Fundamentals of Gears

MEMS1029 – Mechanical Design II

Fundamentals of Gears

MEMS1029

FUNDAMENTALS AND

NOMENCLATURE



Why gears?

• Usually used as part of a transmission to convert

high-speed/low-torque to low-speed/high-torque

– Speed=voltage (cheap)

– Torque=current (expensive)

www.lego.com


Technical Info

• AGMA

– American Gear Manufacturer’s Association

– “Fill your mind with the latest gear research!” at the annual mtg.

• Machinery’s Handbook

– Every design engineer should have a copy

• Gear design textbooks

– Just scan Amazon


Where to get more info?

• Stock Drive Products (www.sdp-si.com)

• PIC Design (www.pic-design.com)

• WMBerg (www.wmberg.com)

• Boston Gear (www.bostongear.com)

• Lots of vendors, all with technical literature,

design guidelines, and application engineers

who live gearing.


Customized gearing design

• Standard gears should be a first choice, but custom gears

can be made in any shape.

– See examples and technical papers at AKGears,

(www.akgears.com)

• New manufacturing techniques have allowed almost any

shape you can design to be made (but maybe at a cost…)


Range of sizes and applications

• Little ones are on the

nanoscale (carbon

nanotubes).

• Common in MEMS

applications on the

microscale.

• Larger sizes in ships,

telescopes,

drawbridges, power

generation, …


Conjugate Action• Mating gear teeth are

similar to cams, and said to have conjugate actionwhen designed to produce a constant angular velocity ratio when meshing.

• Involute profile almost universal for gear teeth.

• Line of action (common surface normal at point of contact) always intersects O-O at a constant pitch point P.

• Radius of each circle is called the pitch radius.


Basic Law of Gearing

• A common normal (the line of action) to the tooth profiles at their point of contact must, in all positions of the contacting teeth, pass through a fixed point on the line-of-centers called the pitch point

• Any two curves or profiles engaging each other and satisfying the law of gearing are conjugate curves, and the relative rotation speed of the gears will be constant.

– (Slocum, Fundamentals of Design, Ch. 6)


Generation of an Involute


Constructing an Involute


Gear Layout

• Pressure angle φ is usually 14.5, 20, or 25 degrees.


Pressure Line

• Pressure line is line of contact – line forces act on.

rb=r cosφ


Nomenclature

• Pitch circle: Theoretical circle upon which all

calculations are based.

• Pitch diameter: Diameter of pitch circle.

• Circular pitch: Distance from a point on one tooth

to a corresponding point on the adjacent tooth

measured on the pitch circle.

• Module: Ratio of pitch diameter to number of

teeth (pitch diameter in mm, metric only)

• Diametral Pitch: Ratio of number of teeth to the

pitch diameter (teeth per inch, english only)


Nomenclature


Basic Relationships

pP

mN

dp

N

dm

d

NP

pitchcircular

mm diameter,pitch

mm module,

in diameter,pitch

teethofnumber

inchper teeth pitch, diametral

p

d

m

d

N

P

• Governed by AGMA and ANSI standards.


Tooth Action

• When properly

designed, gears

have points of

rolling contact

all with the

pressure line as

a constant

surface normal.


Rack and Pinion

• Gear radius is infinite (straight-walled teeth)

• Converts rotary to linear motion, but without mechanical advantage of a screw mechanism.

pb=pc cosφ


Internal (Ring) Gear

• Centers of rotation are on same side of the pitch point.


Backlash

• Backlash is the amount by which the width

of a tooth space exceeds the the thickness of

the engaging tooth.

– Felt as “slop” or lost motion on direction

reversals.

– Anti-backlash gears available for special cases.


Contact Ratio

• Contact ratio mc measures the average number of

teeth in contact. Prefer > 1.2

cosp

Lm

p

q

p

qqm

abc

trac


Interference• The base of the gear

tooth profile is not a

perfect involute, any

contact on this

noninvolute portion of

the flank is refererred

to as interference.

• Interference can be

eliminated by

undercutting, but this

weakens the tooth

profile.


Forming Gear Teeth

• Milling, shaping, hobbing for basic profile.

• Grinding, lapping, burnishing for finishing.

• Manufacturing and inspecting gears

generally requires specialized equipment

and techniques beyond a typical machine

shop (at least for volume production)


Gear Shaping

• Pinion (or rack) cutter

reciprocates while

being fed into the

correct depth.

• Both cutter and gear

are rotated slightly

after each cutting

stroke and cycle

continues.


Gear Hobbing• “Gear Hob” is a spiral

cutting tool shaped like a

worm.

• Hob and blank are rotated

at the proper angular

velocity ratio while hob is

fed across the face of the

blank.

• Requires specialized tools.


Materials

• Gears are made from almost every

engineering material

– Steel used where tooth strength is required.

– Aluminum can be used with proper surface

treatment.

– Brass and bronze used with worm-gears for

wear-in over time.

– Plastics and reinforced composites very

common, especially in consumer applications.


Four basic types

• Spur gears

• Helical gears

• Bevel gears

• Worm gears


Types of Gears – Spur Gears

• Spur gears have teeth

parallel to axis of

rotation and transmit

motion between

parallel shafts.

• Remember Legos?


Types of Gears – Helical Gears

• Helical gears have teeth inclined to the axis of rotation so teeth engage gradually

– Less noisy

– More contact area

– Can be used with non-parallel shafts

– But…generate thrust loads that must be designed for in bearing support structure.


Helical Gears• Shape is an involute

helicoid, formed by

unwrapping a spiral.

• Gears contact gradually,

starting at a point and

extending to a line with

full engagement.

– Spur gears contact all at

once with line contact.

• Transmission is smooth

and quiet, but axial thrust

is generated.


Double Helical Gears

• Helixes come together in a V-shape.

• Also known as herringbone gears.

• Axial thrust force is cancelled, extremely useful for heavy loads and high speeds.

• Can be made from separate gears in an assembly.


Pitch in Helical Gears

anglehelix

pitchcircular axial

pitchcircular transverse

pitchcircular normal

tan

cos

x

t

n

tx

tn

p

p

p

pp

pp

• Angularity of the teeth complicates definitions.


Pressure Angle in Helical Gears

anglehelix

angle pressuredirection transverse

angle pressuredirection normal

tan

tancos

t

n

t

n

• Angularity of the teeth complicates definitions.


Types of Gears – Bevel Gears

• Bevel gears have teeth formed between conical surfaces; usually used to transmit power between intersecting shafts.

• Straight or spiral toothed.

• Called hypoid if the shafts are offset and non-intersecting.


Straight Bevel Gears

• Used to transmit

motion between

intersecting shafts.

– Usually 90 deg, but not

always.

• Pitch measured at the

large end of the tooth

P

G

G

P

N

N

N

N

tan

tan


Types of Gears – Worm Gears

• Worms and worm

gears strongly

resemble screws.

– Transmission ratios

can be very high.

– Can be left-handed or

right-handed threads.


Worm Gears

• Used to generate extremely high transmission ratios.

• Customary to specify lead angle λ on the worm, and the helix angle ψG on the gear.

• Also stage axial pitchon worm and transverse circular pitch on gear.


Tooth Systems

• Standard tooth shapes created by the

American Gear Manufacturers Association

(AGMA).

– See text for reference.

– Vendors are better source of info.

• Allow use of standardized components and

replacements (spares).

GEAR TRAINS



Gear Trains

• Ratio of input to output speed proportional to

number on teeth on different gear combinations.


Calculating Transmission Ratio

• Same equations can be used with pitch diameters instead of numbers of teeth.

• Covention is e positive if last gear rotates in same direction as first gear.

FL enn

e

nN

N

N

N

N

Nn

nN

Nn

numbersth driven too ofproduct

numbers tooth driving ofproduct

2

6

5

4

3

3

26

2

3

23


Compound Gear Trains

• Given a specified

transmission ratio,

start by determining

the number of stages,

the break into

proportions for each

stage.

• Pitch diameters follow

from number of teeth.


Compound Gear Trains

• Arthur Ganson, “Machine with Concrete” on exhibit at MIT Museum (Cambridge, MA).

• 12 sets of 50:1 worm gear reducers.

• Driving motor (not shown) turns at 212 RPM.

• Final gear embedded in concrete block.

• How long until final gear makes one theoretical turn?


Compound Reverted Gear Trains

• Input and output shafts are collinear, but this

forces distances between the shafts to be the same.

d2/2+d3/2=d4/2+d5/2 or N2+N3=N4+N5

Example


Shaft a rotates at 600 rpm. Find the speed and direction of shaft d.

Example


CW rpm06.4760051

4

0784.051

4

60

20

17

8

40

20

dn

e


Planetary Gears

• Planetary gear trains always contain a sun gear, a

planet carrier (or arm), and a planet gear.

• Systems have two degrees of freedom, meaning

that they need two inputs for constrained motion.

Reuleux Collection

FORCE ANALYSIS



Forces along line of action

• Angled line of action leads to force trying to push gears

apart…provide means of support in your design.

www.lego.com


Force Analysis – Spur Gearing

• Generate free body diagrams of each gear.

– Keep consistent sign convention.


Force Analysis – Spur Gearing

power2/

torque2

load dtransmitte32

dWTH

Wd

T

FW

t

t

t

t

• Resolve forces into radial and tangential components.


Power Calculations

• Transmitted power

depends on gear

diameter and gear

speed.

– Watch units in

calculations, especially

with English units.

• Rolling contact leads

to high efficiencies,

usually > 98%. rev/min speed,gear

mm diameter,gear

Wpower,

N load, dtransmitte

60000

UnitsSI

n

d

H

W

dn

HW

t

t


Force Analysis – Bevel Gearing

sintan

costan

ta

tr

av

t

WW

WW

r

TW

• Convention is to assume that all forces act at the midpoint of the tooth (close enough).


Force Analysis – Helical Gearing

coscos

tan

tan

sincos

coscos

sin

n

t

ta

ttr

na

nt

n

WW

WW

WW

WW

WW

WW

• Assumed point of force application is in the pitch plane

and in the center of the gear face.


Force Analysis – Worm Gearing

cotcos

tancos

sincoscos

sin

cossincos

f

f

fWW

WW

fWW

n

n

n

z

n

y

n

x

• Worm gears are all sliding contact (not rolling)

– efficiency is a function of geometry and coefficient of friction


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Fundamentals of Gears

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