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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-.
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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
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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
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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
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'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
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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
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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
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*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
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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
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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
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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
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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?
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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
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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
$
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,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
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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
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+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
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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
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C-.?
or
>C-2?
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+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
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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
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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
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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