Chapter 2 Gear Drive-4

8/10/2019 Chapter 2 Gear Drive-4

1/39

(1) Features of helical gear

Contact line leans at an angle with respect to the axis of gear (Left/Right)

There are more teeth engaged simultaneously, a greater contact ratio

Smooth transmission, low noise, high load capacity

Normal module mn1=mn2

Normal pressure angle n1=n2

Helical angle 1=

2

(2) Engagement conditions

1. Introduction

2.7.3 Design of helical gear
http://localhost/var/www/apps/conversion/tmp/scratch_2/0808/%E6%9C%BA%E6%A2%B0%E8%AE%BE%E8%AE%A1/%E7%AC%AC%E4%B8%80%E7%AB%A0%20%20%E7%BB%AA%E8%AE%BA/%E6%96%9C%E9%BD%BF%E5%9C%86%E6%9F%B1%E9%BD%BF%E8%BD%AE%E4%BC%A0%E5%8A%A8.exe


2/39


3/39

Directions of force component

Ft, Fr

Axial force Fa

Identical to the spur gear

For driving gear, Fa1can be decided by Right/left hand rule.

Decided by Direction of helical lineand rotational direction.

Left hand for left hand helical, Right hand for right hand helical.

If bending direction of fingers matches with the rotational direction,

tip of thumb points to the direction of Fa1.

Ft1

Fr1Fa1

Driving

gear

For driven gear, direction of Fa2is opposite to that of Fa1.


4/39

Relationship between force components

n1 n2F F r1 r 2F F a1 a 2F F t1 t 2F F

Example

n1

n2

1

2

Top view

Right hand

helical

Left hand

helical

Fa2Fa1

Ft1

Ft2

Fr2

Fr1

n2

n1

Side view

Fa1

Fa2 Ft1Ft2

Fr2

Fr1


5/39

Principles:

3. Contact strength of helical gear

Module = Normal module of helical gear mn

Pressure angle = Normal pressure angle of helical gear n

Number of teeth = Virtual number of teeth of helical gear zv= z /cos3

Normal force = Normal force of helical gear Fn

2

1

1cos

ddv

3

11

cos

TTv

cos

bbv u

z

z

z

zu

v

v

v

1

2

1

2

(1) Considering the strength of an equivalent virtual spur gear

(2) Contact line has an angle with respect to axis, which is advantageous to

reduce contact stress. So we introduce a coefficient of helical angle Z.


6/39

For spur gear

For helical gear

1H E H 2

1

2 ( 1)KT uZ Z Z

bd u

1H E H 2

1

2 ( 1)KT uZ Z Z Zbd u

cosZ Coefficient of

contact ratio

Under same condition, H_Helical < H_Spur

So, Helical gear has a greater load capacity than spur gear.

Coefficient of local

area at pitch point

(Fig. 2-18)

Coefficient of helical angle

0.75 ~ 0.88Z

1

1

cos

nm z

d

b

The greater number of teeth, the less value of Z.

tt

b

HZ

tancos

cos22

Coefficient of elasticity

(Table 2-15)


7/39

1H E H HP1 HP22

1

2 1min[ , ] (MPa)

KT uZ Z Z Z

bd u

For checking contact strength

H E 2 13

1

HP d

2 1( ) (mm)Z Z Z Z KT udu

HP1 HP2min[ , ] d 1b d

For designing diameter of pitch circle

See Table 2-14


8/39

For design of geometries

H E 2 13

1

HP

Z Z Z Z 2 1( ) (mm)

d

KT ud

u

Recalculating d1, d2, accurate to the three decimal places

First computing d1 , then specifying other geometries:

Number of pinion teeth z1(soft surface, closed gear drive20~40hard

surface, closed gear drive or open gear drive1725)

Initial helical angle , most commonly in range of 10~ 15

Computing mn= d1cos/z1Rounded up to a basic value and mn1.5 forpower driving

Computing center distance a=(d1+ d2)/2=mn(z1+ z2) / (2cos)Rounded up to

an integer

Recalculating= cos1 [mn(z

1+ z

2) / 2a ]Accurate to second

add

2

21

cos

nzm

dAttention: Satisfying


9/39


10/39

For checking the bending strength:

1F1 Fa1 Sa1 FP1

1 n

2(MPa)

KT

Y Y Y Y bd m

2

1 Fa Sa3n 2

d 1 FP

2 cos (mm)

KT Y Y Y Ym

z

1F2 Fa2 Sa2 FP2

1 n

2(MPa)

KTY Y Y Y

bd m

The greater value

between pinion and gear

For designing the normal module:


11/39


12/39

2.7.4 Design of bevel gear

Gear drive with vertically intersecting axes

Teeth on a conical shaped surface

Uneven distribution of load along the tooth width

b

We only investigate bevel gear of straight teeth,

and axis angle is 90

.

1. Features of bevel gear drive

Significant vibration and noise, only for low speed transmission, v5m/s

Tooth profile: Straight teeth, helical teeth, curved teeth

Geometrical factors on large end are standard

For convenience of computing, we assume

Normal force Fn, a concentration force on the pitch cone at the midface of teeth.

Strength of bevel gearStrength of virtual spur gear at the midface of teeth.

0.5b

Fn

Large end


13/39

2. Geometries of straight bevel gear and its virtual spur gear

1) Geometries of straight bevel gear

Module at the larger end m is standard value,Dia. of pitch circle

2211 ; mzdmzd

Ratio of teeth number uz2 / z1d2 / d1

Cone angle of pitch circle

uu 21 tan;/1tan

Cone distance2

1 15.0 udR Coefficient of

tooth widthRb/R

Dia. of pitch circle at the midface of teeth

1Rm1 )5.01( dd

Module at the midface of teeth mm )5.01( Rm

dm1
http://localhost/var/www/apps/conversion/tmp/scratch_2/0808/%E6%9C%BA%E6%A2%B0%E8%AE%BE%E8%AE%A1/teaching/%E6%9C%BA%E6%A2%B0%E8%AE%BE%E8%AE%A1/2007%E6%9C%BA%E6%A2%B0%E8%AF%BE/%E5%BC%A0%E8%80%81%E5%B8%88%E9%83%A8%E5%88%86%E7%94%B5%E5%AD%90%E6%95%99%E6%A1%88/3%E7%AB%A0%EF%BC%8D%E9%BD%BF%E8%BD%AE%E4%BC%A0%E5%8A%A8%E8%AE%BE%E8%AE%A1/t3%EF%BC%8D21a.jpghttp://localhost/var/www/apps/conversion/tmp/scratch_2/0808/%E6%9C%BA%E6%A2%B0%E8%AE%BE%E8%AE%A1/teaching/%E6%9C%BA%E6%A2%B0%E8%AE%BE%E8%AE%A1/2007%E6%9C%BA%E6%A2%B0%E8%AF%BE/%E5%BC%A0%E8%80%81%E5%B8%88%E9%83%A8%E5%88%86%E7%94%B5%E5%AD%90%E6%95%99%E6%A1%88/3%E7%AB%A0%EF%BC%8D%E9%BD%BF%E8%BD%AE%E4%BC%A0%E5%8A%A8%E8%AE%BE%E8%AE%A1/t3%EF%BC%8D21a.jpghttp://localhost/var/www/apps/conversion/tmp/scratch_2/0808/%E6%9C%BA%E6%A2%B0%E8%AE%BE%E8%AE%A1/teaching/%E6%9C%BA%E6%A2%B0%E8%AE%BE%E8%AE%A1/2007%E6%9C%BA%E6%A2%B0%E8%AF%BE/%E5%BC%A0%E8%80%81%E5%B8%88%E9%83%A8%E5%88%86%E7%94%B5%E5%AD%90%E6%95%99%E6%A1%88/3%E7%AB%A0%EF%BC%8D%E9%BD%BF%E8%BD%AE%E4%BC%A0%E5%8A%A8%E8%AE%BE%E8%AE%A1/t3%EF%BC%8D21a.jpghttp://localhost/var/www/apps/conversion/tmp/scratch_2/0808/%E6%9C%BA%E6%A2%B0%E8%AE%BE%E8%AE%A1/teaching/%E6%9C%BA%E6%A2%B0%E8%AE%BE%E8%AE%A1/2007%E6%9C%BA%E6%A2%B0%E8%AF%BE/%E5%BC%A0%E8%80%81%E5%B8%88%E9%83%A8%E5%88%86%E7%94%B5%E5%AD%90%E6%95%99%E6%A1%88/3%E7%AB%A0%EF%BC%8D%E9%BD%BF%E8%BD%AE%E4%BC%A0%E5%8A%A8%E8%AE%BE%E8%AE%A1/t3%EF%BC%8D21a.jpghttp://localhost/var/www/apps/conversion/tmp/scratch_2/0808/%E6%9C%BA%E6%A2%B0%E8%AE%BE%E8%AE%A1/teaching/%E6%9C%BA%E6%A2%B0%E8%AE%BE%E8%AE%A1/2007%E6%9C%BA%E6%A2%B0%E8%AF%BE/%E5%BC%A0%E8%80%81%E5%B8%88%E9%83%A8%E5%88%86%E7%94%B5%E5%AD%90%E6%95%99%E6%A1%88/3%E7%AB%A0%EF%BC%8D%E9%BD%BF%E8%BD%AE%E4%BC%A0%E5%8A%A8%E8%AE%BE%E8%AE%A1/t3%EF%BC%8D21a.jpghttp://localhost/var/www/apps/conversion/tmp/scratch_2/0808/%E6%9C%BA%E6%A2%B0%E8%AE%BE%E8%AE%A1/teaching/%E6%9C%BA%E6%A2%B0%E8%AE%BE%E8%AE%A1/2007%E6%9C%BA%E6%A2%B0%E8%AF%BE/%E5%BC%A0%E8%80%81%E5%B8%88%E9%83%A8%E5%88%86%E7%94%B5%E5%AD%90%E6%95%99%E6%A1%88/3%E7%AB%A0%EF%BC%8D%E9%BD%BF%E8%BD%AE%E4%BC%A0%E5%8A%A8%E8%AE%BE%E8%AE%A1/t3%EF%BC%8D21a.jpghttp://localhost/var/www/apps/conversion/tmp/scratch_2/0808/%E6%9C%BA%E6%A2%B0%E8%AE%BE%E8%AE%A1/teaching/%E6%9C%BA%E6%A2%B0%E8%AE%BE%E8%AE%A1/2007%E6%9C%BA%E6%A2%B0%E8%AF%BE/%E5%BC%A0%E8%80%81%E5%B8%88%E9%83%A8%E5%88%86%E7%94%B5%E5%AD%90%E6%95%99%E6%A1%88/3%E7%AB%A0%EF%BC%8D%E9%BD%BF%E8%BD%AE%E4%BC%A0%E5%8A%A8%E8%AE%BE%E8%AE%A1/t3%EF%BC%8D21a.jpghttp://localhost/var/www/apps/conversion/tmp/scratch_2/0808/%E6%9C%BA%E6%A2%B0%E8%AE%BE%E8%AE%A1/teaching/%E6%9C%BA%E6%A2%B0%E8%AE%BE%E8%AE%A1/2007%E6%9C%BA%E6%A2%B0%E8%AF%BE/%E5%BC%A0%E8%80%81%E5%B8%88%E9%83%A8%E5%88%86%E7%94%B5%E5%AD%90%E6%95%99%E6%A1%88/3%E7%AB%A0%EF%BC%8D%E9%BD%BF%E8%BD%AE%E4%BC%A0%E5%8A%A8%E8%AE%BE%E8%AE%A1/t3%EF%BC%8D21a.jpghttp://localhost/var/www/apps/conversion/tmp/scratch_2/0808/%E6%9C%BA%E6%A2%B0%E8%AE%BE%E8%AE%A1/teaching/%E6%9C%BA%E6%A2%B0%E8%AE%BE%E8%AE%A1/2007%E6%9C%BA%E6%A2%B0%E8%AF%BE/%E5%BC%A0%E8%80%81%E5%B8%88%E9%83%A8%E5%88%86%E7%94%B5%E5%AD%90%E6%95%99%E6%A1%88/3%E7%AB%A0%EF%BC%8D%E9%BD%BF%E8%BD%AE%E4%BC%A0%E5%8A%A8%E8%AE%BE%E8%AE%A1/t3%EF%BC%8D21a.jpghttp://localhost/var/www/apps/conversion/tmp/scratch_2/0808/%E6%9C%BA%E6%A2%B0%E8%AE%BE%E8%AE%A1/teaching/%E6%9C%BA%E6%A2%B0%E8%AE%BE%E8%AE%A1/2007%E6%9C%BA%E6%A2%B0%E8%AF%BE/%E5%BC%A0%E8%80%81%E5%B8%88%E9%83%A8%E5%88%86%E7%94%B5%E5%AD%90%E6%95%99%E6%A1%88/3%E7%AB%A0%EF%BC%8D%E9%BD%BF%E8%BD%AE%E4%BC%A0%E5%8A%A8%E8%AE%BE%E8%AE%A1/t3%EF%BC%8D21a.jpghttp://localhost/var/www/apps/conversion/tmp/scratch_2/0808/%E6%9C%BA%E6%A2%B0%E8%AE%BE%E8%AE%A1/teaching/%E6%9C%BA%E6%A2%B0%E8%AE%BE%E8%AE%A1/2007%E6%9C%BA%E6%A2%B0%E8%AF%BE/%E5%BC%A0%E8%80%81%E5%B8%88%E9%83%A8%E5%88%86%E7%94%B5%E5%AD%90%E6%95%99%E6%A1%88/3%E7%AB%A0%EF%BC%8D%E9%BD%BF%E8%BD%AE%E4%BC%A0%E5%8A%A8%E8%AE%BE%E8%AE%A1/t3%EF%BC%8D21a.jpg


14/39


15/39


16/39

Ft2=Ft1 Fr2=Fa1 Fa2=Fr1

n2

Fr2

Fa2

Fr1Ft1

Fa1

n1

Ft2

Example problem

1

2


17/39

Force checking

For design

1H H E HP2 3

R R 1

4

0.85 (1 0.5 )

KTZ Z

d u

2H E 13

1 2

HP R R

4( )

0.85 (1 0.5 )

Z Z KTd

u

4. Contact fatigue strength of bevel gear

Considering the equivalent virtual spur gear at the midface of teeth.

Ignoring the influence of contact ratio.

Effective face width

beH0.85b


18/39

Discussions

ZE, ZH, HP, identical to the process of spur gear

Often u 5, limiting the diameter of bigger bevel gear, easier to manufacture

After specifying d1 , other geometries needs to be decided

Initialz

1

Computing m=d1/z1,

rounded up to standard value

Calculating d1= mz1, z2, d2, u , et al2

1 15.0 udR

Dont round up R !

Often b1

b2, easier to assemble and coincide the vertex

mm)5.01(85.0

43

2

RR

1

2

HP

HE1

u

KTZZd

For design

Dia. of pitch circle at the large

end of pinion


19/39

Spur gear:

5. Bending fatigue strength of bevel gear

1

1F Fa Sa3 2

d

2KTY Y Y

m z

1F Fa Sa FP2 3 2 2

1

4

(1 0.5 ) 1R R

KTY Y

m z u

1 Fa Sa3

2 2 2FP1

4

(1 0.5 ) 1R R

KT Y Ym

z u

For checking

For design Virtual number of teeth v /cosz z

Bigger value

In the like manner, considering the bending fatigue strength of virtual gear.

Ignoring the influence of contact ratio, using the geometries of virtual gear,depending on

A d bl d i d i b l t i l t


20/39

Example:

A double gear reducer is driven by an electrical motor.

Inputting power P17kW.

Inputting rotational speed n11000r/min.

Speed ratio of the first stage i4.6

Unidirectional, slight impact.

Life expectancy 10 years, 8h per day.

Try to design the first gear drive.

1. Requirements

Medium speed and medium load.

2. Design plan

Plan A: Helical gear

Either soft surface or hard surface is OK.

Using helical gear for smooth movement.

Plan B: Helical gear

45 steel, soft surface

45 steel, hard surface

First gear drive

Second gear drive


21/39

3. Design process

1

Material and heat treatment

45 45Gear

45 45Pinion

H.T. of pinion H & T

Norm.

Case hardening

H.T. of gear

Table 2-11, middle valueHB of pinion 230HBS 50HRC

190HBS 50HRCHB of gear

1170Contactstress

1limH 580Fig. 2-242limH 550 1170

Case hardening

HB of pinion > HB of gear

Table 2-11

2Allowable contact stress and bending stress

Fig. 2-26

Bending

stress

1limF 220 340Fig. 2-30

Fig. 2-282limF 210 340

Design contents Design principles Plan A Plan B


22/39

load cycle 1N )830010(100016060 11 tanN 14.4108

2N iNN /12

3.13108300 day per year

Coefficient of

contact life

1NZ

Fig. 2-271

2NZ 1

Coefficient of

bending life

1NY

Fig. 2-321

1NY 1

Design content Design principles Plan A Plan B

Min. safety

coefficient

minHS Table 2-14 1

Table 2-14minFS 1.4

Allowable

contact stress

1HP minH1N1limH1HP / SZ 580 1170

2HPminH2N2limH2HP / SZ 550 1170

Allowable

bending stress

1FP minF1NST1limF1FP / SYY 314 486

2FP minF1NST1limF1FP / SYY 300 486

2STY

Pitting not

allowable

Line 1


23/39

3Specifying initial parameters

12Initial helical angle

Precision grade Table 2-1 Grade 8

Coe. of face width d Table 2-14 0.9 0.5

No. of teeth 1z Closed, soft z120~40

Closed, hard z1

17~25 26

Modification

vA KKKKK 1.898 2.234

1.25 1.25Where

K

K

Assuming the linear

speed v9m/s

2z

21z2z1iz14.6 119 97

Load coefficient

0 0

AK

Table 2-7, slight impact

vK Fig. 2-6, Kv1.02~1.2 1.1 1.1

Table 2-9, soft1~1.2hard1.1~1.35 1.15 1.25

Design content Design principles Plan A Plan B

K Table 2-8 K1.2~1.4 1.2 1.3

In the range of 10~15


24/39

4Design and calculation

Principles

Coe. of elasticityEZ

2.45

0.8Coe. of contact ratio

HZCoe. of local area

189.8

Z

Table 2-15

ZZ

0.75~0.880.989

HP2


25/39

Normal module nm

71.986mm

222.36mmaCent distance

162350N.mm

11o33'21' '

26

12cos986.71cos

1

1n

z

dm

Accurate helical

angle

Torque 1T1000

17109550109550

3

1

13

1

n

PT

Dia. of Pitch circle

Design content Design principles Plan A

2.71mm

Rounded up to 3

12cos2

)11926(3

cos2

)( 21n

zzma

Rounded up to

222

2222

)11926(3cos

2

)(cos 121n1

a

zzm

3

2

16.4

16.4

9.0

162350898.12

550

989.08.045.28.189

d


26/39

Face width

v

b272mm

Checking bending

strength

79.61mm

1vz 27.65

60000100061.79

100060

11

ndv

Virtual teeth

number

Accurate dia. of

pitch circles1d "'zm 213311cos/263cos/

o

1n

4.17m/sOK!

3v cos

zz

364.39mm2d "'213311cos/1193 o

Satisfyingd1d2/2a

65.7161.799.01d2 db

b178mmmm)10~5(21 bb

Line speed

2vz 126.54

1FaY 2.6Coe. of tooth

profile 2FaY 2.2

Design content Design principles Plan A

Fig. 2-20

FPSaFa

1n

1

F

2YYYY

dbm

KT

Rounded upb


27/39

Coe. of contact ratio

Y 0.89

Checking bending

stress

1.61

92.0~85.0Y

Coe. of stress

modification

Sa1Y

1.82Sa2Y

85.0~65.0Y

Coe. of helical angle

The endclosed, soft face

89.075.061.16.2

61.79372

162350898.12

Design contents Design principles Plan A

Fig. 2-21

Y 0.75

Bigger, bigger Y

F1Bending

strength OK!MPa100 MPa3141HP

61.16.2

82.12.2100

Sa1Fa1

Sa2Fa21F

YY

YY

F2Bending

strength OK!

MPa3002HPMPa96


28/39

22.44

Design by

bending

strength

Virtual number

of teeth

mmcos2

3

FP

SaFa

d

2

1

2

1

n

YY

z

YYKTm

v1z o33

1v1 12cos/21cos/ zz

103.65v2z o33

2v2 12cos/97cos/ zz

1FaY 2.76Coe. of tooth

form 2FaY 2.22Fig. 2-20

1.55Coe. of stress

modification

Sa1Y

1.79Sa2YFig. 2-21

Design contents Design principles Plan B

Coe. of contact

ratio

Y 0.8992.0~85.0Y

85.0~65.0Y

Coe. of helical

angle

Y 0.8The more teeth,

the less value

Closed, hard

surface


29/39

Comparingbending strength

0088.0

486

55.176.2

FP1

Sa1Fa1

YY

0082.0486

79.122.2

FP2

Sa2Fa2

YY

P

G

Considering

pinion

Normal modulus

nm 3

FP

SaFa

d

2

1

2

1

cos2

YY

z

YYKT

32

o2

0088.05.021

89.08.012cos162350234.22

mm70.2

165.87mmaCenter distance

12cos2

)9721(75.2

cos2

)( 21n

zzma

Rounded to 166


Rounded up, mn2.75mm


30/39

12o12'6' 'Accurate helical

angle1662

)9721(75.2cos

2

)(cos 121n1

a

zzm

59.08mmAccurate pitch

diameter1d "'zm 61212cos/2175.2cos/

o

1n

272.92mm2d "'61212cos/9775.2 o

Face width

v

b230mm

60000

100008.59

100060

11

ndv

3.09m/s

OK!

54.2908.595.01d2 db

b

1

35mmmm)10~5(21bb

Linear speed

b

Checking

contact stress MPa

12

HP21

1

HEH

u

u

bd

KTZZZZ

Coe. of elasticity EZ

2.45HZ

Coe. of local area

189.8Table 2-15 MPa

Fig. 2-18



31/39

0.82Coe. of contact ratio

Z

Z

Helical gear Z0.75~0.88

0.989HP2HP11170MPa 1170MPa

u Speed reducer ui 4.6

"'Z 61212coscos

Coe. of helical angle

Allowable contact stressHP

Ratio of teeth


Checking contact

stress

u

u

bd

KTZZZZ

122

1

1HE

H

6.4

6.5

08.5930

162350234.22989.082.045.28.189

2

MPa1170MPa1089 HP

Conclusion: contact strength is OK!

The end( closed, hard surface)


32/39

Comparison of two design plan

45 45Material

Heat treatment Case

Hard.

P: H&T

G:Norm.

Helical angle

No. of teeth26 21

79.61 59.08Pitch diameter

mm

1d

2d364.39 272.92

3 2.75Modulusmm

Face width

mm

78 35

Center distancemm 222 166

Design contents Plan A Plan B

12o12'6' '11o33'21' '

1z

2z

119 97

nm

1b

2b 72 30

a

Analysis:

Plan A: soft surface, easy

to manufacture, low cost.But structure size larger.

Plan B: hard surface,

complicated tomanufacture, high cost.

But smaller structure size,

light weight, high

strength.

Plan A for common use

Plan B: for great load

and compact size

2 8 Other Considerations of Gear Drive Design


33/39

1. Efficiency of gear drive

2.8 Other Considerations of Gear Drive Design

Power loss of gear drive

Friction loss

Oil resistance loss

Bearing friction loss

Efficiency of closed gear drive = 123

1Efficiency in mesh, decided by precision grade;

2Churning loss;

3Bearing efficiency.

Average efficiency of closed cylinder gear drive, 0.96-0.99;

Average efficiency of closed bevel gear drive, 0.94-0.98;

Most

significant

2 Lubrication of gear drive


34/39

2. Lubrication of gear drive

Teeth in mesh Relative sliding Friction and wearing

Functions of lubrication:

Cooling

Anti-corrosion

Vibration absorbingNoise reduction

Types of

lubrication

12 15m/sv

15m/sv

Churning

Injection

Oil sump

3 Structure design


35/39

By strength design:

Such as center distance,

module, helical angle

To specify pattern and size of spoke and hub

To specify geometrical parameters

By structure design

(decided by dia. of addendum circle)(1) Gear shaft

The distance from dedendum to root of gear e

Cylinder geare


36/39

(2) Solid gear

Dia. of addendum circle da200mm

Forged

gear blank


37/39

(3) Web

Dia. of addendum circle da200~500mm

Forged

gear blank


38/39

4. Spoke

Dia. of addendum circle da500mm

cast iron or cast steel


39/39

Homework-9

Specify the helical direction of Gear 1 and Gear 2. Mark the directions of each force components.

Tips: Both Gear 2 and Gear 3 exert axial forces on Shaft II. We can

ff t th ft if th h li l di ti

Chapter 2 Gear Drive-4

Documents

Transcript of Chapter 2 Gear Drive-4