Chapter 13 Gears - Generalmechfamilyhu.net/downloads/MACHINE DESIGN 2/SLIDES... · Pitch diameter /...

1

Chapter 13Gears - General

Dr. Hitham TlilanDr. Hitham Tlilan

November 18, 2008 [email protected] 1

What are Gears for?• Reducing or increasing speed• Reducing or increasing torque

• Transmitting power around corners over Transmitting power around corners, overdistances

• We want (at various times)– Efficiency– Reliability

A


– Accuracy– Smoothness

– Quiet operation

2

13-1 Types of Gears1- Gears with

Parallel Shafts• Spur gears (Fig.13.1)(i l di k d i i )(including rack and pinion)Simple to make

• Helical gears (Fig.13.2)– Gradual engagement reduces

impact– Quieter

Fig. 13-1


– Quieter– Introduces thrust

– have an involutes tooth profile – on a cutting plane perpendicular

to the axisFig. 13-2

13-1 Continued

13.1 ContinuedSpur Gear


Helical GearThe inclined tooth in the Helical gear

develops thrust load and bending coupleSometimes the Helical gear used to transmit motion between non-

parallel shafts

3

2-Nonparallel Shafts• Bevel gears– Intersecting shaft axes

Straight toothed

13-1 Continued(Fig. 13.3)

– Straight-toothed– Spiral• Hypoid gears– Nonintersecting axes• Involute profiles areobserved on a cutting

Fig. 13-3


gplane perpendicular tothe axis

Straight bevel gear Spiral bevel gear

Worm and Worm Gear or Wheel• Perpendicular shafts,nonintersecting• Worm shaft in plane of worm wheel

13-1 Continued

• Worm resembles a screw, can have right or left hand thread

• Single enveloping:straight worm, curved wheel

• Double enveloping: both are curved


both are curved

Fig. 13-4

4

13-2 NomenclatureThe distance measured on the pitch circle from a point of a tooth to a corresponding point on the adjacent tooth.p =tooth thickness + width spacep p

Module = m=Pitch diameter / No. of teeth

Diametral Pitch= P =No. of teeth / Pitch diameter


-Theoretical circle upon which calculations based-Its diameter is the pitch diameter -The pitch circle of two mating gears are tangent

to each other

Whole depth = ht =Addendum + dedendum

13-2 Continued

2)-(13 [mm]

1)-(13 h][teeth/inc

NdmModule

dNPPitchDiametral

==

==

di tit hteeth of No.:

4)-(13

3)-(13

)([ ]

dN

pPN

dpPitchCircular

N

π

π

=

==


diameter pitch :d

5

Example• A gear with 20 teeth and a pitch diameter of 2inches has a circular pitch = ?N =20 teethd =2 inp = 2 π/20 = 0.1 π in ,or

= 0.1(25.4) π = 2.54 π mmWhat is its diametral pitch?P = 20/2 = 10 teeth/in.Its mod le is?


Its module is?m = d/N = 2(25.4)/20 = 0.1mm

13-3 Conjugate ActionWhen the tooth profile, or cams, are designed so as to produce a constant angular velocity ratio during meshing, these are said to have conjugate action.

Pitch point

Fig 13 6

Line of action


Fig. 13-6

Pitch circles

6

• Conjugate action: engaging bodies rotate at a constant angular velocity ratio• Involute: one curve producing a conjugate action• An involute curve is produced by unwinding a string from a base circle

13-4 Involute Properties

• Two involutes engaging have conjugate action• Base circles are tangent to a line crossing center to center line at the pressure angle• Intersection of pressure line with center line creates pitch point on both pitchcircles.• Increasing center distance changes pitch radii but Fi 13 7


Increasing center distance changes pitch radii but does not destroy conjugateaction

Fig. 13-7


7

13-5 Fundamentals• Speed ratios = ratio of pitch circle radii

• An integral number of teeth is preferred

5)-(13 1

2

2

12211 r

rwwwrwrV =⇒==

An integral number of teeth is preferred• Pitch of gears must be the same (diametral pitch = p = N/d )• Pressure angle of mating gears should be the same, usually 14 ½° , 20 or 25° deg.• Final definition of pitch circle and pressure angle depends on center to center distance


• “Driver” / “Driven” gear distinction determines which face carries load• Initial contact occurs at angle of approach• Final contact occurs at angle of recess

• Contact below the base circle does not produce conjugate action• Gear shape below base is radial or undercut• Contact point moves along pressure line• At the pitch circle the contact is pure rolling• Sliding occurs before and after pitch circle

13-5 Continued

g p

φ Line of

November 18, 2008 [email protected] 14Fig. 13-10 Fig. 13-12

of action

φ

8

13-5 ContinuedThe radius of the base circle =rb= r cosφ (13−6)

abclearenceP

b dedendumP

aaddendum

−=

====. , 2511

abclearence

Pppitchcircular π

==


One gear may be straight: rack and pinion– Base pitch– Circular pitch

13-5 Continued

p = p cos φ (13 7)pb = pc cos φ (13−7)

Pbφ


φPc

9

Example 13-1GivenNp=16 toothNG=40 toothDiametral pitch=P=2 =°=

=+

=+

=

====

====

====

142

2082

202

40

82

16

5712

then,cosand 20Since

in centers-2 bewteen distance The

in diameter pitchGear

in diameter pitch Pinion

in .pitchCircular

Gp

GG

pp

rr

ddP

Nd

PN

d

Pp

b φφ

ππ

a=1/P, b=1.25/Pφ =20°

=′+′

+

=°==

=°==

25142

2

409202

20

7632028

(1) in .

then, in, 0.25 by increase to need we Since

(b)

in .cos)(radius baseGear

in .cos)(radius base Pinion

,

Gp

Gp

dd

dd

r

r

Gearb

pininb

b φφ

November 18, 2008 [email protected] 17°=′=

=′=′

==′

′

56222

357201438

4016

2

.))(

(cos

in . in .

(2) and (1) Eqs. from

(2)

therefore change,not does ratio velocity The

p

-1

Gp

G

p

G

p

d

r

dd

dd

d

d

pinionbφ

13-6 Contact Ratio

Fig. 13-15

8)(13ratioContact

recess of arcapproach of arcaction of Arc +=+=

qqq

t

ra


9)-(13 1.2 cos

cotact. in teeth of pairs ofnumber average the

8)-(13 ratioContact

≥=

==

ϕpLm

pqm

abc

tc

10

13-7 Interference• Interference occurs when contact occurs outside the involute shape of the tooth• Problem occurs when Np << Ng

d N i lland Np is small• Can avoid problem by avoiding large ratios or by using a fine pitch• Alternative is to accommodate interference with undercutting of tooth shape• Undercutting weakens the gear


Undercutting weakens the gear tooth

( ) 10)-(13 sin

ratio)gear 1-to-(1

gear and pinionspur on teeth of No.smallest the

φφ

22 3114kN

N

p

p

++=

=

13-7 Continued

( )

( )is ceinterferenwithout pinionspur on teeth of No.smallest the Therefore;

unity thangreater is ;

is,that pinion, than teeth more hasgear mating the if

)(sin

φφ2

2

6

k

mNNm

p

GG

p

==


( ) 11)-(13 sin)(sin)(

φφ

222 21

212 mmmm

kN p +++

=

11

12)(13

ceinterferenwithout rackawithoperate pinionspur For kN 4

=

13-7 Continued

13)-(13 sin

sin

is free- ceinterferen isthat pinion specified a withgear largest The

12)-(13 sin

2

2

φφ

φ

p

pG

p

NkkN

N

N

244

2

22

2

−

−=

=



12

13-9 Straight Bevel Gear

-Used to transmit motion between intersecting shafts-The pitch is measured from the large end of the tooth

Fi 13 20

the large end of the tooth.-The circular pitch and pitch diameter are calculated in the same as that for spur gear.


Fig. 13-20

GearGPinionPNNΓ

NN

P

G

G

P

: ,:

14)-(13 tan tan

apex theat meating cone pitch the by defined are angles pitch The

==γ

13-9 Continued

.

ionapproximat sTredgold'

conethe back ojected onevel is prwhen the b

, adius rack-cone rl to the badius equahaving a rr gearof the spu

s that the same al gear is th of beveof the teethe shape

b

pitchcircular the :

teeth ofnumber virtual the:

15)-(13

p

N

pr

N

jp

b

′

=′π2

The standard straight – tooth bevel gears are cut usingPressure angle of 20°,


The standard straight tooth bevel gears are cut usingPressure angle of 20 , unequal addenda and dedenda, and full depth teeth

This increase the contact ratio, avoid undercut, andincrease the strength of the pinion

13

13-10 Parallel Helical Gear

anglehilexthe)( distance thepitchcircular transverse the

16)-(13 cos )( distance thepitchcircular normal The

=

==

===

ψ

ψ

acptpnp

ae

t

anglehilex the =ψ

17)-(13 tan

)( distance thepitch axial The

ψtp

xp

xpad

=

==


18)-(13 cos

pitch diametral normal The, since,

ψ

π

tn

n

nn

PP

PthenPp

=

=

=

13-10 Continued

rotation of direction the in angle pressure the :direction normal the in angle pressure the :

19)-(13 tan

tan cos

t

n

t

n

φφ

φφ

ψ =

N

teeth ofnumber actual the :

teeth ofnumber virtual the:

20)-(13 cos

N

N

NN

′

=′ψ3


14

( )cos sin2

the smallest No. of teeth of a helical- spur pinion that

run without interference with the same 4 1 1 3 (13-21)

p

G

N

NkN ψ φ

=

= + +

13-10 Continued

( )sinsin2 1 1 3 (13 21)

6

if the mating gear has more

pN tt

φφ

= + +

teeth than pinion, that is,

Therefore; the smallest No. of teeth on pinion without interference is

GG

p

Nm mN

= =


( )cos ( )sin( )sin

2 22

2 1 2 (1 2p

kN m m m tm t

ψ φφ

= + + ++

13-22)

13-10 Continued

ceinterferenwithout rackawith operate pinionFor k4

24)-(13 sincos

cossin

is free- ceinterferen isthat pinion specified a withgear largest The

23)-(13 sin

cos

2

2

Nk

ktNN

t

kN

p

G

p

φψ

ψφ

φψ

24

4

24

222

2

−

−=

=


sin cos tNk p φψ 24

15

13-11 Worm Gear

The helix angle for worm gear is quite different from that of the helical gears.

The helix angle on theFig. 13.24

The helix angle on the worm (ψW) is quite large, because of this we it usually specify the lead angle (λ),

ψW + (λ) = 90°

The helix angle on the The circular pitch


gear is very small (ψG)

axis. worm the containing plane a on the from measured diameter; pitchgear the :

25)-(13

G

tGG

d

pNdπ

=

por, transverse circular pitch

13-11 ContinuedThe worm pitch diameter (dW) should be the same as the that of the hob, which is used to cut the worm. For best power transmission (dW) must be

87508750

26)-(13 .

..

713

87508750 CdCW ≤≤

28)-(13 λ tan

27)-(13 Wx

dL

NpL

π=

=

November 18, 2008 [email protected] 30angle lead the :λpitch axial the :

lead the :

x

W

p L

dπ

16

13-13 Gear Train3

Driven2

Pinion

rev/minor nsrevolustio

29)-(13

:gearset anyFor

=

==

n

nddn

NNn 2

3

22

3

23

diameter pitch teeth ofnumber

==

dN

For the gear trainFor the gear trainFig (13-27)

6gearofspeedTh


Fig. 13-27(a)

6gear of speed

26 nNN

NN

NNn

The

6

5

4

3

3

2−= Gears 2, 3, and 5 are driversGears 3, 4, and 6 are driven

For spur and helical gears the CCW direction of rotation is the For spur and helical gears the CCW direction of rotation is the positive direction of rotation using (right hand rule)positive direction of rotation using (right hand rule)

13-13 Continued

equationthisinusedbecandiameterspitchThe

30)-(13 value train Theoth umber driven toproduct of

ooth umber driving tproduct ofe ==

gearlast the of speed the :31)-(13

sence opposite in rotatesgear last the if -, sence same the in rotate gearsfirst the andlast the if ,

equationthisinusedbecandiameters pitch The

L

FL

nenn

e

=⎩⎨⎧+

=


gearfirst the of speed the :Fn

17

13-13 ContinuedPlanetary Train

32)-(13valuetrainThe AL nne −==

November 18, 2008 [email protected] 33(rev/min) arm the of speed the :(rev/min)gear first the of speed the :(rev/min)gear last the of speed the :

32)-(13 valuetrain The

A

F

L

AF

nnn

nne

−==

value train TheAF

AL

nnnn

oth umber driven toproduct ofooth umber driving tproduct ofe

−−

===

Planetary Train -\Continued


18

I

vB=r5 n5

Example 13-4

A BI

vA=r2 n2

I: Instantaneous center


I: Instantaneous center

4

2255

4

2255

44 222 r

nNnNr

nrnrr

vvn AB +=

−−=

−=

)(

13-14 Force analysis- Spur Gear

33)13(Fi2i ii t3bt dfth

load. dTransmitte :(a)

F

WFW

t

t

tt 32=

November 18, 2008 [email protected] 3633)-(13 ,

usefulnot radial the while component, useful real the is load tangential the

33)-13(Fig.2pinionagainest 3gear byexerted force the :

tWdT

isF

F

r

t

2

32

32

=

19

13-14 continued

2

2

ddtheTT

if

a

=

== 2(pinion)gear on (a)shaft by exerted torque becomes 33)-Eq.(13 then, 2,gear with dealing are we

d

35)-(13 )(

is units SI in

34)-(13 33000

power The

ft/min velocity line pitch The

d H

tW

VtWH

ndV π

31060

12

=

==

==

November 18, 2008 [email protected] 37rev/min speed, :

mm diameter,gear :kW Power, :

kN load, dtransmitte :

)(

ndH

tWd nt π

13-15 Force analysis- Bevel Gear


20

13-15 Force analysis - Helical Gear



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Dr. Hitham Tlilan 45

Involute face crown gear:


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Chapter 13 Gears - Generalmechfamilyhu.net/downloads/MACHINE DESIGN 2/SLIDES... · Pitch diameter /...

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