through the angle to P the ollower arm will mo#e to

T0 KT will ,e perpendicular to KP: hence1 angle KT @ 0 But angle KT is the

pro7ection o the real angle descri,es ,y the ollower0 Tn is the real

component o the motion o the ollower in a direction parallel to AB and line

AB is the intersection o the planes o the dri#erGs and the ollowerGs planes0

+he true angle descri,ed ,y the ollower while the dri#er descri,es theangle can ,e ound ,y re#ol#ing KT a,out AB as an a%is into the plane o

the circle AJB0 +hen KR @ the true length o KT and RKJ @ @ the true

angle that is pro7ected as angle KT @ 0

Fow


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tan @ RmKm

and

tan @ TnKn

But

Tn @ Rm

&ence

+hereore

tan @ costan

+he ratio o the angular motion o the ollower to that o the dri#er is ound as

ollower ,y dierentiating a,o#e e/uation remem,ering that is constant

liminating 1


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)imilarly can ,e eliminated1

According to the a,o#e e/uations when the dri#er has a uniorm angular

#elocity the ratio o angular #elocities #aries ,etween e%tremes o cos and

.cos0 +hese #ariations in #elocity gi#e rise to inertia orces tor/ues noise

and #i,ration which would not ,e present i the #elocity ratio were constant0

E0502 Dou,le "ni#ersal Loint

By using a dou,le 7oint shown on the right in *igure E-C the #ariation o

angular motion ,etween dri#er and ollower can ,e entirely a#oided0 +his

compensating arrangement is to place an intermediate shat 3 ,etween the

dri#er and ollower shats0 +he two or6s o this intermediate shat must lie in

the same plane and the angle ,etween the rst shat and the intermediate

shat must e%actly ,e the same with that ,etween the intermediate shat and

the last shat0 $ the rst shat rotates uniormly the angular motion o the

intermediate shat will #ary according to the result deduced a,o#e0 +his

#ariation is e%actly the same as i the last shat rotated uniormly dri#ing the

intermediate shat0 +hereore the #aria,le motion transmitted to the

intermediate shat ,y the uniorm rotation o the rst shat is e%actly

compensated or ,y the motion transmitted rom the intermediate to the last

shat the uniorm motion o either o these shats will impart through the

intermediate shat uniorm motion to the other0

"ni#ersal 7oints particularly in pairs are used in many machines0 Kne

common application is in the dri#e shat which connects the engine o an

automo,iles to the a%le0

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