1

Gears

SOLO HERMELIN

Updated: 26.11.06Run the Power Point

http://www.solohermelin.com

2

Antikythera Mechanism (cc. 87 B.C.)

Antikythera mechanism is an ancient mechanism,made from bronze in a wooden frame, is believed tobe an analog computer designed to track the movementsof heavenly objects. The device the motion of sun, moonAnd Mercury, Venus, Mars, Jupiter and Saturn theCelestial body known to Ancient Greeks.

http://www.voyager.in/Antikythera_mechanism

Discovered in 1900in a shipwreck of theGreek island ofAntikythera.

SOLO

3

Antikythera (87 B.C.(SOLO

Original Reconstruction 1

Reconstruction 2Front view

Reconstruction 2Back vew

4

Antikythera (87 B.C.(SOLO

36842.1319

254

32

127

24

48

38

64 ==××

Moon to Sun ratio In 19 years, the moon goes through 235 phase cycles

from fool moon to crescent moon and back to fool moonand passes through the zodiac 254 times.

5

GearsSOLO

6

Spur Gears

Spur gears are used to transmit rotary motion between parallel shafts.

SOLO

7

Pitch Circles

P

O 1

O 2

R 1

R 2

ϕ

Pressure

Line RB1

RB2

BaseCircles

Spur Gear Nomenclature

Pitch circles – Theoretical circles upon which all calculations are based with centers O1 and O2 and diameters D1 and D2. The two pitch circles are tangent at

the pitch point P.

Pressure Line – The line passing through P and making an angle φ with the tangent to the pitch

circles along which the tooth of one gear presses the tooth of the second gear.

Base circles – Circles tangent to pressure line with centers O1 and O2 and diameters

DB1 and DB2.

ϕϕ cos&cos 2211 DDDD BB ==

Circular pitch p – The distance measured along the pitch circle from a point on one tooth to the

corresponding point on the adjacent point. Since we assume that the two

gears always maintain contact during rotation the circular pitches are equal.

N1, N2 – Number of teeth on the two gears.

ω1, ω2 – Angular rates of the two gears.

2211 ωω NN =

2

2

1

1

N

D

N

Dp

ππ ==

1

2

2

1

2

1

ωω==

N

N

D

D

SOLO

8

Spur Gear Teeth

Spur Gear Tooth shape must be such that contact between the tooth of one gear to thecorresponding tooth of the second gear iscontinuously maintained until the contact occurs with the adjacent tooth.

If the shapes of the teeth on the two gearshave this property they are called conjugates.

From the figure we can see that during therotation of the driver the contact begins atpoint A and continues until it ends at point B.

The segment AB is on the pressure line.

Among infinite possibilities of conjugateshapes the one that is almost exclusively usedin the gear design is the involute.

SOLO

9

Spur Gear Teeth

Among infinite possibilities of conjugateshapes the one that is almost exclusively usedin the gear design is the involute.

The involute of a circle (base circle) is obtainedby an imaginary string IA (see Figures) wound on

the circle and then unwounded (IC, IB(, while holding it taut.

Base circleO

β

I

θ ψA

β1BR

β1BR

r

x

y

Cβ

Cθ Cr

( )θ,rB

( )CCrC θ,

( )( )βββ

βββcossin

sincos

1

1

−=+=

B

B

Ry

Rx

The involute equation

The involute has the following properties:

1. All lines normal to the involute are tangent to the base circle

2. The radius of curvature of the involute at a point P (r,θ) is given by ρ=RB1β. The base circle is the locus of the center of curvature

of the involute.

SOLO

10

Spur Gear Teeth

In the same way

Point A on the involute has the radius RA and thethickness tA along the circle RA.

SOLO

AB

AR

BR

bRAφ

Bφ

2/Bt2/At

O

E

D

FG

toothinvolute

base circle

B

involute

OG

BG

OG

DGDOG φtan===∠

∩

BBBB invDOGDOB φφφφ =−=−∠=∠ :tan

AAA invDOA φφφ =−=∠ :tan

Point B on the involute has the radius RB and thethickness tB along the circle RB.

A

A

AA

A

R

tinv

R

tDOADOE 2

121

+=+∠=∠ φ

B

B

BB

B

R

tinv

R

tDOBDOE 2

121

+=+∠=∠ φ

−+= BA

B

ABB invinv

R

tRt φφ

22

11

Width ofspace

Face width

Top land

Addendumcircle

Tooth

thicknessAddendum

Dedendum

Dedendum

circle

Clearencecircle

Clearence

Pitch

circle

Face

Flank

Bottom la

nd

Spur Gear Teeth SOLO

Addendum – the radial distance between the top land of the tooth and the pitch circle

Dedendum – the radial distance between the bottom land of the tooth and the pitch circle

Addendum circle – the circle passing through the top land of the tooth

Dedendum circle – the circle passing through the bottom land of the tooth

Clearance circle – the circle tangent to the addendum of the mating gear

12

Spur Gear Force Equation SOLO

If no backlash R1θ1 = R2θ2

21

21 θθ

=

R

R

2

2

1

1

N

D

N

Dp

ππ ==

Mating condition of the teeth on the two gears is

Circular path of gear 1 = Circular path of gear 2

1

2

1

2

1

2

N

N

D

D

R

R ==2

1

22

1

21 θθθ

=

=

N

N

R

R

The force applied by gear 1 on gear 2 is Fn . Since the tooth surface shape is an involute and the force at the point of contact is normal to the surface, it will be tangent to the base circle that defines the involute. Therefore Fn will always be on the pressure line and tangent to both base circles.

ϕϕ cos

2

23'

cos

1

1

21 R

b

rLawsNewton

n

R

b

r

R

TF

R

Tth

==

22

12

2

11 rrr T

N

NT

R

RT ==

Pressure line

Base circle

Pitch circle

Pitch circleBase circle

P B

Pinion(driver)

Gear(driven )

C

O1

ϕBegin

contact

EndcontactA

nF

nF

1bR1R

2bR

2R

ϕ

ϕ

Tr1, Tr2 are the reaction moments onthe two gears

13

Spur Gear Equations of Motion SOLO

Rotor

Stator

Gimbal

Gears

Motor

Body

1θ

1R

2R

2θ2T

1rTBω

( )Bbn JRFT ωθ −=− 1111

( )Bbn JTRF ωθ +=− 2222

1

2

2

1

θθ

=

N

N

we obtain

multiply the second equation by and add both equations using 2

1

2

1

2

1

b

b

R

R

R

R

N

N ==

BJN

NJJ

N

NJT

N

NT ωθ

−−

+=

− 2

2

1112

2

2

112

2

11

BJN

NJJ

N

NJTT

N

N ωθ

−+

+=−

1

1

2221

2

1

2221

1

2

multiply the second equation by and subtract from first2

1

2

1

2

1

b

b

R

R

R

R

N

N ==

Bbn JN

NJJ

N

NJT

N

NTRF ωθ

+−

−−

+= 2

2

1112

2

2

112

2

1112

Bbn JN

NJJ

N

NJT

N

NTRF ωθ

+−

−+

+= 1

1

2221

2

1

221

1

2222

1

2

N

N×

1

2

N

N×

14

Dedendum circle

Pressure line

Base circle

Pitch circle

Addendum circle

Addendum circlePitch circleBase circle

Dedendum circle

Angle ofapproach

Angle ofrecess

Angle ofapproach

Angle ofrecess

PA

B

Pinion(driver)

Gear(driven )

ϕ

O2

O1

ϕ

ϕ

EndcontactBegin

contact

Spur Gear Teeth SOLO

15

Bevel GearsSOLO

Worm gears are used to transmit rotary motion between intersecting shafts.

16

Angular Bevel GearsSOLO

Γ - Pitch angle

Σ - Shaft angle

α - Addendum angle

δ - Dedendum angle

ΓO - Face angle

ΓR - Root angle

21 Γ+Γ=Σ

( ) 2221 sincoscossinsinsin ΓΣ−ΓΣ=Γ−Σ=Γ

ΣΣ−

ΓΓ=

ΓΣΓ

sin

cos

sin

cos

sinsin

sin

2

2

2

1

22

1

tan

1cos

sin

sin

sin

1

Γ=

Σ+

ΓΓ

Σ

AO – Cone distance

D – Pitch diameter

0

11 2

sinA

D=Γ0

22 2

sinA

D=Γ

2

2

1

1

N

D

N

Dp

ππ ==

Circular path of gear 1 = Circular path of gear 2

1

2

1

2

1

2

N

N

D

D

R

R ==2

1

22

1

21 θθθ

=

=

N

N

R

R

2

1

2

1

sin

sin

D

D=ΓΓ

22

1

tan

1cos

sin

1

Γ=

Σ+

Σ D

D

2

12

cos

sintan

NN+Σ

Σ=Γ

17

SOLO

Γ - Pitch angle

Perpendicular Shafts Straight Bevel Gears ) Σ = 90° - Shaft angle (

α - Addendum angle

δ - Dedendum angle

ΓO - Face angle

ΓR - Root angle

AO – Cone distance

D – Pitch diameter

2

2

1

1

N

D

N

Dp

ππ ==

Circular path of gear 1 = Circular path of gear 2

1

2

1

2

1

2

N

N

D

D

R

R ==2

1

22

1

21 θθθ

=

=

N

N

R

R

2

12

cos

sintan

NN+Σ

Σ=Γ1

22tan

N

N=Γ2

11tan

N

N=Γ90=Σ

18

Bevel GearsSOLO

19

Bevel GearsSOLO

Spiral Bevel Gears

20

Crossed Shaft Helical GearSOLO

Helical Gears are used to transmit rotary motion between parallel and nonparallel shafts.

21

t

s

F

F=φtan

n

st FF

φψ tancos= ψ

φφcos

tantan n= N

Dp

π=

ψπψ coscosN

Dppn ==

Helical GearSOLO

22

Normal circular path of gear 1 = Normal circular path of gear 2

2

22

1

11 coscos

N

D

N

Dpn

ψψ ==

2211 ωω NN = 11

22

1

2

2

1

cos

cos

ψψ

ωω

D

D

N

N ==

21 ψψ ±=Σ

iψ - Gear i helix angle

iN - Gear i teeth number

iD - Gear i diameter

iω - Gear i angular velocity

Crossed Shaft Helical GearSOLO

23

Worm and Worm Gear

If a thread on a helical gear makes a complete revolutionon the pitch cylinder, the resulting gear is called a worm.

1

tanD

L

πλ =

λ - lead angle

ψ - helix angle

D1 – worm pitch diameter

N1 – number of threads

p1 – worm circular pitch

1

11 N

Dp

π=

px – worm linear pitch

λλ cossin1 xpp =

L - lead of the worm

λπλ tantan 1111 DpNpNL x ===

Worm

SOLO

λ

1ψ1D

λ

1ψ1D

xp

Different threadsN1 = 3

λ

111 pND =π

1p

xpNL 1=xp

Thread 1

Thread 2 Thread 3

1

11 N

Dp

π=

Unwrapped completerevolution of the worm

24

Worm and Worm Gear

The mating gear for a worm is called a worm gear orworm wheel.

The mating condition of the worm gear with the worm is thatworm axial pitch = worm gear circular pitch

2ppx =

2

22 N

Dp

π=1N

Lpx =

1

tanD

L

πλ =

λπ

tan2

1

22

1

D

D

D

L

N

N ==

SOLO

iψ - Gear i helix angle

iN - Gear i teeth number

iD - Gear i diameter

iω - Gear i angular velocity

25

Worm and Worm GearSOLO

Worm gears are used to transmit rotational motion between nonparallel and nonintersecting shafts.

26

Worm and Worm Gear

1

tanD

L

πλ =

SOLO

xp

x

p

R =1

11θ

λθθ tan111

11 Rp

pRx x == 111 2

tan θπ

λθ L

Rx ==

Rotation of the worm by an angle θ1 will move the pointon the worm that made contact with the gear in the wormaxis direction by x.

2

22

p

R

p

x

x

θ= 22θRx =

22θ Rx =

λθ tan11 Rx = 2

1

2

1

1

2 tanN

N

R

R == λθθ

This movement by x is equivalent to a rotation θ2 of the worm gear, such that:

λπ

tan2

1

22

1

D

D

D

L

N

N ==

λ

1ψ1D

λ

1ψ1D

xp

Different threadsN1 = 3

λ

111 pND =π

1p

xpNL 1=xp

Thread 1

Thread 2 Thread 3

1

11 N

Dp

π=

Unwrapped completerevolution of the worm

27

Worm and Worm GearSOLO

2

22

1

11 coscos

N

D

N

Dpn

ψψ ==

22

1

22

11

2

1

1

2 tancos

cos

D

L

D

D

D

D

N

N

πλ

ψψ

θθ ====

λπψλψ −==212

Normal circular path of worm=Normal circular path of worm gear

1

tanD

L

πλ =

λ - lead angle ψ - helix angle

λcosxn pp =Normal circular path of worm

28

Worm and Worm Gear

Static Equations

1DnF

λ

11,θTworm

2pnF

worm gear

2RFn - contact force

- lead angleλ - pressure anglein normal plane

nφ

nφ

λ

nnF φcos

nF

λφ coscos nnF

nnF φsinλφ sincos nnF

nφ

λφ coscos nnF

nnF φsin

λnnF φcos

nF

λφ sincos nnF

x

yz

nFµλµ cosnF

λµ sinnF

nFµλµ sinnF

λµ cosnF

( )λµλφ cossincos11 −= nnr FRT

( )λµλφ sincoscos22 += nnr FRT

22 ,θT

0&1 11 >θθ x

SOLO

Worm efficiency calculation

( )λµλφ sincoscos22 += nnr FRT

( )λµλφ cossincos11 −= nnr FRT

λµλφλµλφ

cossincos

sincoscos

1

22

1

−+=

n

n

r

r

T

TRR

Without friction

λλ

µ

sin

cos

0

1

22

1

=

=

r

r

T

TRR

λµλφλµλφ

λλ

η µ

cossincossincoscos

sincos

1

22

1

0

1

22

1

−+== =

→

n

n

r

r

r

r

WGW

T

TRR

T

TRR

0tancos

cotcos >+−=→ θ

λµφλµφη

n

nWGW

( )0>θ

or

29

Worm and Worm Gear

Static Equations

1DnF

λ

11,θTworm

2pnF

worm gear

2RFn - contact force

- lead angleλ - pressure anglein normal plane

nφ

nφ

λ

nnF φcos

nF

λφ coscos nnF

nnF φsinλφ sincos nnF

nφ

λφ coscos nnF

nnF φsin

λnnF φcos

nF

λφ sincos nnF

x

yz

( )λµλφ cossincos11 += nnr FRT

( )λµλφ sincoscos22 −= nnr FRT

22 ,θT

nFµ

λµ sinnF

λµ cosnF

nFµ

λµ cosnF

λµ sinnF

0&1 11 <θθ x

SOLO

or

Worm Gear efficiency calculation

( )λµλφ sincoscos22 −= nnr FRT

( )λµλφ cossincos11 += nnr FRT

λµλφλµλφ

sincoscos

cossincos

2

11

2

−+=

n

n

r

r

T

TRR

Without friction

λλ

µ

cos

sin

0

2

11

2

=

=

r

r

T

TRR

λµλφλµλφ

λλ

η µ

sincoscoscossincos

cossin

2

11

2

0

2

11

2

−+== =

→

n

n

r

r

r

r

WWG

T

TRR

T

TRR

0cotcos

tancos <+−=→ θ

λµφλµφη

n

nWWG

( )0<θ

30

Worm and Worm Gear

Equations of Motion (no Friction(

1DnF

λ

1Tworm

2pnF

worm gear

2RFn - contact force

- lead angleλ - pressure anglein normal plane

nφ

nφ nnF φsin

λ

nnF φcos

nF

λφ sincos nnF

λφ coscos nnF

nφ

λφ coscos nnF

nnF φsin

λnnF φcos

nF

λφ sincos nnF

x

yz

SOLO

( ) λφθω sincos1111 nnBx RFTJ −=+

( ) 2222 coscos TRFJ nnBy −=+ λφθω

1

2

2

1

2

1 tanθθλ

==

R

R

N

N

( ) ( ) λθωλθω tantan2

1212

2

1211 R

RTT

R

RJJ ByBx −=+++

22

11

2

1211

2

2

121 T

N

NT

N

NJJ

N

NJJ ByBx

−=

++

+ ωωθ

or

we obtain

multiply the first equation by and add both equations

BxBynn JN

NJJ

N

NJT

N

NTRF ωωθλφ

11

2221

2

1

221

1

222 coscos2

−+

−+

+=

multiply the second equation by and add both equations using

λtan2

1

R

R

1

2

1

2

tan N

N

R

R −=−λ

BxBynn JJN

NJ

N

NJT

N

NTRF ωωθλφ

122

112

2

2

112

2

111 sincos2 −

+

−−

+=

2

1

N

N×

211

221

1

22

2

1

212 TT

N

NJJ

N

N

N

NJJ ByBx −

=+

+

+ ωωθ

1

2

N

N×

31

Worm and Worm Gear

1T

λψ =1

xp 2p

np

xnnF φcos

nnF φcos

λφ coscos nnF λφ coscos nnF

λφ sincos nnF

worm

worm geertooth

worm geeraxis

worm geer

λφ sincos nnF

22 ,θT

1,θ

λφ sincos nnF

λφ coscos nnF

nnF φcos

nnF φsin

λ1θµ signFn

nnF φsin

λφ sincos nnFnnF φcos

λφ coscos nnF

1θµ signFn

λ

λλ

zy

SOLO

Equations of Motion (including Friction)

( ) ( )θλµλφθω signRFTJ nnBx cossincos1111 +−=+

( ) ( ) 2222 sincoscos TsignRFJ nnBy −−=+ θλµλφθω

( ) ( )λ

θµλθωλθωcos

tantan 1

2

1212

2

1211

signRF

R

RTT

R

RJJ n

ByBx −−=+++

λθµωωθ

cos1

2

121

2

1211

2

2

121

signFR

N

NTT

N

NJJ

N

NJJ n

ByBx −

−=

++

+

or

we obtain1

2

2

1

2

1 tanθθλ

==

R

R

N

N

multiply the second equation by and add both equations using

λtan2

1

R

R

λθµωωθ

sin2

211

22

1

212

2

1

222

signFR

TTN

NJ

N

NJ

N

NJJ n

ByBx −−

=+

+

+

1

2

N

N×

32

Roller ScrewSOLO

33

34

Gear BoxesSOLO

35

Differential GearsSOLO

36

SOLO Gears

References

1. Wolfram Stadler, “Analytical Robotics and Mechatronics”, McGraw-Hill, 1995

2. Hamilton H. Mabie and Fred W. Ocvirk, “Mechanism and Dynamics of Machinery”, SI Version, Third Edition, John Wiley & Sons, 1978

3. Joseph E. Shigley, Charles R. Mischke, Richard G. Budynas, “Mechanical Engineering Design”, Seventh Edition, McGraw-Hill, 2004

Antikythera References

1. Ivars Peterson, “Newton’s Clock, Chaos in the Solar System”, Freeman, 1993

http://en.wikipedia.org/wiki/Antikythera mechanism2.

January 4, 2015 37

SOLO

TechnionIsraeli Institute of Technology

1964 – 1968 BSc EE1968 – 1971 MSc EE

Israeli Air Force1970 – 1974

RAFAELIsraeli Armament Development Authority

1974 – 2013

Stanford University1983 – 1986 PhD AA