MECH103 Mechanisms and Dynamics of Machinery
• Mechanisms: Introduction to Gear Trains
Jaguar 3.8 Litre, 6 cylinder
Textbook: Chap 9
1
Spur Gear
Axis of the gear
Spur gear: gear with radial teeth parallel to its axis
2
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 the
gear into which it is meshed
Rack & Pinion, Bevel Gear
Pinion
Rack
A mechanism in which a toothed wheel
(pinion) engages a notched bar (rack) to
convert rotary motion into linear motion
-Allow bi-directional drive
- rack-and-pinion steering in cars
3
Either of a pair of toothed
wheels whose working surfaces
are inclined to nonparallel
axes. Example: differential in
automobile
Differential: a device that allows a difference in velocity (and displacement)
between two elements
Helical Gears
4
Helical gear: a gear that has the teeth cut at an angle to the
center line of the gear. This kind of gear is useful because there is
no chance of intermittent tooth-to-tooth operation because there are
at least two teeth engaged at any time. It can operate quieter than
spur gear. Helical gears are either right- or left-handed.
Worm Gear & Harmonic Gearing
Worm wheel
(worm gear)
Worm
A coarse, spiral shaped gear cut on a shaft. It is
used to engage with and drive another gear or
portion of a gear. As used in the steering
gearbox, it often engages the cross shaft via a
roller or by a tapered pin.
Ultra low backlash gear technology
with medium-high reduction ratios for
accurate bi-directional repeatability,
high efficiency and power to weight.
Very high gear ratio is possible in small package
5
Allow one directional drive: worm → worm wheel
Good for motion control: robotics
Herringhbone Gear & Gear Train
Herringbone Gears (double helical gears): two
helical gears operating together and so placed
that the angle of the teeth form a "V" shape;
cancel out end-thrust forces. → no thrust
bearing is needed
6
http://content.scvs.tpc.edu.tw/top1/chap10/htm/chap10-12.htm
Differential Gear & Planetary Gear Train
7
Kinematics of Gears
v A = v O + v A / O = 0 + ω1 × rA / O
= ω1k × r1i = ω1 r1 j
v A = v P + v A / P = 0 + ω 2 × rA / P
= ω 2k × (− r2 i ) = −ω 2 r2 j
ω1r1 = −ω 2 r2 or
Fundamental law of gearing:
Angular vel. Ratio = constant throughout the mesh
8
ω2
r1
=−
ω1
r2
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
9
Simple gear set
O/P
I/P
Here, gears 3 & 4 are rigidly connect, as are gears 5 & 6
ω5
n4
ω3
n2
ω7
n6
=−
=−
=−
ω4
n5
ω2
n3
ω
n
6
7
Clearly ω4 = ω3 and ω6 = ω5
ω7 ω3 ω5 ω7
n2 × n 4 × n6
=
=−
ω2 ω2 ω4 ω6
n3 × n 5 × n 7
10
The − sign is necessary to take into account the change in direction of rotation.
Reverted gear train
ω1
ω2
Used in automotive transmission:
- compact, save space
Revert = go back to a previous state
11
Compare:
Internal gear
ω 3 r2 n2
= =
ω 2 r3 n3
The + sign is used here to take into
account the direction of rotation.
12
Example: Find the speed reductions possible for the
transmission
the power is transmitted
through gears 0–4–5–6–
10–12.
If gear 3–4 slides to the left (disengaging 4 from 5) and gear 1–2 to the left
(engaging 1 and 9), then power is transmitted through 0–1–9–6–10–12
13
Example: Find the speed reductions possible for the
transmission
If gear 3–4 slides to the left
(disengaging 4 from 5) and
gear 1–2 to the left (engaging
1 and 9), then power is
transmitted through 0–1–9–6–
10–12
Note: There are 8 possible speed
reductions.
14
Example: Find the gear reductions in the automotive transmission
Low gear: gear 3 meshes with
gear 6, power flows 1–4–6–3.
Second gear: gear 2 meshes
with gear 5, power flows 1–4–
5–2.
High gear: gear 2 is shifted so
that the clutch teeth on the
end of gear 2 mesh with the
clutch teeth of gear 1.
(Direct drive results.)
Reverse gear: gear 3 is shifted
to mesh with gear 8, power
flows 1–4–7–8–3.
http://auto.howstuffworks.com/transmission.htm
http://auto.howstuffworks.com/sequential-gearbox.htm
15
Example: Find the gear reductions in the automotive transmission
power flows 1–4–6–3
⎛ ω out ⎞
ω
⎟⎟ = 3
⎜⎜
⎝ ω in ⎠low ω1
=
⎛ ω out ⎞
⎟⎟
⎜⎜
⎝ ω in ⎠ 2nd
power flows 1–4–5–2.
ω
= 2
ω1
=
16
ω 4 ω5 ω 2
⋅ ⋅
ω1 ω 4 ω 5
⎛ n ⎞ ⎛ n ⎞
= ⎜⎜ − 1 ⎟⎟(1)⎜⎜ − 5 ⎟⎟
⎝ n4 ⎠ ⎝ n2 ⎠
14 25
= ⋅ = 0.564
31 20
ω 4 ω6 ω3
⋅ ⋅
ω1 ω 4 ω 6
⎛ n ⎞ ⎛ n ⎞
= ⎜⎜ − 1 ⎟⎟(1)⎜⎜ − 6 ⎟⎟
⎝ n4 ⎠ ⎝ n3 ⎠
14 18
= ⋅
= 0.301
31 27
Example: Find the gear reductions in the automotive transmission
power flows 1–4–7–8–3.
⎛ ω out ⎞
ω
⎜⎜
⎟⎟ = 3
⎝ ω in ⎠ rev ω1
=
ω 4 ω 7 ω8 ω 3
⋅ ⋅ ⋅
ω1 ω 4 ω 7 ω8
⎛ n1 ⎞ ⎛ n7 ⎞⎛ n8 ⎞
= ⎜⎜ − ⎟⎟(1)⎜⎜ − ⎟⎟⎜⎜ − ⎟⎟
⎝ n4 ⎠ ⎝ n8 ⎠⎝ n3 ⎠
14 14
=− ⋅
= −0.234
31 27
17
Planetary gear train
Example: Find the output angular velocity ω2 for the planetary gear train shown
when the input angular velocity is ω4 = 50 rad/sec counterclockwise.
annulus
sun
planet
arm
Program: 9-33.wm2d
18
Note that gear 2 and
arm 4 are not joined.
n2 = 40
n3 = 20
using the tooth relationship to replace the radii,
Substituting back into the other equation gives
19
Example: Find the gear ratios for the model T Ford gearbox
P2
P1
I/P
2: On
O/P
Gearbox : Integral with the engine. Foot
operated 2 speed and reverse epicyclic
transmission foot-brake, 1908 for 19 yrs
S2
S1
9 million were made!
http://www.t-ford.co.uk/car.htm
Textbook p.507
20
Low gear for the model T Ford
ωin
ωout
Replacing the radii by the number of teeth
on the appropriate gears →
21
Example: Reversed Gear case
P2
P1
I/P
1:On
O/P
Gearbox : Integral with the engine. Foot
operated 2 speed and reverse epicyclic
transmission foot-brake, 1908 for 19 yrs
S2
S1
9 million were made!
http://www.t-ford.co.uk/car.htm
Textbook p.507
22
Reverse gear for the model T Ford
ωin
ωout
Note the negative sign indicating a change in direction
23
Reverse on a Car
http://www.innerauto.com/innerauto/anim/trans.html
24
Model T Ford, 1912 Landaulette
25
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
26
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.
27
Profile of the Involute Profile
Pressure angle = the angle between Line of Action (common normal) and the direction
of velocity at the pitch point (has been standardized: 14.5°, 20°, 25°)
Line of action: A line normal to a pair of mating tooth profiles at their point of contact
28
Involute in Action
Pitch circle=rolling cylinder circle
Addendum: the amount of tooth that sticks out above the pitch
circle
29
video from http://www.howstuffworks.com
Nomenclature
Figure 11-8
30
pc =
πd
N
Pitches, Etc.
pc =
circular pitch (mm, in.)
base pitch (mm, in.)
31
N
pb = pc cos φ
pd =
diametral pitch (teeth/in.)
module (mm/teeth)
πd
m=
d
N
N
d
Minimum # of Teeth
minimum # of teeth to avoid undercutting with gear and rack
N min
32
2
= 2
sin φ
φ = pressure angle
Cycloid curve for cycloidal gear
Commonly used in watches and clocks
x = aθ − a sin θ
y = a − a cos θ
http://mathworld.wolfram.com/Cycloid.html
33
Anton’s Calculus (7th): p.93
Involute curve for involute gear
x = a(cosθ + θ sinθ )
y = a(sinθ − θ cosθ )
Commonly used in all kinds of power
transmission systems
http://mathworld.wolfram.com/CircleInvolute.html
34
Anton’s Calculus (7th): p.782
Origin of involute curve
x = a (cos θ + θ sin θ )
y = a (sin θ − θ cos θ )
35
Rack & Pinion
36
Bevel Gear
37
Worm Gear
38
Gear Train
39
Automotive Differential Gear
40
Manual Transmission
Low gear
41
High gear
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 tooth
center. (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 differentials.
42
Herringbone gears examples
from D.O. James Gear Manufacturing Co.
http://www.linngear.com/products/highlights/infosheets/g-3.html
43
Comparision between Helical Gear and Herringbone Gear
44
Bevel Gear: based on rolling cones
Apices must be
conincident
Incorrect arrangement
45
Correct arrangements
Spiral bevel Gear & Hypoid Gear
Spiral bevel Gears
Hypoid Gears are similar in their general
form to bevel gears. However, Hypoid
Gears operate on non-intersecting axes.
(Hypoid = a contradiction of hyperboloid)
46
Hypoid Gear: based on hyperboloids of revolution
Rolling hyperboloids of
revolution
47
Automotive hypoid final drive gears
(General Motors, Detroit, MI)
Hyperboloids: quadric surface generated by rotating a hyperbola around its
main axis (http://mathworld.wolfram.com/Hyperboloid.html)
48
Example: Automotive steering
49
http://auto.howstuffworks.com/automatic-transmission6.htm
50
Example:Cordless Screw Driver
51
Example:Cordless Screw Driver
52
Cordless Screw Driver Gear Trains
53
Planetary gear train
Example: Find the output angular velocity ω2 for the planetary gear train
shown when the input angular velocity is ω4 = 50 rad/sec counterclockwise.
annulus
sun
Note that gear 2 and
arm 4 are not joined.
planet
arm
rs ω s = ( rs + r p )ω arm − r p ω p
54
nsω s = ( ns + n p )ω arm − n pω p
0 = ( rs + rp )ω arm + rpω p
n2 = 40
n3 = 20
0 = ( ns + n p )ω arm + n pω p
using the tooth relationship to replace the radii,
Substituting back into the other equation gives
55
Mechanism in Cars
56
How Automatic Transmissions Work
http://auto.howstuffworks.com/automatic-transmission.htm
http://auto.howstuffworks.com/automatic-transmission18.htm
57
How Automatic Transmissions Work
58
How Automatic Transmissions Work
Planetary Gear Sets
Hydraulic System: transmission fluid via Oil Pump through the Valve
Body 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 be
and prevent it from leaking out.
The Torque Converter which acts like a clutch to allow the vehicle to
come to a stop in gear while the engine is still running.
The Governor and the Modulator or Throttle Cable that monitor
speed and throttle position in order to determine when to shift.
59
How Automatic Transmissions Work
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)
60
Harmonic Gearing
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 teeth
formed along the outer circumference of the cup's opening. The gear's output shaft is
attached to the bottom of the flex spline.
Circular Spline:
The circular spline is a rigid internal gear with teeth formed along its inner
circumference.
http://www.hds.co.jp/HDS_hp_english/english/products/index.html
http://www.harmonicdrive.net/reference/operatingprinciples/
61