Gears
1
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
MECE 304 Mechanical Machine Elements-Gears
LECTURE NOTES- MECE 304 Mechanical Machine Elements
Chapter 8- Gears
(Notes from: Chapter 13, Budynas R.G., Nisbett J.K., Shigley’s Mechanical Engineering Design, Mc Graw
Hill, 8th Edition and special notes)Hill, 8th Edition and special notes)
2
13-1 Types of gears
1-Spur gears2- Helical gears3-Bevel gears4-Worm gears
MECE 304 Mechanical Machine Elements-Gears
3
Spur Gears
MECE 304 Mechanical Machine Elements-Gears
4
Helical Gears
MECE 304 Mechanical Machine Elements-Gears
5
Bevel Gears
MECE 304 Mechanical Machine Elements-Gears
6
Worm Gears
MECE 304 Mechanical Machine Elements-Gears
7
MECE 304 Mechanical Machine Elements-Gears
8
Fig 13-5 Nomenclature of a spur gear teeth
13-2 Nomenclature
The pitch circle is the theoretical circle upon which all calculations are based; its diameter is the pitch diameter
The circular pitch p is the distance from a point on one tooth to a corresponding point on an adjacent tooth.The module m is the ratio of the pitch diameter to the number of teeth. It is the index of tooth size in SI. The diametral pitch P is the ratio of the number of teeth to the pitch diameter. P is used with U.S. units.
MECE 304 Mechanical Machine Elements-Gears
9
P is used with U.S. units.Addendum is the radial distance between top land and pitch circleDedendum is the radial distance between pitch circle and bottom land.
P=N/d 13-1
m=d/N 13-2
p=π d/N=π m 13-3
pP=π 13-4
where P=diametral pitchN=number of teethd= pitch diameter, inm=module, mmd=pitch diameter, mmp=circular pitch
The relation between P and m are as follows
MECE 304 Mechanical Machine Elements-Gears
10
The relation between P and m are as follows
Diametral pitch=25.4 / Module
Metric Gears Geometrical Relations (without profile modification)(yellow color are controlling parameters of the gears)
MECE 304 Mechanical Machine Elements-Gears
Parameter Description Formula Type or
unit
Ø Pressure angle — degree
m Module — millimeter
Z Number of teeth — integer
p
Pitch of the teeth,on a straight
generative rack. π * m millimeter
11
S
Circular tooth thickness, measured
on the pitch circle. p / 2 millimeter
ha
Addendum = height of a tooth
above the pitch circle. ha=m millimeter
hf
Dedendum = depth of a tooth
below the pitch circle. (For
m>1.25) hf = m * 1.25 millimeter
r Radius of the pitch circle. m * Z / 2 millimeter
ra Radius of the outer circle. r + ha millimeter
rf Radius of the root circle. r - hf millimeter
rb Radius of the base circle. r * cos( Ø ) millimeter
rc Radius of the root concave corner. 0.38 * m millimeter
13-3 Congugate action
When the tooth profiles are designed so as to produce constant angular velocity ratio during meshing these are said to have congugate action
MECE 304 Mechanical Machine Elements-Gears
12
Fig 13-6 When contacting surfaces of cam A and follower B are involute profiles, congugate action produces constant angular velocity ratio
13-4 Involute properties
MECE 304 Mechanical Machine Elements-Gears
Base circle:The circle on which the involute is generated
13
Fig 13-7 (a) generation of an involute, (b) involute action
13-5 Fundamentals
MECE 304 Mechanical Machine Elements-Gears
14
Fig 13-8 Construction of a involute curve
Fig 13-9 Circles of a gear layout
MECE 304 Mechanical Machine Elements-Gears
15
ω1/ ω2=r2/r1 13-5
Fig 13-9 Relation of base circle to pitch radius by pressure angle:rb=r cos Ø
MECE 304 Mechanical Machine Elements-Gears
16
Fig 13-12 Tooth action
MECE 304 Mechanical Machine Elements-Gears
17
Fig 13-13 Rack and pinion
MECE 304 Mechanical Machine Elements-Gears
18
pb=pc cos Ø 13-7
where pb is base pitch and pc is circular pitch
Fig 13-14 Internal gear and pinion
MECE 304 Mechanical Machine Elements-Gears
19
Addendum circle of internal gear lies inside the pitch circle.
Compound gears: Compound gears are gears attached to each other and rotating around the same axis.
MECE 304 Mechanical Machine Elements-Gears
20
Backlash: Is the amount by which the width of a tooth space exceeds the thickness of the engaging tooth measured on the pitch circles
MECE 304 Mechanical Machine Elements-Gears
21
13-6 Contact ratio
AP+PB=qt (arc of action)
Contact ratio=mc=qt/p (gears to be designed with mc>1.2) 13-8mc indicates the number of gears in contact or
mc=Lab/p*cosØ
MECE 304 Mechanical Machine Elements-Gears
22
Fig 13-15 Definition of contact ratio
13-7 Interference
The contact of tooth profiles that are not congugate are called interference
**In gear interference, the involute tip of the driven gear tends to dig out the non-involute flank of the driver.
MECE 304 Mechanical Machine Elements-Gears
23
driver.**Interference can be eliminated by using more teeth on the gears or byusing a larger pressure angle. The demand for smaller pinions with fewer teeth favors the use of a 25°pressure angle.**As a rule of thumb, use at least 14 teeth in a pinion.
13-8 Forming of Gear Teeth
Machining: Milling, Shaping, HobbingFinishing: Shaving, Burnishing, Grinding, Lapping
13-9 Straigth Bevel GearsTransmits motion betweeen intersecting shafts
MECE 304 Mechanical Machine Elements-Gears
24Fig 13-20 Terminology of bevel gears
tan γ =NP/NG
tan Г =NG/NP
13-10 Parallel Helical Gears
Transmit motion between parallel shafts. Helix angle is the same for the 2 gear set but hands are different (rigth hand and left hand helix)
MECE 304 Mechanical Machine Elements-Gears
25
Fig 13-21 An involute helicoid
Parallel Helical Gears
MECE 304 Mechanical Machine Elements-Gears
pn=ptcosψ
px=pt/tanc
Pn=Pt/cosψ
Cosψ=tanØn /tanØt
where
26Fig 13-22 Nomenclature of helical gears
where
px=axial pitch
pt= transverse circular pitch
pn=normal circular pitch
Pn=normal diametral pitch
Pt=transversal diametral pitch
ψ= helix angle
13-12 tooth systemsIs a standard that specifies the relationship involving addendum, dedendum, working depth, tooth thickness and pressure angle
Table 13-1 Standard and Commonly Used Tooth Systems for Spur Gears
MECE 304 Mechanical Machine Elements-Gears
Tooth System Pressure Angle Ø deg. Addendum a Dedendum b
Full depth 20 1/Pd or m 1.25/Pd or 1.25m 1.35/Pd or 1.35m
22 1/P or m 1.25/P or 1.25m
27
221/2 1/Pd or m 1.25/Pd or 1.25m 1.35/Pd or 1.35m
25 1/Pd or m 1.25/Pd or 1.25m 1.35/Pd or 1.35m
Stub 20 0.8/Pd or 0.8m 1/Pd or m
13-12 tooth systems
Table 13-2 Tooth sizes in general uses
MECE 304 Mechanical Machine Elements-Gears
Diametral Pitch
Coarse 2, 21/2, 21/2, 3, 4, 6, 8, 10, 12, 16 Fine 20, 24, 32, 40, 48, 64, 80, 96, 120, 150, 200
28
Fine 20, 24, 32, 40, 48, 64, 80, 96, 120, 150, 200
Modules
Preferred 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 10, 12, 16, 20, 25, 32, 40, 50 Next Choice 1.125, 1.375, 1.75, 2.25, 2.75, 3.5, 4.5, 5.5, 7, 9, 11, 14, 18, 22, 28, 36, 45
13-12 tooth systemsTable 13-4 Standard Tooth Proportions for Straigth Helical Gears
MECE 304 Mechanical Machine Elements-Gears
Quantity Formula Quantity Formula Addendum (a)
[ ]n
n
mP
00.1
External gears:
Dedendum (b) [ ]n
n
mP
25.125.1
Standard center dis.
2
dD +
Pinion pitch diameter
ψψ coscos
npp mN
P
N
Gear outside diameter
D+2a
29
ψψ coscosnP
diameter
Gear pitch diameter
ψψ coscos
nG
n
G mN
P
N
Pinion outside diameter
d+2a
Normal arc tooth
thickness
−−
22
n
n
n
n
Bm
B
Pπ
π
Gear root diameter Pinion root diameter Internal gears:
D-2b d-2b
Pinion base diameter d cosØt Center distance
2
dD −
Gear base diameter D cosØt Inside diameter D-2a Base helix angle tan-1[tanψ cosØt Center distance D+2b
13-13 Gear trains
If a pinion 2 is driving a gear 3 thann3=(N2/N3)n2=(d2/d3)n2 13-29
where n=rpmN=number of teethd=pitch diameter
MECE 304 Mechanical Machine Elements-Gears
30
Fig 13-27 A gear train
.
numberstoothdrivenofproduct
numberstoothdrivingofproducte
....
....= 13-30
nL=e*nF 13-31nL=speed of last gear, nF=speed of first gear
Notes:**Gear ratio between two gears should not exceed 6 (may be up to 10 in certain cases).
This should also be extended to train values of gear trains such that:
e ≤ 35 for a 3-gear system
MECE 304 Mechanical Machine Elements-Gears
31
e ≤ 35 for a 3-gear systeme ≤ 150 for a 4-gear system
** Gear ratio should not be an integer number. Otherwise the same set of teeth will come in contact repeatedly, thus generating an unwanted wear
Planetary Gear trains: Gear trains in which some of gear axis rotate about axis of the other gears
MECE 304 Mechanical Machine Elements-Gears
32
Fig 13-30 A planetary gear train
Fig 13-31 A gear train on the arm of a planetary gear train
Planetary Gear trains:
From figure 13-31n23=n2-n3 and n53=n5-n3
If we divide the 2 equations side by side
MECE 304 Mechanical Machine Elements-Gears
3553
nn
nn
n
n
−
−=
33
3223 nnn −
3
3
nn
nne
F
L
−
−= 13-32
13-14 Force analysis-Spur Gears
Power (H) transmitted is :H=T*ω 13-33
To convert to customary units:
V=π d*n 13-34(V=π d*n/12)
Where V= pitch line velocity mm/sec (ft/min)d=gear diameter mm (inch)
MECE 304 Mechanical Machine Elements-Gears
34
d=gear diameter mm (inch)n=gear speed rev per sec (rpm)
Wt=(60000H)/π d*n 13-36(Wt=33000*H/V)
where Wt=transmitted load kN (lbf)H=power kw (hp)d=gear diameter, mm n=speed rpm(V=pitch line velocity, ft/min)
Force analysis-Spur Gears
MECE 304 Mechanical Machine Elements-Gears
35
Fig 13-32 Spur gears free body diagrams
13-15 Force analysis Straigth Bevel gears
Wt=T/rav 13-37where T is the torque, rav is the pitch radius at the midpoint of tooth for the gear under consideration
Wr=Wt*tanØ *cosγWa=Wt*tanØ *sinγ 13-38
MECE 304 Mechanical Machine Elements-Gears
36
Fig 13-35 Bevel gear tooth forces (see also Fig 13-20)
13-16 Force analysis Helical Gearing
MECE 304 Mechanical Machine Elements-Gears
Wr = W sinØn
Wt = W cosØn cosΨ
Wa = W cosØn sinΨ
where W= total force
W =radial component
37
Fig 13-37 Tooth forces acting on a rigth hand helical gear
Wr=radial component
Wt=tangential component (transmitted load)
Wa = axial component (thrust load)
Notes
**Helical gears subject the shaft bearings both radial and axial loads
(see following figures for the direction of axial loads)
**The initial contact of helical gear teeth is a point which extends into a line as the teeth come into more engagement. It is this gradual
engagement of teeth and the smooth transfer of load from one tooth to another which give helical gears the ability to transmit heavy loads at high speeds.
MECE 304 Mechanical Machine Elements-Gears
38
high speeds.
**When the thrust load is objectionable, it may be desirable to use double helical gears (herringbone)
Direction of Axial Force in Helical Gear Systems
MECE 304 Mechanical Machine Elements-Gears
39