differential gear box design

8/22/2019 differential gear box design

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Mechanisms: Introduction to Gear Trains

MECH103 Mechanisms and Dynamics of Machinery

Jaguar 3.8 Litre, 6 cylinder

Textbook: Chap 9


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Spur gear: gear with radial teeth parallel to its axis

Backlash: the amount of "play" or clearance between two parts. For gears, it

refers to how much one gear can be moved back and forth without moving thegear into which it is meshed

Spur Gear

Axis of the gear


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A mechanism in which a toothed wheel

(pinion) engages a notched bar (rack) toconvert rotary motion into linear motion

Either of a pair of toothedwheels whose working surfaces

are inclined to nonparallelaxes. Example: differential inautomobile

Differential: a device that allows a difference in velocity (and displacement)between two elements

Rack & Pinion, Bevel Gear

Rack

Pinion

-Al low bi-directional drive

- rack-and-pinion steering in cars


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Helical gear: a gear that has the teeth cut at an angle to thecenter line of the gear. This kind of gear is useful because there isno chance ofintermittent tooth-to-tooth operation because there are

at least two teeth engaged at any time. It can operate quieter thanspur gear. Helical gears are either right- or left-handed.

Helical Gears


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A coarse, spiral shaped gear cut on a shaft. It isused to engage with and drive another gear orportion of a gear. As used in the steeringgearbox, it often engages the cross shaft via aroller or by a tapered pin.

Very high gear ratio is possible in small package

Allow one directional drive: worm worm wheel

Ultra low backlash gear technology

with medium-high reduction ratios foraccurate bi-directional repeatability,high efficiency and power to weight.

Good for motion control: robotics

Worm Gear & Harmonic Gearing

Worm

Worm wheel

(worm gear)


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Herringbone Gears (double helical gears): two

helical gears operating together and so placedthat the angle of the teeth form a "V" shape;

cancel outend-thrust forces. no thrust

bearing is needed

Herringhbone Gear & Gear Train

http://content.scvs.tpc.edu.tw/top1/chap10/htm/chap10-12.htm


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Differential Gear & Planetary Gear Train


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Kinematics of Gears

jik

rvvv

1111

/1/ 0

rr

OAOAOA

==

+=+=

jik

rvvv

2222

/2/

)(

0

rr

PAPAPA

==

+=+=

2211 rr =2

1

1

2

r

r=

or

Fundamental law of gearing:

Angular vel. Ratio = constant throughout the mesh


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How can the radii r1 and r2 be related to the number of

teeth on each gear?

Assume that the gears must have the same circular pitch

n1 = teeth on gear 1

n2 = teeth on gear 2

Pitch: the distance between a point on one gear tooth and the same

point on the next gear tooth


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Simple gear set

The sign is necessary to take into account the change in direction of rotation.753

642

6

7

4

5

2

3

2

7

nnn

nnn

==

I/PO/P

Here, gears 3 & 4 are rigidly connect, as are gears 5 & 6

Clearly 4 = 3 and 6 = 53

2

2

3

n

n=

5

4

4

5

n

n=

7

6

6

7

n

n=


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Reverted gear train

Used in automotive transmission:

- compact, save space

1

2

Revert = go back to a previous state

Compare:


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Internal gear

The + sign is used here to take intoaccount the direction of rotation.

3

2

3

2

2

3

n

n

r

r

==


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Example: Find the speed reductions possible for the

transmission

the power is transmittedthrough gears 04561012.

Ifgear 34 slides to the left (disengaging 4 from 5) and gear 12 to the left

(engaging 1 and 9), then power is transmitted through 01961012


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Example: Find the speed reductions possible for thetransmission

If gear 34 slides to the left

(disengaging 4 from 5) and

gear 12 to the left (engaging

1 and 9), then power is

transmitted through 0196

1012

Note: There are 8 possible speed

reductions.


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Example: Find the gear reductions in the automotive transmission

Low gear: gear 3 meshes withgear 6, power flows 1463.

Second gear: gear 2 meshes

with gear 5, power flows 1452.

High gear: gear 2 is shifted sothat the clutch teeth on the

end of gear 2 mesh with theclutch teeth ofgear 1.

(Direct drive results.)

Reverse gear: gear 3 is shifted

to mesh with gear 8, powerflows 14783.

http://auto.howstuffworks.com/sequential-gearbox.htm

http://auto.howstuffworks.com/transmission.htm


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power flows 1463

power flows 1452. 301.027

18

31

14

)1(3

6

4

1

6

3

4

6

1

4

1

3

==

=

=

=

n

n

n

n

lowin

out

564.020

25

31

14

)1(2

5

4

1

5

2

4

5

1

4

1

2

2nd

==

=

=

=

n

n

n

n

in

out


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power flows 14783.

234.02714

3114

)1(3

8

8

7

4

1

8

3

7

8

4

7

1

4

1

3

rev

==

=

=

=

n

n

n

n

n

n

in

out


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Planetary gear trainExample: Find the output angular velocity 2 for the planetary gear train shownwhen the input angular velocity is 4 = 50 rad/sec counterclockwise.

Note that gear 2 and

arm 4 are not joined.sun

annulus

planet

arm

Program: 9-33.wm2d


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using the tooth relationship to replace the radii,

Substituting back into the other equation gives

n2 = 40

n3 = 20


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Example: Find the gear ratios for the model T Ford gearbox

Gearbox : Integral with the engine. Footoperated 2 speed and reverse epicyclictransmission foot-brake, 1908 for 19 yrs

9 million were made!

http://www.t-ford.co.uk/car.htm

Textbook p.507

I/P O/P

P1

S1

P2

S2

2: On


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Low gear for the model T Ford

Replacing the radii by the number of teethon the appropriate gears

in

out


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Example: Reversed Gear case

Gearbox : Integral with the engine. Footoperated 2 speed and reverse epicyclictransmission foot-brake, 1908 for 19 yrs

9 million were made!

http://www.t-ford.co.uk/car.htm

Textbook p.507

I/P O/P

P1

S1

P2

S2

1:On


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Reverse gear for the model T Ford

Note the negative sign indicating a change in direction

in

out


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Reverse on a Car

http://www.innerauto.com/innerauto/anim/trans.html


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Model T Ford, 1912 Landaulette


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Towards the Involute Profile

A belt connecting the two cylinders

The involute is a curve traced by a point on a taut, inextensible

string as it unwinds from a circle.

http://www.ies.co.jp/math/java/calc/en-circum/en-circum.html


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The Involute Profile

Involute curve: created by tying a pencil to the end of a string and wrapping the string

around a cylinder. Hold tension in the string as you unwind it from the cylinder. The

curve drawn by the pencil as it moves out from the cylinder is an involute curve.


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Profile of the Involute Profile

Line of action: A line normal to a pair of mating tooth profiles at their point of contact

Pressure angle = the angle between Line of Action (common normal) and the directionof velocity at the pitch point (has been standardized: 14.5, 20, 25)


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Involute in Action

video from http://www.howstuffworks.com

Pitch circle=rolling cylinder circle

Addendum: the amount of tooth that sticks out above the pitchcircle


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Nomenclature

Figure 11-8

N

dpc

=


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Pitches, Etc.

circular pitch (mm, in.)

base pitch (mm, in.)

diametral pitch (teeth/in.)

module (mm/teeth)

N

dpc

=

coscb

pp =

d

Npd =

N

dm =


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Minimum # of Teeth

minimum # of teeth to avoid undercutting with gear and rack

2min

sin

2

=N = pressure angle


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cos

sin

aay

aax

=

=

Antons Calculus (7th): p.93

Cycloid curve for cycloidal gear

Commonly used in watches and clocks

http://mathworld.wolfram.com/Cycloid.html


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)cos(sin

)sin(cos

=

+=

ay

ax

Antons Calculus (7th): p.782

Involute curve for involute gear

Commonly used in all kinds of power

transmission systems

http://mathworld.wolfram.com/CircleInvolute.html


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35)cos(sin

)sin(cos

=

+=

ay

ax

Origin of involute curve


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Rack & Pinion


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Bevel Gear


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Worm Gear


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Gear Train

A t ti Diff ti l G


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Automotive Differential Gear


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Manual Transmission

Low gear High gear


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Gear Types Grouped According to Shaft Arrangement

Straight bevel: These are like spur gears, the teeth have no helix angle

Spiral bevel gears: Teeth have a spiral angle which gives performance improvements much

like helical gears

Zerol bevel gears: Teeth are crowned, so that tooth contact takes place first at the toothcenter. (Zerol Bevel Gears are Spiral Bevel Gears with a spiral angle of zero)

Hypoid gears: Similar to spiral bevel gears, but connect non-parallel shafts. The pitch

surface of this gear is a hyperboloid of revolution (rather than a cone, the pitch surface in

bevel gears). It is stronger, operate quietly, used for higher reduction ratios. Hypoid gears

are found in auto dif ferentials.

Herringbone gears examples


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Herringbone gears examples

http://www.linngear.com/products/highlights/infosheets/g-3.html

from D.O. James Gear Manufacturing Co.


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Comparision between Helical Gear and Herringbone Gear

Bevel Gear: based on rolling cones


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Bevel Gear: based on rolling cones

Incorrect arrangement Correct arrangements

Apices must be

conincident

S i l b l G & H id G


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Hypoid Gears are similar in their generalform to bevel gears. However, Hypoid

Gears operate on non-intersecting axes.

(Hypoid = a contradiction of hyperboloid)

Spiral bevel Gears

Spiral bevel Gear & Hypoid Gear

Hypoid Gear: based on hyperboloids of revolution


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Rolling hyperboloids of

revolutionAutomotive hypoid final drive gears

(General Motors, Detroit, MI)

Hypoid Gear: based on hyperboloids of revolution

Hyperboloids: quadric surface generated by rotating a hyperbola around itsmain axis (http://mathworld.wolfram.com/Hyperboloid.html)


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Example: Automotive steering


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Example: Automotive steering


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http://auto.howstuffworks.com/automatic-transmission6.htm

Example:Cordless Screw Driver


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Cordless Screw Driver Gear Trains


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Cordless Screw Driver Gear Trains

Planetary gear train


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Example: Find the output angular velocity 2 for the planetary gear train

shown when the input angular velocity is 4 = 50 rad/sec counterclockwise.

Note that gear 2 andarm 4 are not joined.

sun

annulus

planet

arm

pparmpsss rrrr += )(


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using the tooth relationship to replace the radii,

Substituting back into the other equation gives

n2 = 40

n3 = 20

pparmpsss nnnn += )(

pparmps rrr ++= )(0

pparmps nnn ++= )(0

Mechanism in Cars


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Mechanism in Cars

How Automatic Transmissions Work


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http://auto.howstuffworks.com/automatic-transmission.htm

http://auto.howstuffworks.com/automatic-transmission18.htm



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Planetary Gear Sets

Hydraulic System: transmission flu id via Oil Pump through the ValveBody to control the Clutches and the Bands in order to control the planetary

gear sets.

Seals and Gaskets are used to keep the oil where it is supposed to beand prevent i t from leaking out.

The Torque Converterwhich acts like a clutch to allow the vehicle tocome to a stop in gear while the engine is sti ll running.

The Governor and the Modulatoror Thrott le Cable that moni torspeed and throttle position in order to determine when to shift.



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1. Provides automated control of vehicle launch

(starting the vehicle from a stop)

2. Selects the desired gear ratio

3. Shifts to the desired gear ratio

4. Modifies the engine's speed/torque

5. Transmits power efficiently (helps provide good fuel

economy)

Harmonic Gearing


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Wave Generator:

The wave generator is an oval-shaped cam. It is mounted onto the motor shaft

Flex Spline:

The flex spline is a thin, cup-shaped component made of elastic metal, with teethformed along the outer circumference of the cup's opening. The gear's output shaft isattached to the bottom of the flex spline.

Circular Spline:

The circular spline is a rigid internal gear with teeth formed along its innercircumference.

http://www.hds.co.jp/HDS_hp_english/english/products/index.html

g

http://www.harmonicdrive.net/reference/operatingprinciples/

differential gear box design

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