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Chapter 13Gears - General
Dr. Hitham TlilanDr. Hitham Tlilan
November 18, 2008 Tlilan@hu.edu.jo 1
What are Gears for?• Reducing or increasing speed• Reducing or increasing torque
• Transmitting power around corners over Transmitting power around corners, overdistances
• We want (at various times)– Efficiency– Reliability
A
November 18, 2008 Tlilan@hu.edu.jo 2
– Accuracy– Smoothness
– Quiet operation
2
13-1 Types of Gears1- Gears with
Parallel Shafts• Spur gears (Fig.13.1)(i l di k d i i )(including rack and pinion)Simple to make
• Helical gears (Fig.13.2)– Gradual engagement reduces
impact– Quieter
Fig. 13-1
November 18, 2008 Tlilan@hu.edu.jo 3
– Quieter– Introduces thrust
– have an involutes tooth profile – on a cutting plane perpendicular
to the axisFig. 13-2
13-1 Continued
13.1 ContinuedSpur Gear
November 18, 2008 Tlilan@hu.edu.jo 4
Helical GearThe inclined tooth in the Helical gear
develops thrust load and bending coupleSometimes the Helical gear used to transmit motion between non-
parallel shafts
3
2-Nonparallel Shafts• Bevel gears– Intersecting shaft axes
Straight toothed
13-1 Continued(Fig. 13.3)
– Straight-toothed– Spiral• Hypoid gears– Nonintersecting axes• Involute profiles areobserved on a cutting
Fig. 13-3
November 18, 2008 Tlilan@hu.edu.jo 5
gplane perpendicular tothe axis
Straight bevel gear Spiral bevel gear
Worm and Worm Gear or Wheel• Perpendicular shafts,nonintersecting• Worm shaft in plane of worm wheel
13-1 Continued
• Worm resembles a screw, can have right or left hand thread
• Single enveloping:straight worm, curved wheel
• Double enveloping: both are curved
November 18, 2008 Tlilan@hu.edu.jo 6
both are curved
Fig. 13-4
4
13-2 NomenclatureThe distance measured on the pitch circle from a point of a tooth to a corresponding point on the adjacent tooth.p =tooth thickness + width spacep p
Module = m=Pitch diameter / No. of teeth
Diametral Pitch= P =No. of teeth / Pitch diameter
November 18, 2008 Tlilan@hu.edu.jo 7
-Theoretical circle upon which calculations based-Its diameter is the pitch diameter -The pitch circle of two mating gears are tangent
to each other
Whole depth = ht =Addendum + dedendum
13-2 Continued
2)-(13 [mm]
1)-(13 h][teeth/inc
NdmModule
dNPPitchDiametral
==
==
di tit hteeth of No.:
4)-(13
3)-(13
)([ ]
dN
pPN
dpPitchCircular
N
π
π
=
==
November 18, 2008 Tlilan@hu.edu.jo 8
diameter pitch :d
5
Example• A gear with 20 teeth and a pitch diameter of 2inches has a circular pitch = ?N =20 teethd =2 inp = 2 π/20 = 0.1 π in ,or
= 0.1(25.4) π = 2.54 π mmWhat is its diametral pitch?P = 20/2 = 10 teeth/in.Its mod le is?
November 18, 2008 Tlilan@hu.edu.jo 9
Its module is?m = d/N = 2(25.4)/20 = 0.1mm
13-3 Conjugate ActionWhen the tooth profile, or cams, are designed so as to produce a constant angular velocity ratio during meshing, these are said to have conjugate action.
Pitch point
Fig 13 6
Line of action
November 18, 2008 Tlilan@hu.edu.jo 10
Fig. 13-6
Pitch circles
6
• Conjugate action: engaging bodies rotate at a constant angular velocity ratio• Involute: one curve producing a conjugate action• An involute curve is produced by unwinding a string from a base circle
13-4 Involute Properties
• Two involutes engaging have conjugate action• Base circles are tangent to a line crossing center to center line at the pressure angle• Intersection of pressure line with center line creates pitch point on both pitchcircles.• Increasing center distance changes pitch radii but Fi 13 7
November 18, 2008 Tlilan@hu.edu.jo 11
Increasing center distance changes pitch radii but does not destroy conjugateaction
Fig. 13-7
November 18, 2008 Tlilan@hu.edu.jo 12
7
13-5 Fundamentals• Speed ratios = ratio of pitch circle radii
• An integral number of teeth is preferred
5)-(13 1
2
2
12211 r
rwwwrwrV =⇒==
An integral number of teeth is preferred• Pitch of gears must be the same (diametral pitch = p = N/d )• Pressure angle of mating gears should be the same, usually 14 ½° , 20 or 25° deg.• Final definition of pitch circle and pressure angle depends on center to center distance
November 18, 2008 Tlilan@hu.edu.jo 13
• “Driver” / “Driven” gear distinction determines which face carries load• Initial contact occurs at angle of approach• Final contact occurs at angle of recess
• Contact below the base circle does not produce conjugate action• Gear shape below base is radial or undercut• Contact point moves along pressure line• At the pitch circle the contact is pure rolling• Sliding occurs before and after pitch circle
13-5 Continued
g p
φ Line of
November 18, 2008 Tlilan@hu.edu.jo 14Fig. 13-10 Fig. 13-12
of action
φ
8
13-5 ContinuedThe radius of the base circle =rb= r cosφ (13−6)
abclearenceP
b dedendumP
aaddendum
−=
====. , 2511
abclearence
Pppitchcircular π
==
November 18, 2008 Tlilan@hu.edu.jo 15
One gear may be straight: rack and pinion– Base pitch– Circular pitch
13-5 Continued
p = p cos φ (13 7)pb = pc cos φ (13−7)
Pbφ
November 18, 2008 Tlilan@hu.edu.jo 16
φPc
9
Example 13-1GivenNp=16 toothNG=40 toothDiametral pitch=P=2 =°=
=+
=+
=
====
====
====
142
2082
202
40
82
16
5712
then,cosand 20Since
in centers-2 bewteen distance The
in diameter pitchGear
in diameter pitch Pinion
in .pitchCircular
Gp
GG
pp
rr
ddP
Nd
PN
d
Pp
b φφ
ππ
a=1/P, b=1.25/Pφ =20°
=′+′
+
=°==
=°==
25142
2
409202
20
7632028
(1) in .
then, in, 0.25 by increase to need we Since
(b)
in .cos)(radius baseGear
in .cos)(radius base Pinion
,
Gp
Gp
dd
dd
r
r
Gearb
pininb
b φφ
November 18, 2008 Tlilan@hu.edu.jo 17°=′=
=′=′
==′
′
56222
357201438
4016
2
.))(
(cos
in . in .
(2) and (1) Eqs. from
(2)
therefore change,not does ratio velocity The
p
-1
Gp
G
p
G
p
d
r
dd
dd
d
d
pinionbφ
13-6 Contact Ratio
Fig. 13-15
8)(13ratioContact
recess of arcapproach of arcaction of Arc +=+=
qqq
t
ra
November 18, 2008 Tlilan@hu.edu.jo 18
9)-(13 1.2 cos
cotact. in teeth of pairs ofnumber average the
8)-(13 ratioContact
≥=
==
ϕpLm
pqm
abc
tc
10
13-7 Interference• Interference occurs when contact occurs outside the involute shape of the tooth• Problem occurs when Np << Ng
d N i lland Np is small• Can avoid problem by avoiding large ratios or by using a fine pitch• Alternative is to accommodate interference with undercutting of tooth shape• Undercutting weakens the gear
November 18, 2008 Tlilan@hu.edu.jo 19
Undercutting weakens the gear tooth
( ) 10)-(13 sin
ratio)gear 1-to-(1
gear and pinionspur on teeth of No.smallest the
φφ
22 3114kN
N
p
p
++=
=
13-7 Continued
( )
( )is ceinterferenwithout pinionspur on teeth of No.smallest the Therefore;
unity thangreater is ;
is,that pinion, than teeth more hasgear mating the if
)(sin
φφ2
2
6
k
mNNm
p
GG
p
==
November 18, 2008 Tlilan@hu.edu.jo 20
( ) 11)-(13 sin)(sin)(
φφ
222 21
212 mmmm
kN p +++
=
11
12)(13
ceinterferenwithout rackawithoperate pinionspur For kN 4
=
13-7 Continued
13)-(13 sin
sin
is free- ceinterferen isthat pinion specified a withgear largest The
12)-(13 sin
2
2
φφ
φ
p
pG
p
NkkN
N
N
244
2
22
2
−
−=
=
November 18, 2008 Tlilan@hu.edu.jo 21
November 18, 2008 Tlilan@hu.edu.jo 22
12
13-9 Straight Bevel Gear
-Used to transmit motion between intersecting shafts-The pitch is measured from the large end of the tooth
Fi 13 20
the large end of the tooth.-The circular pitch and pitch diameter are calculated in the same as that for spur gear.
November 18, 2008 Tlilan@hu.edu.jo 23
Fig. 13-20
GearGPinionPNNΓ
NN
P
G
G
P
: ,:
14)-(13 tan tan
apex theat meating cone pitch the by defined are angles pitch The
==γ
13-9 Continued
.
ionapproximat sTredgold'
conethe back ojected onevel is prwhen the b
, adius rack-cone rl to the badius equahaving a rr gearof the spu
s that the same al gear is th of beveof the teethe shape
b
pitchcircular the :
teeth ofnumber virtual the:
15)-(13
p
N
pr
N
jp
b
′
=′π2
The standard straight – tooth bevel gears are cut usingPressure angle of 20°,
November 18, 2008 Tlilan@hu.edu.jo 24
The standard straight tooth bevel gears are cut usingPressure angle of 20 , unequal addenda and dedenda, and full depth teeth
This increase the contact ratio, avoid undercut, andincrease the strength of the pinion
13
13-10 Parallel Helical Gear
anglehilexthe)( distance thepitchcircular transverse the
16)-(13 cos )( distance thepitchcircular normal The
=
==
===
ψ
ψ
acptpnp
ae
t
anglehilex the =ψ
17)-(13 tan
)( distance thepitch axial The
ψtp
xp
xpad
=
==
November 18, 2008 Tlilan@hu.edu.jo 25
18)-(13 cos
pitch diametral normal The, since,
ψ
π
tn
n
nn
PP
PthenPp
=
=
=
13-10 Continued
rotation of direction the in angle pressure the :direction normal the in angle pressure the :
19)-(13 tan
tan cos
t
n
t
n
φφ
φφ
ψ =
N
teeth ofnumber actual the :
teeth ofnumber virtual the:
20)-(13 cos
N
N
NN
′
=′ψ3
November 18, 2008 Tlilan@hu.edu.jo 26
14
( )cos sin2
the smallest No. of teeth of a helical- spur pinion that
run without interference with the same 4 1 1 3 (13-21)
p
G
N
NkN ψ φ
=
= + +
13-10 Continued
( )sinsin2 1 1 3 (13 21)
6
if the mating gear has more
pN tt
φφ
= + +
teeth than pinion, that is,
Therefore; the smallest No. of teeth on pinion without interference is
GG
p
Nm mN
= =
November 18, 2008 Tlilan@hu.edu.jo 27
( )cos ( )sin( )sin
2 22
2 1 2 (1 2p
kN m m m tm t
ψ φφ
= + + ++
13-22)
13-10 Continued
ceinterferenwithout rackawith operate pinionFor k4
24)-(13 sincos
cossin
is free- ceinterferen isthat pinion specified a withgear largest The
23)-(13 sin
cos
2
2
Nk
ktNN
t
kN
p
G
p
φψ
ψφ
φψ
24
4
24
222
2
−
−=
=
November 18, 2008 Tlilan@hu.edu.jo 28
sin cos tNk p φψ 24
15
13-11 Worm Gear
The helix angle for worm gear is quite different from that of the helical gears.
The helix angle on theFig. 13.24
The helix angle on the worm (ψW) is quite large, because of this we it usually specify the lead angle (λ),
ψW + (λ) = 90°
The helix angle on the The circular pitch
November 18, 2008 Tlilan@hu.edu.jo 29
gear is very small (ψG)
axis. worm the containing plane a on the from measured diameter; pitchgear the :
25)-(13
G
tGG
d
pNdπ
=
por, transverse circular pitch
13-11 ContinuedThe worm pitch diameter (dW) should be the same as the that of the hob, which is used to cut the worm. For best power transmission (dW) must be
87508750
26)-(13 .
..
713
87508750 CdCW ≤≤
28)-(13 λ tan
27)-(13 Wx
dL
NpL
π=
=
November 18, 2008 Tlilan@hu.edu.jo 30angle lead the :λpitch axial the :
lead the :
x
W
p L
dπ
16
13-13 Gear Train3
Driven2
Pinion
rev/minor nsrevolustio
29)-(13
:gearset anyFor
=
==
n
nddn
NNn 2
3
22
3
23
diameter pitch teeth ofnumber
==
dN
For the gear trainFor the gear trainFig (13-27)
6gearofspeedTh
November 18, 2008 Tlilan@hu.edu.jo 31
Fig. 13-27(a)
6gear of speed
26 nNN
NN
NNn
The
6
5
4
3
3
2−= Gears 2, 3, and 5 are driversGears 3, 4, and 6 are driven
For spur and helical gears the CCW direction of rotation is the For spur and helical gears the CCW direction of rotation is the positive direction of rotation using (right hand rule)positive direction of rotation using (right hand rule)
13-13 Continued
equationthisinusedbecandiameterspitchThe
30)-(13 value train Theoth umber driven toproduct of
ooth umber driving tproduct ofe ==
gearlast the of speed the :31)-(13
sence opposite in rotatesgear last the if -, sence same the in rotate gearsfirst the andlast the if ,
equationthisinusedbecandiameters pitch The
L
FL
nenn
e
=⎩⎨⎧+
=
November 18, 2008 Tlilan@hu.edu.jo 32
gearfirst the of speed the :Fn
17
13-13 ContinuedPlanetary Train
32)-(13valuetrainThe AL nne −==
November 18, 2008 Tlilan@hu.edu.jo 33(rev/min) arm the of speed the :(rev/min)gear first the of speed the :(rev/min)gear last the of speed the :
32)-(13 valuetrain The
A
F
L
AF
nnn
nne
−==
value train TheAF
AL
nnnn
oth umber driven toproduct ofooth umber driving tproduct ofe
−−
===
Planetary Train -\Continued
November 18, 2008 Tlilan@hu.edu.jo 34
18
I
vB=r5 n5
Example 13-4
A BI
vA=r2 n2
I: Instantaneous center
November 18, 2008 Tlilan@hu.edu.jo 35
I: Instantaneous center
4
2255
4
2255
44 222 r
nNnNr
nrnrr
vvn AB +=
−−=
−=
)(
13-14 Force analysis- Spur Gear
33)13(Fi2i ii t3bt dfth
load. dTransmitte :(a)
F
WFW
t
t
tt 32=
November 18, 2008 Tlilan@hu.edu.jo 3633)-(13 ,
usefulnot radial the while component, useful real the is load tangential the
33)-13(Fig.2pinionagainest 3gear byexerted force the :
tWdT
isF
F
r
t
2
32
32
=
19
13-14 continued
2
2
ddtheTT
if
a
=
== 2(pinion)gear on (a)shaft by exerted torque becomes 33)-Eq.(13 then, 2,gear with dealing are we
d
35)-(13 )(
is units SI in
34)-(13 33000
power The
ft/min velocity line pitch The
d H
tW
VtWH
ndV π
31060
12
=
==
==
November 18, 2008 Tlilan@hu.edu.jo 37rev/min speed, :
mm diameter,gear :kW Power, :
kN load, dtransmitte :
)(
ndH
tWd nt π
13-15 Force analysis- Bevel Gear
November 18, 2008 Tlilan@hu.edu.jo 38
20
13-15 Force analysis - Helical Gear
November 18, 2008 Tlilan@hu.edu.jo 39
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21
November 18, 2008 Tlilan@hu.edu.jo 41
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22
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November 18, 2008 Tlilan@hu.edu.jo 44
23
Dr. Hitham Tlilan 45
Involute face crown gear:
Dr. Hitham Tlilan 46
24
Dr. Hitham Tlilan 47
Dr. Hitham Tlilan 48
25
Dr. Hitham Tlilan 49
Dr. Hitham Tlilan 50
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