FOUNDATION of MECHANICS 1FOUNDATION of MECHANICS 1 · Bevel gears Worm-gear Spur gears Helical...

FOUNDATION of MECHANICS 1FOUNDATION of MECHANICS 1

Presentation08: Gears

Outline

• Introduction: power transmission with constant transmission ratio; parallel axes; incident axes; skew axes.

• Elements of geometry and kinematics of spur gears: conjugate profiles; primitive curves; fundamental law of gearings; generation of conjugate profiles by envelop; involute curve; main definitions and properties of gears; spur p; ; p p g ; pgears; rack; gear manufacturing (mention).

• Helical gears: geometry and kinematics; main definitions and properties.Helical gears: geometry and kinematics; main definitions and properties.

• Incident axes transmission: spherical motion; bevel gear geometry & kinematics.

• Skew axes transmission: worm-gear geometry & kinematics; other solutions.

INTRODUCTION

Transmitting power between twoTransmitting power between twoshafts with a constant transmissionratio is one of the most commonproblems in designing machines,that can be solved with a numberof different solutions, i.e. using:

• Linkages

Fl ibl d i• Flexible devices

• Wheels

Friction wheelsDriven wheel

Driving wheel

o Friction wheels

o Gears

INTRODUCTION

Gears and Gearings

Bevel gears Worm-gear

Spur gears Helical gears

INTRODUCTION

Gears and Gearings

Simple gearings

Shaft 3

Simple gearings(gear trains)

Shaft 2

Shaft 1

Planetary (or epyciclic)gearings

(car differential gearing)

ELEMENTS OF KINEMATICS

(21)Conjugate profiles

1 tn(21)

(21) (21)M

M Mt

t

v

v v2

1

M2 (21)

Mn v 0

(21)(2)

(1)Mv

(21) (1)(2)

M

C12

3O1 O2

(21)Mv

(2)Mv

(21)

2 2

(1)(2)

( )M MM

M O

vω

vv

3

12 12 12

(21) (2) (1)0C C C v v v( )C O

1 1( )M O ω

12

(2)2

(1)

( )

( )C 12 23C C

C C

v ω

v ω

2 12 1

1 12 2

( )( )C OC O

12

( )1 ( )C 12 13C C v ω


Fundamental Law of Gearing

1 n

M

1 2t

2M

(21)Mv

12 1( )( )C OC O

pitch point

C12

3O1 O2

Mv12 2( )C Opitch point

line of centres C12

3

line of action (line of contact)

The transmission ratio of two mating gears is constant if, and only if, the locationof their relative centre of rotation is stationary with respect to the absolute centres.y p


Primitive curves

2

1

3O O

M ≡ C12 12

(21) (21)

(2) (1)

0M C v vO1 O2 (2) (1)

M Mv v

2 12 1( )C O 2 12 1

1 12 2

( ) .( )C O constC O

Primitive curves of the motion are the geometrical loci of the relative centre of rotation (seen by the two links respectively). The relative motion between the links can be described by the rolling motion of the primitive curves.In the case of a gear pair, the primitive curves are two circles (pitch circles).

ELEMENTS OF GEOMETRY&KINEMATICS

Generation of conjugate profiles *

Envelope of a family of curves in the plane is a curve that is a curve tangent to each member of gthe family at least in one point.

Generation of conjugate profiles by Cs2

s

σ1

ε

μ

enveloping proper curves.

The envelope of line μ for a rolling

s1

σ2

ρ2

of line ε over the two primitive curves σ1 and σ2 generates two conjugate profiles. The conjugate

R2

profiles are involute curves.

Being circles the two primitive curve the transmission ratio is

* see also:- Paul B., Kinematics and dynamics of planar machinery, Prentice-Hall, New Jersey, 1979.

curve, the transmission ratio is constant.

- Doughty S., Mechanics of Machines, John-Wiley & Sons, New York, 1988.


Involute curve *

ρ1

K1

C

Involutecurve

M

ρ1ρ2

K2a

KL

K1

ρBase circle

K2K1

* see also:- Paul B., Kinematics and dynamics of planar machinery, Prentice-Hall, New Jersey, 1979.

K- Doughty S., Mechanics of Machines, John-Wiley & Sons, New York, 1988.

K2


Involute as a gear tooth *

R

αCs2ρ1

σ1μ

R12

ρ R R′

α

C

s1α

ρ2α

ε

α

σ2

R2

* see also:Doughty S Mechanics of Machines- Doughty S., Mechanics of Machines,

John-Wiley & Sons, New York, 1988.


Involute as a gear toothO2

1 1 1

' '

K M K L

K M K L

2 2 2

' '

K M K L

K M K L

2

2 1 1 1

1 1' '

K M K L

MM L L

2 2 2

2 2' '

K M K L

MM L L

K2L'2 L2

M'

1 1 1 1 2 2 2 2' 'L L L L

C121O C R

K1

ML1L'1

M

1

C12

α2

2 12 2 ( )O C R

1 12 1 cos( )O C R

O

1

1

2 2 1

α

1R

2 12 2 cos( )

O11 1 2

2R


Involute as a gear tooth

HH'

'LL 'MM

LL' 'HH R

'HH R 1'HH '

HH RLL

1

cos

'HHMM

R cos( )R

O


Variation of the centre distance

α α′

ρ

C′

ρ2

ρ1

C

a Δa

a′

'2 1 1 1

'

R RR R

1 2 2 2R R


Involute as a gear tooth

Pitch circles are the primitive curves of motion and are defined for a gear pair

Base circle is the evolute curve of the involute profile. It is a geometrical characteristic of a certain geardefined for a gear pair. characteristic of a certain gear.


ODefinitions and properties

O2

2

2

K2L'2 L2

MM'

α

K1M

L1L'1

1

αα

Pressure angle

O1

1

Line of action (or contact)O1


Definitions and properties

Doughty S., Mechanics of Machines, John-Wiley & Sons, New York, 1988.



11 12 21 22'MM L L L L

O2

1 1 2 2 2

1 22 2Z Z K2

L22 L21

M'

2

1 2Z Z

2K1

ML11L12

M

1

Z

pb2

Base Pitch

O

1

1

O1


O2Definitions and properties

11 12 21 22H H H HR R

1 1 2 2R R

2 2R R K2

2

2

1 21 2

R RZ Z

2H22

H21

HH

K2

RZ

p 2 Circular Pitch

H11H12

K11

cosppp

p m m:= Module

O

1

cosppR

pb O1



Re

R eiRi

R

R R

Ri

R Re

i

R R eR R i

e

i

R R eR R i

2.25h e i m


The rack

R

σ1

μ

p0 = m0Cs1

ε

s2

n

= =

0

2

= 2.

5 m

0

== linea di riferimentoPitch (or reference) line

h = =


Gear manufacturing *

Gears are generally manufactured by means of tool machines (gear cutting machines) that create the tooth profiles through the generation method, by exploiting the relative motion of pure rolling between the two primitive surfaces (cutting primitives) for enveloping involute tooth profiles.

* see also:Paul B., Kinematics and dynamics of planar machinery, Prentice-Hall, Newplanar machinery, Prentice Hall, New Jersey, 1979.


Gear manufacturing

Cutting gears through the generation method can be easily understood by thinking the gear to be cut as made of mouldable material (e.g. Plasticine) and the cutter gear or rack as rigid (e.g. made of steel).

In gear cutting machines there are three important motions apart from the rolling of theimportant motions, apart from the rolling of the cutting primitive surfaces:

• the cutting motion (e.g. in the case of a cutting rack it is a reciprocating translational motion along the direction of the teeth);

• the feeding motion (to make the cutter• the feeding motion (to make the cutter remove material by little steps)

• the positioning motion (e.g. to take the gear back to the initial position of the rack stroke).


Gear manufacturing

Cutting machines can be divided into:

• Cutting-slotting machines• Cutting-slotting machines, where the tools is provided with a reciprocating translational motion (e.g. cutting rack, cutting gears, F ll h )Fellows gear-shaper);

• Cutting-hobbing machines with a milling cutter (hob)with a milling cutter (hob) provided with a cutting motion which is rotational and continuous.


Gear manufacturing

Cutting machines can be divided into:

• Cutting-slotting machines• Cutting-slotting machines, where the tools is provided with a reciprocating translational motion (e.g. cutting rack, cutting gears, F ll h )Fellows gear shaper);

• Cutting-hobbing machines with a milling cutter (hob)with a milling cutter (hob) provided with a cutting motion which is rotational and continuous.


Gear manufacturing

Other methods (less significant):Other methods (less significant):

• machining the tooth profiles using modular milling cutters;using modular milling cutters;

• generation of internal gears by means of broaching machines.


Properties

'2 1 1 1

'1 2 2 2

R RR R

α α′

ρ1

C′

O O ′

1 2' ' '

1 2

R R a

R R a

Primitive radii of cuttingρ2

ρ1

CO2

O1 O1

1 2

' '1 1

R a

Primitive radii

aa′

Δa

''

2 1aR

Primitive radii of motion

' ' ' ' '1 2 1 2 1 2

'

( ) cos( ) ( ) cos( ) cos( ) cos( )

( ) ( )

R R R R a aa

P l f ti'cos( ) cos( )a

Pressure angle of motion


Propertiesp = m

α α′

ρ1

p0 = m0

= =

OO1 O1′

C′

ρ2

h =

2.5

m0

== linea di riferimentoPitch line

O2C

0 02p R mZ Cutting

module/circular pitch

a

a′

Δa

0cos( ) cos( )2

m ZR '0 '

cos( )cos( )

m m

Motion module

'' ' '

2

cos( ) cos( )2

m ZR ' '

cos( )p m

Motion circular pitch


Contact RR2

K2

2

N2

N2 C

K2

A2B

N2

C

N1B1

CA1

2B2

α

α

1

N1

1

R1

K1

1

N N

O1

1

1 2

1 1 2 2

N N

A B A B

Length of contact

Arc of contact 1 2

1 1 cos( )N NA B

1


Contact

For a uniform transmission of motion, it is necessary that when a tooth pair is loosing contact another tooth pair is already meshing. This means that the

cos( )A B p N N p p

arc of action must be greater that the circular pitch:

1 1 1 2 cos( ) bA B p N N p p

The ratio between the arc of action and the circular pitch is called contact

A B

ratio and must be greater than 1

1 1 1A Bp

generally 1 2


Interference

The length of contact is determined by the intersection of the line of action with the two addendum circles. If the

t t t f thcontact occurs out of the segment K1K2 the action is non-conjugate. This situationnon conjugate. This situation is referred to as interference, and must be avoided.


Interference

Avoiding interference means setting an upper bound to the value of the two gear addendum radii (Re1,2):

2 2 2 21 1 2 2sin ( ) sin ( )e b e bR R a R R a

K1

Re2lim

Rb1

O1 O2

C

N1

Re1lim

O1 O2

Rb1

K

N2

K2


Interference

2 21 1 1 sin ( )e bR R e R a

2 22 2 2 sin ( )e bR R e R a

For standard gears:

1 2 1 2( 1.25 )e e e m i i i m

1 2

1 2

2 2R RmZ Z

An upper bound for e (i.e. for m) entails a lower bound for Z1 and Z2:

1 2Z Z

Z Z Z Z 1 1min 2 2minZ Z Z Z


Interference

R2The condition is more critical (implying a tighter limitation) for2

KR2 + e2 lim

(implying a tighter limitation) for “big” driven gears (having a high pitch radius R2).

N2

K2p 2)

C e2 lim

K11


InterferenceR R The condition is more critical

(implying a tighter limitation) for ll i i (h i l

R2

K2

R2 + e2 lim

small pinions (having a low number of teeth Z2).

N2

C

2

e2 lim

B1

CA1

K

2 lim

The most critical condition is therefore the pair rack-pinion.

R

K1O'1t e e o e t e pa ac p o

R1

O1


Interference

12 2R

It can be demonstrated that for a pinion-gear pair:

1

1 1min 2 2 22 2 1 1 2

2 22 sin 1 1 2 sin

Rz zR R R R R

For a rack pinion pair:

Zmin

= 20° = 1/2

Zmin

For a rack-pinion pair:

120 ( )R R

R

2

min 0 22

2 2limsin1 1 2 i

R

z

2 sin1 1 2 sin


Interference

Undercutting is the consequence of interference wheninterference when manufacturing gears with the generation method g(envelop). If the gear to be cut has a number of teeth smaller than Zmin, the tool removes material from the blank where it should notblank where it should not.

min

Standard gear:

20 17Zpinion with Z = 8

min20 17Z

FOUNDATION of MECHANICS 1FOUNDATION of MECHANICS 1 · Bevel gears Worm-gear Spur gears Helical...

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