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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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Example: Find the gear reductions in the automotive transmission
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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Example: Find the gear reductions in the automotive transmission
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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)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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Example:Cordless Screw Driver
Example:Cordless Screw Driver
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Example:Cordless Screw Driver
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
How Automatic Transmissions Work
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How Automatic Transmissions Work
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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.
How Automatic Transmissions Work
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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/