lecture notes ent345 mechanical components design...

ENT345 Mechanical

Components Design Sem 1-

2015/2016

Dr. Haftirman

School of Mechatronic Engineering

1

LECTURE NOTES

ENT345

MECHANICAL COMPONENTS DESIGN

Lecture 6, 7

29/10/2015

SPUR AND HELICAL GEARS

Dr. HAFTIRMAN

MECHANICAL ENGINEEERING PROGRAM

SCHOOL OF MECHATRONIC ENGINEERING

UniMAP

COPYRIGHT©RESERVED 2015

AGMA

The American Gear Manufacturers

Association (AGMA) has for many years

been the responsible authority for the

dissemination of knowledge pertaining to

the design analysis of gearing.

ENT345 Mechanical

Componets Design Sem 1-

2015/2016

Dr. School of Mechatronic Engineering 2

The Lewis Bending Equation

Wilfred Lewis introduced an equation for

estimating the bending stress in gear teeth in

which the tooth form entered into the

formula.

The equation, announced in 1892, still

remains the basis for most gear design

today.

ENT345 Mechanical


2015/2016


The Lewis Bending Equation

To derive the basic Lewis equation refer to Figure,

which shows a cantilever of cross-sectional

dimensions F and t, having a length l and a load

Wt, uniformly distributed across the face width F.

The section modulus: I/c = Ft2/6

The bending stress (σ ).

ENT345 Mechanical


2015/2016


ENT345 Mechanical


2015/2016


The Lewis bending equations

2

23 6

6)12(

2

/ Ft

lWFt

t

Ft

c

I

cI

M t

l

tx

t

l

x

t

4

2/

2/

2

62

34

3/2

14

6

4

1

4

1

6/

16222

xx

l

tF

W

ltF

W

Ft

lW ttt

ENT345 Mechanical


2015/2016


The Lewis bending equations

Fpy

pW

p

xy

xpF

pW

t

t

3

2

3

2

3

2

,

xPY

FY

PW

yYp

P

t

y is The Lewis form factor

Y means that only the bending of the tooth is considered and

that the compression due to the radial component of the force is neglected.

Wr

Wt

Values of the Lewis form factor

Y

ENT345 Mechanical


2015/2016


Dynamic effects

When a pair of gears is driven at moderate or high

speed and noise is generated, it is certain that

dynamic effects are present.

If a pair of gears failed at 500 lbf tangential load at

zero velocity and at 250 lbf at velocity V1, then a

velocity factor, designated Kv, of 2 was specified

for the gears at velocity V1.

ENT345 Mechanical


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ENT345 Mechanical


2015/2016


Dynamic effects

Kv= the velocity factor

V = the pitch-line velocity in

ft/min

SI units

)(78

78

)(50

50

)(1200

1200

)),(600

600

profilegroundorshavedV

K

profileshapedorhobbedV

K

profilemildorcutV

K

profilecastironcastV

K

v

v

v

v

)(56.5

56.5

)(56.3

56.3

)(1.6

1.6

)),(05.3

05.3

profilegroundorshavedV

K

profileshapedorhobbedV

K

profilemildorcutV

K

profilecastironcastV

K

v

v

v

v

FY

PWK t

vFmY

WK t

v

The metric versions

Example 14–1

Shigley’s Mechanical Engineering Design

Example 14–1


X

X

Example 14-1

A stock spur gear is available having a module of 4 mm, a

44 mm face width, 18 teeth, and a pressure angle of 20°

with full-depth teeth. The material is AISI 1020 steel in as-

rolled condition. Use a design factor of nd = 3 to rate the

power output of the gear corresponding to a speed of n

= 25 rev/s and moderate applications.

Solution

The term moderate applications seems to imply that the

gear can be rated by using the yield strength as a criterion

of failure. AISI 1020 steel in as-rolled, from Table A-18,

Sut = 380 MPa and Sy = 210 MPa.

ENT345 Mechanical


2014/2015

Dr. Haftirman


12

Example 14-1

Solution

A design factor of 3 means that the allowable

bending stress is 𝑆𝑦

𝑛𝑑=

210

3.5= 60 𝑀𝑃𝑎

The pitch diameter of d = Nm = 18 (4) = 72 mm.

The pitch-line velocity is

V = πdn= π(0.072) 25 =5.65487m/s

The velocity factor (Eq 14-6b):

𝐾𝑣 =6.1+𝑉

6.1=

6.1+5.65487

6.1= 1.92703

ENT345 Mechanical


2014/2015

Dr. Haftirman


13

Table 14-2, Y=0.309 for 18 teeth.

The tangential component of load Wt

𝑊𝑡 =𝐹𝑌𝜎𝑎𝑙𝑙

𝐾𝑣𝑃⇒ 𝜎𝑎𝑙𝑙 =

𝑆𝑦

𝑛𝑑=

210

3.5= 60 𝑀𝑃𝑎

𝑃 =𝑁

𝑑⇒

𝑁

𝑑=

1

𝑚

𝑊𝑡 =𝐹𝑌𝜎𝑎𝑙𝑙

𝐾𝑣𝑃=

𝑚𝐹𝑌𝜎𝑎𝑙𝑙

𝐾𝑣

=4𝑚𝑚 44𝑚𝑚 0.309 60 𝑁/𝑚𝑚2

(1.92703)= 1693.30𝑁

The power that can be transmitted is

Hp=Wt V= (1693.30 N)(5.65487 m/s)= 9575.391W

ENT345 Mechanical


2014/2015

Dr. Haftirman


14

√

Example 14–2


𝑚 = 4 𝑚𝑚 ⇒ 𝑃 =25.4

𝑚=

25.4

4= 6.35

𝑚 = 3 𝑚𝑚 ⇒ 𝑃 =25.4

𝑚=

25.4

3= 8.47

Example 14–2


Fatigue Stress-Concentration Factor

A photoelastic investigation gives an estimate of fatigue stress-

concentration factor as


Surface durability

Wear is the failure of the surfaces of gear

teeth.

Pitting is a surface fatigue failure due to

many repetitions of high contact stresses.

Scoring is a lubrication failure, and

abrasion, which is wear due to the presence

of foreign material.

ENT345 Mechanical


2015/2016


Surface durability

To obtain an expression for the surface-contact

stress, we shall employ the Hertz theory. The

contact stress between two cylinders may

computed from the equation;

pmax =largest surface pressure.

F= force pressing the two cylinders together.

l = length of cylinders.

ENT345 Mechanical


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Cylindrical Contact Stress

Two right circular cylinders with length l and

diameters d1 and d2

Area of contact is a narrow rectangle of width

2b and length l

Pressure distribution is elliptical

Half-width b

Maximum pressure


Fig. 3−38

Surface durability

Half-width b is obtained from

The surface compressive stress (Hertzian stress)

ENT345 Mechanical


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Surface durability

The radii of curvature of the tooth profiles at the

pitch point are

An elastic coefficient Cp

ENT345 Mechanical


2015/2016


Example 14–3


SPUR GEAR BENDING

Based on ANSI/AGMA 2001-D04 (US. Customary units)


Fig. 14–17

SPUR GEAR WEAR

Based on ANSI/AGMA 2001-D04 (US. Customary units)


Fig. 14–18

AGMA equations

Two fundamental stress equations are used in the

AGMA methodology, one for bending stress and

another for pitting resistance (contact stress).

In AGMSA terminology, these are called stress

numbers, as contrasted with actual applied stress

(σ).

ENT345 Mechanical


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AGMA Bending Stress

equations

ENT345 Mechanical


2015/2016


AGMA Contact Stress

equations

ENT345 Mechanical


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AGMA Strengths

AGMA uses allowable stress numbers rather than strengths.

We will refer to them as strengths for consistency within the

textbook.

The gear strength values are only for use with the AGMA stress

values, and should not be compared with other true material

strengths.

Representative values of typically available bending strengths are

given in Table 14–3 for steel gears and Table 14–4 for iron and

bronze gears.

Figs. 14–2, 14–3, and 14–4 are used as indicated in the tables.

Tables assume repeatedly applied loads at 107 cycles and 0.99

reliability.


Bending Strengths for Steel Gears


Bending Strengths for Iron and Bronze Gears


Bending Strengths for Through-hardened Steel Gears

Shigley’s Mechanical Engineering Design Fig. 14–2

Bending Strengths for Nitrided Through-hardened Steel Gears


Bending Strengths for Nitriding Steel Gears


Fig. 14–4

Allowable Bending Stress


Allowable Contact Stress


Nominal Temperature Used in Nitriding and Hardness Obtained


Table 14–5

Contact Strength for Steel Gears


Contact Strength for Iron and Bronze Gears


Contact Strength for Through-hardened Steel Gears


Geometry Factor J (YJ in metric)

Accounts for shape of tooth in bending stress equation

Includes

◦ A modification of the Lewis form factor Y

◦ Fatigue stress-concentration factor Kf

◦ Tooth load-sharing ratio mN

AGMA equation for geometry factor is

Values for Y and Z are found in the AGMA standards.

For most common case of spur gear with 20º pressure angle, J can be read directly from Fig. 14–6.

For helical gears with 20º normal pressure angle, use Figs. 14–7 and 14–8.


Spur-Gear Geometry Factor J


Helical-Gear Geometry Factor J

Get J' from Fig. 14–7, which assumes the mating gear has 75 teeth

Get multiplier from Fig. 14–8 for mating gear with other than 75

teeth

Obtain J by applying multiplier to J'


Modifying Factor for J


Fig. 14–8

Surface Strength Geometry Factor I (ZI in metric)

Called pitting resistance geometry factor by AGMA


Elastic Coefficient CP (ZE)

Obtained from Eq. (14–13) or from Table 14–8.


Elastic Coefficient


Dynamic Factor Kv

Accounts for increased forces with increased speed

Affected by manufacturing quality of gears

A set of quality numbers Qv define tolerances for gears

manufactured to a specified accuracy.

Quality numbers 3 to 7 include most commercial-quality gears.

Quality numbers 8 to 12 are of precision quality.

The AGMA transmission accuracy-level number Av is basically the

same as the quality number.


Dynamic Factor Kv

Dynamic Factor equation

Or can obtain value directly from Fig. 14–9

Maximum recommended velocity for a given quality number,


Dynamic Factor Kv


Overload Factor KO

To account for likelihood of increase in nominal tangential load

due to particular application.

Recommended values,


Surface Condition Factor Cf (ZR)

To account for detrimental surface finish

No values currently given by AGMA

Use value of 1 for normal commercial gears


Size Factor Ks

Accounts for fatigue size effect, and non-uniformity of material

properties for large sizes

AGMA has not established size factors

Use 1 for normal gear sizes

Could apply fatigue size factor method from Ch. 6, where this size

factor is the reciprocal of the Marin size factor kb. Applying

known geometry information for the gear tooth,


Load-Distribution Factor Km (KH)

Accounts for non-uniform distribution of load across the line of

contact

Depends on mounting and face width

Load-distribution factor is currently only defined for

◦ Face width to pinion pitch diameter ratio F/dp ≤ 2

◦ Gears mounted between bearings

◦ Face widths up to 40 in

◦ Contact across the full width of the narrowest member



Face load-distribution factor



Cma can be obtained from Eq. (14–34) with Table 14–9

Or can read Cma directly from Fig. 14–11




Fig. 14–11

Hardness-Ratio Factor CH (ZW)

Since the pinion is subjected to more cycles than the gear, it is

often hardened more than the gear.

The hardness-ratio factor accounts for the difference in hardness of

the pinion and gear.

CH is only applied to the gear. That is, CH = 1 for the pinion.

For the gear,

Eq. (14–36) in graph form is given in Fig. 14–12.


Hardness-Ratio Factor CH


Hardness-Ratio Factor

If the pinion is surface-hardened to 48 Rockwell C or greater, the

softer gear can experience work-hardening during operation. In

this case,


Fig. 14–13

Stress-Cycle Factors YN and ZN

AGMA strengths are for 107 cycles

Stress-cycle factors account for other design cycles

Fig. 14–14 gives YN for bending

Fig. 14–15 gives ZN for contact stress


Stress-Cycle Factor YN


Fig. 14–14

Stress-Cycle Factor ZN


Fig. 14–15

Reliability Factor KR (YZ)

Accounts for statistical distributions of material fatigue failures

Does not account for load variation

Use Table 14–10

Since reliability is highly nonlinear, if interpolation between table

values is needed, use the least-squares regression fit,

Shigley’s Mechanical Engineering Design Table 14–10

Temperature Factor KT (Yq)

AGMA has not established values for this factor.

For temperatures up to 250ºF (120ºC), KT = 1 is acceptable.


Rim-Thickness Factor KB

Accounts for bending of rim on a gear that is not solid


Fig. 14–16

Safety Factors SF and SH

Included as design factors in the strength equations

Can be solved for and used as factor of safety

Or, can set equal to unity, and solve for traditional factor of safety

as n = all/


Comparison of Factors of Safety

Bending stress is linear with transmitted load.

Contact stress is not linear with transmitted load

To compare the factors of safety between the different failure

modes, to determine which is critical,

◦ Compare SF with SH2 for linear or helical contact

◦ Compare SF with SH3 for spherical contact


Summary for Bending of Gear Teeth


Fig. 14–17

Summary for Surface Wear of Gear Teeth


Fig. 14–18

Example 14–4


Example 14–5


Comparing Pinion with Gear

Comparing the pinion with the gear can provide insight.

Equating factors of safety from bending equations for pinion and

gear, and cancelling all terms that are equivalent for the two, and

solving for the gear strength, we get

Substituting in equations for the stress-cycle factor YN,

Normally, mG > 1, and JG > JP, so Eq. (14–44) indicates the gear

can be less strong than the pinion for the same safety factor.


Comparing Pinion and Gear

Repeating the same process for contact stress equations,

Neglecting CH which is near unity,


Example 14–6


Example 14–7


Example 14–8


Assignment 1

Page 773-774

Problem:

14-15

14-16

14-17

14-18


ENT345 Mechanical


2015/2016

Dr. Haftirman


113

Thank you

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