M E C H AN I S M

ROBERT M cARDL E IfiEOWN , B .S .

ENGINEER , INDUSTRIAL COMMISSION OF W ISCONSINFORMERLY ASSOCIATE PROFE SSOR OF MACH INE

DESIGN, UNIVERSITY OF W ISCONSIN

SE COND E DITIONFIFTH IMPRESSION

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M cGRAW—H IL L BOOK COM PANY,IN C .

NEW YORK:370 SEVE NTH AVE NUE

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COPYR IGHT,1912

,1921 ,

PRINTE D I N

PREFACE TO SECOND EDITION

The purpose Of this revision has been to make more clear

those points that have been the most difficult for the studen ts .

The defini tions and rules have been italicized as it has been

our experience that the student in going over the text for the firsttime is at a loss to know just what are the most important points .

Very l ittle new subj ect matter has been added as it is felt that

the size Of the text is all that can be covered in the time usual ly

al lotted to this subject,and it has been the writer ’s experience

that it is better to cover the entire text than to omit certain

parts in Order to complete the course .

The number Of problems has been doubled and most Of the

illustrations for those problems suitable for drafting room prae

tice have been enlarged to make them more clear than they were

in the first edition .

E xperience has shown us that the class and drafting room work

can be more closely coordinated if the time al lowed for the course

be made one recitation and three two- hour drafting periods per

week,using the third drafting period for a lecture or second

recitation when necessary to keep ahead Of the drafting room work .

The writer wishes to express his appreciation for the very

helpful assistance Of his former col league P rofessor P . H . Hyland

Of the Department of M achine Design,University of W isconsin ,

for his suggestions as to changes,preparation Of drawings , and

assistance in reading proof.R. M eA. KEOWN .

M ADI SON , WI s .,

July, 1921 .

8 69 1 521

PREFACE TO FIRST EDITION

The writer ’s aim has been to cover the subj ect Of mechanism

as briefly,simply and clearly as possible . The text is designed

for a half year ’s work Of one l ecture , one recitation and four

hours ’ drafting work per week .

NO especial claim to originality Of subj ect matter can be made,

nor has the writer found need for a special effort in this direction .

The arrangement and method Of treatment are new,and these

the author bases upon satisfactory results Obtained in the work

Of his classes in the Coll ege Of E ngineering Of theUn iversity OfWisconsin .

After a brief discussion of motions and velocities,linkages are

taken up,as they are comparatively easy for the student to

understand,and simple problems can be given out whil e the sub

ject is yet new to him .

Cams are taken up in detail , as they form a part Of the

subj ect in which the student needs considerable practice in

order to work out original problems,and practically al l cam

problems are original .

The involute system Of gearing is taken up before the cycloidal

system,because it seems to the writer easier for the student to

grasp . H aving once become famili ar with the involute system,

the student can more readily understand the cycloidal . Further

more,the involute system is in more general use at the present

P roblems are given at the end Of each chapter . Some of these

are designed especial ly for drafting- room work,and the necessary

instructions as to scale,position of drawing on the sheet

,method

of procedure and time,etc . ,

are given . These few problems are

not intended to exhaust the subj ect,but rather to serve as a

guide for instructors in giving the drafting- room work in con

nection with the recitations .

Quotations have been made from the various authorities,and

PRE FACE

credit given where the quotations appear . The writer w ishes

to express his thanks to the several authors from whom he has

made quotations,the manu factu r ing compan ies who have fur

nished cuts and to al l others who have aided in the preparation

Of the manuscript .

R. M CA. KEOWN .

M AD I SON , W I S .

,

August, 1912 .

CONTENTS

P REFACE

CHAPTER I

M OTIONS AND VE L OC ITI E s

Design Of a machine—M otion— Forms Of motion —P lane motionRotation— Translation— H e l ical motion— Spherical motion— PathVelocity — Linear velocity— Variab le l inear velocity— Angu larvelocity— L inear ve locities of points in the same body—Relationbetween l inear and angu lar ve locity— P rob lems .

CHAPTER I I

INSTANTANE OUS CENTERS , KINEMATIC CHA INSM echanisms— M odes Of connection— Instantaneous motionCentro—Location of the centro in a single body— Location ofcentros in three bodies moving re lative ly to each other— Kinematic chain— Location and number Of centros in a simp le kinematicchain— Location and number Of centros in a compound kinematicchain— P rob lems .

CHAPTER I I I

SOLUTION OF RELATIVE LINEAR VE L OC ITI E s BY CENTRO M ETHOD .

Re lative l inear ve locity— Open four- l inked mechanisms—Crossedlink mechanisms— S l ider crank mechanisms— P rob lems .

CHAPTER IV

VELOC ITY D IAGRAM SLinear velocity diagrams— P o lar ve locity diagrams— Velocitydiagram Of engine crank and crosshead—Comparison Of ve locitydiagrams—Variab le motion mechanisms— Variab le rotary motionmechanism —Whitworth qu ick- return motion mechanism—Oscillating arm qu ick- retu rn motion mechanism— P rob lems .

ix

CONTE NTS

CHAPTER V

P ARALLEL AND STRAIGHT-LINE M OTION M ECHANISMS

Parall e lograms— Parallel ru ler—Roberval balance—Draftin g boardparalle l mechanisms— Straight—line motion mechanisms— Watt ’sstraight- line motion—Scott Russel l straight- l ine motionsThompson indicator motion— Tchebicheff ’s approximate straightl ine motion—Robert ’s approximate straight- l ine motion— P eauce ll ier ’s exact straight- l ine motion— Bricard ’

s exact straight- l inemotion— P rob lems .

CHAPTER VI

CAMSDefi n ition— Cam and fo l lower— Contact between cam and fo l lowerTheoretical cam curve—Working cam curve—Base circle—M otionsused for cam curves— Unifo rm motion— Harmonic motion— Un iformly accelerated motion— Uniformly retarded motion— Cams w ithOffset fo l lowers— Flat face fo llower— Invo lute cams— M ethod oflaying out an invo lute cam— Cams w ith oscillating followers— P roblems— P ositive motion cams— Cams of con stant diameter— M ain and

return cam—Cams Of constant breadth— Cyl indrical cams— Layou tOf cyl indrical cam w ithout developing cyl inder— Layou t Of cyl indricalcam on deve loped surface— Cy l indrical cams w ith oscil lating follower— Inverse cams— P rob lems .

CHAPTER VI I

GEARINGFriction gearing—Ro l l ing cy l inders—Velocity ratio— Groovedcyl inders— Evans ’ friction cones— Velocity ratio— Se ller ’s feeddisks— Velocity ratio— Bevel cones— Velocity ratio— Brush p lateand whee l—Velocity ratio— Spu r gears— Velocity ratio— Defi n i

tions of tooth parts— Invo lu te system—Spur gears— Invo lu te rackand p inion— Involu te annu lar gear and pinion— Interference of involute teeth E ffect of changing the distance betw een centers Ofinvolute gears— Interchangeab le invo lu te gears— Standard sizesOf teeth for gears— Laying out a pair of standard involute gearsThe stub tooth— M ethod Of designing the stub tooth— D imensionsof stub tooth gear teeth —Advantages of t he stub tooth— Cycloidalsystem—Cycloid— E p icycloid— Hyp ocycloid— A p p l i c a t i O n ofcycloidal curves to gear teeth— Cycloidal spu r gears— Cycloidalrack and pinion— Cycloidal rack teeth w ith straight flankS ~ —Cycloida lannu lar gear and pinion— Limiting size of annu lar

,

gear pinionInterchangeab le cycloidal gears— Standard diameter Of ro l l ing circlefor cycloidal gears— Comparison of involu te and cycloidal systemsP in gears —Stepped and hel ical gears— Approximate method oflaying out tooth cu rves— Cu tting spu r and annu lar gears— Interchangeab le gear cu tters— Conj ugate methods Of cu tting teethFellow ’s gear shaper—Gear bobbing— P rob lems .

CON TE NTS

CHAPTER VIII

BEVEL GEARS, WORM AND WORM WHEEL

Beve l gears—Ve locity ratio— M iter gears—Crown gears— Layingout the teeth Of beve l gears— Tredgold

’

s method— Shop draw ing ofbeve l gears— Cutting bevel- gear teeth— Approximate methodsAccurately cut bevel gear teeth— Bilgram bevel-gear p lanerWorm and worm whee l—Velocity ratio— P itch—S ystems Of teethused— Tooth contact— L ength Of worm— Cu tting the worm-whee lteeth— E l liptical gears— Ve locity ratio— P rob lems .

CHAPTER IX

GEAR TRAINS

Value Of a train— Id le gear and direction Of rotation—Gear trainfor thread cutting— P rob lems .

CHAPTER X

BELTING

Ve locity ratio and directional rotation— Crowning pu l leys—Tightand loose pu l leys— M u le pu l leys—Be lts for intersecting shafts— Beltsfor shafts that are neither paralle l nor intersecting— Length Of beltsStepped cones for same length of belt— Rope driving— Chain driving—Renold silent chain—M orse rocker joint chain— P rob lems .

CHAPTER X I

INTERMITTENT M OTIONS

Ratchet gearing— M u ltip le-paw l ratchet—Weston ratchet—Armstrong universal ratchet— Doub le-acting ratchet—Reversing paw lMasked ratchet— Silent ratchet— P osit ive ratchets— Magneticratchet— Jack ratchets—Clutches— P ositive clutches— Frictionclutches .

INDEX

xi

ME CH AN I SM

CHAPTER I

M OTION S AND. vnnoci jgi iajs33 1

0‘j

The science Of M echanism treats Of the design and constru ct ion

Of machinery .

“A machin e i s a combin ation of resi stan t bodi es so arrangedthat by thei r mean s the mechan i cal forces of n atu re can be compelled

to produ ce some efi ect or work accompan i ed by certain determi n atemotion s ,

” Reu leaux . In general it may be said that a machin e i s

an assemblage of fi xed and movi n g parts i n terposed between the

source of power and the work for the pu rpose of adapti ng on e to the

other .

1 . Design Of a M achine — In the design Of a machine there

are three distinct parts to be considered:Fi rst. —The general outl ine is sketched without any regard to

the detailed proportions Of the ind ividual parts,and from this

sketch or skeleton,by means Of pure geometry alone

,the dis

placement,velocity

,and acceleration Of each Of the moving parts

can usual ly be accu rately determined .

S econd — The forces acting on each part must be determined,

and each part given its proper form and dimensions to withstand

these forces .

Thi rd — H aving designed the machine,the dynamical effects

Of the moving parts can be accurately determined .

The first part belongs to the subj ect known as the Kinematics

Of M ach ines,the second to the Design Of M ach ine P arts

,and the

third to the Dynamics Of M ach ines .

This course w i l l treat only of the first part deal ing with the

motions Of the machine parts,and the manner of supporting and

2 ME CHAN ISM

gu iding them without regard to their strength . Th is is some

times cal led P ure M echani sm or the Geometry Of M ach inery .

S ince M echanism is a study Of relative motion it wil l be wel l to

d iscu ss the different k inds Of motion .

2 . M otion .—M otion is a change Of position . The change Of

pos ition can be noted only with respect to the position Of some

other body wh ich is at rest,or assumed to be at rest

,or w ith

respect to some body,the mot ion of whi ch is known or assumed

to be known . That is,the motion is purely a relative one .

Two bodies may be at rest relat ively to each other bu t in

motion relat ively to a third body,as for example

,two car wheels

fastened tb.,the sam

’

e asp» t av’e no motion w ith respect to eachother

,butmaybe In motibn relatively to the truck or track .

In problems deali n gi wit‘htmachin ery the motions Of the various

parts are usual ly taken(

with reference to the frame Of the machine .

This is not always the case,however

,as sometimes it is easier to

compare the motion Of one part Of a mach ine d irectly with that

Of some other moving part,as for example the number Of revolu

tions that the cylinder Of a print ing press makes to each revolu

t ion Of the kn ife for cutt ing Off the sheets .

3. Forms of M otion .

— I n Order to be of any use in machine

constru ction,the motions mu st be completely control led or con

strained . M ost Of the motions used are , or can be reduced to on e

Of three forms, plan e, heli ca l and spheri ca l .

4. P lane M otion .

— P lane motion is the most common as wel l

as being the most s imple . In order to have plane motion a ll

poi n ts i n a phi ne secti on of the bodymust remain i n that plan e and

all poi n ts outsi de that section wi l l move in para l lel planes .

In Fig . 1 , if the lower face abcd Of a cube is kept in contact with

FIG . 1 .

a flat table top,al l points in the lower face wil l move in a plane ,

and al l points similarly located in any other plane sect ion w i l l

MOTION S AND VE LOCITIE S 3

move in parall e l planes , no matter what path the cube describesin moving from on e position to another .

Thus if we know the motions of two points in a body that has

plane motion , the motions of other points in the body are also

known .

P l ane mot ion is always found as rotation or tran slation,or a

motion that can be redu ced to a comb inat ion Of rotat ion and

translation .

5. Rotation .

— I n rotation all poin ts in the body move i n

ci rcles,that i s

,they remai n at a fi xed di stan ce from a right line

ca lled the axi s of rotation . For example,shafts

,pul leys

,fly

wheels, etc.

,have a motion Of rotation .

6. Tran slation .

— Translation may be divided into two c lasses,

Recti li near Translation and Cu rvi linear Translation . A body

has recti li n ear tran slation when all poi n ts in i t move i n straightl ines

,as the carriage Of a lathe

,piston of an engine

,or spindle of

a dril l press .

The crank pin of a locomotive driving wheel is an example

Of curvil inear translation . It moves in a circ l e around the

center of the wheel and at the same time moves alon g the

track .

7 . H e lical M otion .

~ -The path traced by a poin t movi n g at a

fi xed di stan ce from an axi s an d wi th a un iform motion along the

axi s i s a helix,and a point moving in su ch a path is said to have

hel ical motion .

P erhaps the most common example Of hel ical motion is that Of

a screw being turned into a nut . All points in the screw,except

those in the axis,have helical motion . Both limits Of hel ical

motion are plane motion,that is

,if the pitch Of the screw is zero

the resu lting motion is rotation,while if the pitch is infinity , the

motion w i l l be that Of translation .

8. Spherical M otion .

—Spherical motion may be defined as

the motion of a body movi ng so that every poin t i n i t remain s at a

con stan t di stan ce from a cen ter of motion ,but does not remain i n a

plan e.

9. P ath .-A point in changing from one position to another

traces a l ine called‘

its path. The path may be Of any form .

The path traced by any point on a pul ley revolving on its shaft

is a circle,but a path n eed not be continuous , as for example the

path traced by a rifle bul let .

4 ME CHAN ISM

In general , the motion Of a body is determined by the paths Of

three Of its points not in a right l ine . If the motion is in a plane,

two po ints are sufficient , and if rectil inear, one point determines

the motion .

10 . Velocity .— I n addi tion to know ing the path and d irection

Of a moving body,there is another element necessary to com

pletely determine its pos it ion , and that is its veloci ty.

H eretofore we have not cons idered the time necessary fer a

body to complete a certa in motion .

Veloc ity is measu red by the relation between the space passed

over and the time occup ied in traversing that distance . It is

expressed numer ical ly by the number Of un its Of distance passed

over in one unit Of time,as m i l es per hour

,feet per minute , inches

per second,etc . It is the rate of moti on Of a po int in space .

1 1 . L inear Ve locity — When a motion is referred to a

point in the path of a body,its veloc ity is expressed in l inear

measure and is cal led its li n ear veloci ty.

Vel oc ity is uniform when equal spaces are passed over in equal

units Of time ; that is , the di stan ce vari es as the time.

L et V= velocity; S = total distance passed over ; T= t ime

occup ied .

Then if we know the distance passed over by a body and the

time it occupied in pass ing over that d istance , the velocity,S

VT"Thus if a body w i th a uniform veloc ity passes over a

distance Of 50 ft . and occup ies 5 minutes in doing so , it has a

velocity Of5

75

9: 10 ft . per m inute .

If the velocity and the t ime occupied are known , the space

passed over,S V><T,

while if the space and velocity are known ,

the time,T g.

12 . Variable L inear Velocity .

— A body whi ch has a moti on

that i s accelerated or retarded i s sai d to have a vari able lin ear veloci ty.

Such a velocity is not constant , but when cal cu l ating the velocity

at any point in its path,its veloc ity is assumed to be constant for

that instant . Thus a tra in start ing from rest and g radual ly

increas ing its speed unti l it attains a velocity Of 60 m i l es per hour ,has a var iable velocity . After the train had gone a d istance of

one mile from its starting-point,it might have a velocity Of 15

M OTION S AND VE LOCITIE S 5

miles per hour , meaning that if its velocity rema ined constant , it

would go 15miles in the next hour .

13. Angular Ve locity The angu lar veloci ty of a poi n t

i s the n umber of un i ts of angu lar space whi ch wou ld be swept over in

a un i t of time by a line joi n ing the given poin t, wi th a poin t outsi de

i ts path, about whi ch the angu lar veloci ty i s desi red.

The angular space is measured in circu l ar measu re,or it is the

ratio of the arc to the radi u s .

The un it angle or the radian is one subtended by an are equal

in length to the radius,r

H ence in on e c ircumference there are 27r = 6 .2832 rad ians ,

360°

O

or a radian

In engineering practice,angular velocity is usual ly expressed

in revolutions per unit of time .

Thus if a point S in Fig . 2 makes n revolut ions per unit Of

time,its angular ve loc ity

,V

° = 27rn

radians per unit Of time . This is the

angle that a line j oining S with the

center of the pul ley wil l pass through

per unit of time .

All points in the pul ley have the

same angu lar velocity,since they al l

pass through the same angle in a

unit of time .

14. L inear Ve l ocities of Al l P oints in

the S ame Body.—The lin ear veloci ti es FI G . 2 ,

of all poin ts in the same bodyare di rectly

proportional to thei r di stan ces from the cen ter of rotation of the

body.

Al l points in the pul ley Of Fig . 2 having the same radiiw i l l have the same l inear velocity

,but the greater the radius

the larger the l inear velocity . Thus if the point M on the

pu l ley has a rad ius r,the V“

M = 27rrn while the V“

S = 27a .

V"S V

“

M = 27 a 27rrn or

V S 2 1a RV

“

M 21rrn r

0 M E CHAN ISM

15. Relation between L inear Velocity and Angular Velocity .

The angular velocity of a point in a rotating body is 27rn radians

per unit of time and the linear velocity of a point in the same body

is 27rrn or the an gular velocity is the same as the l inear velocity

of a po int in the body havin g a un i t

We may then say that the angular

rad iu sThl s

g ives the angular velocity in radians

and to reduce it to revolutions,divide

by 27r.

This relation is Often used and should

be remembered . The angu lar veloci ties

of two poin ts i n difi eren t bodi es having

the same linear veloci ties,but difi eren t

radi al di stan ces from thei r cen ters ofrotati on

,are i nverselyproportiona l to thei r

radi i .

L et P and S ,Fig . 3

,be the two points in A and B respectively

in which R is the radius Of P from its center of rotation and r

the radius Of S from its center Of rotation .

FIG . 3.

L et V

V°

A V°B

V°

A V°

B

V-

P = V—S .

P roblems

1 . An engine piston makes 400 Sin gle strokes per minute ; the flywheel ison the crank shaft . What is the l inear velocity of the crank pin if thelength of the crank is 15 in ‘

7

https://www.forgottenbooks.com/join

8 IWE CHAN I S M

13. I n the hand- operated O il sw itch shown,through what angle mu st

.

thehandle AC move in order to comp letely open or close the sw itch?

P rob lem 13.

Data:L inks A BC BD D E F FG’GH H I HJ JK

Length 3i"8 Ga ze

"3 a"4~

2"

Al iH ubs A B D E F G I J K

D iameter g”g"g""gr ;

P IIIS,diameter %

I I

126

"1

5

6

1 1“If"

1 153 1

55

N OTE — M ake a fu l l size ink draw ing and leave off al l dimensions .

the l inks in one extreme position,and represent the other extreme by

l ines only . Time, 6 hours .

14. I n the Tabor engine indicatorplot the path Of B for a motion of 31l

in . for A along the straight l ine ab.

Data:D iameter of rol l at B 1

3

5

CD CF= 1

DE l i"

.

N OTE — M ake a pencil skeletondraw ing

,scale 4 in . 1 in . Take points

on l ine ab in . apart an d fi nd the corresponding positions Of B an d F.

Time,4 hou rs .

15. Design the circu it-breaker soP rob lem 14 ,

that it w il l make or break contact fora difference Of 137} in . in the water

level in the tank . Locate the extreme positions of both bal ls and of all thelevers . M arb le slab for c ircu it-breaker 12 in . square .

N OTE .

-M ake an ink draw ing hal f size, and design the different partswithout regard to strength but so that they w i l l look of the proper pro

M OTIONS AND VE LOCITIE S 9

portions The tank need not be drawn to scale,although the difference

in water level must be drawn to the same scale as the sw itch mechanism .

Do not show the rheostat,motor or pump .

Cir cu i t B re aker

M otor

Rh eos t a t

P rob lem 15.

The statement of the prob lem to be p laced in the upper right-handcorner of the sheet and underneath the draw ing

,AUTOMATIC FLOAT SW ITCH .

Time 6 hours.

CHAPTER I I

INSTANTANE OUS CE NTERS ,KINE MATIC CHAINS

16. M echan isms .

— A mechanism or train Of mechanism is the

term applied to a portion of a machine where two or more part s

are combined so that the motion of the first compels the motion

of the others according to a law depending on the natu re of the

combination . The two parts conn ected together are known as

an elemen tary combi nation,so that a train Of mechanism consists

of a series of elementary combinations .

If a part is considered separately from the others it is at liberty

to move in the two opposite directions and w ith any veloc ity,as

the crosshead of an engine that is not connected to the connecting

rod .

Whee ls, shafts and rotatin g parts gen erallv are so connected

with the frame of the machine that any given point is compel led ,when in motion

,to describe a circ le around the axis

,and in a

plane perpendicular to it . S l idin g parts are compelled by fixed

guides to describe straight lines,other parts to move so that

points in them describe more compl icated paths and so on .

These parts are connected in successive order in various ways

so that when the first part in the series is moved,it compel s the

second to move,which again gives motion to the third

,etc . The

various laws of motion Of the different parts of a train are affected

by the mode of connection .

17 . M odes of Connection .

—Connection between the different

parts Of a machine may be made in any Of the fol l owing ways:

1 . By direct contact

Turning pairs— connected l inks .

Sl ide connector— crosshead .

Cams w ithou t rol ls .

Friction contact —friction cyl inders .

Rol l in g and sl iding contact— toothed gears .

10

Q

P

P‘

Q

<5

INS TANTANE OUS CE N TERS ,KI NE MA

’

I’

I C CHAIN S 1 1

a . Cams w ith rol ls .

b. Rigid l inks .

0 . Flexib le connectors—ropes,belts and

chains .

2 . By intermediate connectors

a . E lectricity and magnetism .

3. Withou t material connectors b. Gravity .

c. Centrifugal forcefi

governor .

The first two methods are perhaps the most important in this

subj ect,and wil l be discussed later on .

18. Instan taneous M otion .

— When a body changes its position

its motion at any instant may be said to be its instan tan eousmotion .

As an i l lustration take the car wheel A,moving along the track

B,Fig . 4 . For an instant there is contact between the wheel and

the track along a line at C perpen

dicu lar to the paper and paral l el to

the axis of the wheel . The wheel

then rotates about C which becomes

an element Of both the wheel and

the track for an instant,and this

motion of the wheel is its instan

tan eous motion . The line through C

and perpendicular to the paper is its

i n stan taneous axi s .

If instead of considering the whole wheel,we take a section

made by passin g a plane through it perpendicular to the axis,it

wil l cut the line through C in a point which is called the i n stan

tan eous cen ter or cen tro.

19. The relative motion between A and B is the same whether

we consider the track stationary and the wheel revolving about

it,or whether the wheel is considered stationary and the track

revolving .

20 . Cen tro.

—A cen tro is a poi n t common to two bodi es havi ng

the same lin ear veloci ty in each,or i t i s a poin t in one body about

which the other tends to rotate.

As an il lustration of the first part of the definition , the tangent

point Of the two wheels A and B,Fig . 3

,is an example . It is a

point common to the bod ies A and B and having the same linear

velocity in each . The second part Of the definition is i l lu strated

by the points in the frame C of Fig . 3,about which A and B rotate .

FIG .

12 M E CHAN ISM

21 . Location of the Cen tre in a S ingle Body .— I n any ri gi d

body movi ng abou t an axi s,the di rection of moti on of any poi n t i s

perpen di cu lar to a li n e joi n i ng the poi n t w i th the axi s ; conversely,

the axi s of rotation wi ll i n tersect a li n e drawn perpen di cu lar to themotion of anypoi n t i n the body, and lying i n the plan e of the motionof the poi n t.

An i l lu stration of a special case of this statement is the motionOf a po int on the r im Of a fly wheel . Its instantaneous direction

of mot ion is tangent tO the r im ,or perpendicular to a l ine j oin in g

the po int w ith the ax is Of rotation .

A general i l lustration is as fol lows:L et the points M and N,

Fig . 5, have the d irections of mot ion shown by the arrows , both

motions lying in the same plane .

L ines drawn through M and N

and perpendicular to their respective directions of motion

,wi l l

,if

they intersect,l ocate the cen tre

,

or instantaneou s center of the two

points,and Of the body as a whole .

In Fig . 5these l ines intersect at 0 .

To determine the direct ion of mo

t ion Of any other po int as P,draw

a l ine through P ,perpendicu l ar to

the l ine P O.

In some cases the cen tre can

not be located,as when the l ines

drawn perpendicular to the direct ions Of motion co incide or when

they are paral le l . In the latter case the cen tre is at an infinite

distance away and the body has a motion Of translation .

22 . L ocation of Cen tres in Three Bodies M ovin g Re lative ly

to E ach Other .

— Any three bodi es movi n g relatively to each other

have but three cen tros,and these cen tros li e on the same strai ght li n e.

In Fig . 6,l et A

,B and C be the three bod ies that move rela

tively to each other . Assume that the body C is held stationary ,and that A and B are fastened by pin j oints to C at ac and be

respectively .

The number and names Of the cen tres are found by taking the

bod ies in combinations of two,using each on e with each Of the

others,as AB

,AC and BC . The -bod ies are u sual ly represented

by cap ital letters and the cen tres by l ower case letters,so that

INS TANTANEOUS CE N TERS ,KINE MATICS CHAIN 13

the cen tres will be ab, ac and be. It makes no difference in wh ich

order the letters are taken as ab or ba for the relative motion of A

about B is the same as the relative mo

tion of B about A (Art . 19)The only motion that A can have is

that of rotation about ac,while the only

mot ion Of B is that of rotation about

bc. This locates two Of the cen tres,and

the third on e ab is common to the bodies

A and B and has the same l inear velocity

in each .

Assume that ab l ies at the point 0 .

When 0 is considered as a point in A,it

has a radius from its center Of rotation

o—ac,and moves in the are 1—2 . I ts

direction of motion is perpendicular to

its radius,or alon g the l ine m—n .

When 0 is considered as a point

in B, it has a radius o—bc and moves FI G 6 .

in the are 3—4 . Its direction of mo

tion is perpendicu lar to its radius,or along the line p—q .

Thus We have the point 0 moving in two different directions

at the same time,which is impossible . The lines m—n and p—q

must be parallel or they must coincide . They cannot be paral le l

since they must both pass through the same point,hence they

must coincide,and the only time that they wil l coinc ide is when

the radii o—ao and o—bc l ie in the

same straight l ine .

23. KinematicChain .

—A ki n

emati c chai n i s a combi n ation

of rigid bodi es or links so con

n ected that the motion of each

i s completely con trolled by and

depends upon the motion s and

posi tion s of each of the others .

Figs . 7,8 and 9 are examples . By hold ing one Of the links

stationary, any motion given to one will cause a certa in definite

motion in each of the others . The l inks can be of any shape

whatever as long as their shape does not cause them to interferewith the motions of any of the other links .

FIG . 7 .

14 M E CHAN IS M

Fig . 7 is a s imple fou r- l inked kinematic chain consisting of

four turn ing pairs , while Fig . 8 is a s imp l e chain of three turning

and on e sl iding pair . Fig . 9 is a compound chain . A compound

FIG . 8 .

kinematic chain i s on e in whi ch on e or more of the links has more

than two join ts . In Fig . 9 the l inks B and E have three j oints

or are each connected to three other l inks .

2 4. L ocation and Number of Centres in a Kinematic Chain .

To find the centres in a kinematic chain write the names of the

FIG . 9 . FIG . 10 .

links in a row,and undern eath each

,its combination s with each

of the others . The number of centres in any kinematic chain

M ,where N is the number of l inks in the chain .

In the four- l ink cha in of Fig . 10,the number of centres wil l be

401

51 ) — 6 centres . The ir n ames can be found as stated above .

A B C D

ab be at

ac bd

ad

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16 IlI E CHAN I SM

25. Location of Centres in a Compound Kinematic Chain .

P roblem — L ocate al l of the centres in the compound kinematic

chain shown in Fig . 12 . There bein g six l inks,there wil l be

—1 )2

15 cen tres . M ake a l ist of the cen tres as fol lows

A B C D E F

ab bc cd de efac bd ce dfad be cfae bfaf

FIG . 12 .

The centros ad,be

,of , ef , ce and cd are at the link j oints and

are found by inspection . The centres ab, ac and bd are found as

in Fig . 1 1 . By striking out the known cen tres in the above li st,

the cen tres ae, be, bf , cf , de and df remain to be located .

P roceed as expla ined in Art . 24 and it wil l be found that

according to Art . 22,

L inks AE C and AEF will locate centro ae

L inks CFA and CEE wil l locate centro cf ,L inks DEA and DEC w i l l l ocate centro df ,L inks DEA and DE C w i l l l ocate cen tre de,L inks BEC and BFD wil l locate centro bf ,L inks BEA and BBC will locate centre be.

INS TANTANE OUS CE NTERS , KINE MATIC CHAINS 17

P roblems

1 . P rove that in three bodies moving relatively to each other,their

three centros l ie on the same straight l ine .

2. Assume the positions of the l inks in a simple Crossed- l ink chain and

locate al l of the cen tres .

/3. Locate all of the cen tres in Fig . 29 for the position of the l inks shown .

4. Locate al l of the cen tres in Fig . 9 for the position of the l inks shown .

5. Combine Fig . 7 and Fig . 8 by inverting Fig . 7 making the l inks A and

B common,thus forming a compound kinematic chain an d locate al l of the

centres .

6. L ocate all of the cen tres in Fig . 30 for the position of the l inks shown .

7 . L ocate al l of’

the centres in the Watts straight- l ine motion mechanismfor the position of the l inks as shown in Fig . 41 .

/8. Locate al l of the cen tres in Fig . 43 for the position of the l inks shown .

/9 . Locate all of the cen tres in Fig . 44 for the position of the l inks shown .

10. L ocate al l of the centres in the Thompson Indicator M otion M echanism for the position of the l inks shown in Fig . 46.

CHAPTER I II

SOLUTION OF RE LATIVE LINE AR VE LOCITIE S BY TH E CE NTROM ETHOD

26. Relative L inear Velocity .— I n comparin g the linear veloci ty

of any poin t i n on e link wi th the lin ear veloci ty of a poi n t i n any

ether li n k,the compari son mu st be made through a poi n t common

to both of them .

In Fig . 13 let A and B be two bodies , held in contact with each

other by the frame C,wh ich consider stat ionary . The cen tres of

A,B and C are ab

,ac and be. If C

is stat ionary the cen tres ac and bc

wi l l be stationary s ince they are the

points in C about which A and B

respective ly tend to rotate ; and ab

is a po int common to A and B having'

the same l inear veloc ity in each .

The common poin t i s named fromthe two li n ks i n whi ch the poi n t bei ngconsidered i s located .

L et J be a point on A whose

radius and l inear velocity are known,

and let K be a point on B,whose

radius is known and whose velocityit is des ired to find . The poin t J

has a radius J—ac and revolving J about its center will not

affect its velocity as lon g as its radiu s is not chan ged . Revolve

J around to the line of centers and lay off a line m—n to any

convenient scale to represent its l inear velocity .

The velocities of al l points in the same l ink or body are directly

proportional to their distances from the center of rotation (Art .

so that the linear velocity of ab having a radius ao—ab is

directly proportion al to the linear velocity of J, having the radius

18

FIG . 13.

S OL UTI ON OF RE LATI VE LINE AR VE LOCITIE S 19

J—ac. The velocity of ab can now be found by similar triang l es

and is represented by the line ab—o.

S ince the point ab is common to the bodies A and B ,its linear

velocity will be the same in each , although its radius , when con

sidered as a point in B is not the same as it was when considered

as a poin t in A . The linear velocities of K and ab are directly

proportional to their radii .

Complete the triangle bc—ab—o by drawing the line bc—o,and

revolve K around its center to the line of centers at q . Draw

through q the line q—s, paral l el to ab—o. Then from similar tri

angles,if ab—o is the linear velocity of ab

, g—s to the same scale will

be the linear velocity of K ,

or if m—n is the l inear ve

locity of J then q—s is the

linear ve locity of K to the

same scale .

Bear in mind that fromthe linear veloci ty of the given

poin t, the lin ear veloci ty of the

common poi n t must fi rst be

found and from that fi nd the

linear veloci ty of the requ i red

poin t.

27 . Open Fou r - l i n ked

M echan isms — Veloci ties ofP oin ts in Adjacen t L inks

L et A ,B

,c

,D

,Fig . 14, be

FI G . 14 .

an open four- linked mech

an ism in which it is required to find the V"K

, a point in the link

C,knowing the V

‘

J ,a point in the link B, for the position s

of the links shown and with the l ink A stationary .

It is only necessary to consider the links containing the points

and the stationary link . These are A,B and C ,

the centres bein g ,

A B C

ab bcW)

as

The centre be is the common point and is named from the

links which contain the poin ts . The centres ab andgyc are the

20 M E CHAN ISM

centros about which the links B and C rotate . The three centros

ab,ac and bc determine the l ine of centers .

m 72

L ay Off from J, a length of l ine to any convenient scale,to

represent the V"J (this line should be laid off perpendicular to

the radius of J) .

Bymeans of sim ilar triangles , find the V‘

bc.

S ince the l inear velocities of all points in the same link are

directly proportional to their instantaneous radii , revolve K about

pzc in to the line of centers and after completing the triangle of

which V‘

bc is one side,lay off the V

'

K .

28. Veloci ti es of P oi n ts in Opposi te L inks — G iven the V J,

Fig . 15, to find the V"K with the link A stationary .

FIG . 15.

The links that must be considered are the two links containing

the points and the station ary l ink , which in this case are A ,B

and D,having centros ab, ad and bd. S ince the l ink A 18 station

ary,the cen tres of A wil l be stationary

,viz .

,C729 and

gal, and bd

is the common point,or the point common to B and D ,

and having

the same linear velocity in each .

Revolve J around into the line of centers and lay off a line to

any convenient scale to represent its velocity,then by means of

similar triangles find the l inear velocity of the common poin t bd.

From this,find the linear velocity of K

,after revolving it around

into the l ine of centers .

29. Crossed L ink M echan isms . Veloci ties of P oi n ts i n Adja

cen t. L i nks .

— L et it be required to find the V_K

,Fig . 16

, having

given the V_J with the link B stationary .

The links to be taken into consideration are A , g and C,

having centros ab, ac and bc, of which aband be are stationary .

m m z

L ay off a l ine to represent the VT", and by means of s imilar

SOL UTION OF RE LATI VE LINEAR VE LOCITIE S 21

triangles find the V'

ac, the common point . Then revolve Kabout be into the li ne of centers and by means of another set of

similar triangles find the V—K as shown .

30. Veloci ties of P oin ts i n N on -adjacen t L inks — L et A ,B

, C

and D,Fig . 17

,be a crossed link mechanism in which it is desired

to find the V"

K,a point in the link D , having given the V

”

J a

poin t in the link B ,with the link A stationary .

The links that need to be considered are , $12, B and D ,

having

centros ab, g/cy

l,and bd, in which ab and ad are stationary and bd

w

is the common point . Notice that bd is located at the intersec

tion of the li nks A and C ,but of course there is no j oint at that

point,for if there were , the mechanism would be as one rigid link ,

as there could be no motion of any of the links .

FIG . 16 .

The link B rotates about the centre ab,and as J is a point in

the l ink B , it will rotate about ab.

Revolve J into the line of centers and lay off a line to repre

sent its linear velocity . From the V"

J,find the V

"

bd by similar

triangles . K rotates about the centre ad as ad is the point in A

about whi ch D rotates . Revolve K aroun d:ad into the line of

centers and find its linear velocity by similar triangles .

31 . S l ider Crank M echan ism. Veloci ties of P oin ts i n Adjacen tL i nks — L et Fig . 18 be a sl ider crank mechanism in which the

V"

K is desired , having given the V"

J with the link A stationary,

The l inks to be considered are A , B and C,having centros ab, ac

,a m m

22 M E CHAN ISM

and be, of which ab and ac are stationary and be is the common

point .

L ay off a line representing the V'

J and by sim i l ar triangles

find the V'

bc. Then revolve K about ac to the l ine of centers

and find its linear velocity .

FIG . 18 .

N otice that the above figure represents the crank ,connecting

rod and crosshead of an ordinary steam engine ; the frame and

crosshead guides being represented by the l ink A . Fig . 22,

page 26 .

FIG . 19 .

32. Veloci ties in Non - adjacent L inks — In the slider crank

mechanism of Fig . 19 l et the points J and K be in the li nks B and

D respectively and the link A stationary .

The centres of the links é B and D areawl) , and bd . Revolve

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24 M E CHAN ISM

5. (a) L ocate al l of the cen tres .

(b) In the beam engine mechanism as Shown,w ith the l ink A stationary

and the velocity of the crosshead F equal to 4 ft . per second,fi nd the revo

lutions per minute of the crank w ith‘ the crank angle as shown .

Data:L ength 1 2 3 4 51

P rob lem 5.

NOTE .

—Make a full - size draw ing,using penci l l ines only. Draw circles

of 335 in. diameter at al l pin joints . P lot the velocity vector to the scale1 in .

=3 ft .

6. (a ) Locate all of the cen tres .

(b) I n the valve gear mechanism as shown, what is the l inear velocityof the valve in feet per minute

,when the eccentric is making 15R.p .m . for the

l inkage position when the angle 2—1—4 =30 degrees .

Data:Length 1—2 = i”; 2 3 4 5

P roblem 6 .

N OTE — Make a fu l l -size draw ing, using penci l l in es only. Draw circlesof ya; in .

diameter at all pin joints. P lot the velocity vector 1 in . 5 ft .Time

,2 hours .

7 . In the Thompson indicator motion mechanism,Fig . 46

, assume thevelocity of the p iston rod connection E and fi nd the velocity of the penci lpoint T.

j8. I n Fig . 43, assume the velocity of the point T and find the velocity Of

/a. point in the middle of the l in k B .

CHAPTER IV

VE LOCITY DIAGRAM S

33. Velocity Diagrams .

- I t is often desirable to find the

velocity of a point for more than on e instant,and this may be

done by means of velocity triangles . There are two ways of repre

senting the velocity,depending upon whether the point in ques

tion has a motion of translation or rotation,the former being

shown by a li n ear velocity diagram and the latter by a polar

velocity diagram .

34. L inear Velocity Diagram.

— A linear velocity diagram is

one in which the velocities are plotted by using rectangular

coordinates .

In the several positions of the block A,shown in Fig . 20

,l et

a length of line represent the instantaneous velocity of a point in

FIG . 20 .

the block , for each of these positions each line being drawn to the

same scale . Revolve each velocity around the point,perpon

di cular to its direction of motion , and through each of the points

cut off on the perpendiculars , draw a smooth curve , which is the

velocity diagram .

To find the linear velocity of the block at any intermediate

position ,draw a l ine through the diagram perpendicular to the

direction of motion,and the length of li ne cut off between the

upper and lower sides of the shaded portion will be the linear

velocity for that position .

26 M E CHAN ISM

35. P olar Velocity Diagram .

— The polar velocity diagram

differs from the linear velocity diagram in that pol ar coordinatesare used instead of rectangular

coOrdinates .

In Fig . 21 let A be a crank

revolving about the center 0,and

let the velocity of the outer end

in its variou s positions he repre

sented by a length of l ine,each

of these lengths being drawn to

the same scale . About the point

in its several positions,revolve

the velocity into the radius andFI G . 2 1 . draw a smooth curve through

the points cut off on the radii .The velocity at any intermediate point can then be assumed

to be equal to the length of radial l ine cut off between the sidesof the diagram .

36. Velocity Diagram of E ngine Crank and Cresshead .

— I n

Fig . 22 l et 0 be the crank shaft , 0A the crank , AB the connectin g

FIG . 22 .

red,and B the crosshead of an ordinary steam en gine . If we

know the length and revolutions per minute of the crank,its

l inear veloc ity can be found from the equat ion

(Art .

Then if the l ength of the connecting rod is also known the

var ious instantaneous velocities of the crosshead can be found .

VE LOCITY DIAGRAM S 27

The instantaneous velocities of the p iston wil l be the same as

those of the crosshead .

The angular velocity of the crank can be assumed to be con

stant throughout the revolution .

Draw the crank pin circle,Fig . 22

,and divide it into anv con

ven ien t number of parts as 12,corresponding to different crank

positions . Find the extreme crosshead positions by taking 0 as

a center and radii,length of connecting rod plus length of crank

,

and length of connecting rod minus length of crank,giving M

and N respectively . For the intermediate positions of the cross

head take a radius equal to the length of connecting rod and

centers at 1,2,3,4 12 on the crank pin circ le and strike

arcs cutting the center line of the crosshead in B,2 1 , 31 , 4 1 and

L ay off a length of line,drawn to any convenient scale to

represent the V"

of the cran k pin A,and revolve it around into

the radius to get a point on the polar diagram .

S ince we assumed the angular velocity of the crank constant

for all parts of the revolution,the linear velocity of any point

will also be constant,and the polar diagram becomes a circle

,the

radius of which is the length of the crank plus the length of line

representing the l inear velocity of the crank pin . Knowing the

direction of motion of each end of the connecting rod, its instan

taneous center F can be found by drawing lines perpendicular to

the direction of motion and their intersection will give the instan

taneous center or centro (Art .

The l inear velocity of the crosshead end of the connecting rod

wil l be to the linear velocity of the crank pin end directly as their

instantaneous radii , or

or V_B X V

_A

Instead of locating the instantaneous center of the connecting

rod for each of its positions , we may draw through the various

points on the polar diagram,

3’ lines parallel to

the connecting rod in its several positions,and the lengths of lines

cut off on the perpendiculars drawn through the crosshead in its

various positions g ives the linear velocities of the crosshead _B

2 1 etc . This is true because a plane passed through a tri

ang l e paral lel to the base cuts off proportional parts from the

28 M E CHAN ISM

other sides , and since the part cut off on one Side will be the V"A

,

the part cut off on the other side wi l l be the V_B

After locating the points 1 3”

6 draw a smoothcurve through them . This curve will be the linear velocity dia

gram for the crosshead . The curve above and below the l ine issymmetrical

,so that it is not necessary to draw the lower portion

.

37 . Comparison of Velocity Diagrams — I n order to compare

the l inear velocity of the crosshead with that of the crank pin so

that the difference in their velocit ies can be seen at a glance,they

can be superimposed,one on the other .

In Fig . 23draw a circl e of radius equal to the linear velocity ofthe crank pin

,and divide it into the same number of equal parts

as the crank pin circle of Fig . 22,in

th is case 12 . L ay off on the first crank

position,0 1 =B 1

” from Fig . 22 and onthe second crank position

, 02

etc .

, al l the way around , then draw a

smooth curve through the points

1'

3”

12” giving the velocity diagram

of the crosshead . The distance 1”—1 ’ is

the amount of l inear velocity of the

FIG . 23. crank pin in excess of that of the

crosshead . In some parts of the revo

lutien the linear velocity of the crosshead is greater than that of

the crank pin . This occurs when the instantaneous radius of the

crosshead is greater than that of the crank pin .

38. Variable M otion M echan ism.

— I f the horizontal center

line of the crosshead does not pass through the center of the crank

as in Fig . 24,the linear velocity diagram for the crosshead wil l not

be the same for the forward and back stroke . L et 0 be the center

of the crank OA and MN the center line of the crosshead travel .

The points M and N can be found by taking 0 as the center and

radii equal to the sum and difference of the lengths of conn ecting

rod and crank respectively . Draw a line through M and O,

cutting the crank pin circ le at R,and through N and O

,cutting

the crank pin circ le at S . R and S then wil l be positions of thecrank pin when the crosshead is at each end of its stroke .

If the crank moves in a clockwise direction as: indicated by the

arrow,the crosshead wil l move from M to N while the crank is

going through the an gleROS,which is l ess than The cross

VE LOCITY DIAGRAM S 29

head moves from N to M while the crank moves through the

angle SORwhi ch is more than so that if the linear velocity

of the crank pin is constant throughout the revolution,the cross

head will have to move from M to N at a faster rate than it does

when travelling from N to M,and this is shown by the linear

velocity diagram of the crosshead , the area of the curve above

the horizontal line MN being greater than the area of the curve

below the line .

The construction of the velocity diagram is similar to that of

Fig . 22 .

39. Variable Rotary M otion M echan ism.—By using the same

combination of lin ks as shown in Fig . 22 , but by making the crank

stationary,and using the frame as on e of the moving links and also

changing the preportions of the links another variable motion

mechanism can be obtained .

L et A,B

, C and D of Fig . 25 be the links , with A stationary .

Assume that the end of the revolving arm B , which is fastened

to the sli ding block,has a constant velocity , represented by the

length of line L—R,laid off at right angles to B . To find the l inear

ve locity of K ,a point in the link D

,when the mechanism is in the

position shown by the full lines .

30 M E CHAN ISM

The velocity ofK can be found by the method of instantaneous

centers as was done in a previous chapter, but a more simple way

where the velocity is to be found for a number of different positions

is as follows .

The length of line L—R represents the velocity of the sl iding

block C,turning about the center ab. To find the velocity of C

turning about the cen

ter ad,resolve L—Rin to

two components,one

L —S at ri ght ang les toD ,

and the other R- S par

a llel to D . The length

of l ine L —S represents

the velocity at which

the block is turning

about the center ad,

while R—S ,is its slidin g

velocity alongD . Know

ing the turning veloci

ty of the S l i ding block

around the center ad,

we can find the turning

velocity of K around

the same center by

means of similar tri

angles,by drawing the

l ine ad—P through S,

and K—P paral lel toFIG 25. L—S . The length of

l ine K—P then reprosents the linear velocity of K to the same scale that L—Rrepre

sents the linear velocity of the end of B , since linear velocities ofpoints in the same link are directly proportional to their radii .In the dotted positions of the links is shown the velocity of

K for the same length of line representing the linear velocity

of L .

The l ink B is cal l ed the con stan t radi us arm Since its length

does not change and the center about which it rotates is called

the con stan t radi us arm cen ter . The link D is cal led the variableradi us arm, since its radius from the sl iding block to the center ad

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32 M E CHAN ISM

With A as a center and a'

radius equal to the length of the con

n ecting rod , draw the arc mn ,and with the same radius and center

B,draw the arc rs .

L ocate O,the center of the variable radius arm by trial

,on the

line CD,by taking 0 as the . center of a circle , the circumference

of which will be tangent to the arcs mn and rs .

Through A and O draw a l ine intersecting the arc mn at K ,

and one through B and O intersecting the arc rs at K ’. Then OK

and OK ’ are the positions of the variable radius arm when the

ram is at the extreme ends of its stroke A and B respectively .

Draw a line KK’ and through 0 ,perpendicular to KK ’ draw

the line CC of indefinite length . L ocate the constant radius arm

center G on this l ine at the given distance from O , the variable

radius arm center .

N ext locate the positions of the constant radius arm when the

ram is at each end of its stroke .

In this problem the velocity ratio is 2 1 or it takes twice as

long for the forward stroke as the return stroke,and since the con

stant radius arm revolves at a constant velocity , it will take 240°

of revolution for the forward stroke and 120°

for the return stroke .

VE LOCITY DIAGRAM S 33

With G as a center lay off two lines symmetrical with 0G,

making an angle of 120° with each other . These lines will inter

sect the l ines through 0A and B0 at L and L ’ respectively,giving

the positions of the constant radius arm when the ram is at the

ends of its stroke .

With G as a center and radius CL describe a circle,which wil l

be the path of the end of the constant radius arm .

Divide the variable radius arm circle up into any convenient

number of equal parts , as 12 , symmetrical with the line 0G,to

represent different positions of the variable radius arm and find

the corresponding positions of the ram by taking centers at 1,2, 3

12,and a radius equal to the length of the Connecting rod

and striking arcs cutting the l ine AB .

S ince the velocity of the constant radius arm is constant foral l parts of the revolution we may represent it by a line of any

convenient length . L et L 1—Rrepresent the velocity of the end ofthe constant radius arm

,laid off at right angles to the radius .

To find the V"K

,resolve L l—Rinto two components (Art . one

L l —S perpendicular to the variable radius arm K 10 and the other

R—S,paral l el with it . The component L 1—S is the linear velocity

with which L 1 is turning about the center 0 , and the V"K 1 which

is turn ing about the same center,is found by similar triangles

and is equal to K l—P . Revolve P around P ' to get a point on the

polar velocity diagram and through P ’ draw a line parallel to the

connecting rod,getting the point 9 on the linear velocity diagram

of the ram .

Repeat the process for each of the other points by using the

same length of line to represent the linear velocity of the con

stant radius arm,resolve it into its two components

,find the

velocity of the j oint of the connecting rod and variable radius

arm,and from that the velocity of the ram .

After completing the diagram it will be noticed that the veloc

ity for the forward stroke of the ram is very much more constant

than for the return stroke .

The minimum stroke of the ram is obtained by movin g the end

of the connecting rod K 1 , toward 0 until the ram moves through

the required minimum stroke . The radius OK 1 wil l not , however ,be one- half of the stroke because 0 does not lie on the line

AB .

It will be seen from the figures illustrating this mechanism that

34 M E CHAN ISM

the constant radius arm and the variable radius arm,both describe

complete circles .

42 . O scillating Arm Qu ick Return M otion M echan ism.

— By

using the same combination of links as in the Whitworth quick

return motion mechanism,but with different proport ions , a

mechanism known as the oscil l ating arm quick return motion

can be obtained .

L et Fig . 29 represent it in the same manner that Fig . 26 repre

sented the Whitworth quick return motion . N ote that the

distance between . the fixed centers is much greater and that the

variable radius arm D is much longer than the constant radius

arm, so that while the constan t radius arm revolves in a com

plete circle , the variable radius arm oscillates . The angle through

FIG . 29 .

which it oscillates is found by draw ing lines from the variable

radius arm center, tangent to the circle described by the end of the

constant radius arm .

E xcept for the difference noted , the method of laying out the

velocity diagram of the ram is similar to that of the Whitworth

quick return motion .

P roblems

1 . Make sketches show ing the essential difference between the Whitworth qu iek return motion mechanism and the oscil lating arm qu ick returnmotion mechanism .

2. I s the velocity diagram of the crosshead, Fig . 22, symmetrical about

a vertical axis? Why?

3. Make a sketch show ing how the velocity diagram for the crosshead isobtained when its path of travel is above the center of the crank .

VE LOCITY DIAGRAM S 35

4. (a) L ay out the velocity diagram of the ram for the oscil lating arm

(b) How many R.p .m . must H make in order that the maximum cu ttingspeed be 60 ft . per minu te? Diameter of G= 12". Diameter of H :4"

P rob lem 4 .

Data:L ength of oscil lating arm C = 20 Length of l ink B = 9

Maximum length DE =4i”

. Time ratio (maximum stroke ) =5:3.

N OTE — Make a skeleton penci l draw ing,half s ize . L ay off the l ine

DErepresenting the velocity of D (not half s ize) .

P ut no statement of prob lem or data on s heet,but this title

L AYOUT OF VELOC ITY D IAGRAMS

FOR

OSC ILLATING ARM QU ICK RETURN M ECHANISMTime, 8 hours .

5. (a) L ay out the ve locity diagram of for the oscil lating armshaper .

(b) I f C makes 50 R.p .m . cutting speed?

36 M E CHAN ISM

Data:Diameter of gear = 14 diameter of pinion c=4~§strokes 1:2 . DE = 5

”

P roblem 5.

NOTE —Make a pencil draw ing,half s ize . Lay off line representing the

l inear velocity of the pitch point of the (not half size ) . The

sub- title is the same as for P rob lem 4 . Time, 8 hours .

6. (a) L ay out the l in ear velocity diagram of the crosshead in the crank,

connectin g rod and crosshead mechanism .

P rob lem 6 .

(b) Lay out the linear ve locity diagram of the crosshead in the crankand Scotch crosshead mechanism .

VE LOCITY DIAGRAM S 37

Data:Length OR=3 length RC 12 hub O diameter .

H ubs Rand C diameter ; pins Rand C 1% diameter .

P in 0 Take the velocity vector equal to l é” .

NOTE —Make a ful l-Size pencil draw ing, superimposing the l inear velocitydiagrams one over the other . Time, 3 hours .

7 . Lay out the l inear velocity diagram of the ram of the shaper, using

the Whitworth qu ick return motion mechanism .

P rob lem 7 .

Data:L ength OB 3 length DA l ength OD lengthBC 12 Take the length of the velocity vector equal to 1 }N OTE .

—Make a fu l l- size pencil draw ing . Sub- title similar to that ofP rob lem 4. Time

, 8 hours .

8. Assume the l inear velocity of the b lock C (Fig . 29) and find the l inearvelocity of a point in the middle of the l in k E .

9. Assume the l inear velocity of the ram F (Fig . 29 ) and fi nd the linearvelocity of the s l iding b lock C .

10. What is the essential difference between the Whitworth qu ick returnmotion mechani sm (Fig . 26) and the oscil lating arm qu ick return motionmechanism (Fig .

1 1 . A p lank 10 ft . long lean s against the wal l . The foot of the plank isdrawn out at a constant rate of 30 ft . per minute . At the instant the footof the p lank is 6 ft . from the wal l

,fi nd (a ) at what velocity the upper end

is moving, (b) at what velocity the middle point of the p lank is moving andin what direction

, (c) what point in the p lank has the least velocity , its amountand direction .

12. Referring to Fig . 23, why are the l inear velocity diagrams of thecrosshead not symmetrical about the vertical center l ine 3’

CHAPTER V

PARALLE L AND STRAIGHT-LIN E MOTION M E CHANISM S

43. P aral lel and S traight- l ine M otion M echan isms — A

paral lel motion mechanism is a combination of links so constructed

that when one point in it moves in any path , another point w i l l

move in a paral lel path . A straight- line motion mechanism is a

combination of links so constructed that some point or points in

one of the l inks w i l l trace a straight l ine .

44. P aral lelogram .

-The paral l elogram is one of the most sim

ple mechanisms for giving an exact parallel motion . It con

sists of four links,the lengths of opposite ones being equal .

Fig . 30 is a paral l elogram in which D,the j oint of the two

l inks CD and AD,is fixed .

FIG . 30 . FIG . 31 .

Take a point P on the link AB extended and draw the li ne PD,

cutting the l ink BC at T,then in the triangles P BT and PAD .

or BT XAD=a constant .

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40 M E CHAN ISM

cutter revolving at a high speed , is made to produce the similar

pattern reduced or enlarged as the case may be.

1

FIG . 34.

P arallel Ruler.—Fig . 35 shows the application of the

parallelogram to the parallel ruler for drawing parallel l ines .

46. Roberval Balance.

— By addi ng a fifth link to the parallelogram

,a double parallelogram is obtained in which an appl ication

FIG . 35. FIG . 36 .

is found in the druggists ’ balance of Fig . 36. By placing the support midway between the scale pan uprights

,the load wil l be the

same on each pan when they are at the same level .

47 . Drafting Board P aral lel M echan isms .— There are many

of these devices for guiding the straight edge with a paral lelmotion

,onl y a few of which will be shown here .

In Figs . 37 and 38 the straight- edge is made so that it proj ects

beneath the board at the ends , as wel l as being on tep .

In Fig . 37 one cord is fastened to the straight- edge where it

proj ects beneath the board at a ,carried up and around one pulley

,

FIG . 37 .

1 For a description of an engraving machine see M achinery of April , 191 1 ,page 602 .

PARALLE L AND S TRAIGHT-LINE M OTION .ME CHAN ISM S 41

then diagonally down across the board around a second pulley

and fastened to the straight- edge at b. Another cord beginning

at b is carried up and around a pulley then diagonally across the

board around a second pulley and up to a . Fig . 38 is somewhat

similar except that there are two pulleys each at a and b and the

cords are fastened to the corners of the board .

FI G . 38 .

In Fig . 39, six pulleys are required , but the cord is continuous .

Fig . 40 shows the Un iversal Drafting M achine .

”This machine

FIG. 39.

takes the place of the tee- square,scales, triangles and protractor .

The mechanism consists of two sets of parallelograms , so arranged

that in no matter what position they are placed , the scales will be

parallel to their original posi

tions . The head to which the

scales are connected is mov

able about its center,thus

allowing any angle ofthescaleswith the horizontal

,the most

common angles being obtained

by a pin dropping into holes,

and the others by means of

a Vernier,the head being

claInped by a thumb screw .Fm40»

42 M E CHAN ISM

48. Snaight- line M otion M echan isms — A straight- line motion

mechanism is on e that wil l cause some point in the mechanism to

move in a straight l ine without the aid of guides .

It was necessary to use them originally when it was des ired to

make some part move in a straight l ine , as for example , the cross

head of an engine,because machine tools were not far enough

developed to make plane surfaces .

M ost of the se-called straight- l ine motions are only approxi

FI G . 41 .

mate , while there are a fewthat are mathematical ly cor

rect .

49. Watt’s S traight- lineM otion .

— Thi s is the best

known and most widely used

of al l the straight- line motions .

In Fig . 41, it is shown in its

simplest form . In the figure

the connecting l ink BC is

shown perpendicular to ABand CDwhen they are paral lel

,

but while this is the best preportion , it is not necessary that they

be so. The tracing point T is located on BC,where a line j oining

A and D intersects it .

Fig .

’

42 shows the appl ication of the Watt straight- line motionin combination with the form of parallelogram shown in Fig . 32,

FIG . 42 .

to a walking beam engine . 0 is the center of the walking beam ,

F, a point on the frame and H ,

the crank shaft hearin g . The

PARALLE L AND S TRAIGHT-LINE M OTION M E CHAN ISM S 43

l inks CO,CD and DE compose the Watt motion mechanism

,the

point T, moving in a straight vertical line , while the links OA ,

AE,ED and DC make up the elements of the parallelogram .

The point T is found as in Fig . 41 and E is taken on a line

drawn through 0 and T. In practice CC and FD are usual ly

taken as on e- half of 0A . A pump rod is usually attached to li nk

CD at T as shown .

50. S cott Russell S traight- line M otions .— In this mechanism

shown in Fig . 43there are two links A and B and the sl iding block

FIG . 43.

C. B is one- half the length of A and is fastened to A at its middle

point . Take a center at the intersection of A and B and a radius

of length B ; strike an are whi ch will pass through the points T,

O and C ,so that the angle TOC is a right angle , and Twill move in

a straight line . This is an exact straight- line motion , but it

requires the sliding block moving between plane guides .

Fig . 44 shows a modification of the mechanism ,in which the

link D is substituted for the sliding block . In this case , the

longer the arm D,the more nearly will Tmove in a straight line .

This is sometimes known as the grass- hopper motion .

”

FIG . 44. FIG . 45.

In Fig . 45 is shown a second modification of the mechanism

44 M E CHAN ISM

that will give an approximate straight line . In thi s case,the arm

E is substituted for D of the previous figure .

51 . Thompson Indicator M otion — Fig . 46 shows the appl ica

tion of the Scott Russell motion to a Thompson steam engine indicater .

1 The pencil at T, which traces the diagram on an oscillat

ing drum,is guided by a Scott Russell straight- line motion

, con

sisting of the links OA , AT and CD . The l ength of link BE is

determined by drawing a l ine through 0 and T,and noting the

point E,where it cuts the center line of the cyl inder . If instead

of the link CD,the l ink FE were used , the linkage would then be

FIG . 46.

the parallelogram shown in Fig . 32 . This lin k is impossible , how

ever, as it would cut the side of the steam cyl inder of the indicator .

While the pencil point does not move in an exact straight lin e

TG,the error is sli ght .

52 . Tch ebi ch eff ’ s ApproximateS traight- l ineM otion .

-Thismechanism

devised by P rof. Tchebicheff of P etro

grad,is shown in Fig . 47 . The proper

tions of the links are such thatAB CD

=5,BC= 2 and AD= 4. The tracing

point Tin the middle of BCmoves in an

FIG . 47 .

approximatestraight lin eparallel toAD .

1 For various other forms of indicator mechanisms , together w ith theproportions as made by the various manufacturers, see Machine Design i

"Part 1 , by Forrest ‘R. Jones .

PARALLE L AND STRAIGHT-L INE M OTION M E CHAN ISM S 45

53. Robert’s Approximate S traight- l ine M otion — Fig . 48

shows thi s mechanism which con

sists of two equal l inks AB and

DE and a triangular- shaped l ink BTD

in whichBT=TD AB ,and BD

A"EThe tracing point T c lose ly fol lows

the straight lin e AF .

54. P eaucell ier ’s E xact S traight

line M otion .

— This mechanism ,Fig .

49,consists of seven movable links and on e stationary link . The

proportions are such that AO BO AC BC BT AT,and CD

FIG . 48 .

FIG . 49 .

D0 . Then 0 ,C and T wil l always lie on the same straight li ne

,

and

6132 —6? =B

—

T2—T

_

E

or transposmgCE =DE

2 4 T?

(OE +TE ) (OE TE )

OTXCC

Therefore OTX 0C is a constant .

Now suppose that the l inkage is moved until C

let T’ be the new position of T.

Then we have or

OC’GT

CC OT’

46 M E CHAN ISM

And OD =BC =DC ’ so that O , C and C’ l ie on a semi—c ircle which

makes the angle OCC ’ a right angle .

In the triangles COC’ and TOT’

,the angle TOT’ is common

,

and s ince the sides are proportional,the triang l es are sim i lar .

Thus since the angle OCC’. is a right angl e

,the angl e TT’

O is

al so a right angle .

In the same way it can be shown for any other position of the

linkage , that the line TT’ is perpendicular to OT’

,and therefore

Tmoves in a straight l ine .

In applying this motion to eng ines,the piston rod is fastened

to T and this takes the place of the u sual crosshead and gu ides .

If the link DC is not made equal to D0,T wil l describe a c ir

cular arc .

1

CD

O_

D< 1 ’ the are described by T wi l l be concave toward 0

,

CDwhi l e If

CD> 1 ’ the

FIG . 50 .

P roblems

1 . W ith the arrangement of the l inks of a parallelogram as shown inFig . 30

,let DA = 10 in . ; DC =6 in . ; CT=4 in . Wil l the copy be larger or

smal ler than the original and how much?

2 . P rove that a point in the P eaucell ier cel l moves in an exact straight.l ine .

3. Make sketches of two approximate and two exact straight- l ine motionmechanisms .

4. Make sketches of as many approximate straight- l ine motionmechanismsas you can .

5. Make a sketch of the Thompson Indicator motion mechanism .

1 For the proof of this see “ K inematics of M achines, by R. J . Durley .

2 For these proportions and the proof see “ K inematics of Machines ,” by

R. J . Durley,page 94 .

be convex toward 0 .

55. Bricard ’s E xact S traight

line M otion .

— I f the li nkage ofFig . 50 is made of the proper

t ions 2 such that AC=DF= a ; CT

TD =b; AF= c ; E G=BE g?»be

Ft”the pomt T W l ll describe a

straight l ine perpendicular to and

bisecting AF .

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CHAPTERVI

CAM S

56. Defi n ition of Cam.

— A cam i s common ly a pla te or a

cylinder that tran smi ts motion to i ts follower bymean s of i ts edge or

a groove cut i n i ts surface. The cam is a very important part in

the construction of many machines,especial ly those of the auto

matic type su ch as pr int ing presses , screw machines and shoe

machinery . Their shapes are as numerous as the uses to which

they are put,so that only some of the most common forms wil l be

considered .

57 . Cam and Fol lower .

—The cam mechan ism u sual ly consistsof two elements

,the cam and the fol lower . The cam is usual ly

the driver .

58. Contact between Cam and Fol lower .

- Theoreti ca lly the

con tactbetween a cam and i tsfollower i s a poi n t or a li n e (see Fig .

This is impracticable for wear,so a rol l is general ly used as in Fig .

52 . The’

re ll turns on a pin that is rigidly fastened to the foll ower,

which gives rol l ing contact between the rol l and the face of the

cam and a sl iding contact between the rol l and the pin . In this

way nearly al l the wear wil l be on the p in which is easily replaced .

Rol l s are ord inar i ly from inch to 2 inches in diameter . When

the rise is smal l and the load is l ight,a flat face fol lower , as shown

in Fig . 53may be used .

59 . Theoretical Cam Curve .

—The theoreti cal cam curve (oftencalled the pi tch li n e) i s the cam cu rve that wou ld give the follower thedesi red motion

,if the con tact between the cam and thefollower were a

poi n t or a line. See Fig . 51 . In this case the theoretical curve

and the working curve would be identical .

60 . Working Cam Curve .

—The working curve is the curve

which gives the des ired mot ion to the fol lower when contact

between the cam and the fol l ower is made by other than point or

line contact . This is accomplished by the use of a roll follower

or by a flat- face follower .

CAM S 49

61 . Base Circle.— The base circ le i s a circle with its center

at the center of the cam shaft and a radius equal to the shortest

distance to the theoretical cam curve . The cam is la id out from

FIG . 51 . FIG . 52 .

the base circle . In choosing the size of the base circle it should

be taken large enough to insu re an easy motion to the fol lower,

the larger the base c ircle , the easier the follower’s motion wil l be

for any given rise in any given angle of rotation of the cam,as

shown in Fig . 54 .

FIG . 53. FI G . 54 .

L et 0 be the center of the cam shaft and the angle through

which the cam turns equal 30° to raise the rol l fol l ower inch .

It can be seen that the l ine d—e representing the r ise when using

the base circ l e of radiusRis more gradual than the rise representedby the line b—c, using the base circ le of radius r . Al so note that

the pressure ang le a is greater than the pressure angle a 1 .

50 M E CHAN ISM

On the other hand,the larger the base circle

,the larger the

resulting cam wil l be,and the designer often has to make the cam

smal l in order to get it into a certain space , so that it is largely a

matter of judgment as to how large to make the base circle,but

it shou l d be large enough to leave plenty of stock around the cam

shaft .

When u sing a smal l base circle and a large rise of fol lower in

a small angle of revolution of cam ,a difficulty that is apt to be

encountered is shown in Fig . 55. The theo

retical curve is shown by the dot two- dash

l ine,and the working curve is found

,if

using a rol l fol lower,by taking centers of

the rol l on the theoretical curve and strik

ing arcs,then draw ing a smooth curve

tangent to these arcs . This curve is theworking curve . At the top of the cam

shown in the figure,the distance from the

center of the rol l to the work ing curve,

represented by A,is more than the radiu s

of the rol l,so that the fol lower wou l d not

be raised to the preper height by the differ

ence between the radiu s of the rol l and the

distance A .

The remedy for this is to use a l arger base circle or a smal ler

rol l or both .

62 . M otions Used for Cam Curves — The motions most commonly u sed in the layout of cams are un iform,

harmon i c,un iformly

accelerated and un iformly retarded.

63. Uniform M otion .

— As appl ied to the motion of the cam

fol lower,un iform motion mean s equal ri ses of follower i n equa l

i n terva ls of time, the time being measured by div isions or degrees

on the base circle .

The heart- shaped cam of Fig . 56 is an example of this kind .

In this cam the base c ircl e has a radius OA and the fol lower

is to be ra ised a d istance 08 in 180 ° of revolution of the cam w ith

a uniform motion,and to drop a d istance 08 in 180° of revolut ion

w ith a un iform motion .

The method of l aying out the curve is as fol lows

D ivide the sem i—c ircumference of the base circ le into any number

of equal parts,in th is case 8

,and after l aying out the rise .08 on any

CAM S 51

of the radial lines,divide it into the same number of equal parts

as the semi- circumference of the base circle , and number the

points on the rise to correspond to the divisions into which the

base circ le is divided .

The radial lines dividing the base circle can be assumed to be

the different posit ions of the center line of the fol lower revolving

around the cam,as it is not possible to revolve the cam on the

paper . With a center 0 and radius 0 1 strike an arc intersectin g

0 1’ at 1 ’ with the same center and a radius 0 2 intersect 0 2 ’ at 2 ’

lower

FIG . 56 .

Do this for al l the divisions of the rise , then draw a smooth curve

through the points 3’

8’ which wil l give the theoreti ca l

curve for one- half of the cam . S ince the second half of the cam

is the same,if the second half of the base circ le is divided into the

same number of equal parts as the first half,the same points on

the radial l ine can be used for finding the points on the theoretical

cu rve .

If the fol lower were of the knife- edge type shown in Fig . 51,

the curve ju st found would also be the worki ng curve,but where

a rol l fol lower is used,it is necessary to find the working curve by

taking a radiu s equal to the radius of the roll,and with centers on

52 M E CHAN ISM

the theoretical curve strike arcs , then draw a smooth curve tangent

to these arcs which will give the working curve .

It wil l be seen from this,that for the same theoretical curve

,

an infinite number of working curves can be obtained by changing

the diameter of the roll .

64. H armonic M otion .

— The projection on the diameter of a

ci rcle of a poi n tmovi n g wi th un iform veloci ty i n the ci rcumferen ce i ssai d to have harmon i c moti on . The application of this motion tothe fol lower of a cam can perhaps best be seen by working out a

simple problem .

P roblem.

—Design a disk edge cam that will give a reciprocat

ing fol lower a harmonic rise of 4 in . in 180° of the cam ’s revolution,

a harmoni c drop of 4 in . in the

next and a rest or

dwel l during the remaining

angle . Cam to revolve coun ter

clockwise .

L ay out the base circle of

radi us 00,Fig . 57 , and divide

the 180° angle through which

the cam is to revolve during

the harmonic rise of the follower

into any convenient number of

equal parts, as 8

,and draw

radial l in es through these points .

On one of these radial l ines layoff 08

” from the base c ircle

equal to 4 in . Take 08 as a

di ameter and construct a semi

circle . Divide the circumfer

ence of the semi- circle into

the same number of equal

parts as the 180° of revolution,

FI G ' 57 thus getting the points

3"

Through the points

1 2 3 8 drop perpendiculars to the diameter of the

semi- circle to get the points a , b, c g . The distances 0a ,ab

,

bc, etc. ,

will be the rise of the fol lower for equal angles through

whi ch the cam revolves . With 0 as a center and radii 0 a ,0b,

0 c strike arcs cutting 0 1 , 0 2 , 03, etc. , at 1 , 2 , 3, etc.

CAM S 53

Draw a smooth curve through 1 , 2 , 3, etc .,whi ch wil l be the

theoretical curve .

The next angle through which the cam turns for the 4 in .

harmonic drop is Divide the 90° into any convenient

number of equal parts . S ince the drop is the same amount,as

the rise,if the 90° is divided into the same number of equal parts

as the rise,the same construction for the harmon ic motion can

be used .

During the remaining angle the follower is to remain at rest,

which means that during that part of the revolution the follower

does not move toward or away from the center of the cam, so

that the theoretical curve is the arc of a circle with a center at 0and a radius 0 0, which in this case is the radius of the base circl e .

The working curve is found in the same way as was done in the

previous case .

65. Un iformly Accelerated M otion — Thi s i s motion wi th a

un iformly i n creasing veloci ty. It is the easiest motion that can

be given to the fol lower of a cam .

When the follower starts from rest , the formula S = éaT2 is

appl icable,where S = space passed over ; a = acceleratien (a con

stant) and T= time.

Taking the successive angles through which the cam turns asthe units of time

,the space passed over in

1 unit of time,S = -

a = k (where k is a . Difference 1k

2 units of time,S = 2a=4k 3k

3 units of t ime,S =4eu = 9k 5k

4 units of time , S = 8a = 16k 7k

5 units of time , S 12 —a = 25k . 9k

So that if the angle through which the cam turns,and the total

rise of the fol l ower during which it has uniformly accelerated

motion are known,by dividing the angle up into any number of

units,we can solve for k.

For example,if the total rise is 3 in . and there are five un its of

time in the angle,

in . L ay off from the base circle

on the line at the end of the first un it of time in . and on the

second line in .

,on the third

in .

, on the fourth in .,and on the fifth line

in . These are the Various distances that the

54 M E CHAN ISM

follower has been moved away from the base circ le at the different

intervals of time .

Instead of calculating this amount each time,a g raphical

method is often used and is much more simple .

It will be seen that in the series 1k—4k—9k—16k- 25k,etc .

,the

distance that the point has moved from the starting place at theend of successive units of time

,that the distance moved du ring

each successive unit of time is in the series 1—3—5—7—9,etc .

In order to apply this to the fol l ower of a cam ,a simple example

wil l be used .

space moved through during eache Un i t of T ime lK-3K-5K-7K-9K- 11K

FIG . 58 .

L et AB ,Fig . 58

,be the distance that a reciprocating follower

is to be raised with a uniformly accelerated motion while the cam

turns through the angle AO6. Divide the angle AOG into any

convenient number of equal parts as six , and on a l ine AC of

indefinite length lay off un its in the series 1—3—5—7—9—1 1,using as

many numbers in the series as there are div isions in the angle,in

thi s case six . Through the end division and through B draw a l ine ;parallel to this line and through the other points on AC draw l ines

intersectin g AB at 2’ and This gives divisions on

AB that are proportional to those on AC .

With 0 as a center and radii 0 2’

0B strike arcs

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56 M E CHAN ISM

points on the diameter , giving the points 1 1 , 2 1 , 31 , 41 , 51 and 61 .

With 0 as a center and a radius 0 1 1 , strike an are cutting the first

division of time at 1 , and so on for al l of the points , always using

the center 0 . S ince the fol lower is to remain at rest during the

next the theoretical cam -curve will be the arc of a circle for

with 0 as a center and a radius 061 .

FIG . 59 .

The cam has now turned through which leaves 180° to

complete the revolution . Divide the 180° into any convenient

number of equal parts,as eight

,and as the drop is to

’

be uni form,

also divide the 3 in . into eight equal parts . Then with 0 as a

center,find points on the theoretical curve as before .

68. The Flat Face Fol lower.

— I n any of the foregoing cams ,the flat face follower can be substituted for the rol l follower and

the method of laying out the cam is exactly the same up to the

point of laying out the working curve .

Using the same data as were used in Fig . 59, let us design the

CAM S 57

cam for a flat face follower , the face of the fol l ower being perpen

dicular to its center line .

L et us assume that the theoretical curve has already been

found . To get the working curve draw lines l a,2b

, 30, etc.

,Fig .

60, perpendicular to the center line of the follower in its various

positions . After these lines are drawn in they will form a series

FI G . 60 .

of triangles , whi ch are shown cross- hatched in the figure . Draw

a smooth curve tangent to the middle point of the base of each of

these triangles , which will be the working curve .

If it is impossible to draw the curve tangent to all of the lines

without crossing some of the others , the curve cannot be found,

and a larger base circle must be taken .

The length of the flat face of the follower is found as follows

58 M E CHAN ISM

With the dividers , measure the greatest distance from the point

Where the theoretical curve crosses the center line of the follower,

as at C in Fig . 60, to the point D where the working curve is tan

gent to the follower face,thi s distance is the necessary length of

the fol lower face to be laid offi

to the right of the center line of the

follower . The length of the fol lower face to the left of the center

line of the follower is found in a similar manner and is represented

by the distance E F in the figure . These distances wil l be the

lengths of the fol l ower face required on the

right -hand and left -hand side of the center

l ine of the follower respectively .

69 . Involute Cams — I nvolute cams are

se-ca lled because the cam curve i s usua l ly ofi nvolute form. The foll ower is general ly

raised in less than half of the revolution of

the cam and is brought back to the starting

point by gravity or springs . Fig . 61 shows

such a cam . One use for them is on ere

crushers in stamp mil l s . S everal cams are

placed on the same shaft and so arranged

that they will raise their respective rods at

different times during the revolution of the

shaft . The cam may be so designed thatFI G:61 the fol lower is raised twice for each revolution

of the shaft .

70 . M ethod of L aying Out an Involute Cam.— An i nvolute i s

the curve generated by a poi n t in a taut stri ng as i t i s unwrappedfrom a cylinder, or by a poi n t on a straight edge that i s rolled on a

cyli nder .

The circle from which the involute is developed is called the

base ci rcle.

Divide the angle d) , Fig . 62,through which the involute is to

be developed into any number of preferably equal parts,as Six

and tangent to the base circle through these points draw the linesl a

,2b

, 30 6f . Where the lengths of divisions on the base

circle are not too great 1 the chord can be taken as the length of

the arc . L ay off the chord 01 en l a getting the point a . Twice

1 I f the lengths of divis ions are taken equal to one- tenth the diameterof the base circle

,they w il l be accurate to about one- thousandth of an inch

per step.

CAM S 59

the chord 01 on 2b, getting the point b, etc .,then draw a smooth

curve through the points a,b,c,

etc . , which w i l l be the invo

lute curve .

L ength of arcThe equation for the invol ute i s qb Radius of base c i rc ’

angle ct being in circular

measure .

In applying this to a

cam fol lower , the amount

that the follower is raised

is equal to the length of

the arc .

P roblem. Through how

many degrees must an in

volute cam turn in order to

raise its follower 3” if the

diameter of the base circle

from which the involute is

developed is

e ( in degrees)

The di stan ce between the cen ter of the cam and the cen ter line ofthe follower should be equal to the radi us of the base ci rcle

, so that

con tact between the cam and follower wi ll be on the cen ter lin e of thefollower . The force exerted on the follower will then be more

nearly all used in raising it .

CAM S W ITH OSC ILLAT ING FOLLOWERS

71 . Cams w ith O scillating Fol lowers .

— An osci l lating follower

is one in which points in it move in arcs of circles instead of in

straight lines . Fig . 63 illustrates a roll follower cam of this type,

while Fig . 64 shows a flat face or tangent follower cam .

The method of laying out a cam of this kind is somewhat more

difficult than that for the reciprocating fol lower .

L et the angle ABC ,Fig . 65

,be the angle through which an

oscil l ating follower is to be raised with a harmonic motion , while

the cam turns through an angle of the follower to remain

60 M E CHAN ISM

at rest for 150° of revolution

,and to drop with harmonic motion

in the remaining angle . In this figure 0A is the radius of the

FIG . 63. FIG . 64 .

bas e circle , AB the radius of the follower arm ,and the arc AC

the path of the center line of the roll as the foll ower is raised .

FIG . 65.

As the center of the roll moves along AC with a harmonic

motion which cannot be laid out directly on the are,it is neces

CAM S 61

sary to rectify the are on a straight line and lay out the harmonicmotion on that . L ay off on AD a length A6 equal to the arc AC .

With A6 as a diameter lay out a semi- circ le and divide it into any

convenient number of equal parts as six then proj ect these points

on the diameter , getting the points 1 , 2 , 3, 4, 5 and 6 . L ay off

these points on the are, so that A l’ =A 1

,= 12

,= 23

,etc.

Through the points 3’

C and the center of the follower

B draw l ines intersecting the lin e 0D at 2 1 , 31 , 41 , 51 and 61 .

Divide the first 120° of the cam ’s circumference into six equal

parts to correspond with the parts into which the semi- circle was

divided .

W ith 0 as a center lay off 0 1 = 0 1 1 , 0 2 = 0 2 1 , 03 =031

0 6”

0 61 . If the rol l had moved along the line AD,instead

of along the are,the points 3

”6” would he points on

the theoretical cam curve,but since it moved on the arc and the

lines 03”

0 6” represent different positions of the

l ine 0D,it is necessary to make a correction for the distance that

the center of the roll is away from the straight line .

Thi s can be done by drawing through the points 1 2 3

lines making the same angles with 0 1 03

0 6 that 1 1B ,2 1B ,

31B 61B make with 0D and laying off

3”c=313

’6”f= 6 l C .

2

A smooth curve drawn through the points a,b, c, d, e and f will

give the theoretical cam curve . Taking points on this line and a

radius equal to the radius of the roll the working curve can be

found . The remaining part of the cam curve is found in a similar

manner .

If a flat -face follower is to be used the method is the same up

to the point of finding the working curve,except that it will be

necessary to draw the lines l ”a ,2’b, 3

”e etc.

,much longer

so that they will form triangles as was done in Fig . 60 .

72 . There are a number of other methods of laying out this

cam whi ch will give the same results ; on e of these is shown in Fig .

1 1 1 and 2 1 are not shown on the figure on account of the smal lness of thedraw ing

,but they w i l l be at the intersection of the l ines through l ’B

,2

’B

and 0D .

2 The reason for making the correction can be seen if the student w il ltake a piece of tracing paper or cloth and p lacin g it over the figure

,mark

the center 0 . Then trace any l ine as 05” and 5”e. Revo lve 05” around0 until it coincides w ith 0D,

when the po int e w i ll fal l on

62 M E CHAN ISM

66. This method requires a mu ch larger sheet of paper for the

layout,but is less laborious if the faci l ities are at hand . Assume

the data as in Fig . 65 and

after locat ing the center of

the osc i l l ating arm B take a

radius OB,and with 0 as a

center,str ike an arc . Divide

the ang l e through which the

cam is to turn to give the

desired rise to the fol lower

up into six equal parts,and

through these po ints draw

tangents to the base circ le,

intersecting the arc of radiu s

0B at B l , B2 ,~ B3 Be.

With these points as centers

and a rad ius AB,draw arcs

from the base circle out

ward .

The points 4’

and 5’on the arc AC are

found as in Fig . 65; with 0

as a center and radii 0 1 ’

0C strike arcs , getting the points 3”

which wil l be pomts on the theoretical curve .

FIG . 66 .

P roblems

1 . L ay ent the theoretical and working curves for a disk-edge cam thatw i l l raise a reciprocating rol l fo l lower 4 in . w ith harmonic motion in 180

°

of the cam ’s revo lution ; the fol low er to remain at rest for and to dropw ith uniform motion to the Starting point in the remaining angle . Cam torevolve clockw ise .

2 .Lay out the theoretical and working curves for a disk-edge cam that

w il l raise a reciprocating ro l l fo l lower 3 in . w ith uniform motion in 120°

of revolution ; the fol lower to remain at rest for and to drop 3 in . w ithharmonic motion in the remaining angle . Cam to revolve counter-clockw ise .

Center l ine of fo l low er 1 in . to left of cam shaft center .

3.Show the theoretical and working curves for a disk-edge cam that

w il l raise a reciprocating flat- face fo l low er 5 in . w ith uniform motion in120

° of the cam ’s revo lu tion ; the fol low er to remain at rest during the nextand to drop w ith harmonic motion to the starting point in the next

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64 M E CHAN ISM

P O S ITIVE M OTION CAM S

73. P ositive M otion Cams .

—A posi tive moti on cam is on e

that does depend the force of gravi ty, springs or any otherexterna l mean s to bring the follower back to i ts i n i tial posi tion .

A common method of mak

ing a positive motion cam

where a rol l fol lower is used,is

to have the roll move in a

groove in the side of a disk or

plate as is shown in Fig . 67 .

74. Constan t D i am e t e rCam .

— A con stan t diameter cam

i s one i n whi ch the di stan ce be

tween cen ters of rolls, measuredthrough the cen ter of the cam

shaft, i s con stan t. It has the dis

advantage that any desired l aw

of motion can be l aid out for

but the second 180° being

laid out to cor respond . The layout of a cam of this type is

shown in Fig . 68. L ay out the first 180° of the cam for the

desired law of motion , get

ting the points 1,2,3

8 on the theoretical cam

curve . The distance from

the starting point through

the center of the cam shaft

to the point 8 (not shown

in Fig ) , wil l be the dis

tance between the centers

of the rol l s . W ith this

length as a radius and

FIG . 67 .

centers at 1,2,3 7

strike arcs intersecting their

respective radii d r a w 11 FIG . 68 .

through the center of the

cam . These intersections will give points on the theoretical curve

for the second half of the cam .

CAM S 65

This type of cam can be used for heavier work than the groove

cam a l ready mentioned .

75. M ain and Return Cam .

—As the name implies,this con

sists of two cams . The main cam is laid out for any desired law

of motion for a complete revolution or L ay out the points

1 , 2 , 3 16 on the theoretical curve , Fig . 69,for any desired

law of motion and choose a distance that the rolls are to be apart .

With this distance as a radius and centers at 1 , 2 , 3 16 strike

arcs on the radii diametrically Opposite the points , which wil l give

points on the theoretical curve of the second cam. The working

curves may then be found after choosing rolls of any convenient

diameter .

In this case as in that of the constant diameter cam,the

fol lower is generally constructed so that it envelops the shaft

and is guided by a square bushing . This cam combination is

particularly desirable for heavy work and slow speeds .

76. Con stant Breadth Cam .

—A con stan t breadth cam i s on e

in which the di stan ce between para llel faces i s con stan t. It

is used with a flat- face fol lower , and as in the case of the constant

diameter cam,can be laid out for any desired law of motion

,for

180° only . Assume that the points 1

,2,3 8, Fig . 70

,are

points on the theoretical cam curve . The distance 08 determ ines

the distance between the parallel sides of the follower faces .

Through the points 1 , 2, 3 8 draw l ines perpendicular to

66 ME CHAN ISM

the radial lines . and on the opposite side of the cam center locate

the points 5’

1’ by using 1

,2,3 7 as centers and

a radius 08 . Through the

points 3’

7’ draw

lines paral lel to those drawn

through 1,2,3 8 . To

obtain the working curve draw

a smooth curve tangen t to the

middle points of the triangles

formed by these perpend icular

l ines, as was done in Fig . 60 .

It wil l be noted that the

above construction is similar

to that of the constant diam

eter cam up to the point ofFI G 70 finding the working cam curve .

Fig . 7 1 is a special type of

constant breadth cam used on light mechan isms such as sewing

machines .

L ines j oining the three points of the cam form an equilateral

triangle . The sides are arcs

of circ les w ith centers at the

points of the trian gl e,and

radii equal to the chordal

distances between the points .

77 . Cyl indrica l Cams .

These cams may be used

for either recti l inear or oscil

lating mot ion .

The motion is obtain ed

by a groove in the cam as inFig . 72 or by fastenin g strips

on the surface of the cylinder

as in Fig . 73. The l atter case

is used on automatic machines

su ch as screw machines for

operating the turrets and wire feeds , where it is desired to

do different kinds of work . The same cyl inder can be used by

making a different adjustment of the strips , instead of using a

new cam complete .

FIG . 7 1 .

CAM S 67

Cylindrical cams can be laid out in several different ways,

one of which is to lay out the cam w ithout developing the cylin

drical surface , and another is to develop the surface before laying

out the curve . A problem wil l illustrate each of these two methods .

FIG . 72 .

78. L ayout of Cy l indrical Cam without Devel oping Cylinder.

P roblem — L ay out the theoretical curve for a cylindrical cam

that wi l l move a reciprocating fol lower a distance of 12 in . to the

right with a uniform motion in 1 7} turn s of the cam ; the follower

to remain at rest for i turn ; to move 12 in . to the left with a uni

FIG . 73.

formly accelerated and uni form ly retarded motion in one turn,

and to remain at rest for turn,thus bringing the fol lower back

to the startin g poin t .

L ay out the end and side views of the cylinder , Fig . 74, and

divide the circle representing the end of the cyli nder into any

convenient even number of equal parts,as eight . Divide the

12 in . on the side View into the same number of equal parts as

68 MECHAN ISM

there are equal divisions in l i revolutions on‘the end view , or

ten,and number all of the divisions to correspond . Through

the points on the side View draw lines across the cylin der , per

pendicu lar to its axis . Then through the points 1 , 2 , 3 8

on the end view ,draw li nes paral lel to the axis of the cylinder until

they intersect the lines drawn through the points 1 , 2 , 3 8

on the side view,thus gettin g the points etc.

Assume that the top of the cyl inder moves away from you as

you look at the side elevation ; then the curve wil l pass across the

FIG . 74.

front of the cylinder and disappear at and will be on the far

side until the point 8 is reached , when it will again pass in front .

From 2’ to A the fol lower is to remain at rest

,so the curve can

have no motion along the cylinder .

During the n ext revolution the fol lower is to move 12 in . to

the left with uniformly accelerated and uniformly retarded motion .

From A draw a line AB of indefinite len gth and l ay off on this

line eight divisions in the series 1—3—5—7—7—5—3—1,using any con

ven ien t unit .

Through B and the point C’,12 in . away from A draw a line .

P arallel to BC and through the points on AB draw lines inter

CAM S 69

secting AC . Through these intersections on AC and perpen

dicu lar to the axis of the cyli nder draw l ines , and proj ect across

from the points on the end V iew to get points on the theoreticalcurve as before .

From this point the fol lower is to remain at rest for revolu

tion of the cyl inder, so that the theoretical curve will be drawn

from C to 8,perpendicu l ar to the axis .

79 . L ayout of Cyl indr ical Cam on Developed S urface .— I h

order to lay out this same cam on a developed surface,l ay

out the circumference and length of the cylinder,as in Fig . 75,

and divide the circumference into the same number of equal parts

as there are divisions in the end view of Fig . 74, or eight .

— C i rcumferen ce of Cyl in der

FIG . 75.

Divide the length of the cylinder up,according to the law of

motion desired,and obtain points on the theoretical curve as

shown on the figure .

In cams of this kind where the groove crosses itse l f, the part

that fol l ows in the groove,which in disk- groove cams is a rol l

,

but in cyl indrical cams is the frustum of a cone,must be of special

shape .

It is general ly made oblong and of such a length that the lead

ing end wil l be across the intersecting groove and wel l entered in

the original groove before the widest part of the fol lower has

reached the intersecting groove

In order to obtain best results with a cylindrical cam the

angle that the theoretical curve on the developed cylinder makes

70 MECHAN ISM

with a perpendicular to the axis of the cam should not be more

than If the angle exceeds this, make the diameter of the

cylinder larger .

80. Cyl in drical Cam with O scillating Fol lower.

— The layout

for a cam of this type is shown in Fig . 76. The rad ius of the fol

lower is 0A and turns through the angle AOB . The other data

are the same as in the previous case . It wil l be seen that the ele

ments into which the cyl inder is divided,instead of being paral lel

to the axis as in the case of the reciprocating fol lower , are arcs of

circles,of radii 0A and with centers on a l ine through 0 perpen

dicular to the axis of the cyl inder . Other than this,there is no

Circumfer en ce of Cyl in der

FIG . 76 .

difference between the methods of laying out the oscillating and

reciprocating follower cams .

81 .Inverse Cams — Ah inverse cam i s one i n which the roll i s

on the driver and the groove i n thefol lower . Fig . 77 show s a cam of

this type .

The inversion is made to avoid dead points that would

sometimes occur when using an ordinary cam combination . It

has the disadvantage that the motion can be laid out for only

It is l argely used for l ight mechanisms such as sewing

machines .

Fig . 78 represents a S cotch crosshead which is a special

case of an inverse cam with a reciprocating fol lower , and often

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72 ME CHAN ISM

Fig . 78, when the drivin g arm had passed through the angle AO1with its initial position , the fol lower would have been raised a

distance but since the follower is to be raised but 1 in . in that

angle , the groove must be curved upward the difference between1 1

’ and 1 inch,or the distance 1 ’ 1 1 . The point 1 1 can be found

directly by laying off on the line from 1,1 in .

When the driving arm passes through the angle 10 2,the fol

lower is to be raised in .,and since it has al ready been raised 1

inch , l ay off 1% in . on getting the point 2 1 . During the next

angle the fol l ower remains at rest,so 31 wil l be in . down

from 3.

Whil e the fol lower is passing through the angle 304,the fol

lower drops in .,and as we already had a raise of 1—5in .

, the net

raise laid off from 4 wil l be in .

A 2 in . raise during the next 40° makes in . to be laid off

from 5. During the last 20° the follower remains at rest , so 232in .

wil l also be laid off from 6,to get the point 61 .

The points 51 and 61 in this problem fal l bel ow the original

position of the groove at A,but since the fol lower is 2% in . higher

than it was there,the groove must be the same amount lower for

the driving arm is horizontal in both cases . Draw a smooth

curve through the points 1 1 , 2 1 , 31 61 to obtain the theoret

ical curve . From 2 1 to 31 and from 51 to 61 the fol lower remains

at rest so these parts of the curve should be arcs of circles of radii

0A,and a center on the center line of the fol l ower .

The working curve is found by choosing a diameter of rol l and

with centers on the theoretical curve draw circles the diameter of

the rol l,then draw curves tangent to them .

During the last 180° of revolution of the driving arm , the roll

passes through the groove in the opposite direction , and causes

the follower to go very much below its original position . If this

is not desirable,the ends of the groove can be left Open and the

roll leave the groove during the last

P roblems

1 . Design an oscillating arm cam to raise the sl iding block 4 in . w ithharmonic motion in 180

° of counter-clockw ise revo lution of the cam ; theb lock to remain at rest for and to drop 4 in . w ith a uniform motion inthe remaining angle .

CAM S 73

Data: BC =4§ CE =8 ~§Diameter of base circle = 6Diameter of cam rol l 1 1

33

Diameter of cam ro ll pin i”

.

I f fl a t fiance Iol lowu i s

speci f i ed make l ike th is .

S ection at abP rob lem 1 .

Nore —Make a fu l l- size ink draw ing and leave off all dimensions . Dividethe 180° angle into 12 equal parts and the 135° angle into 10 equal parts .

Time,6 hour s .

2 . Make a ful l- size ink draw ing of the belt- shifter cam . The shif terarms to move w ith uniform acceleration and retardation abou t their centers0 and 9 while the cam turns abou t its center f . Diameter of cam ro l ls 1 in .

D iameter of shifter arm studs -g in . 0=angle through which shifter armsmove .

Prob lem 2 ,

74 MECHAN ISM

NOTE .

-Find the centers 0 and g on a l ine parallel to the pul ley shaft andmidway between the outer ends of the shifter arms . The angle efg =39, and

f is located by draw ing through c, downward to the right, a l ine makin g an

angle of w ith the horizontal , and a l ine through 9 upward to the rightand also making an angle of 1 7130 w ith the horizontal . The poin t f is at thein tersection of these l ines .

Divide cfg into three equal parts . W ith f as a center an d a radius fcstrike the arc cd

,thus locating d . Lay off dce= 0. W ith 0 as a center

and a radius cd strike the are de. Then d and e are the two extreme position sof the cam rol l . The movement of the other rol l is the same . D ivide theangle eflc into six equal parts an d the are de into correspondin g parts forun iform ly accelerated and uniformly retarded motion . Time

, 6 hours .

3. Lay out the theoretical and working curves for a main an d returncam that w il l give a reciprocating fol low er a harmonic rise of 6 in . in 90

°

of revolution,to remain at rest for and a uniform drop durin g the

remaining angle . Cam shaft revo lves clockw ise .

4. Lay out the theoretical and working curves for a constant diametercam that w il l raise a fol lower 8 in . w ith harmonic motion in rest for

an d a uniform rise of 3 in . in the next Cam revo lves clockw ise .

I f the diameter of the base circle is 6 in .,what is the correct distance between

the rol ls?5. Make a sketch show ing the theoretical curve for a cy l indrical cam

that w il l give a reciprocating fol lower a harmonic motion of 8 in . to the leftin on e revolution of the cam ; to remain at rest for revolu tion ; to moveuniformly to the left 2 in . in revolution

,and a uniform motion back to

the starting point in the least number of revo lu tions . Top of cam to turnaway from you as you look at side elevation .

6. Lay out the theoretical and working curves for a cy l indrical cam

that w il l give its oscil lating fol lower a harmonic motion from A to C in

revolution ; the fol low er to remain at rest for revolution ; to return frcmC to A in revo lution w ith uniform motion, and remain at rest for revolu

tion .

P rob lem 6 .

CAM S 75

Diameter of base of ro l l in .

Depth of groove in .

N OTE .

— Make the views as shown,in ink and fu l l size . P ut no statement

of the prob lem on the sheet . The student ’s attention is called to the cor

rection to be made in finding the bottom l ines of the groove in the cy l inder .

The center of the rol l passes through the axis of the cy l inder when the ro l l isat A or C only . When the rol l is at B ,

its center l ine is paralle l to 0D and adistance BD from it . Time

, 8 hours .

7 . M ake a sketch show ing the theoretical curve for an inverse cam

that w il l raise its reciprocating fol low er 1 in . during the first 30 ° revo lu tionof the driving arm ; to raise in . during the next to rest during the next

to raise 2 in ; during the next to drop I i in . during the nextand to raise L } in . during the next Driving arm 6—5in . long and tu rnscounter- clockw ise .

8. Design an osci l latin g fo l lower cam to lower the rod CD 2 in . w i thuni formly accelerated an d uniform ly retarded mo tion in 120

° of clockw iserevo lution of the cam ; the rod CD to dwe l l for 120° and to rise 2 in . w ithuniformly accelerated an d uniformly retarded motion .

P rob lem 8 .

Data:D iameter of base circle = 5 length of arm AB length ofarm BC =3

Diameter of ro l l 1 é”

.

Diameter of ro l l pin g"

.

Diameter of hub at B 1—3D iameter of pin at BD iameter of pin at C

N or a — Make a ful l-size ink draw ing and leave off dimensions Time ,6 hours .

76 MECHAN ISM

9. Design a cy l indrical cam that w i l l move a clutch 1 in . to the l eftalong the shaft w ith harmonic motion in i revo lution . Shaft to turn clockw ise .

P rob lem 9 .

NOTE .

— The pin P is moved down by a system of levers operated by apedal ; when the pin P drops down below the cam ,

the clutch is forced along theshaft to the right by springs (not shown) engaging w ith the other half ofthe clutch . The shaft then turns and as the

'

pin P engages w ith the cam ,

the clutch is disengaged . Make a ful l -s ize pencil draw ing, show in g thedeveloped surface of the cam w ith all dimensions . Time

,4 hours .

10 . Design a main and return cam that w il l move its fo llow er 3 in . tothe left w ith uniform motion during 135

° of the cam ’s revo lu tion . Restfor return the fol lower to the starting point during the next 135° ofthe cam ’

sr evolution w ith uniform motion and rest for the rema ining portionCam revo lves counter- clockw ise .

P rob lem

Data:Diameter of base circle=6” .

Diameter of shaft 1

D iameter of hub= 2Diameter of ro lls 1

Distance between rol ls =7

N OTE - P lace cam shaft center in the centersize ink draw ing . Time , 4 hou rs.

CAMS 77

11 . Design a constant diameter cam that w i l l move its fol lower 4 in . tothe left w ith uniform ly acce lerated and uniform ly retarded motion in 135° of

revo lution an d to rest for 45° of revo lution . Cam revo lves clock

P rob lem

Data:Base circle 5 diameter ;H ub 2 5

” diameter .

Shaft l1 1"diameter .

Rol ls 1 1 ” diameter .

No'rE .

—Make ink draw ing . P lace the center l ine of the cam 13% in . fromthe right border l ine and 7 in . from the top border l ine . Time

,4 hours .

CHAPTER VI I

GEARI NG

82. There are three separate cases for which gears can be used .

a . Constant velocity ratio .

Fi rst. To connect parallel shafts .

b. Variable veloc ity ratio .

S econd. To connect inter Constant velocity ratio .

sectin g shafts . Var iable velocity ratio .

Thi rd. To connect shafts that are neither paral le l nor inter

sectin g .

The first two cases with a constant velocity rat io are the most

common .

FRICTION GEAR ING

83. Friction Gearing — This may be sal d to be gearing in which

the motion or power that is transmitted depends upon the friction

between the surfaces in contact . In this case the power that can

be transmitted at a constant velocity rat io is l imited,for as soon

as sl ipping occurs,the velocity rat io changes .

84. Rol ling Cylinders — This is the most simple friction gear

combination and is used to transmit motion between paral lel

shafts w ith a constant velocity rat io . Fig . 80 shows such a com

bination .

Velocity Ratio .

— I t has been stated before (Art . 15) that the

angular velocities of two points in two bodies having the same

l inear velocities,but different radial d istances from their centers

of rotation,are inversely as their radii . Thu s if the two cyl inders

of Fig . 80 rol l together without sl ipping , their revolution s are

inversely as their radii . In order to have a constant velocity

ratio,there must be pure roll ing contact between the cyl in ders .

This means that the consecutive points or elements on the sur

face of one must come in con tact with the su ccessive points or

elements on the surface of the other in their order , or in other

words there must be no slipping .

78

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80 IVI E CHAN I S M

is 4 in . and that of the smal l ends 2 in .,when the belt is at the

large end of the driver, the revolutions of the driven wi l l be twice

those of the driver , while if the belt is at the smal l end of the driver

and the large end of the driven , the revolutions of the driven wi l l

be one- half those of the driver .

It can al so be seen from the figure that the d irection of rota

tion,or the di rectional relation is opposite in the two shafts . If

it is desired that they turn in the same direction,a smal l wheel

can be substituted for the belt . Thi s wheel should be covered

w ith some soft material,

such as rubber or l eather,

to make good frictional

contact . This combina

tion is used on machine

tools,su ch as grinders and

speed lathes,for obtain

ing variable speeds .

87 . S el lers ’ Feed Disks .

— Thi s is al so a type of

friction gear for connect

ing paral l e l shafts to ob

tain a variable veloc ity

ratio and is shown in

Fig . 83. The disks A

and C are on the shafts

to which it is required to

give the desired velocity

ratio .

Two disks B ,having their inner surfaces sl ightly convex

,fit

loosely on the same shaft and engage the sides of A and C near

their r ims,between the convex surfaces .

On the shaft with the d isks B,between the frame and the back

of the disks , are hel ical springs wh ich tend to pu sh the disks

together and exert pressure on the s ides of A and C .

Vel ocity Ratio .

— I f A i s the driver,the revolutions of

C can be increased by mov ing the intermediate disks B,to

ward A .

This combination derives its name from the fact that it was

first u sed on the feed mechanisms of machine tools bu i lt by the

William Sel lers Company of P hiladelphia .

FIG . 83.

GEARING 81

88 . Bevel Cones .

—These cones of which Fig . 84 i s an il lustra

tion are used for transmitt ing motion between shafts that intersect . The cones need not be of

the same size,and the shafts need

not intersect at a right an gle .

Vel ocity Ratio .— I f the veloci

ty ratio is as the bases of

the cones must be the same

diameter . In order to lay out a

pair of cones,

for any velocity

ratio and shaft angle,the method FI G 84

is as fol lows:Assume that the driver is to make 5 revolutions while the

driven makes 7 and that the shaft an gle is L et 0A,Fig 85

,

be the axis of the driver and 0B that of the driven . At any point

as a on CA draw a perpendicular l ine ab and on this l ine lay off 7

units ; also at any point as c on 0B draw the perpendicular l ine

cd,and on it lay off 5 units

of the same size as those laid

off on ab. Through b and d

draw lines paral lel to 0A and

0B respectively ; through their

intersection at C, draw the

l ine 00 . This l ine will be an

element of the two cones,and

cones of any diameter maybe

drawn as indicated in the

dotted outli nes,by using C0

as the common element .

89 . Brush P late andWheel .— This form of friction gear

ing shown in Fig . 86 is for the

purpose of transmitting vari

abl e velocity ratio between

shafts that intersect at right

angles . It consists of a large

disk A and a smal l wheel B,which can be moved across the face

of A ,by means of a feather key in the shaft of B . The smal l wheel

is usual ly made of some softer mater ial than the disk which is gener

ally of iron , in order to secure more friction between the surfaces .

FIG . 85.

82 MECHAN ISM

The small wheel should be the driver so that in case the load

becomes excessive and there is sl ipping,the smal l wheel w i l l be

worn down an equal amount al l

around . If the large disk were

the driver and the load became

excessive , a flat spot wou l d beworn on the smal l wheel

,wh ich

would then have to be trued up .

Ve locity Ratio .

—The veloc

ity ratio depends upon the di

ameter of the smal l wheel and

the efi ective diameter of the

disk,that is

,the d iameter of

the path that the smal l wheel

makes on the disk .

The brush plate is used on a number of forms of sensitive dri l l s

and 1s useful in other cases for light loads where slight variations

in speed are desired .

TOOTHED GEARING

90. Spur Gears — I n order to transmit more power than can

be obtained with the rol l ing cylinders of Fig . 80 l et us assume that

FIG . 87 . FIG . 88 .

proj ections are put on cylinder A ,Fig . 87

,paral lel to the axis

,and

corresponding grooves in cylinder B . Between the grooves on B ,

put proj ections,and corresponding grooves on A . The cylinders

wil l then appear as in Fig . 88.

The end view of the orig inal cyl inders wil l be imaginary circ les ,and are cal l ed the pi tch ci rcles

,and their po int of tangency is the

pi tch poin t.

GEAR"NG 83

Velocity Ratio .

— The velocity ratio of a pair of gears wi l l be

inversely as the radii of the pitch point of the pitch circles . I n

order that a pai r of gears tran smi t a con stan t veloci ty ratio, thei r

tooth cu rves must be such that a normal to the common tangen t of the

teeth at the poi n t of con tact wi ll a lways pass through the pi tch poi n t.

This statement is cal l ed the fundamental l aw of gearing . The

velocity ratio in this case wil l be constant for any fraction of a

revolution , or , in other words , the pitch circ les have pure rol ling

contact (Art .

Two gears the centers of which are at A and B,Fig . 89

,and

pitch point B'

have a pair of teeth in contact at T as shown .

FIG . 89 .

Their tooth curves . are so formed that a normal to the common

tangent MN through T also passes through the pitch point P .

The same relative motion between the tooth curves is obtained

by holding one gear stationary and revolving the other gear and

the frame,as by holding the frame stationary and revolving the

gears . Suppose that the gear whose center is at B is held station

ary and the one with a center at A revolved . Then the direction

of motion of the contact point T of A is at right angles to P T, or

along the line MN,and therefore in instantaneous contact with the

contact portion of B .

Suppose that the cu rves are so formed that the common

normal passes through any other point P ’ instead of P and let A

be revolved as before in the direction indicated by the arrow:

84 MECHAN ISM

then A and B wil l break contact , because the contact point T of A

moves toward M along the line M N as before,which is not the

direct ion of the contact portion of B . If A is revolved in the

opposite direction,the contact point T of A wil l move toward N ,

and the tooth curves wil l interfere,hence the fundamental law of

gearing as stated above .

91 . Defi nitions of Tooth P arts .

-The addendum ci rcle i s the

circ le bound ing the ends of the teeth . See Fig . 90 .

The worki ng depth ci rcle is the circle below the pitch c irc l e,a

distance equal to that of the addendum circle above the pitch

circle .

FIG . 90 .

The dedendum ci rcle or root ci rcle is the c irc le bounding the

bottoms of the teeth .

The clearan ce is the d istance between the work ing depth and

dedendum c irc les .

The face or addendum of the tooth is that part of the tooth

lying between the pitch and addendum circles .

The flank of the tooth is that part of the tooth lying between

the pitch and dedendum c ircles .

The thi ckn ess of tooth is its width measu red on the p itch c ircle .

The wi dth of space is the space between the teeth measu red on

the pitch circ le .

Backlash is the difference between the thickness of a tooth and

the space into which it meshes,measured on the p itch c irc les .

P i tch di ameter is the diameter of the p itch c ircle , and is the

GE ARING 85

diameter that is used in making all calculations for the size of

gears .

P i tch is the measure of the size of the teeth . There are two

kinds of pitch in general use for cal culat ing gear teeth sizes , ci r

cu lar and diametral .

Ci rcu lar pi tch is the distance in inches between similar points

of adj acent teeth measured along the pitch circ le . It is the sum

of the thickness of tooth and width of space , measured . on the

pitch l ine .

As there mu st be a whole number of teeth on the circumfer

ence of a gear , i t is necessary that the circumference of the pitch

circle divided by the circular pitch be a whole number . Thus it

is seen that if the circular pitch is a whole number,the diameter

of the pitch circle wil l usual ly be a number containing a fraction .

L et P’

circular pitch in inches

L et D = pitch diameter

L et T= number of teeth

Then TP ’ = 1rD or T—W

g;P

’ = 1% in .

,T=36

,the pitch diameter D in . very

Thus if

nearly .

Diametral pi tch is the number of teeth on the gear per inch of

diameter of the pitch circle .

L et P diametral pitch

L et T= number of teeth

L et D pitch diameter

Then

TT= PD ; D

D

In a gear having 64 teeth and a pitch diameter of 16 in the

diametral pitch P M eaning that for every inch of diam

eter of the pitch circle,there are 4 teeth on the gear .

D iametral pitch is more general ly used than circular pitch for

the reason that the pitch diameter comes out in equal fract ions,

and the pitch is deduced from the d iameter and n umber of teeth

w ithout having to use the constant 1r,which is necessary when

using circumferences .

86 M E CHAN IS M

It is sometimes desirable to convert from one pitch to the other .

This can be done as fol lows

In a given gear by the circu l ar pitch system T—w

l

lj, while if

using the d iametral pitch,T= PD .

DSo that we may write 135, PD .

In the same gear by the two systems , the pitch diameter D

wil l be the same,and as it appears in both sides of the equation

may be canceled out .

Then

So that ci rcu lar pi tct iametral pi tch= 7r.

The ang le of action is the angle through which a gear turns

while a tooth on it is in contact w ith its mate on the other gear .

The are of action is the arc subtend ing the angle of action .

The ang le of approach is the angle through which a gear turns

from the beginning of contact of a pair of teeth until contact

reaches the pitch point .

The ang le of recess is the angle through which a gear turn s

while contact in a pair of teeth is from the pitch point until the

teeth . pass out of contact .

Ang le of Approach+Angle of Recess=Ang le of Action .

There are two systems of generating tooth curves in generaluse that wil l g ive a constant velocity ratio— the i nvolute system

and the cycloidal system .

P roblems

1 . In a brush p late and wheel combination,the diameter of the smal l

wheel , which is the driver , is 4 in .,an d makes 60 R.p .m . The minimum

effective radius of the brush p late is l % in .

,and its maximum effective radius

is 15 in . What are its revolu tions in each case provided there is no sl ipping?2 . L et P = diametral pitch

,P

’ = circu lar pitch,T= n umber of teeth and

D=pitch diameter .

1 . Given P = 10 ; D =5. Find T.

2 . Given D= 15; P’ = 2 . Find T.

3. G iven T=48 ; D =4% . Find P ’.

4 . G iven T=3O. Find P .

f 3. I n two Spur gears the driver tu rns five times while driven turns threetimes . C ircu lar pitch in . Number of teeth in driven 30 . Find distancebetween centers an d number of teeth in driver .

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88 M E CHAN ISM

the tracing point Twill trace the involute aT and at the same

time it wi l l trace the involute dT as it is wrapped on B . The

dotted portions of the curves Tb and To are the parts of

the involutes that will be traced as the cylinders are revolvedfurther .

It is a property of the involute that a normal to the curve

is tangent to the base circle . The two curves have a common

normal at the point of contact T,which is tangent to both base

circl es . This line which is the locus of al l the points of contact is

called the line of action .

93. Spur Gears .

—L et A and B,Fig . 93

,be the centers of two

spur gears with pitch radi i AP

and BP respectively .

The radii of the base circles

from which the involutes are

developed are obtained by

drawing the li ne of action ab

and dropping perpendiculars

to it from the centers of the

g ears . The lengths of these

perpendiculars are the radii of

the base cir c les .

The ang le of obli qu i ty a,i s

the ang le that the lin e of action

makes wi th a tangen t to the

pi tch ci rcles drawn through the

pi tch poi n t.

If the angle of obl iquity

is zero the base circ les wil l

coincide with the pitch circ les,

and the teeth of the gears wil l

act only at the pitch point . The angle of obl iquity can be taken

any angle more than zero,but in practice for the standard in

volute tooth it is usually taken such that its sine is which

corresponds to an angle of approximately

94 . In the figure , l et the gear with its center at A be the driver

and revolve c lockwise as shown . There should be no contact

between the teeth of A and B until the point a , where the line of

action is tangent to the base circle of the driver , is reached ,as this is the point where the generation of the involute begins .

FIG . 93.

GEARING 89

The l imit of contact on the other side of the pitch point is b, where

the line of action is tangent to the base circle of the driven gear .

Contact will be along the line ab, beginning at a and ceasing at b

provided the teeth of the driven are long enough to reach the point

a,and the teeth of the driver of sufficient l ength to reach the

point b.

If the teeth of the gears are not long enough for contact to begin

at a and cease at b,it wil l begin where the addendum circle of the

driven cuts the l ine of action and cease where the addendum circle

of the driver cuts the l ine of action . Contact between a pair of

teeth shou ld be continuous between the beginning and end of

contact along the l ine ab.

Con tact between the teeth of any pai r of gears that are i n mesh

a lways begin s between the driver ’s flank and the follower ’s face and

ceases between the driver ’s face and the follower ’s flank.

For a pair of gears having equal arcs of approach and recess

to have continuous contact , that is , not to have one pair of

teeth cease contact before a second pair have begun con tact,

the angle PAa = a,Fig . 93

,should n ot be less than

15

2?

where

T equals the number of teeth in the smal ler gear .

It is desirable to have more than one pair of teeth in contact

at on e time,or to have the pi tch arc less than the arc of acti on .

Thus if the pitch arc is one- half the arc of action,two pairs of

teeth wil l be in contact at one time .

360 180°

T T

Taking a as 15° the least number of teeth that can be used is

11

235

912 teeth

,which is taken as the smallest standard inter

The maximum pitch angle equals 2PAa 2a

changeable gear . S ince there cannot be a fractional tooth on a

gear , if the number of teeth does not come out a whole number ,the next higher whole number must be taken .

95. Involute Rack and P in ion .— I f on e of the spur gears of

a pair is enlarged until its diameter is infinite,the combination

will be that of a rack and pinion,shown in Fig . 94, since a rack is

the same as a spur gear of infinite d iameter .

L et AP ,Fig . 95

, be the radius of the pitch circl e of the pinion

in a rack and pinion combination . The pitch circle of the rack

being of infinite diameter is therefore a straight line.

90 M E CHAN IS M

L et ab be the line of obliquity , then Aa w ill be the radius of

the base circle of the pinion .

If the pinion is the dr iver and turning as shown , contact can

not begin before the point,

a,the tangent point of the pinion ’s

base circle with the l ine of obl iquity , is reached , thus limiting the

length of the rack teeth . S ince the base l ine of the rack is tangent

to the l ine of obliqu ity at infinity , the length of the pinion teeth

can theoretical ly be of infinite length , but practical ly they are

limi ted by thei r becomi ng poi n ted.

A point on the line abwil l generate the rack and pinion teeth ,as in the case of spu r gears (Art . As the pinion turns in the

direction of the arrow ,the rack wil l move to the left with the

FI G . 94. FIG . 95.

same linear velocity as a point on the pitch circ le of the pinion,

which is greater than the linear velocity of a point on the base

AP

Aa'

When the generating point reaches b, the point a on the rackwil l have reached a

’

,for in the triangles PAa and aba

’

,ab is

perpendicular to Aa and aa' is perpendicular to AP . Al so

aa’AP

ab Aa’

which is equal to the angle AaP ,is a right angle . Therefore

the outline of the rack tooth is a straight line perpendicular to the

l ine of obl iquity .

It wil l be noted that the directional relation between the rack

and pinion is not constant , but depends upon the length of the

rack .

circle of the pinion in the ratio:

so that the triangles are similar and the angle a’ba

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92 M E CHAN ISM

begin before the tangent point of the pinion ’s base circle with the

line of action is reached,the teeth of the annular gear cannot

pass inside the point a . Ba wil l then be the shortest radius for

the addendum circle of the annular gear .

The l ength of the pinion teeth , as in the case of the rack and

pinion combination,are limited only by their becoming pointed .

In the figure,contact begins where the addendum circl e of the

driven cuts the l ine of action and ceases where the addendum

circ le of the driver cuts the l ine of action .

It wil l be noted that the sides of the teeth on the annular gear

are concave instead of convex , as in the case of spur gears . The

annular gear tooth is exactly the same as the space between the

teeth of a spur gear of the same angle of obl iquity,pitch and

number of teeth,with the exception of clearance top and bottom .

97 . Interference of Involute Teeth .— I f the teeth of the gears

of Fig . 93 are long enough to pass outside of the points a and b

there wil l be interference between this extra length of the teeth

and that part of the flank of its mate , lying ins ide the base c ircle ,unless the flank is hol lowed out , or the extra length of face rounded

off. The latter method is general ly used as the obj ection to hol

lowing out the flank is that it weakens the tooth . Th is part of

the tooth lying between the base and work ing depth circles is

general ly made radial .

98. E ffect of Chang ing the Distance between Centers of

Involute Gears — I n involute gears,the tooth curves being

developed by a point on a line tangent to the base circles,if the

centers of the gears are drawn apart , the same involutes w i l l be

generated and the gears wil l sti l l transmit a constant velocity

ratio,even though the pitch circles are not tangent .

The gears can be drawn apart until the teeth just touch at

their ends,but in this case the backlash wil l be excessive .

The centers may also be moved toward each other so that the

pitch circ les overlap,the l imit being when the backlash is reduced

to zero .

This is a desirable property of involute gears as it al low s gears

to be used on shafts that are not the exact center distances apart,

as calculated by the pitch diameters .

The path of action being along a straight l ine , the pressure on

the bearings is constant and tends to keep the centers as far apart

as possible , and prevent ratt l ing if there is any l ooseness in the

GE ARING 93

bearings . This property of involute gears used to be considered

obj ectionable , but with gears and bearings of modern design the

pressure is not excessive .

99 . Interchangeable Involute Gears .

— I n order that involute

gears may be interchangeable,that is , any gear of a set mesh

properly with any other gear,the p itch and angle of obl iquity of

one must be the same as the pitch and angle of obliquity of the

mating gear .

100 . S tandard S izes of Teeth for Gears .

— I n the spur gears

thus far discussed,nothing has been said about standard sizes of

tooth parts . Two gears can be made to mesh properly together

even though some of the parts are not the same size . For differ

ent pitches there are standard lengths and thicknesses of teeth,

however,and those adopted by the Brown Sharpe M anu

facturing Company and given in Tables A and B on pages 94 and

95 are almost universally used in this country . Fig . 98 shows

Circu lar Pitch

FIG . 98 .

parts of the teeth referred to in each column of the tables . The

sizes of the parts apply to other systems of teeth as well as to

the involute system .

101 . To L ay Out a P a ir of S tandard Involute Spur Gears .

P roblem — L ay out the tooth curves for a pair of involute spur

gears of 24 teeth and 16 teeth respectively ; 2 diametral pitch and15

° angle of obl iqu ity .

24 teeth,2 diametral pitch = 12 in . p itch diameter for large gear .

16 teeth,2 d iametral pitch = 8 in . pitch d iameter for smal l gear .

94 M E CHAN ISM

TABLE A .

—GEAR WHEELS

Table of Tooth Parts— Diametral P itch in First Column

Diam Thickness Addendum Working Depth of Wholeetral C ircu lar of tooth or depth of space depth ofpitch

,pitch

,on pitch 1

” tooth,

below pitch tooth,

l ine , P l ine,

z s S +f

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96 M E CHAN ISM

Draw the pitch circles Fig . 99 with radii AP and BP equal to4 in . and 6 in . respectively

,and through P draw the l ine of obliq

u ity ab having the given angle of obl iqu ity with the common tan

gent to the pitch circles through the point P . Drop perpen dicu

l ars from the centers A and B,cutting the l ine of obl iquity in a

and b respectively . Then Aa and Bbwil l be the radii of the base

circ les wh i ch can now be drawn .

From Table A on page 94 find the amount that the adden

dum,dedendum and working depth circ les differ from the pitch

circ le,and draw them in .

Divide the pitch circ le of the smal ler gear into 16 equal parts

and that of the larger gear into 24 equal parts , which w i l l give

the circular pitch . When no backl ash is al l owed,the thickness of

the tooth,and the width of space measu red on the p itch circle

wil l be the same . Bisect the circular pitch on each of the gears

which wil l give 32 equal d ivisions on the pitch circ le of the smal l

gear and 48 on the large one.

At any point on the base circle of each gear develop an involute

,and draw in the curves between the base and addendum

GEARING 97

circles through alternate points on the pitch circles . This wil l

give on e side of all the teeth on each gear . The curve for the

other side of the teeth is the reverse Of the one j ust drawn .

1

Draw the part of the teeth lying between the base and working

depth circ les radial , and put in a smal l are or fi l let between the

working depth and dedendum circles . Thi s fi l let shou ld be as

large as possible without passing outside of the working depthcircle .

STUB TOOTH GEAR

102 .

— The S tub Tooth .

— I n 1898 the Fellows Gear Shaper

Company adopted the stub form of tooth having the angle of

obliquity This gives a shortened and thicker tooth at the

root as compared with the standard involute tooth .

‘40S tan dard

volute Tooth

O

20 S tub Tooth

FIG . 100 .

103. M ethod of Design ating the S tub Tooth .—The stub

tooth is designated by the use of two diametral pitches , such as 157 .

The numerator being used to calcu late the pi tch diameter and the

n umber of teeth, while the denominator is used for calculating the

addendum and dedendum.

1 A convenient method Of transferring the original invo lute to draw in

the tooth curves is by means of a temp let . This can be made of a thin pieceOf soft wood . P ut a fi ne needle through the wood, near one end, and thecenter of the gear

,then cut away the wood to conform to the deve loped

tooth outl ine . Then sw ingin g the temp let about the center draw in one

side of al l the teeth . Draw the other sides of the teeth by turning the templetover and keeping the same centers . The same templet can be used fortwo gears only when they have the same number Of teeth and angle ofobl iqu ity .

98 ME CHAN ISM

TABLE C .

—D IMENSIONS OF STUB TOOTH GEAR TEETH

(The Fel lows Gear Shaper Company S tandard )

D IMENSIONS IN IN CH E S

Addendum,

The clearance in this system is made greater than in the standard geartooth sys tem and is equal to the addendum d ivided by 4 .

104. Advantages of the S tub Tooth .—The advantages

the stub tooth are as fol lows:1 . Greater strength .

2 . Wear on the teeth more un iform .

3. L onger l ife .

4 . L ess distort ion in hardening .

5. Greater production ( l ess metal to be cut away) .

P roblems

1 . M ake a sketch of a pair of involute spur gears,assuming the angle

of ob l iqu ity and diameters of the pitch and addendum circles . Show thepath of contact and skeleton teeth where contact begins and ceases only .

2 . M ake a sketch Of two invo lute spu r gears of unequal diameter, w iththe smaller gear driving clockw ise . Assume the angle of ob l iqu ity and theaddendum circles so that there w i l l be no angle of approach

,but a maximum

angle Of recess .

3. M ake a sketch show ing the path Of contact in a rack and pinioncombination, involute system ,

w ith the rack driving to the left . Assumethe angle of ob l iqu ity and addendum l ines .

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100 ME CHAN ISM

N OTE — No statement of prob lem or data . Only INVOLUTE ANNULARGEAR P INION . I nk draw ing . Time

, 3 hours .

8. Draw the correct tooth ou tl ines for two spur gears that are in mesh .

Stub tooth 20 ° system . Show the teeth going into and out of action .

I t'

2

u ity Smal l gear is driver and revolves counter- clockw ise .

N OTE .

— This problem is for the purpose of comparing the standardinvo lu te tooth outl ine w ith that of the stub tooth 20

° toothoutl ine . I t is suggested that

,Since the data other than the type of tooth is

the same as the data u sed for P rob lem 5,that this draw ing be made on

transparent paper and the two systems be compared by superimposing on e

draw ing over the other and the difference noted . Time,5 hours .

The pitch used in this problem is not found in the tab le for the stubtooth gear

, and the proportion shou ld be taken from Table A on page 94 .

9 . Draw the correct tooth curves for an invo lute rack and pinion thatare in mesh and show the teeth going into and out of action .

Data:Number of teeth in the pinion 42 .

D iametral pitch Angle of ob l iqu ityRack is driving to the left .

Data:N umber of teeth 36a nd 2 1 . Diametral pitch Angle of obl iq

N OTE .

— Make rack horizontal and the center Of the pinion in the centerof the sheet right and left an d I i in . from top border l ine . Make templetand check for contact . Fu l l- size ink draw ing . Time

,4 hours .

10. Draw the correct tooth curves for an invo lute annu lar gear and pinionthat are in mesh .

Data:Number of teeth in gear 72 .

Number of teeth in pinion 42 .

Diametral pitch 2; angle of ob l iqu ityP inion is driving clockw ise .

N OTE — Make center l ine of gears horizontal and in the center Of thesheet w ith the pinion to the left . Fu l l- size ink draw ing . Time

, 3 hours .

1 1 . Draw the correct tooth curves for a standard involute gear of 12 teethand for a stub tooth gear Of 12 teeth .

Data:P itch diametersShaft diameter ; face of gear 3Key

N OTE .

-P lace gears on center l ine of sheet top and bottom . P enci ldraw ing . Cut and check templets . Draw ing fu l l size . Time

,5 hours .

12 . Two spur gears are in mesh . I t is desired to design the teeth so thatthe addendums w i l l be a maximum . Show the outl ines of the teeth goinginto an d out of action .

Data:Number of teeth in gears 40 and 18 .

Diametral pitch 2 ; angle of obl iqu ityLarge gear is the driver and revolves counter-clockw ise .

J 0 J’

J J J

GEARI‘

N’

GW ? 10 1J J J

J .)J

”0

y .

NOTE .

—Make a fu l l- size pencil d’

ra’

vvmg on bbn tér l ine ’of ’ th’e Sheet topand bottom .

Show angles of approach and recess and compare the angle of action w iththe angle that wou ld be found if the teeth w ere of standard length . Time

,

4 hou rs .

CYCLO IDAL SYSTEM

105. Cycloidal System.

-The cycloidal system,as the name

impli es has tooth curves of cycloi da l form. It is an older system

than the involute,but is being replaced by it for many classes Of

work .

Before taking up the tooth outlines,a brief discussion of the

construction of the different cycloidal curves wil l be given .

106. Cycloid .

—A cycloid i s the curve generated by a poin t on

the ci rcumferen ce of a ci rcle rolling on a straight lin e. L et the

circle Of radius 0T,Fig . 101

,be the rolling or generating circle ,

which rolls on the straight line T6’ .

Divide the semi- circumference of the circle up into any con

venien t number Of equal parts as six ,and lay Off on the straight

line T1 ’ = T1 , and

Through 5’and 6

’ draw perpendiculars to T6’,

and through the points 1,2,3, 4 and 5 on the semi- circle draw

lines paral l el to T6’,intersectin g the diameter of the semi- circle

in 1 1 , 2 1 , 0 ,41 , and 51 , and the perpendiculars in 1 4

"5"an d 6

On 1 1 1 from 1 lay Off 1 a = l 1 1 ; on 2 12 from 2” lay Off

2 b= 22 1 , etc . Through the points a,b,c,d,e,and 6” draw a

smooth curve , which will be the cycloid traced by T as the circle

rol ls on the straight line.

102:I

‘

irE'

CHAN I S M

107 . Bpicydo’

idfui

WI

’

ei ‘

e’

picbOid i s the curve traced by a

poin t on the ci rcumferen ce of a ci rcle as i t rolls on the outsi de of a

second ci rcle, ca lled the di rectin g ci rcle. L et 0T,Fig . 102

,be the

rad ius of the rol l ing circle and AT, the radius of the d irecting

c irc l e .

Divide the semi- circumference of the rol l ing circl e into any

convenient number of equal parts as six,and lay Off from T

,

divisions of the same length on the directing circ l e . Through

FIG . 102 .

the points 1’ 6’ just found and through the center

A draw l ines .

W ith A as a center and rad i i A 1 , A2 , A3, A6 draw arcs

intersecting the diameter of the rol l in g circ l e in 1 1 , 2 1 , 31 , 51 ,

and ,the rad ial l ines in 3 On from 1”

l ay Off 1”a = 1 1 1 ; on 2 2 1 , lay Off 2 b= 22 1 ; on 3 31 , lay Off

3"a=331 , etc .

Through the points a,b,c,d,e and 6 (11‘a a smooth cu rve

which w i l l be the epicycloid .

If the rol l ing circl e is larger than the d irect ing circle and rol l ed

i n ternally on it , an ep icycloid wil l sti l l be generated .

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104 M E CHAN ISM

the radius of a circle that rolls on the outside of

and traces the epicycl oid Pm. CP is the radius Of a

FIG . 105.

rolls on the inside of the pitch circ le and traces the hypocycloid P n .

Pm wil l be the face and P n the flank of a gear tooth which is shownshaded .

GEARING 105

1 10. Cycloidal Spur Gears .

—L et AP and BP,Fig . 106

,be

the pitch radii and CP and DP the radii of the rolling circles of

two cycloidal spur gears that are in mesh .

The rol ling circle with its center at C,when rol led on the

inside of the lower pitch circle generates the hypocycloid P n ,and

when rolled on the outside of theupper pitch circl e generates the

epicycloid P r . These curves are the flank Of the lower gear and

FIG . 106 .

the face of the upper gear respectively,and are generated simul

taneously.

The rollin g circle with a center at D when rol led on the inside

Of the upper pitch circle generates the hypocycloid Pm,and when

roll ed on the outside of the lower pitch circle generates the epicy

cloid P s . These curves are respectively the flank and face of the

teeth of B and A ,and are also gen erated simultaneously .

The tooth outlines that are generated simultaneously are the

parts of the teeth that are in contact,which is the face Of on e

tooth with the flank of its mate on the other gear .

106 M E CHAN ISM

If the addendum circ les are as shown and the upper gear is the

driver turning in a counter- c lockwise direction,contact w i l l beg in

at a where the addendum circl e Of the driven cuts the rol l ing circle

of the driver . It wil l fol low along the upper rol l ing c irc le until

the pitch po int is reached . At this point contact wil l change

from the driver ’s flank and fol lower’s face

,to the driver ’s face

and fol lower ’s flank, where it w i l l fol low along the roll ing c irc le of

the driven to the point bwhere the addendum circle Of the driver

cuts the rol l ing circl e of the driven , where contact between that

pair Of teeth w i l l cease .

From the figure it can be seen that the length of the face of thedriver ’s teeth govern s the angle of recess, and the length of the face ofthe driven gear

’

s teeth govern s the ang le of approach.

Thus,if n o ang le of approach i s desi red, the teeth of the driven

gear w i l l have n o faces , and the driver’

s teeth wi ll requ i re noflanks .

An example of th is is sometimes found in the rack and '

pin ion on

planer beds .

It w i l l also be noted that if the d iameters of the rol l ing c irc les

are made equal to the pitch radii of their respective gears,that

the flanks of the teeth wil l be radial .

1 1 1 . Cycloidal Rack an d P inion .

— I n the rack and pinion

combination shown in Fig . 107 let AP be the radius of the pitch

circle of the pinion,and C and D the centers Of the roll ing circles .

The upper rol l ing circl e when rol l ed on the inside Of the pitch

circ l e of the pinion generates the hypocycloid Pm,the flank of the

pinion teeth,and when rol l ed on the pitch l ine of the rack gen

erates the cyclo id P s , the face of the rack teeth . The lower rol l ing

circ le when rol led on the outside of the pinion ’s pitch circl e gen

erates the epicycloid P r , the face Of the pinion teeth , and generates

the flank of the rack teeth when roll ed on the lower side of the

pitch l ine of the rack , which is the cyclo id P n .

If the pin ion drives clockwise as shown,contact w i l l beg in at

a on the rol l ing circle of the pinion,and cease at b on the rol l ing

c irc le of the rack .

1 12 . Cycloidal Rack Teeth with S traight F1anks .

— I n order

to have straight flanks on the rack teeth,the rol li ng ci rcle of the rack

wi l l be of i nfi n i te di ameter and wi ll coi n cide wi th the pi tch lin e. The

faces of the pinion teeth,which are developed by this circle , wil l

therefore be involutes . The disadvantage Of this form of tooth is

that al l Of the wear on the rack teeth is between the p itch and

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108 M E CHAN ISM

equal to the diameter of the directing circle . On account Of this

fact,the difference between the pitch circles of a cycl oidal annular

gear and pinion combination must be equal to the sum of the

diameters of the rol l ing circles or else there wil l be what is cal l ed

secondary contact between the teeth .

1

If the pitch circles differ by the sum of the diameters of the

rolli ng circl es , and the gears are standard , interchangeable gears ,they will differ by twelve teeth .

Circles

FIG . 108 .

P rovided one of the gears have no faces on its teeth,they need

differ by but six teeth ?

115. In terchangeable Cycloidal Gears — I n the different pairs

of cycloidal gears thus far considered , the rolling circles have

not been taken of the same diameter, and it is possible to lay out

the teeth of two gears having the same pitch that will mesh prop

erly together even though their rolling circles are not the same

size . If , however , the gears are to be interchangeable , that is , any

gear mesh with any other of the set , they must , besides having the

same pitch,have their tooth curves developed by the same size

1 For a fu l l discussion of this property of the cycloidal annu lar gear andpinion

,see P rofessor M acCord

’

s“ Kinematics

,

” also a “ Treatise on GearWheels,

” by George B . Grant, M . E .

2The reason for this is discussed in Art . 1 16,

GEARING 109

rolling circles . In Fig . 109 if the gears,A

,B and C are all of the

same pitch , and their tooth curves developed by rolling circles of

the diameter shown , B will mesh properly with A and C,but A

and C wil l not mesh with each other .

1 16. S tandard Diameter of Rolling Circle for Cycloidal

Gears — The standard diameter of rolli n g ci rcle i s one that wi ll

give radi al flanks on a gear of twelve teeth. I n order to have radi al

flanks on the teeth,the diameter of the rolli ng ci rcle must be one-half

that of the pi tch ci rcle. (Art

This does not mean that all the gears of the set will have radial

flanks,but on lythe 12- tooth gear .

For example,the standard diameter rolling circle for a 64- tooth

4- pitch gear would be the same as that for a 12- tooth of the same

FIG . 109 .

pitch,or l f in . pitch diameter

,and in .

= 1% in . diameter of

rol ling circle .

Geo. B . Grant says 1 The standard adopted by the manu

facturers Of cycloidal gear cutters is that having radial flanks on

a gear Of fifteen teeth,but is not and should not be in use for

other purposes . If any change is made,it should be in the other

direction, to make the set take in gears Of ten teeth .

It must be borne in mind that the standard adopted does not

limit the set to the stated minimum number of teeth , but that it

simply requires that the smaller gears have weak under curved

teeth .

1 “ Treatise on Gear Wheels , page 41 .

1 10 M E CHAN ISM

1 17 . Comparison of Involute and Cycloida l Systems — I n thetwo systems

,the advantages are practical ly al l in favor Of the

involute tooth . Some of them are as fol lows:1 . The distance between the centers can be changed w ithout

affecting the velocity ratio:2 . The path of contact is a straight l ine

,thus keeping a

constant pressu re on the axes .

3. Tooth curves of single curvature .

4. Involute rack teeth have straight s ides,thus cutter is

easily made .

5. Wear on involute teeth more uniform than in cyc loidal .

6 . L ess cutters requ ired to cut a complete set of gears .

The cyc loidal system is the older of the two,and for those

using it,the expense of changing over to the involute system

would be large,but the main obj ection would be that for repa irs

on machines al ready equipped w ith cycloidal gears,cutters of

that system must be kept on hand .

P roblems

/1 . Construct an ep icycloid for a 3- in . generating circle and an 8- in .

d irecting circle .

Construct hypocycloids using a 12- in . di recting circle and generatingcircles of 4 in . and 7 in . respectively .

3. I n a cycloidal spur gear combination of unequal size,assume the

diameters of the pitch,addendum and rol l ing circles

,an d Show the path of

contact for the small gear driving clockw ise .

4. I n a cycloidal rack and pinion combination w ith the pinion drivingcounter-clockw ise

,assume the diameters of pitch

,addendum and ro l l ing

circles so that there w i l l be no angle Of recess , and show path of contact .

5. What are the standard diameters of rol l ing circles that wou ld beused to generate the tooth outl ines for each of the fol low ing gears?

1 . 28 teeth 2 pitch 4 . 86 teeth 4 pitch .

2 . 49 teeth 2 pitch . 5. 4 pitch 20 in . pitch diameter .

3. 100 teeth 12 % in . pitch d iameter . 6 . 50 teeth 1 in . pitch .

6. Draw the correct tooth curves for two spur gears that are in mesh ,cycloidal system . Cut temp lets to lay out the tooth curves and checkthem for contact . Show the teeth going into and out of action .

Data:N umber of teeth in gears 36 and 2 1 .

Diametral pitchD iameter of ro l l ing circles 4 in .

Thickness of rim below tooth bottoms,

C ircu lar p l tCh(=g)2

Smal l gear to drive and revo lve clockw ise .

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1 12 M E CHAN ISM

small number of pins can be used on a gear to correspond to a

smal l number of teeth,but they have not the disadvantage of

having weak teeth . Fig . 1 10 shows a pin wheel of six teeth

meshing with a toothed gear . The dotted outl ine shows the

teeth Of gear if the pins had no diameter . The circular pitch must

of course be the same in each gear,or the arc P 1 ’ must equal the

arc P 1 . The maximum radius Of addendum circle is found by

drawing a stra ight l ine P 1 and find ing where it cuts the circum

ference of the p in at a . The rad iu s of the addendum circl e wil l

be from the center Of the gear to a .

FIG . 1 10 .

Sometimes rolls are placed on the pins in order to make the

action between the teeth more nearly a rol ling one. In this case

the pins are secured between two circular plates , and the gear is

then known as a lanwrn wheel . This type is common in watch

and clock mechan isms , except that no rol l s are used .

The pins shou ld always be on the driven wheel since then the

action between the teeth is almost all on the recess side . It wi l l

be wholly so if the pins have no diameter .

P in gears are not confined to spu r gears , but can be used in

any combination,and the pins can be on e ither gear , as a p in

annular gear and a toothed pinion .

GEARING 113

STEPPED AND H ELICAL GEARS

119. S tepped and H elical Gems — Smoother action can be

Obtained in a pair of gears by having more teeth in contact at

one t ime . This can be done by lengthening the teeth or by mak

ing the pitch smal ler . By lengthening the teeth,the angle of

action is made greater and since more teeth are brought into con

tact at one time,the pressure on each tooth is decreased

,but gear

teeth are assumed to act as cantilever beams and the teeth are

lengthened at a greater rate than the number of pairs of teeth in

contact is increased, so that the resulting gear is weaker .

By decreasing the pitch , the thickness of the tooth is decreased ,and this is at a greater rate than the pairs of teeth are increased

,

so again the gear is made weaker .

M ore pairs of teeth can be brought into contact at on e time

by placing several gears side by side on the same shaft,and advan c

ing the teeth of each ahead of the on e next to it,by the circular

pitch divided by the number of gears,and then fastening them

secu rely together . Such a gear is called a stepped gear .

If the number of steps is increased infinitely,and the teeth on

each placed ith of the circu l ar pitch ahead Of the one next to it,

where n equals the number Of steps , a line drawn through the

center of the top Of each tooth wil l be a helix , and such gears are

cal led heli cal , spi ral,1or twi sted gears . A pair of such gears are

shown conventionally in Fig . 1 1 1 .

If the power to be transmitted is great , there will be con sider

able end thrust,that is

,one gear wil l tend to slide by the other

along its shaft . This tendency is overcome by placing on the

shaft gears of the same size and Of equal but opposite helix angles .

These gears are represented conventional ly in Fig . 1 12 , and are

known as herring-bon e gears .

There has recently been introduced into this country a systemOf herring- bone gears known as the Wuest gear , named after its

1 The term spiral , as appl ied to gears , suggests a gear the teeth Of whichl ie in a plane

,but the word is used to denote gears the teeth of which have

hel ical form,and while it has common usage the terms

,tw isted or hel ical

wou ld perhaps convey a better idea of the gear . This is suggested byM r . R. E . Flanders in his book on

“Gear Cu tting Machinery .

”

1 14 M E CHAN ISM

inventor . The gear is cut from a sol id piece and the teeth of one

half are advanced one half the circu lar pitch ahead of the other.

1

APPROX IMATE M ETHODS or L AYING OUT TOOTH CURVES

120. Approximate M ethods of L aying Out Tooth Curves .

There are a number Of methods of laying out tooth curves by

u sing templets,rectangular coOrdinates and c ircu l ar arcs . It

is not necessary to use these methods as mu ch s ince gear cut

ting mach inery has been brought to su ch a h igh state of develop

FIG . 1 1 1 . FIG . 1 12 .

ment as it was when the pattern maker or mi l lw r ight had to

lay out the teeth of gears that were usual ly cast .

The Rob inson Templet Odontograph is one of the best knowntemplet methods for laying out tooth cu rves . The Odontograph ,together with the method of forming the s ide of a tooth

,is

i l lustrated in Fig . 1 13. It is necessary to have a table to set

the instrument properly .

The curves of the sides of the templet are logarithmic sp iral s ,one being the evolute of the other .

The holes in the templet are for the purpose of fastening it to a

1 See article on “ The H erring-bone Gear,

” by P . C . Day. Tron saction s

of the Ameri can S oci ety of M echan i ca l E n gi n eers, Vol . 33, page 68 1 .

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116 ME CHAN ISM

TABLE D .

— GRANT ’S INVOLUTE ODONTOGRAPH (STANDARD INTERCHAN GEABLE

TEETH )

Centers on Base Line

M ULTIPLY BY THE CIRCULARPITCH

DIVIDE BY THE D IAMETRAL PITCH

Teeth

Face radius Flank radius Face radius Flank radius

GEARING 117

GEAR CUTTING

121 . Cutting Spur and Ann ular Gears — The general method

for cutting these gears is to use a tool,the outline of the cutting

face Of wh ich is the same as the space between two teeth. Fig .

1 16 shows a tool that can be used in a planer or Shaper for cut

ting spu r and annular gears . The profi le abcd is the shape of the

space between the teeth . The tool can be sharpened across thefront face without changing the profi le

.

Fig . 1 17 shows a rotary cutter for use in a mil l ing machine.

This also has the profi l e abcd the shape Of the space between

the teeth , and the face can be sharpened paral le l to the axis

FIG . 1 16 . FIG . 1 17 .

without changing the profile . This cutter cannot be used asreadily for annular gears as that of Fig . 1 16.

122. Interchangeable Gear Cutters — I n order to use ei ther ofthe two cutters just described to cu t a gear theoreti ca lly correct

, i t i s

necessary to have a difi eren t cutter for each difi eren t n umber ofteeth and pi tch, sin ce the space between the teeth changes wi th each

n umber of teeth. Therefore to cut a complete set of gears of any

one pi tch, from twelve teeth to a rack theoreti ca l ly correct, i t wi l l

take as many dig eren t cutters as there are gears .

It has been found , however , that one cutter can be used tocut several different numbers of teeth of any one pitch

,accurate

1 18 M E CHAN ISM

enough for practical purposes , as in the larger gears the shape ofthe space does not change very much by the addition Of afew teeth .

The following TABLE E shows the cutters used by the BrownSharpe M fg . CO . for gear teeth .

TABLE E

INVOLUTE CUTTERS, CYCLO IDAL CUTTERS,

8 CUTTERS IN EACH SET 24 CUTTERS IN EACH SET

Cutter Teeth Cutter Teeth Cu tter Teeth

In order that standard involute gear teeth may be more accu

rately cut , a set Of fifteen cutters has been introduced , the inter

mediate cutters being designated by half numbers .

TABLE F .

—INVOLUTE CUTTERS,15 CUTTERS TO E ACH SE T

Cutter Teeth Cutter Teeth

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120 M E CHAN ISM

All of the gears cut with this tooth w i l l mesh properly with itand wil l also mesh correctly with each other .

Fig . 1 19 shows a complete gear for a cutter instead of a s ingle

tooth . The gear has cutting edges the same as the single tooth,

and is first fed in to the proper depth,that is

,so that the pitch

circ les are tangent , and then both rotate slightly before beginning

each cutting stroke . Several teeth are in the process Of being

cut at on e t ime , although but one tooth is completed at a t ime .

124. Fel lows Gear Shaper .

— This machine,i l lustrated in

Fig . 120, uses a cutter shaped l ike a pinion , and works on the mold

ing- generating principl e .

FIG . 1 19 .

The cutter and blank are fastened to vertical axes which are

connected so that when one rotates the other is rotated the

proper amount,the two moving so that the ir pitch circ les rol l

together without sl ipping . In addition to this,the spindle

carrying the cutter can reciprocate in the direction along its

axis,so that when ascend ing the teeth of the cutter act as cutting

tools on the blank in the same way as the cutting tool of a shaper .

The cutter spindle , is carried on a cross- rail S O that the ax is of

the cutter can be moved toward that of the blank .

The action of the machine is as fol lows:The cutter A and

blank B of Fig . 12OA are keyed to the ir respective axes,with

ne ither the cutter nor the blank rotating .

During the up- stroke

,the teeth of the cut ter cut metal from

the blank,and on the down- stroke the cutter- spindle is fed

toward the blank axis a smal l amount .

GEARING

FIG. 120.

FIG. 120A.

122 M E CHAN ISM

This feeding in motion is continued at the end of each non

cutt ing stroke unt il the pitch circles are tangent and then ceases .

Helix An g l

FIG . 12 1 .

In place of thi s , at the end of each non -cutting stroke,both the

cutter and blank axes rotate slightly through corresponding

ang l es so that their pitch circles roll together without slipping .

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124 M E CHAN ISM

Fig . 12 1 illustrates the principle , but instead Of a hob,an ordi

nary worm is shown . In a section of the worm along its axis the

teeth have straight sides , the same as involute rack teeth,the

rack being shown in dotted outline .

The teeth of the worm or hob,being of helical form

,l ike a

screw thread,its axis, in order to cut teeth on the blank paral l el

with the axis of the blank,cannot be perpendicular to the blank

,

but must be at an angle Of 90° minus the hel ix angle of the worm .

Fig . 122 shows a gear hobbing attachment on a universal mil l

ing machine,in which the hob is placed on the arbor of the

machine with the hob and blank driven at the desired velocityratio .

The hob is slowly fed across the face of the blank , and when

it has gone all the way across the work is completed .

CHAPTER VII I

BEVE L GEARS, WORM AND WORM WHE E L

126. Bevel Gears .

— I n all of the gears thus far considered,

the elements of the teeth were parallel to each other,and with

the exception of the helical gears,the elements of the teeth were

parallel to the axis of the gear,the tooth outlines being generated

FIG . 123.

by a right line which was the elemen t of a flexible band as in the

involute system,or the element Of a roll ing cylinder as in the

cycloidal system .

In bevel gears,the axes intersect

,the pitch cylinders becom

ing pitch cones,and the elements of the teeth converging at the

po int of intersection Of the shafts .

An idea Of this can be had if it is imagined that alternate

grooves and proj ections are placed on the friction cones of Fig . 84

as was done in Figs . 87 and 88. The cones will then look as in

Fig . 123,which shows a pair of bevel gears .

125

126 M E CHAN IS M

The pitch cones are segments Of a sphere, and in order to

lay out the gears theoretical ly correct , the tooth outl ines need

to be laid out on the surface Of the sphere ,making use of spherical

trigonometry . With the exception of this difference,the bevel

gear differs little in theory f rom the ordinary spur gear .

Bevel gears are always laid out in pairs,and are not in ter

changeable as are gears for paral lel axes .

127 . Ve locity Ratio .

— The velocity ratio for bevel gears is

inversely as the rad i i of the bases of their pitch cones,and the

method Of l aying out a pan Of cones for any desired velocity

ratio was discussed in Art . 88 .

128. M iter Gears .

—These are bevel gears that are of thesame size and in which the shaft angle is The velocity ratio

in a pair of miter gears is therefore as 1 1 .

129. Crown Gears .

—A crown gear is one in which the sides

of the pitch cone make an angle Of 180° with each other . The

base Of the pitch cone is therefore a great circle of the sphere

on which the teeth Of the crown gear is developed . The crown

gear is sometimes cal led the rack of bevel gears .

The involute crown gear has not , however, teeth with straight

s ides as in the involute rack Of spur gears . The crown gear with

straight sided teeth is

known as the octoi d tooth ,

so cal led from the peculiar

figure eight line of action .

This tooth is the invention

of M r . Hugo B ilgram of

P hiladelphia,

and will be

referred to again under

Cutting Bevel - gear Teeth .

130. Spiral Bevel Gears—The Spiral bevel gear has

recently been introduced

by the G l eason Gear

Works , hav ing been deFI G 124 veloped primarily for auto

mob ile drives , but is ap

plicable to any high- speed machinery where quiet running is

desired .

To get the best resu l ts , the angle Of the spiral shou l d be such

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128 M E CHAN ISM

the cones can be rol led back into their original pos itions and theteeth drawn on the p itch cones .

It w i l l be noticed that that part Of the back cones on which the

teeth are la id out d iffers very l ittle from the Spher ical su rface .

Fig . 127 shows one of these cones laid out w ith half of the

teeth drawn in . The teeth in th is figure appear in their true

th ickness I n the plan v iew,but are foreshortened .

FIG . 126 .

The method of constru ction is c learly shown in the figure so

that there is no need of further explanat ion .

132 . Shop Drawing of Bevel Gears — I n making a drawing

for use in the shop,it is not necessary to go into the detai l of

drawing in the tooth curves as in the previous artic le,but a much

more simple drawing together with the necessary angles and

dimensions is all that is required .

BE VE L GEARS, WORM AND WORM WHE E L

P roblem — M ake a complete working drawing of a pair of bevel

gears Of 26 and 20 teeth ; diametral pitch 4 ; involute system ;shaft angle width Of face 1% in . ; diameter of shafts 1 in .

diameter Of hubs 21 in .

M ake axis Of large gear horizontal .

26 teeth,4 pitch = 6% in . pitch diameter .

20 teeth,4 pitch=5 in . pitch diameter .

FIG . 127 .

Draw the axes of the gears 0A and 0B,Fig . 128

,making an

angle of 90° with each other . From 0 on 0A lay Off 2% in the

pitch radius Of the smal l gear and draw CP perpendicular to CA .

On 0B from 0 lay Off 31 in .

,the pitch radius of the large gear and

draw P D perpendicu l ar to 0B . Complete the pitch cones P OC

and P OD and draw the back cones,or at least that part of them

where the teeth are to be laid out .

The angles E and E ’ are cal led the pitch,or center angles .

From the table of tooth parts,page 94, find the addendum

and dedendum Of the teeth and lay them out from C ,P and D

as shown . From the points thus found,draw toward the apex

of the pitch cones and lay off the face of the gears .

130 M E CHAN ISM

The angles that the addendum and dedendum lines make

with the axes Of the gears, g ive the face and cutting angles F

and C respect ively .

The angle of the edge , is T,and is the same as the center or

pitch angle .

The angles U and V in the smal l figure are cal le‘d,respectively

,

the ang l e increment and ang l e decrement . These are the differ

FIG . 128 .

en ces between the p itch ang l e and the face and cutting angles

respect ively .

When the axes of the gears are at right angles , the various

ang l es and the outside diameter can be found as fol lows:In the smal l figure ,L et Wc= pitch rad ius of smal l gear

L et 0W=pitch rad ius of large gear

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132 M E CHAN ISM

the gear is parallel w ith the pitch l ine,or the face Of the cone

instead of being drawn to the apex of the pitch cone is drawn

paral le l with the pitch cone . In cutting the teeth on such a

blank their depth is made constant .

M r . C . H . L ogue says;1 “The mil l ing of bevel gears is now

pract ical ly a thing of the past,except for an occas ional j ob

required for shop use,the generating Of bevel gears having

reached su ch a point that the mil l ing machine cannot compete

either in quality or time .

”

134. Accurately Cut Bevel- gear Teeth .

— Fig . 129 i l lustrates

diagrammatically a method of cutting the teeth of bevel gears in

which a former or templet is u sed .

The templet is made the shape Of the teeth that are to be cut,

and being farther from the apex Of the pitch cone , is made to a

larger scale than the tooth outline . The cutting tool l is c arried

on a frame,the back end Of which is supported by a rol l resting

on the templet .

The cutt ing po int of the tool moves w ith a rec iproca ting

mot ion along the l ine 0A,wh ich is tangent to the templet and

rol l,and passes through the apex Of the p itch cone .

Another method of guiding the rear end of the cutting tool

frame is to move it in a circular are by means of a l ink,instead

Of by using a templet . This is cal led the Odontograph method .

1 American Machinist Gear Book,page 153.

BE VE L GEARS , WORM AND WORM WHE E L 133

135. Bilgram Bevel- gear P lan er .

-This machine bears the

same relation to the bevel gear that the Fellows gear planer does

to the spur gear,in that the teeth are generated . In Art . 123

was discussed a method of making interchangeable gears by

means of a rack tooth,and al l gears that would mesh correctly

with the rack wou l d also mesh correctly w ith each other . In

the B i lg ram bevel - gear planer,the cutting tool corresponds to

the straight- s ided tooth of an octoid crown gear .

FIG . 130 .

The Space between the teeth of bevel gears being of varying

width , however, it is not possible to use a tool the shape of Fig .

1 18,but only the right or left half of it .

The planer i l lu strated in Fig . 130 is automat ic,and consists of

two principal parts , the Shaper wh ich holds and operates thetool

,and the evolver which holds and moves the blank .

The blank must move as a rolling cone,which is done first by

134 M E CHAN ISM

securing its arbor in an incl ined position to a semi- c ircular

hor izontal plate between two uprights . This semi- circular plate

can be oscil lated on a vertical axis passing through the apex of

the blank,and is driven by a worm and worm-wheel combination .

The second part Of the motion is obtained by means of two steel

bands stretched in Opposite directions,one end Of each being

fastened to the frame and the other ends wrapped around a

cone,which corresponds to the pitch cone of the blank .

The tool which is operated by a Whitworth quick- return mo

tion mechanism,cuts on the forward stroke

,and during each return

stroke the tool is raised clear of the blank by means of a cam,

and the blank rotated on e tooth by means of gearing and index

mechanism . All spaces are treated in the same manner,after

which the tool automatical ly adj usts itself for the second cut on

a tooth , and so on until the blank is completed . A sim i l ar tool

is used to finish the other Sides of the teeth .

The inc l ination of the arbor that holds the blank is made

adjustable in order to adapt it to the angle of the desired gear .

This adj ustment must be exactly concentric with the apex of

the blank . Ordinarily a different sized rol l ing cone would have

to be used to cut each different s ized gear,but means have been

devised whereby a limited number of cones are sufficient .

1

P roblems

1 . Make a finished penci l or ink shop draw ing Of a pair Of bevel gears of36 teeth and 27 teeth respectively , 3 pitch and 15

° angle Of ob l iqu i ty . Faceof gears 21 in .

NOTE .

— P ut on al l of dimensions and angles shown in Fig . 128 . N 0 statement of prob lem to go on sheet, but this data. BEVEL GEARS .

Teeth 36 and 27 .

P itch 3 diametral .Invo lute system .

Time, 4 hours .

2 . Make a sketch show ing how you wou ld fi nd the pitch cones for a pairof bevel gears in which the driver made five revo lutions to seven of thedriven

,w ith a 1 15° shaft angle .

3. M ake a finished pencil draw ing of a miter gear of 20 teeth , 2 diametralpitch

,and 15

° angle of ob l iqu ity . Face of gear 2 7} in .

N OTE .

—Make views of gear similar to gear shown by Fig . 127 . P lacegear in center of the sheet top an d bottom ; the point A i in . to right ofleft border l ine and O ’

6 in . left of right border l ine . Time,4 hours .

1 For a more complete description of this machine, see Ameri can M achin i st

of M ay 9 , 1885, and January 23, 1902 .

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136 M E CHAN IS M

while if it were a double- thread worm,the worm wheel would be

rotated two teeth and so on . For example,with a single- thread

worm and a 48- tooth wheel,

it wou ld requ ire 48 revolu

t ions Of the worm to rotate

the wheel once . If the worm

were of double—thread,it wou l d

require 24 revolutions of the

worm for one of the wheel,

while if a triple- thread worm

were u sed 16 revolutions of the

worm wou l d be necessary for

one of the wheel .

Thus it wil l be seen that the

pitch diameters do not enter

into the question .

If the p itch and number Of

teeth of the wheel are g iven ,FIG . 131 . and also the distance between

the shaft centers,the pitch

d iameter Of the worm shou ld be su ch that the two pitch circles

are tangent .

137 . P itch .

—The circular pitch system is ordinari ly used in

making worm and worm-wheel cal culat ions,for the reason that

the worms are usual ly cut in the lathe the same as screw threads,

and lathes are not equipped with the proper change gears for

cutting diametral pitches . Diametral pitch may be used to cut

a worm in the lathe if the proper change Of gears are prov ided .

The proport ion is as fol lows:1

S tud gear

L ead screw gear22

7 x pitch of l ead screw X diam . pitch of worm to be cut

E xample — TO cut a worm of 12 diametral pitch on a l athe

havin g a lead screw 6 threads to the inch .

S tud gear 22 1 1

L ead screw gear 7 X 12 X t 7

1 Treatise on Gearing,by G . B . Grant .

BE VE L GEARS ,WORM AND WORM WHE E L 137

Any change Of gears in the proportion Of 1 1 and 7 will cut the

1worm With an error Of

107000

of an Inch to the thread of the worm .

The Obj ection to the c ircular pitch system for spur gears,

that the distance between the shaft centers wil l be an in con

ven ient fraction unless the circular pitch is inconvenient,does

not apply in th is case however .

FIG . 132 .

138. System of Teeth Used.

—The involute system is used

in worm gearing for the reason that the worm wil l have teeth

of straight sides which are eas i ly cut in the lathe .

Fig . 132 shows two views Of a worm and worm wheel,the left

hand view being a section along the axis of the worm wheel and

the right- hand view a section along the axis of the worm . From

138 M E CHAN IS M

these views it can be seen that the section of the worm in the right

hand view is the same as an involute rack and the section of the

wheel (not whol ly cross- hatched) , in the same view is the same asthat of a Spur gear having the same pitch diameter and number of

teeth as the wheel . The throat diameter of the wheel is the same

as the outside diameter Of a spur gear having the same pitch

and number of teeth . The dimension E on the figure represents

the diameter of the hob that would be used to cut the teeth on

the worm wheel .

139. Tooth Contact — Contact between the worm and wheel

teeth is not very definitely determined but it is general ly bel ieved

to be nearly line contact between any pair of teeth for smal l gears,

although it may have some width,and becomes surface contact

for the larger gears . TO get as much contact as possible,the

worm wheel is constructed so that its sides partial ly envelop the

worm . The included angle between the sides is general ly made

60 degrees or 90 degrees .

140 . L ength of Worm .— The worm is not l imited in length but

should be long enough to provide al l of the contact that can be

Obtained between any pair of teeth . It is Often made longer

than necessary and moved along the shaft when it becomes worn ,to bring a new part of the worm in contact with the wheel teeth .

The curve1 shown in Fig . 133gives the length of worm required

to get the maximum contact when M al

i-degree angle of obliquity

is used . Contact will not be along the whole length of the worm

on its axial section,but on the sides as the wheel partial ly

envelops the worm .

141 . Cutting the Worm Wheel Teeth .

— A worm is first made

in the lathe,the same as the worm that is to mesh with the wheel

,

except that its diameter is twice the clearance greater, then its

sides are fluted similar to a reamer or tap in order to form

cutting edges . This hob and the wheel blank are then mounted

in a milling machine,the hob being fed down into the blank

until their pitch lines are tangent,then both are turned at the

proper velocity ratio . The only difference between hobbing the

worm wheel and the spur gear shown in Fig . 122 is that in the

worm wheel the axis Of the hob is directly over the center l ine

Of the blank, and does not move across it .

1 From data of Browne Sharpe M anufacturing Company .

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140 IVI E CHAN I SM

of the crossed l inks is at the tangent point P , of the ellipses .

If one of the ellipses is held stationary,the other can be rol led

with pure- rol l ing contact around it .

143. Velocity Ratio .

— S ince the two el l ipses are the same

size,the velocity ratio wil l be as 1 :1

,but at any part Of the

revolution the ratio is inversely as the radi i from the fixed foci to

the tangent point P,Of the el l ipses .

Any system of teeth that is practicable for spur gears can

be used for el l iptical gears and the teeth if out with the same

cutter wi l l al l have different profiles . This may resu lt in weak

under- cut teeth at and near the ends of the el l ipses,but can be

remed ied by using one cutter for the ends where the el l ipse has a

smal l rad ius,and another for the sides where the radius is larger .

The blanks are usual ly cut by clamping them together and cuttingboth at one Operation . This insures the mat ing teeth be ing the

same .

E l l iptical gears can be‘

used for quick- retu rn motions such

as slotters and shapers,where al l of the work is done on half

of the revolution,the other half being u sed to get the tool out

of the way and the work ready for the next stroke .

They are not in very general use as they are hard to cut

without the use of spec ial attachments .

P roblems

1 . Make a ful l- size working draw ing of a worm and worm-whee l combination . Involute system .

P roblem 1 .

BE VE L GEARS , WORM AND WORM WH E EL 141

Data:Velocity ratio 32 1 .

Single thread right-hand worm .

C ircu lar pitch g in .

N OTE .

— Mal"e teeth in contact at pitch point . Draw tooth outl ines ofwhee l teeth by means Of Grant ’s Involute Odontograph table

,page 1 16 .

Develop hel ix for outl ine Of worm teeth . No statement of prob lem,but

these data

WORM AND WORM WHEEL .

Teeth in wheel 32 .

Single thread R. H . worm .

C ircu lar pitch in .

Invo lute system .

Time,4 hours .

2 . M ake a fu l l- size working draw ing of a worm and worm wheel combination . Involute system .

Data:Velocity ratio 16 1 .

Doub le thread right- hand worm .

C ircular pitch in .

N OTE — Same as for P rob lem 1 .

3. In a worm an d worm wheel combination the number of teeth in theworm is 80, the velocity ratio 40 1 ; the circu lar pitch is 1 in . ; the ou tsidediameter of the worm is 6 in . What is the distance between centers?

4. M ake a ful l- size working draw ing of a worm and worm wheel combination . Involute system .

Data:Outside diameter of worm 5 in .

Length Of w orm 6 in .

Diameter Of worm shaft 2% in .

Distan ce'

between centers 9 in .

Diametral pitch 2 .

Angle of obl iqu ityTr iple thread R. H . worm .

N OTE — P lace draw ing on the center l ine of the sheet right and left,and the

center l ine of the worm shaft 31 in . down from top . P enci l draw ing show ingall dimensions . No statement of the prob lem . Time 4 hours .

CHAPTER IX

GE AR TRAINS

144. Gear Train s .

— When more than two gears are in mesh,

such a combination i s a trai n of mechan i sm (Art . and i s called

a gear train or train of gears.

It is often desirable to find the revolutions of the last gear of

a train without finding the revolutions of each intermediate gear .

The number of teeth on any two gears that mesh together is

proportional to their respective pitch diameters,and since the

revolutions are inversely proportional to the pitch diameters

(Art . therefore,their revolutions are also inversely pro

portion al to the number Of teeth on the gears .

145. Value of a Train .—By this expression is meant the ratio

of the revolutions of the first and last gear of the train in a given

time .

In Fig . 135 let A,B

, C,D

,E

,and E

'

be the pitch circles of

gears on the shafts,1,2,3,4 and 5

,and having the number of

teeth as shown .

FIG . 135.

The gear A drives B ,B drives C

,and since C and D are on

the same shaft they wil l rotate as one gear . D drives E and E

drives F.

Revolutions of B 80 Revolutions of C 20

Revolutions of A 20’ Revolutions Of B 70

Revolutions of E 25 Revolutions of F 40

Revolutions of D 40’ Revolutions Of E 50

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144 M E CHAN IS M

although they are on the same shaft,s ince the number of teeth

on them is not the same .

In spur gears when there is an Odd number Of intermediate

axes between the first and last,the d irection Of rotat ion of the

first and last axes wi l l be the same,wh i le if there is an even num

ber Of intermediate axes,the direction of rotat ion w i l l be opposite .

If there is one annular gear in the train , it changes the direct ion of rotation of the last gear from what it wou l d be otherw ise .

Thu s in the last problem worked out there are fou r intermed iate

axes and on e annular gear,so that the direction of rotation of the

first and last axes w i l l be the same .

147 . Gear Train for Thread Cutting — Fig . 136 i l lustrates the

gearing found on an ordinary eng ine lathe that is equipped forcutt ing screw threads . On the Sp indle E

,are the gears F

, C and

D and the stepped cone . The gear D is keyed to the spindle and

the stepped cone is free to revolve on the sp indle except when it

is locked to D by means Of a pin .

The gear C is fastened to the stepped cone and revolves w ith it .

A and B are the back gears and are keyed to a shaft that is paral le l

to the spindle . This shaft has eccentric bear ings so that the

back gears may be thrown out of mesh w ith C and D .

When the back gears are “ thrown out,D and the stepped

cone are locked together,and as many speeds can be obtained as

there are steps on the cone , wh i l e w ith the back gears“ thrown

in,

” D and the stepped cone are not locked and the number of

speeds of the spindle is doubled .

F is the spindle gear and is for driving the thread- cutting train .

G and H are tumbl ing gears or tumblers , and are for the purpose

of changing the direction of rotation of the screw M .

Gmeshes with I ,which is the ins ide stud gear, and the arrange

ment is such that F and H may be thrown out Of mesh and G be

made to mesh directly with F. F and I are general ly made the

same size,but when they are not the same , I is usual ly made

twice the size Of F, so that I makes on e- half as many revolutions

as the spindle .

On the same shaft with I is the outs ide stud gear J . M eshing

with J is an“

intermediate K which meshes with the screw gear L

on the end Of the lead screw M . The only gears that it is n ecessary to change in thread cutting are the stud gear J and the screw

gear L . The intermediate K is held on a slotted bracket,not

GEAR TRAIN S 145

shown in the figure,which allows K to be adj usted to accommo

date different Sized gears J and L .

The cutting tool is held in a tool holder which is moved back

and forth along the work by means Of a spl it nut engag ing the

lead screw .

If there are eight threads per inch on the lead screw , then eight

revolutions of the lead screw would advance the cutting tool

one inch,and if the work made four revolutions to eight Of the

lead screw,four threads per inch would be cut on the work .

Dead Cen ter

B ack Gea rs

Con e Pu l ley

S pi n d l embler

Lead S crew

Lead S c rew Gea r

FIG . 136 .

To find the number of teeth in the stud and screw gears when

F and I are of the same size . S ince the gears G,H and K are

idlers they need not enter into the cal cu lations,and since the

spindle and stud shafts make the some number Of revolut ions it

is only necessary to find the revolutions Of the stud shaft.

L et N = number Of threads per inch to be cut on work .

L et n = number of threads per inch on lead screw .

L et T= number of teeth on screw gear .

L et t= number Of teeth on stud gear .

146 M E CHANISM

Then

Revolutions Of stud shaft number of tee th on screw gear

Revolutions of lead screw number of teeth on stud gear

and . T

The following table Of change gears was taken from the plate

on a 16 in . swing lathe,having six threads per inch on the lead

screw and twice as many teeth on the inside stud gear as on the

spindle gear .

TABLE G

Threads Teeth on Teeth on Threads Teeth on Teeth on

to be cut stud gear screw gear to be cut stud gear screw gear

TO obtain a greater range Of threads without add ing more

change gears,the intermediate gear K

, on Fig . 136, might be

compounded, that is, replaced by two gears of different sizes, and

fastened together,one Of them meshing with the stud gear and

the other with the screw gear .

The involute system is used for change gear sets for in th is

system the center distance can be varied without affecting the

velocity ratio .

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CHAPTER x

BE LTING

148. Belts — Belts are flexible connectors and are used for

connecting shafts where the distance is too great for gears,or

where an absolutely constant velocity ratio is not required .

Belting may be divided into two general c lasses,flat and round ;

the former is used on pu l l eys with faces that are cyl indrical ornearly so

,and the latter which is usual ly either of fiber or metal

,

requiring pull eys with grooves or flanges to keep the mm fromrunning Off the pul ley or drum .

149. Velocity Ratio and Directional Relation .

—When two

pul leys are connected by an inelast ic belt of no appreciable thick

ness,all parts of the belt and the surfaces of the pu l l eys have the

same l inear velocity,if there be no sl ipping and therefore their

revolutions are inversely proportional to their rad ii .

This cond it ion is not attained in practice on account of the

elasticity and thickn ess Of the belt . In order to be accurate the

thickness of the belt shou ld be added to the d iameters of

the pull eys in making calcu lations . In th is d iscuss ion,where the

diameter of the pul ley is referred to , it means diameter of pul ley

plus thickness of belt .

When the velocity rat io between two shafts is large , instead ofmaking a single reduct ion

,it may be made in one or more steps

by using a coun ter shaft or jack shaft. This is an intermed iate

shaft on which there are two pu l leys of d ifferent diameters . Thus

if the vel oc ity ratio between two shafts were 1 :40, the ratiom ight be d iv ided up 1 5 and 1 8 .

The method Of connection shown in Fig . 137 is an open belt ,while that shown in Fig . 138 is a crossed belt .

With an open belt both pul leys turn in the same d irect ion,

or the directional relat ion is the same , while in the case of the

crossed belt , the directional relation is opposite .

148

BE L TING

In order that a belt may maintain its position on a pulley,

its cen ter line on the approaching1si de must lie i n a plane perpen

di cu lar to the axi s of the pu lley. If a force is applied to the ap

proaching side Of the belt parallel to the ax is of the pul ley,each

succeeding part of the belt wi l l take up a pos it ion a little farther

along on the face Of the pul ley . A force applied to the recedin g

side of the belt wil l have no effect on that pul ley unless it is great

enough to move the belt bodily . H owever the receding s ide with

respect to one pul ley is the approaching side with respect to the

other pu l l ey .

FIG . 137 . FIG . 138 .

150 . Crown ing P u lleys — Because of the fact that belts do

not remain exactly straight and the pulleys in exact alignment ,some means must be provided for keeping the belts from running

Off,and this can be accomplished by means Of flanges on

,

the

pulleys,or a fork placed on each side of the belt , but the most

common method is to crown the pul ley . By this is meant that the

diameter of the pu l ley is greater in the center than at the edges .

The crown may be either straight or Spherical and its amount wil l

depend upon the width and speed Of the belt,and the distance

between the centers of the pulleys . L ess crown is required the

wider the face Of the pul ley,the greater the speed and the greater

the d istance between the pulley centers .

151 . Tight and L oose Pu l leys .

— When machines are drivenfrom a line shaft

,and it is not desirable to have them run all of

the time,means must be provided so that any machine may be

1 By the approaching side is meant the part of the bel t advancing towardthe pu l ley . The side where the belt leaves the pu l ley is call ed the recedingside .

150 M E CHAN ISM

stopped without stopping the line shaft . Common methods o f

do ing this are by means of clutches,or the tight and loose pul ley

combination . The tight and loose pul leys are two pu l leys pl aced

side by side on the counter shaft,the t ight pul ley being keyed to

the Shaft, and revolving with it , while the loose pul ley is he ld in

place by a col l ar, and is free to revolve on the shaft . When th emachine is thrown out of operation the belt is shifted from the

tight pul ley to the loose pu l l ey . The loose pul ley should be

made Of sl ightly smal ler diameter so that the belt wi l l not be

stretched as tightly when running idle .

er S haft

Too l Rest T ail S tock

FIG . 139 .

Fig . 139 illustrates a method Of connecting a lathe with

the line shaft by means Of belts .

The pul ley on the l ine Shaft,over which the belt runs from the

tight or loose pu l l ey is made w ith a cyl indr ical face , and a w idth

at least equal to that of the tight and loose pul leys comb ined .

152 . M ule P u l leys .

— I n Fig . 140 is Shown a method of com

n ectin g two shafts with a belt when they are too close together

to connect them by the methods of Figs . 137 or 138 .

153. Be lts for Intersectin g Shafts .

—When two intersecting

shafts are to be connected by bel ts, gu ide pu l leys mu st be so

placed that the belt wil l run on the pul ley stra ight (Art .

This is accompl ished in Fig . 141 by using guide pul l eys CC, the

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152 M E CHAN ISM

155. Length Of Belts .- I f the distance between the centers of

two shafts and the diameters of the pu l leys are known,the length

Of belt required for them can be cal culated as fol l ows:

FIG . 142 . FIG . 143.

1 . For Open Belts . See Fig . 144 .

FIG . 144 .

L et L distance between pulley centers in inches .L et R radius Of large pul l ey in inches .

L et r radius Of smal l pul ley in inches .

The total length of belt is the length not in contact with either

pul ley plus the length in contact w ith the large pul ley plus the

length in contact with the smal l pul ley .

L et C and D be the tangent points of the belt and pul leys .

Then 2 CD= length of belt not in contact with the pul leys .

BE LTING 153

L ength Of belt in contact with large pulley

R(7r+26) +2 sin

L ength of belt in contact with small pulley

—r (i r— 20) r 2 sin

—1

Total length Of open belt

2 L 2— (R —r) sin

“ 1

L

For large distances between Shafts and with pul leys of nearly

the same diameter,the third part Of the above equation may

be omitted .

2 . For Crossed Belts. See Fig . 145.

FIG . 145.

Using the same notation as before,the tangent points Of the

belt and the pul l eys are C and D,then 2CD= the length Of belt

not in contact with the pulleys .

L ength of belt in contact with large pul ley

R+ r

L

L ength of belt in contact with smal l pulley

R(7r+20) =R 2 sin“ 1

r 2 sin

154 M E CHAN ISM

Total length of belt

7r+2 Sin

2\/L_ 2 (R+r) (i r+2 sin—1

It will be noticed that for Open belts,the sum and difference Of

the radii Of the pul leys are used,while in the case of crossed belts

onl y the sum of the radii are considered .

156. S tepped Cones for S ame L ength of Belt — Ou most

machine tools the driving pul l ey is usual ly stepped,the belt

running over a simi lar one on the counter shaft , and as the samebe l t must be used on the d ifferent pairs of steps they must be

designed with th is in mind .

1 . For Open Belts — No simple formula has been derived that

can be used for cal cu l ating the size of steps on which an open

belt is to be used,so this problem is general ly solved graphical ly .

An approximate method Often used is that of C . A . Smith1 andis as fol lows:

L et A and B,Fig . 146, be the centers of the shafts , and L the

distan ce between them in inches .

FIG . 146 .

At C half way between A and B , erect a perpendicular CD of

length equal to L (this coefficient was determined experi

mentally) .2

1 See Transactions American Society of M echanical E ngineers, Vol . X ,

page 269 .

2 When the angle between the belt an d center l ine of pu l ley exceeds 18degrees , CD is taken as 0 .298L .

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156 M E CHAN ISM

There are two systems of rope driving in general use, theE nglish or individual rope system

,which has a separate rope

for each groove and the American or single rope system in which

a single rope is carried around al l of the grooves and brought

back from the last to the first by means Of a gu ide pul ley set at

an angle with the other pul leys .

These two systems are shown d iagrammatical ly in Figs . 147

and 148 . With individual ropes,a single rope may break

Second

FIG . 147 .

and the load be carried by the others , which is not the case with

the single rope system . On the other hand,it is difficult to

Obtain un iform tension on al l of the ropes . This is accompl ished

in the s ingle rope system by means of a tens ion pu l l ey and we ight,

so that the wear on the rope is more un iform throughout itslength than on the separate ropes of the individual system .

CHAIN DRIVING

158. Chain Drive.

— The chain drive is general ly used where

the work is heavy and a positive motion is required at a com

BE LTING 157

paratively low speed . The wheels over which the chains run

have teeth to receive alternate l inks of the chain and are cal led

sprocket wheels . The wheel s shown in Figs . 149 and 150 have

First

FIG . 148.

alternate single and double teeth to receive the chain , in other

cases the teeth of the wheels are all alike as in bicycle sprocket

wheels .

FIG . 149 . FIG . 150 .

In Fig . 149 the curve of the tooth may be found by taking a

center at A and a radius AB minus the radius of the end Of the

chain l ink .

158 M E CHAN ISM

To reduce friction the pins are Often provided with rolls as in

pin gearing . This form of cha in is used in automobile drives .

TO find the pitch diameter of an ordinary sprocket wheel,

L et P’

, Fig . 151,be the pitch or length Of One link of the chain .

1

L et T= number of teeth in wheel .

L et c pitch radius of wheel .

L et b = One- half Of the chordal pitch .

and . c

After chains have been used for some time,the pitch may change

on account of the l inks stretching and the p ins becoming worn,

FIG . 151 . FIG . 152 .

so that if the chain originally fitted the teeth , it wil l not later

and the strain may be all carried on a single tooth .

159 . Renold S ilent Chaim— This is the invention of H ans

Renold and is a great improvement in chain drives .

A short section of the chain and sprocket is shown in Fig . 152 .

The chain is bu i l t up of flat pieces which lap by each other

and are fastened together by steel pins . The sides Of the teeth

that come in contact with the wheel as wel l as those of the wheel

are made straight .

It is very satisfactory for motor drives where a positive motion

is desired and the distance between the shafts too smal l for

belting . It also always fits the wheel and takes up the backlash .

1 This can be readily found by taking a piece of chain,measuring its

length and dividing by the number of l inks .

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160 M E CHAN IS M

5. A countershaft 10 ft . from the l ineshaft which revolves 200 R.p .m . isto be driven at the speeds 60

, 80, 100, 120 and 140 R.p .m . Find the diameterof the stepped cone pul leys to drive w ith an Open belt , the smallest pu l leyto be not less than 41 in . diameter .

6. Two stepped cone pul leys are equal and the diameters Of the largestpulleys each 16 in . The maximum and min imum speeds of the driven shaftis 340 and 30 R.p .m . With a crossed bel t , what are the diameters of thepu lleys and fi nd the R.p .m . of the driving cone pul ley shaft?

7 . Two shafts are 4 ft . apart and are to be connected by a chain driveOf 1 in . pitch l inks . The velocity ratio is 5 3 and the smal l sprocket has9 teeth . What are the pitch diameters of the sprockets and what length ofchain is necessary?

CHAPTER X I

INTERM ITTE NT M OTIONS

RATCHET GEAR ING

161 . Ratchet Gearing — This is perhaps the most common

form of intermittent motion and consists of two principal parts,

the ratchet wheel and pawl or deten t.

A simple form is shown in Fig . 154. The center of the ratchet

FIG . 154 .

wheel is at 0,and the driving pawl has a cen ter at A on the arm

0A

The wheel is prevented from turning backward by the pawl

w ith a center at B .

The wheel teeth and pawl should be so shaped that when a

load is applied,the pawl will not tend to leave contact with the

wheel tooth . TO Obtain this result a common normal to the front

face of the pawl and teeth should pass between the pawl and

ratchet wheel centers . CD is such a normal . If the teeth and

pawl were so shaped that the common normal were CD'

,the

pawl would then tend to disengage itsel f from the wheel when

under load .

162 M E CHAN IS M

162 . M ultiple-

pawl Ratchet — In using the pawl w ith center atB

,to prevent backward motion of the wheel

,it will on ly do it after

the wheel has rotated backward far enough for the pawl to engage

the teeth,which may vary from zero to the pitch of the teeth .

This backward motion can be reduced by making the pitch

smal ler,which weakens the teeth , and is not always desirable .

It can also be reduced by plac

ing several pawls side by sideon the pin and making them

of different lengths . In this

way the backward motion can

be reduced to the pitch divided

by the number Of pawls .

By placing a number of

driving pawls side by side and

having them of different lengths,

a very fine feed can be obtained

without decreasing the pitch .

An il lustration Of this is the set works on sawmill log carriages .

H ere a fine feed is required for sawing various thickness bu t

fine teeth would not stand the wear and tear .

Fig . 155 shows this arrangement with two paw l s in wh ich one

is made one—hal f of the

pitch of the teeth longer

than the other .

163. WestonRatchet.— Fig . 156 i l lustrates a

simple ratchet wheel and

pawl as appl ied to the

Weston Ratchet or

S cotch drill . The fric FIG . 156 .

tion Of the drill in the

hole in this case prevents backward motion .

164. Armstrong Universal Ratchet — The Armstrong UniversalRatchet, shown in Fig . 157 , has 12 teeth on the wheel and four

pawls,which engage one at a time . The universal quality of the

tool as claimed by the makers is due only to the fact that the

axis of the two trunnions on which the handle turn s,is at an

acute angle with the axis Of the dri l l , so that a very small motion

Of the end of the handle will turn the dril l .

FIG . 155.

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164 ME CHANI SM

In Fig . 161,if the full movement of the pawl will advance the

wheel,say eight teeth , then the cam b

,which has a slightly larger

d iameter than the outside diameter of the wheel,is cut away on

one s ide so that the pawl can act on the wheel tooth dur ing the

whole angle, and hole number 1 is located in the plate . Then

the cam is revolved on i ts Shaft so that the pawl wi l l act on theteeth during the last seven- eighths of its motion

,and hole number

2 is located , and so on , l ocating the holes so that one tooth wi l l

be cut out each time, ti ll in the ninth hole al l of the teeth are

masked . The width of pawl face must be equal to the face of

ratchet wheel plus the cam face . A mechanism similar to this

is used on some of the posit ive feed engine lubricators and on

printing press inking mechan isms .

FIG . 161 .

168. S i lent Ratchets .

- I t is not always necessary that the

connect ion between the wheel and pawl be of pos itive form as

shown in the foregoing figures,and in some cases a frict ion drive

may be substituted as in Figs . 162,163 or 164 . These are some

times cal led si len t ratchets .

169 . P ositive Ratchets .

—The ratchet mechanisms - so far

d iscussed together with many variations of them are suitable

for feed mechan isms where the mot ions of the driver are not too

rap id ; in such cases the shock at the beg inn ing of mot ion may

be too great,and the inertia of the wheel may be su ch as to cause

it to “ over- travel . In such mechan isms as revolut ion counters

where a definite motion of the fol lower mu st be secured each

time,some means must be employed to prevent over- travel .

A device for this purpose is shown in Fig . 165. The lever to which

IN TERM ITTE N T M OTION S 165

the pawl is attached has a proj ecting beak so formed that when

the pawl first acts on a pin , the end C of this beak passes across

the l ine of mot ion of the p ins and limits the ir mot ion . The curve

CD is an arc with O as a center .

FIG . 163.

FIG . 164 .

170. M agnetic Ratchet. —The magnetic ratchet shown in

Fig . 166 also accomplishes the same resu lt . This consists of a

metal disk A and an arm B that swings about the center of A .

On B is a magnet C , which grips the plate and by a commutator

device is released for the return stroke of the lever .

As soon as the magnet C releases A a second magnet on the

back (not shown) , grips it and prevents any backward movement

of the disk . These ratchets are used on dividing engines .

166 ME CHAN IS M

171 . Jack Ratchets .—I n al l of the ratchet mechanisms so

far d iscussed,a rotary motion was given to the wheel

,but ratchet

FIG . 165.

FIG . 167 . FIG . 168 .

mechanisms can be used to impart a reciprocating motion . Two

examples of this kind are shown in the l ifting j acks of Figs . 167

and 168 . In each case , the pawl a does the lift ing , and b holds

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M E CHAN ISM

In the figure this clutch is shown on a single shaft . Whenthe clutch is “ thrown out” the part A which has a pul ley fastened

FIG . 17 1 .

FIG . 172 .

on its long hub , but not shown in the figure,does not revolve

,

but is made to revolve by the action of the toggle j oint,bringing

the friction surfaces into contact .

INDEX

A

Acce lerated motion, 50, 53, 54Action, angle of, 86

are of, 86l ine of

,87

Addendum circle, 84Angle of ob liqu ity, 88Angu lar velocity, 91 , 107Approach, angle of, 86Approximate methods of laying

gear teeth,1 14

,127

Armstrong ratchet,162

Axis,instantaneou s

,1 1

B

Back cones, 127gears

,144

lash,84

Base circle,49, 87 , 90, 92

Belt,crossed

,148

for intersecting shafts,150

length of,152

open,148

quarter tu rn,151

shifter cam , 73

Belting,148

American system,156

chain drive,156

E ngl ish system,156

M orse rocker joint chain 159

Renold silent cha in, 158rope drive

,155

Bevel cones,81

gears,shop draw ing Of

,128

sp iral,126

Tredgold’

s method of laying ou t,127

Bilgram bevel gear p laner, 133Bricard

’

s straight l ine motion,46

Brush plate and wheel , 81

CCam

,base circle

,49

belt shifter,73

contact w ith fo l lower, 48constant breadth

,65

diameter,64

,77

curves,motions u sed

,50

cyl indrical,66

,67 , 69, 70

definition of,48

dw ell ” or “ rest,52

inverse, 70, 75

invo lute,58

1

main an d return,65, 76pitch l ine

,48

positive motion,64

pressure angle of,49

w ith flat- face fol low er, 49, 56off—set fo l low er, 55oscil lating fol low er, 59, 75ro l l fo l low er, 49, 60

theoretical cu rve,48

working curve,48

Center,instantaneou s

,1 1

Centro,definition of

,1 1

location of,in a single body, 12

Centros,location of

,in three bodi es

moving relative ly to eachother

,12

location of,in compound kine

matic chain,16

Chain drive,156

kinematic,definition of

,13

M orse rocker joint,159

Renold si lent,158

C ircle,addendum

,definition of

, 84

dedendum,defin ition of

, 84

ro l l ing,101

,102

,103

,105, 106, 107,

workin g depth,definition of

,84

C ircu lar p itch,definition of

,85

,131

169

170 INDE X

Clearance, 84

Clutch,cam

,76

cone,167

disk,167

Clutches,167

Comparison of i nvo lute and cycloidalsystems

,1 10

Cones,Evans friction

,79

stepped,144

,154

Conjugate methods of cuttin g gearteeth

,1 19

Counter shaft,148

Crossed be lt,148

Crosshead,S cotch

,70

Crown gears, 126Crowning pu l leys, 149Cutting gear teeth

,1 17 , 1 19, 120

,

Cycloid,construction of

,101

Cycloidal gears,interchan geab le

,108

system of gear teeth,101

Cyl indrical cam,66, 67 , 69, 70

D

Dedendum circle,84

Detent,161

D iagrams, l inear velocity, 25po lar velocity

,26

velocity of engine crank and crosshead

,26

Drafting board parallel mechanisms,

40

Dril l, Weston or Scotch , 162

“ Dwel l,”52

E

E ffect of changing distance betweencenters of invo lute gears , 92

E l liptical gears,139

E picycloid,construction of

,102

Evans friction cones. 79

F

Face of tooth, 84

Fel low s gear shaper, 120Flank of tooth

, 84

Flat-face fo l lower, 49, 54Fol low er

,cam

,48, 70

Friction gears, 78, 79, 80, 81

Friction bevel cones, 81

brush p late and wheel, 81

Evans cones, 79

grooved cy linders, 79

p lain cyl inders,78

Sel lers feed disks,80

G

Gear cutters,interchangeab le

,tab le

of,1 18

cutting,1 17

,1 19 , 120 , 132 , 133

hobbing , 123

shaper,Fel low s ’ , 120

teeth,approximate methods of

,

laying out, 1 14, 1 15, 1 16, 127

contact between, 89l imiting length of

, 90

octoid,126

standard sizes of, 93

w ith straight flanks, 106train

,definition Of

,142

di rection of rotation in ,143

for thread cutting,144

value of,142

Gearing,ratchet

,161

Gears , back , 144bevel

,125

approximate method of layingout, 127

methods of cu tting teeth, 131Bilgram p laner for

,133

shop draw ing for, 128to calcu late ou tside diameter of

,

131

comparison Of invo lute an d cy

cloidal systems of teeth, 1 10crown, 126cycloidal annu lar gear an d p inion,

107

interchan geab le, 108

spu r, 82 , 88, 105

rack and pinion , 106rol l ing circle

,standard diameter

of,109

system,101

,103

definition of tooth parts,84

ell iptical , 139friction

,78, 79, 80, 81

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172 INDE X

M otion, straight l ineTchebicheff

,44

Thompson indicator,44

Watts ’,42

uniform ,50

uniform ly accelerated, 50, 53, 54

retarded,50, 55

O

Ob liqu ity,angle Of, 88

line of, 90, 91Octoid gear tooth, 126, 133Odontograph

, Grant, 1 15, 1 16Robinson

,1 14

W ill is, 1 15Off- set follower, 55Open belt

,148

Oscil latin g fol lower, 59, 70

P

Paralle l motion mechan isms, defi n ition of

,38

drafting board,40

rul er,40

Roberva l balance,40

parallelogram, 38

Path definition of, 3P aw"

,163, 166

P eaucell ier’

s exact straight l ine motion mechanism

,45

P in gears,1 1 1

P inion for annu lar gear,l imiting

size Of,107

P itch,85, 136

are, 89

circle,82

circu lar, 85, 136

cones,127

diameter,84

diametral,85

l ine of cam ,48

P ositive motion cam,64

P ressure angle,49

P rob lems,6,17, 23, 24, 46, 62 , 72, 86,

98, 1 10, 134, 140, 147, 159

Pu l leys,crowning, 149

mu le,150

tight and loose, 149

Q

Quarter turn belt, 151Qu ick return motion, 140oscil lating arm , 34, 35, 36, 37

Whitworth, 31 , 134

R

Rack and p inion,cycloidal

106

invo lu te system, 89, 90

Radian, 5Radius arm

, 30

center, 31

Ratchet gearin g,161

Ratchets,Armstrong universa l , 162

doub le acting,163

jack,166

magnetic, 165masked

,163

mu ltip le paw l , 162positive

,164

silent,164

Weston,162

Recess, angle of, 86Remold silent chain

,158

“ Rest,

”52

Retarded motion,50 , 55

Reversing paw l , 163Roberts straight line motion

,45

Roberval balance,40

Rob inson Odontograph, 1 14RO11 fo llower, 49, 60 , 62 , 63, 64, 65Rol lin g circle, standard diameter, of,

109

Ro lls,friction

, 78

grooved friction,79

velocity ratio of, 78

Root circle, 84Rope drive

,155

Rotation, 3

S

S cotch crosshead, 70Scott Russel l straight l ine motion

,43

Screw cutting gear train, 144lead 144

Sel lers feed disks, 80

Shop draw in g of bevel gears, 128

INDEX 173

Spherical motion,3

Spiral gears, 1 13, 126Sprocket whee l, 157Spur gears

,82

Stepped cones,144

gears,1 13

Stub tooth gear,97, 98, 100

Stud gear,144

Straight line motions,Bricard

’

s,

P eaucel lier’s,45

Roberts,45

Scott Ru sse l l , 43Tchebicheff

’s,44

Thompson indicator, 44Watt ’s, 42

T

Tab les of interchangeab le gearcutters

,1 18

standard tooth parts,94, 95

stub tooth parts,98

Tchebicheff’

s straight l ine motion, 44Theoretical cam cu rve, 48Thompson indicator motion, 44Thread

,mu ltip le

,136

cutting,144

Throat diameter,138

Tight pu l leys,149

Tooth parts,definition of 84

tab les of,94, 95, 98

Translation,3

Tredgold’

s method,127

Tumb ler gears,145

Tw isted gears,1 13

U

Uniform motion,50

Uniformly accelerated motion, 50, 53,54

retarded motion,50

,55

V

Variab le motion mechanisms,28, 29

Velocity, angu lar, 5definition of

,4

diagrams,25

,26

,28

l inear, 4, 5, 1 1 , 18, 25, 29, 32 , 34,35

,36

,37

ratio,belting

,148

bevel gears,79, 125

bru sh p late and wheel,81

effect of idle wheel on,143

e l liptical gears,140

Evans friction cones, 79

gearing, 83

gear tra ins,142

roll ing cyl inders, 78

Sel lers ’ feed disks,80

stepped cones,144

worm an d worm whee l,135

relation between1

l inear and angular

, 6

relative l inear,so lution of prob lems

in,18

variab le l inear, 4

W

Watt ’s straight l ine motion, 42Weston ratchet

,162

Wheel,lantern

,1 12

ratchet,161

sprocket,157

Whi tworth,qu ick return motion

,31

134

W il li s Odontograph,1 15

Working cam curve, 48depth circle

,84

Worm,1 25

,130

length of,138

and worm wheel , 135, 137, 140, 141method of cutting

,138

p itch used,136

Wuest gears,1 13