profile – shifted involute gearing€¦ · profile – shifted involute gearing . 1....

Lecture 5

Profile – Shifted Involute Gearing

1 Interference: For correct tooth action, the points of contacts on two mating

teeth must lie on the involute profiles. If the Addendum of one tooth

is too large, however, contact may occur between the tip of that tooth

and the non-involute portion of the matting tooth between the base

circle and the Dedendum circle. This causes undercutting of the

matting tooth and interference is occurring.

For no interference between the teeth, the first and last points

of contact must lay between the points of tangency F1 & F2, as shown

in fig (2), i.e. the addendum circles must cut the common tangent to

the base circles between F1 & F2. The limiting case occurs when the

addendum circles pass through these points, and since the limiting

addendum for the pinion is larger than that for the wheel, it is

usually interference between the tips of the wheel teeth and the flanks

of the pinion teeth which has to be prevented. This limits either the

maximum wheel addendum or the minimum number of teeth on the

pinion.

2 Methods of Avoiding Interference: One of the following methods may avoid interference between

the tips of the wheels teeth and the flanks of the pinion teeth:

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1 Undercutting the flanks of the pinion teeth or otherwise

modifying their profiles. However, this leads to a weakening of

the tooth and complication of manufacturing. Alternatively, the

tip of the wheel teeth may be modified.

2 Increase the centre distance slightly, so a correct tooth action is

maintained but the pressure angle is increased. This increase

would lead to a higher tooth pressures for a given torque

transmitted and backlash is occurred.

3 Tooth Correction, i.e. modifying the wheel and pinion addenda.

The wheel addendum is decreased by moving the rack – cutter

towards the wheel axis while generating the teeth. On the same

time, the pinion addendum is increased by moving the cutter

away from pinion axis. The reduction in the wheel addendum is

usually made equal to the increase in the pinion addendum to

maintain the same working depth. The pressure angle, centre

distance, and the base circles remain unaltered, but the

thickness of the wheel tooth at the pitch circle becomes less

than (to /2) and that of the pinion becomes greater than (to /2).

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33

Lecture 6 and Lecture 7

Design Procedure for Spur Gears

1 Introduction:

1 Follow the items in tables (22/5 to 22/7) for checking the load capacity of the Spur Gear.

2 Use tables and diagrams for calculations of Spur Gear mentioned in section (22/7).

2 Suggested Steps for Design, see tables (22/5 to 22/10):

1 Indicate Gear Application. 2 Fix the Operating data:

The peripheral velocity along the rolling circle:

311

1 101.19 xdnv =

n1 = nominal rpm of the pinion.

The nominal torque on pinion:

1

11

716n

NM ×=

N1 = nominal horsepower. The face width, b:

2.11

≤bdb

For straddle mounting,

7.01

≤bdb

For overhanging mounting

The nominal peripheral force along rolling circle:

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odM

U3

1 102 ×=

The nominal load intensity:

1ball db

UB⋅

=

db1 = rolling circle diameter.

3 Main Dimensions: The centre distance, a:

221 oo dd

a+

= For no correction

( )

1

2

2

2

1

16

22

11

18.2

10

ZZi

Bai

nNb

mdZ

mdZ

all

o

o

=

⋅+

⋅×=

=

=

The helix angle β = 0 Without correction:

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4 Teeth: Standard Teeth (α = 20°) 5 Tooth Error:

6 Contact Ratio:

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7 Gear Dimensions:

37

8 Load Intensity:

38

9 Factors:

39

40

10 Material:

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11 Lubricant:

12 Safety Factors: Required: SB1, SB2, SG1, SG2, and SF on the meshing teeth

1. Safety factors against Breakage:

21

22

11

11

w

DB

w

DB

qZBwS

qZBwS

⋅⋅=

⋅⋅=

σ

σ

2. Safety factors against Pitting:

1

1

2

22

1

11

+⋅

⋅=

+⋅

⋅=

ii

yBwK

S

ii

yBwK

S

w

DG

w

DG

3. Safety factors against Scoring:

1cos

+⋅

⋅⋅⋅

=i

iyyBw

KS

fe

otestF

β

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13 Full Load Life:

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Lecture 8

Helical Gear Design The profile of the tooth is the involute, and its generation is

similar to that considered in the Spur gears. The tooth is cut by the same methods for Spur gears in spite of that the centreline is inclined by the helix angle β to the cutter feeding direction. 1 Helical Gear Characteristics:

βο = Helix angle at pitch circle.

βg = Helix angle at pitch circle. αon = Pressure angle at normal section. αo = Pressure angle at transverse section. mn = Module at normal section. m = Module at transverse section. Zn = No. of teeth at normal section. Z = No. of teeth at transverse section. don = Pitch circle diameter at normal section. do = Pitch circle diameter at transverse section.

εn = Contact ratio at normal section.

ε = Contact ratio at transverse section.

nng

ononn

g

ono

ogn

ogn

on

onog

o

ono

mZddmZdd

ZZZZ

mm

⋅==⋅==

⋅=

⋅=

⋅=

⋅=

=

222

2121

1

32

231

1

cos&

cos

coscos&

coscos

cos

cossinsincostantan

ββ

ββββ

β

αβββαα

44

gonn

nngn

o mee

mee

βα

βεαπ

ε 221321 coscos

coscos ⋅

+=⋅=

⋅⋅+

=

The overlap contact ratio:

πβ

πβ

ε⋅

⋅=

⋅⋅

=n

oosp m

bm

b sintan

2 Forces on Helical Gears:

The Tangential load = 1

2

odMU ⋅

=

The Radial load = o

rPβα

costan

=

The Axial load = oLP βtan=

The direction of PL depends on the direction of rotating, direction of the Helix, and if a gear is driver or driven.

The design procedure is the same as for the Spur gear. So, to apply the procedure, follow the items on page 125 to 128 with understanding lectures 7 & 8.

45

Lecture 9

Examples on Correction of Gears

46

47

48

49

50

51

52

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Lecture 10 and Lecture 11

Design of Bevel Gears 1 Types, properties, and use: Fig (23/1), p.145 shows different types of bevel gearing:

a) Straight tooth أسنان مستقيمة تمر بالمرآز

b) Helical tooth تمر بالمرآز أسنان مائلة ال

c) Hypoid tooth هايبويد –أسنان منعطفة

d) Displaced bevel gears, hypoid toothed (a) العزامة بالمسافةالمحاور

:مالحظات عامة

).a( واحدة بالمرآز تتقاطع المحاور في نقطة •

δA = 90o : تكون الزاوية بين المحاور عادة •

وفي هذه الحالة . تكون المحاور متخالفة، غير متقاطعة و ال متوازية) d(في النقطة •

:نالحظ ما يلي

و التي تستخدم مثال في المحور الخلفي للسيارة لتقليل ) a(وجود المسافة .1

إمكانية تمـديـد األعمدة من آال ، و الصوت و الضجيج الناتج من الحرآة

.الجانبين ليتم تثبيت األعمدة بشكل أفضل

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.الكفاءة تكون أقل من التروس المخروطية األعتيادية .2

الحرارة الناتجة تكون أآبر نتيجة لإلنزالق الكبير بين األسنان و يجب إستخدام .3

).Hypoid Oil(او ) EP(زيت خاص يرمز له بــ

إلمكان لتقـليل الخسارة الناتجة من قليلة قدر ا) a(يجب ان تكون المسافة .4

لكي يتم تقليل ) a(اإلحتكاك، و يجب إتخاذ دقة خاصة بالتصميم ألختيار

.الصوت و الضوضاء

2 Geometry and Dimensions: According to Fig (23/2), p. 146, various bevel gears 2A to 2D

can theoretically be mated correctly with one another and with a

given gear (1), giving line contact of tooth flanks.

:هذا الشكل يمثل مقدار زاوية المخروط و مسنن التاج و آما يلي

A 90(مسنن خارجي و بزاوية أقل منo (}o90δA < {.

B 90(مسنن خارجي و بزاوية تساويo (} o90δA = {.

C لقطع آلة ا= مسنن التاج}Crown wheel{.

D 90(مسنن داخلي و بزاوية أآثر منo (} o90δA > {.

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2-1 Specification of the cones and their angles: Definitions of rolling cone and pitch cone, Addendum,

Dedendum cone, back cone and supplementary Back cone, see Fig

(23/6), Fig (23/11) and Fig (23/12). Also, see table (23/2) p.156.

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3 Dimensioning and load capacity of bevel gears:

3-1 Determination of Dimensions, (p.155): The recommended values for the choice of the number of teeth,

tooth width, etc. are given in table (23/1).

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58

3-2 Equivalent spur wheels: The calculation of load capacity of straight bevel gears can be

based on that of an equivalent pair of spur wheels as shown in Fig

(23/17):

3-3 Load capacity of the bevel gears (p.157): Load value:

11

161043.1mdn

NU⋅

×= [Kgf]

b

wb

bbe

m

fd

d

dFbb

dde

−=

⋅==

=

1

,sin

,cos

11

1

1

1

11

δ

δ

59

Load intensity for equivalent spur wheel (Be):

311

16

1

1043.1m

d

eee dn

fNdb

UB⋅⋅

⋅×=⋅

= [Kgf / mm2].

Approximate design:

3

1

11 113

all

dm Bn

fNd

⋅⋅

⋅≥ [mm].

3-4 Bearing forces and construction (p.158 & p.159):

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3-5 Calculation Examples:

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