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1 Martin Culpepper, All rights reserved

2.72 Elements of Mechanical Design

Lecture 5 Gears

Image courtesy of Justin Lai


Lecture structure Motivation and overview of gear types

Gear kinematics

Serial gear trains (special case: planetary gear trains)

Gear manufacturing

Gear failure: Bending

Gear failure: Contact


Motivation In your lathe: ???

Critical to understand this machine element and have it in your toolbox


Geared mechanisms Gears transmit power across rotating shafts

Gears can NOT increase power Power loss during transmission in real gears

Gears are used for: Changing direction of rotation Changing torque Changing rotational speed

Gear design/selection based on How are the shafts/gears arranged?

Gear kinematics How much power is transmitted?

Failure strength

Input Output

1


Gears at all scales

Pocket watch movement http://www.timezone.com/library/workbench/workbench631678210214858916

Microgear with pollen and red blood cells Courtesy of Sandia National Laboratories, SUMMiT(TM) Technologies, www.mems.sandia.gov

Tamiya dual gearbox http:// www.pololu.com

South Bend lathe change gears Tractorbynet.com

Cage gear for material processing plant http://www.cage-gear.com

Very Large Array radio telescopes Wikimedia commons

Gear types


Axes of rotation dont have to be parallel! Parallel shafts

Spurs, helical

Intersecting shafts Bevel gears

Neither parallel nor intersecting Hypoids (some sliding contact), spiral bevel, worm


Spur gear set Teeth parallel to axis of rotation Only good for parallel shafts Simple shape = simple design and low cost Noise is sensitive to errors in tooth shape

Helical gear set Teeth inclined to axis of rotation Gradual engagement of teeth = low noise Shaft may or may not be parallel Thrust (axial) loads from teeth reaction forces High speed and high power transmission Tooth-tooth contact force pushes gears around (rotate) & apart along axis

Helical Gears

Spur Gears

Gear types and purposes


Bevel gear set Teeth formed on conical surface Used between parallel or non-parallel shafts (hypoid) For non-parallel shafts, shaft axes intersect at some point Teeth can be straight or spiral away from axis of rotation

Worm gear set Very low transmission ratios (output divided by input) Worm is input and gear is output Sliding between worm-gear leads to high friction losses Non-parallel, non intersecting shafts

Rack and pinion set Rack teeth may be straight or angled w.r.t rack motion Good means to transmit between rotary and linear motion

Bevel Gears

Worm Gear Set

Worm Gear

Rack & Pinion

More gear types and purposes

Gear kinematics: Getting the motions we want


time [sec]

Zout, [rpm]

Constant speed Ideal involute/gear

Non or poor involute

Reality: small imperfections and bending

of gear teeth result in some variation

Assume constant Zin

Output speed of gear train

Want uniform rotary motion Conjugate action: constant angular velocity ratio

Key to conjugate action: use an involute tooth profile

What form of motion do we want?


Rolling cylinders (No slip between cylinders) Circles share common point traveling at velocity:

Pitch circle:

Circle that passes through pitch point

1

2

2

1rr Z

Z2211 rrv***** u u ZZ

Pitch Circle 1 Pitch Circle 2

r1

Z1

Z2

r2

r1

Z1

Z2

v

Pitch Circles Meet @ Pitch Pt. r2

Consequences of conjugate action


Meshing gears must have same pitch!

Ng = # of teeth, Dp = Pitch circle diameter

Diametral pitch, PD:

Circular pitch, PC:

p

gD D

NP

Dg

pC P

ND

P

Instantaneous velocity and pitch

(Used for inch system, given as TPI)

wikipedia

Metric system uses module

Inverse of , units of

mod 2 gear or mod 0.5

No metric gears from McMaster?


Mach. Handbook, 28th ed

To draw a gear: specify the pitch circle diameter and pressure angle,

) cosPB RR)

Pitch Point

Base Circle

Pitch Circle

RP RB

Line of Action

Drawing the involute profile

Common pressure angles: 20,

22.5, 25

Older, less used: 14.5


T' Bn nRL

RB

1 2 3

Pitch Point Pitch Circle

Base Circle

L3

L2

L1

Drawing the involute profile

T'

Unwrap a taut string wound on the base circle

Keep string tangent to base circle Radius varies continuously


I

Pitch Point

Base Circle

Pitch Circle

RP

RB

pqM tc

Mc = Contact ratio qt = arc of action p = circular pitch Lab = Length of line of action

Line of Action

a

b

IcospLM abc

Mc > 1.2 (Shigley)

Contact ratio

Power transmission: 1.4 minimum

Addendum (pinion)

Addendum (gear)


pqM tc

Mc = Contact ratio qt = arc of action p = circular pitch Lab = Length of line of action

IcospLM abc

Mc > 1.2 in order to ensure continuous contact

Contact ratio: close-up

Power transmission: 1.4 minimum


Interference: non-conjugate contact

Variables Np = Minimum number of teeth that can exist without interference k = 1 for full depth teeth k = 0.8 for stub teeth = Pressure angle m = Ratio of the # of teeth on the gear to # of teeth on the pinion

If m = 4 and = 20, then = 16 teeth A 16 tooth pinion will mesh with a 64-tooth gear without interference Any smaller gear set with m = 4 will have interference

If m = 1, = . = If m = 1, = =

II 222 sin)21(sin21 2 mmmmkNP


Interference: diagram

Serial gear train kinematics


Transmission ratio for serial gears A gear train consists of two or more gears in mesh For Large Serial Drive Trains:

22

2

1

11 PD

NDNP

2

1

2

1NN

DD From pitch equation:

Gear train Power in: Tin y Zin Power out: Tout y Zout

in

out teethdriven ofProduct teethdriving ofProduct signproper Z

Z TR

n

in

out

in

out

in

out ...signTR21

ZZ

ZZ

ZZ Important: take direction of rotation into account! (sign)


Serial trains:

Example 1:

Example 2:

in out

? TR

? TR

in

outthdriven tee ofProduct

teethdriving ofProduct signproper ZZ TR

Transmission ratio for serial gears

20

10

in out

drive

driven

driven

drive driven

drive

10 10 20

20


Serial trains:

Example 1:

Example 2:

in out

in out

drive

driven

driven

drive driven

drive

in

outthdriven tee ofProduct

teethdriving ofProduct signproper ZZ TR


21020 TR

12010

1010

1020

TR

1NN

NN

NN

4

3

3

2

2

1

TR

2NN

2

1

TR20

10

10 10 20

20


Example 3: Integral gears in serial gear trains

Gears 2 & 3 are one piece, they rotate together about the same axis

What is TR if Gear 1 = input and 5 = output?

thdriven tee ofProduct

teethdriving ofProduct signproper TR

Gear 5 N5 = 33

Gear 1 N1 = 9

Gear 4 N4 = 67

Gear 3 N3 = 9

Gear 2 N2 = 38

4

1

5

3

2


33

67

38

9 9 IN

OUT


Example 3: Integral gears in serial gear trains

Gears 2 & 3 are one piece, they rotate together about the same axis (same angular velocity)

What is TR if Gear 1 = input and 5 = output?


065.03367

679

389

NN

NN

NN

thdriven tee ofProduct teethdriving ofProduct signproper

5

4

4

3

2

1

TR

Gear 5 N5 = 33

Gear 1 N1 = 9

Gear 4 N4 = 67

Gear 3 N3 = 9

Gear 2 N2 = 38

4

1

5

3

2 33

67

38

9 9 IN

OUT

Planetary gear train kinematics


Z2

Arm

Planet gear Ring

gear

Sun gear

Planet gear

Planet gear

Planetary gear train is analogous to a solar system

Small & large TRs in a compact mechanism

Legend:

Planetary gear trains


How to find TR? 2-stage animation

Sun

Ring gear

Planet gear

Arm

Trai

n 1

Sun

Ring gear

Planet gear

Arm

Trai

n 2

Planetary gear train animation


If we make the arm stationary, then this is a serial gear train:

Ring Gear

Sun Gear

Planet Gear

Arm ring

sun

ring

planet

planet

sun

armsun

armring

sa

ra

NN

NN

NNTR

TR

ZZZZ

ZZ

planetsun

armsunarmplanet

sapa

NNTR

TR

ZZZZ

ZZ

Planetary gear train TR


If the sun gear is the input, and the ring gear is held fixed:

Ring Gear

Sun Gear

Planet Gear

Arm

sunarmoutput

ringsun

ringplanet

planetsun

armsunarm

sara

TRTR

NN

NN

NNTR

TR

ZZZ

ZZZ

ZZ

1

0

Planetary gear train TR

Gear manufacturing


Gear manufacturing - Hobbing


Gear manufacturing - Shaping


Other manufacturing techniques Cold processes Cold drawing

Work hardening

Cold rolled Smooth, work hardened surfaces

Hot processes Sintering

Sintered iron gears in appliances, run quietly, can hold lubricant

Injection molding (polymers) Die casting

Low temp. melting materials -> less load capacity

Extruded


Selection vs. design of gears Why do we care about gear tooth surface finish

What affects the finish on the gear surfaces?

How good could it be?

How much would it cost?

Why do we care about the tooth geometry at the root

What affects the quality of the fillet at the root?

How good could it be?

How much would it cost?

Gear failure


Gear failure Failure modes

Bending failure (e.g. root stresses) Contact fatigue (e.g. pitting)

Failure analysis

Estimating bending/contact stresses

Estimating allowable stresses

Analysis approaches Lewis bending equation

AGMA (American Gear Manufacturers Association)

Root bending failure

Pitting


Photoelasticity: Visualizing stress

Bending Contact


Incredibly uninteresting, plug-chug & non-scientific

AGMA approach: Calculating stresses )..( unitsSUJ

KKFPKKKW Bmdsvotbending V

)..( unitsSUIC

FdKKKKWC fp

msvotpcontact V

Gear failure at the root:

Bending stress


Bending root stress Lewis bending equation (1892)

Model tooth as cantilever Estimate bending stress near the root

;

P: Diametral pitch (1/module)

Y: Lewis form factor ~ to for I = 20o f( # of teeth )

Conservative estimate Implies that one tooth carries the load

Heaviest load occurs mid-tooth, not at root

IcM V 2

6tF

LWt V

YFPWt V


Root stress: dynamic effects How to incorporate dynamic effects

One way of addressing

V = pitch line velocity

Kv depends on fabrication (since a, b, c do too)

This is for English units, for SI is different

cb

v aVaK

YFPWK tV V

For rough estimates


Allowable bending stress These types of plots are associated with conditions

tbtt CHS D


Elements of the equations:

St Allowable bending stress YN Stress cycle life factor KT Temperature factors KR Reliability factors SF AGMA factor of safety

Allowable stresses for:

Unidirectional loading 10 million stress cycles 99 percent reliability

Allowable bending stress )..( unitsSUKK

YSS

RT

N

F

tall V )( unitsSIYY

YSS

Z

N

F

tall

TV

Gear failure

at the surface: Contact (Hertzian)

stress


High cycle failure: Pitting


Equivalent modulus Half contact width Maximum contact pressure

Avoiding high cycle failure: Stress variables

LbWq tS

2

333.0 X2

221

1

21 11

1

Ev

Ev

Ee

Watch out! The book switches meaning of F here (applied force or face width?)

5.0

21

212

ddELddWb

e

tS

Hertzian contact analysis: line contact


Allowable contact stress [ANSI/AGMA 2001-D04 and 2101-D04] cbcc CHS D


Elements of the equations:

SC Allowable contact stress ZN Stress cycle life factor CH Hardness ratio factors for pitting resistance KT Temperature factors KR Reliability factors SH AGMA factor of safety

Allowable stresses for:

Unidirectional loading 10 million stress cycles 99 percent reliability

Allowable contact stress )..(, unitsSUKK

CZSS

RT

HN

H

Callc V )(, unitsSIYY

ZZSS

Z

WN

H

Callc

TV

Summary


Summary Gear types Conjugate action

For uniform speed

Transmission ratio Serial & planetary train

Gear manufacturing Performance vs. cost

Failure Bending vs. contact failure


Design Recommendation Iteration required! Program design equations into your favorite analytical software (Excel,

MathCAD, MATLAB, etc) Excel: you can link values from your design table to Solidworks

dimensions. Learn how to do this! How are you mounting your gears? Here be dragons Dont forget to consider the reaction forces from the gears on the rest of

the structure! CAD: model with addendum, dedendum, and pitch diameter use P.D. for

mating, the others for interference checking (addendum in particular)


References J Shigley, C. Mischke, R. Budynas, K. Nisbett. Shigleys Mechanical

Engineering Design. ASM International. Gear Materials, Properties, and Manufacture. Materials

Park, Ohio 2005 D. W. Dudley. Handbook of practical gear design. McGraw-Hill 1984 K. L. Johnson. Contact Mechanics. Cambridge University Press 1985 E. Oberg, F. D. Jones, H. L. Horton, and H. H. Ryffel. Machinerys Handbook.

28th ed. Industrial Press, New York 2008 P. Lynwander. Gear Drive Systems: Design and Application. Marcel Dekker Inc,

New York 2003.


Appendix A: Spur gear nomenclature

Mach. Handbook

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