Transcript Chapter 13 - Gears and Calculations

Chapter Outline
Shigley’s Mechanical Engineering Design
Types of Gears
Spur
Bevel
Helical
Figs. 13–1 to 13–4
Worm
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Nomenclature of Spur-Gear Teeth
Fig. 13–5
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Tooth Size
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Tooth Sizes in General Use
Table 13–2
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Standardized Tooth Systems (Spur Gears)
Table 13–1
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Standardized Tooth Systems



Common pressure angle f : 20º and 25º
Old pressure angle: 14 ½º
Common face width:
3p  F  5p
p

P
3
5
F
P
P
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Conjugate Action


When surfaces roll/slide
against each other and
produce constant angular
velocity ratio, they are said
to have conjugate action.
Can be accomplished if
instant center of velocity
between the two bodies
remains stationary between
the grounded instant centers.
Fig. 13–6
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Conjugate Action



Forces are transmitted on
line of action which is
normal to the contacting
surfaces.
Angular velocity ratio is
inversely proportional to the
radii to point P, the pitch
point.
Circles drawn through P
from each fixed pivot are
pitch circles, each with a
pitch radius.
Fig. 13–6
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Involute Profile



The most common conjugate profile is the involute profile.
Can be generated by unwrapping a string from a cylinder, keeping
the string taut and tangent to the cylinder.
Circle is called base circle.
Fig. 13–8
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Involute Profile Producing Conjugate Action
Fig. 13–7
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Circles of a Gear Layout
Fig. 13–9
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Sequence of Gear Layout
• Pitch circles in contact
• Pressure line at desired
pressure angle
• Base circles tangent to
pressure line
• Involute profile from
base circle
• Cap teeth at addendum
circle at 1/P from pitch
circle
• Root of teeth at
dedendum
circle at 1.25/P from
pitch circle
• Tooth spacing from
circular pitch, p =  / P
Fig. 13–9
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Tooth Action




First point of
contact at a
where flank of
pinion touches
tip of gear
Last point of
contact at b
where tip of
pinion touches
flank of gear
Line ab is line of
action
Angle of action
is sum of angle
of approach and
angle of recess
Fig. 13–12
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Rack



A rack is a spur gear with an pitch diameter of infinity.
The sides of the teeth are straight lines making an angle to the line
of centers equal to the pressure angle.
The base pitch and circular pitch, shown in Fig. 13–13, are related
by
Fig. 13–13
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Internal Gear
Fig. 13–14
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Contact Ratio
Arc of action qt is the sum of the arc of approach qa and the arc of
recess qr., that is qt = qa + qr
 The contact ratio mc is the ratio of the arc of action and the circular
pitch.


The contact ratio is the average number of pairs of teeth in contact.
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Contact Ratio

Contact ratio can also be found from the length of the line of action

The contact ratio should be at least 1.2
Fig. 13–15
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Interference(Girişim)



Contact of portions of
tooth profiles that are not
conjugate is called
interference.
Occurs when contact
occurs below the base
circle
If teeth were produced by
generating process (rather
than stamping), then the
generating process
removes the interfering
portion; known as
undercutting.
Fig. 13–16
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Interference

For 20º pressure angle, the most useful values from Eqs. (13–11)
and (13–12) are calculated and shown in the table below.
Minimum NP
Max NG
Integer Max NG
Max Gear Ratio
mG= NG/NP
13
14
15
16
17
16.45
26.12
45.49
101.07
1309.86
16
26
45
101
1309
1.23
1.86
3
6.31
77
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Interference

Increasing the pressure angle to 25º allows smaller numbers of
teeth
Minimum NP
Max NG
Integer Max NG
Max Gear Ratio
mG= NG/NP
9
10
11
13.33
32.39
249.23
13
32
249
1.44
3.2
22.64
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Interference





Interference can be eliminated by using more teeth on the pinion.
However, if tooth size (that is diametral pitch P) is to be
maintained, then an increase in teeth means an increase in
diameter, since P = N/d.
Interference can also be eliminated by using a larger pressure
angle. This results in a smaller base circle, so more of the tooth
profile is involute.
This is the primary reason for larger pressure angle.
Note that the disadvantage of a larger pressure angle is an increase
in radial force for the same amount of transmitted force.
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Forming of Gear Teeth

Common ways of forming gear teeth
◦ Sand casting
◦ Shell molding
◦ Investment casting
◦ Permanent-mold casting
◦ Die casting
◦ Centrifugal casting
◦ Powder-metallurgy
◦ Extrusion
◦ Injection molding (for thermoplastics)
◦ Cold forming
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Cutting of Gear Teeth

Common ways of cutting gear teeth
◦ Milling
◦ Shaping
◦ Hobbing
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Shaping with Pinion Cutter
Fig. 13–17
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Shaping with a Rack
Fig. 13–18
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Hobbing a Worm Gear
Fig. 13–19
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Calculation Torque and Revolution
D1 İS pitch
diameter
D2 is pitch
diameter
RESULT:Torque of gear in a gear
pair works together with changes in
proportion to the number of teeth
Fig. 13–19
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Calculation Torque and Revolution
D1 is pitch
diameter
RESULT:Torque of gear
in a gear pair works
together with changes in
proportion to the
number of teeth
D2 İs pitch
diameter
RESULT: Cycles of gears in a pair of gear
which works together changes with
inversely proportional
Fig. 13–19
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Example
P=10 Kw
N=3000 rpm
T=?
N=?
Fig. 13–19
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Example
Calculating Torque
1) P= T1*N1/9550 First find the enter Torque
T1=9550*P/N1 => T1=9550*10/3000 =>
T1=31.8 N-m
2) T1/T2=Z1/Z2 => T2=T1*Z2/Z1 =>
T2=31.8*40/10 => T2=127.2 N-m
3) Due to Z2 and Z3 Gears are on same
shaft T2=T3=127.2N-m
4) T3/T4=Z3/Z4 => T4=T3*Z4/Z3 =>
T4=127.2*50/16 => T4=397.5 N-m
Out Shaft Tork T=397.5N-m
Fig. 13–19
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Example
Calculating Revolution
1) N1/N2=Z2/Z1 => N2=N1*Z1/Z2 =>
N2=3000*10/40 => N2=750rpm
2) Due to Z2 ve Z3 Gears are on same shaft
N2=N3=750rpm
3) N3/N4=Z4/Z3 => N4=N3*Z3/Z4 =>
N4=750*16/50 => N4=240rpm
Out Shaft Revolution N=240 rpm
Ratio of Reduction
i=Input Revolution/OutputRevolution
i=3000/240 => i=12.5
Fig. 13–19
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Example
Pratic Solving
i=i1*i2
i=(Z2/Z1)*(Z4/Z3)
i=(40/10)*(50/16)
i=12,5
Noutput=Ninput/ i
NÇIKIŞ=3000/12.5=240 rpm
Toutput=Tinput*i
P= Tinput*Ninput/9550 => Tinput=9550*P/Ninput
=> Tinput=9550*10/3000 => Tinput=31.8 N-m
Toutput=31.8*12.5
Toutput=397.5 N-m
Fig. 13–19
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Calculations
You must estimate some values.These are
•
•
•
•
Pinion thooth number(Zp)
Main Gear thooth number(Zg)
Presure Angle(f)
Module is main paremater which is variable in
equation, try to find best value in calculations
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Calculations
Slay Makers Equation(interferance control)
k =1 for full depth teeth. k = 0.8 for stub teeth
Slay Makers Equation for Full Sized Gears
Gear Width
b=10*M
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Calculation of Force
P= FS*V(power)
V= п*D*N/60
P=FS* п*D*N/60
Module in terms of:
M=D/Z => D=M*Z
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Calculation of Force
SERVICE FACTOR Cs
Maximum Force From Load (FY)
Fy =FS *CS
GEAR WORKING TIME
LOADING
CASE
Continous
working
to light shock
0.8
8-10
Hour/Da
y
1
1
1.25
1.5
to moderate
shock
1.25
1.5
1.8
1.5
1.8
2.0
to severe shock
3 Hours/Day
24 Hours/Day
1.25
Lewis Equation(Dişin dayanabileceği teğetsel kuvvet)
Ft =σs *b*Y*M
Ft: Maximum Strenght of Teeth(N)
σs : Strenght of the Tooth(N/mm2)
b: Lenght of the Teeth(mm)
Y: Lewis Foctor of Form(Look Table)
M:Module(mm)
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Calculation of Force
Lewis Equation(Dişin dayanabileceği teğetsel kuvvet)
Ft =σs *b*Y*M
Ft: Maximum Strenght of Teeth(N)
σs : Strenght of the Tooth(N/mm2)
b: Lenght of the Teeth(mm)
Y: Lewis Foctor of Form(Look Table)
M:Module(mm)
Number of
theeth
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Calculation of Force
Maximum Force From Load (FY)
Lewis Equation
Ft =σs *b*Y*M
Number of Teeth
Barth Equation (Coming from Lewis)
•
You can add Fatique to Lewis Equation
Ft max =Ft * Cv
=>Ft max =σs *b*Y*M*Cv
Cv : : Speed Coefficient
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Speed Coefficient (CV) Determination
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Iteration
• Must be
• We must find module under this
condition
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Affect of Dynamic Forces From Tooth
Buckinghm Equation
 Under the condution of low speed just load forces exist
 But If the tangential speed more than limits, You must consider dynamic forces
 Buckinghm equtions is used to find dynamic forces.
 FY: maximum tangential force
generated by the load
 Fi: dynamic forces generated by
the speed
 C: Some mistakes coming from
manufacturing
k=0.107*e => Ф=14.50 Full teeth
k=0.111*e => Ф=20 Full teeth
k=0.115*e => Ф=20 Basık (Stub)
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Affect of Dynamic Forces From Tooth
Buckingham Equation
If the pinion and main gear is to be made
of the same material of which the elastic
module (E) is equal to each other. Thus,
from the same steel material and steel
gears Ф = 20 full-length "c" coefficients
occurs as follows.
FY: maximum tangential force
generated by the load
Fi: dynamic forces generated by the
speed
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Affect of Dynamic Forces From Tooth
Buckingham Equation
k=0.107*e => Ф=14.50 Full teeth
k=0.111*e => Ф=20 Full teeth
k=0.115*e => Ф=20 Basık (Stub)
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Affect of Dynamic Forces From Tooth
Buckingham Equation
 Acording to e(mm), manifacturing precision must be
select from table which is given below
The maximum amount of error that may occur in the processing
(mm)
Module
Commercial Gears
Elaborated Gears
Precision machining
M
Grinding Gears
4
0.05
0.025
0.0125
5
0.056
0.025
0.0125
6
0.064
0.030
0.0150
7
0.072
0.035
0.0170
8
0.080
0.038
0.0190
9
0.085
0.041
0.0205
10
0.090
0.044
0.0220
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Affect of Dynamic Forces From Tooth
Buckingham Equation
If the pinion and main gear is to be made of the same material of which
the elastic module (E) is equal to each other. Thus, from the same steel
material and for steel gears Ф = 20 full-length "c" coefficients occurs as
follows.
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In Tooth Corrosion Strength
FW: Abrasion force (N)
DP: pinion gear section circle diameter
(mm)
b: Gear width (mm)
Q: Ratio factor
K: load stress factor (N / mm2)
σ A: Surface strength resistance (N / mm2)
Ф: Gear pressure angle
E: Material elastic modulus (N / mm2)
The surface abrasion resistance is directly
related to the surface hardness σ and BHN
is 2.6487 times the surface hardness.
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Example
Main gear
Coupling
Die rolls
N=310
rpm
Electrical motor
P=20KW
Pinion gear
System works every day between 8 or 10 hour
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Gear reduction ratio:i= NP/NG = 1000/310 => i=3.226:1
Main gear teeth count:ZG =i*ZP = 3.226*31 => ZG =100
ZG /ZP =100/31 =3.226
Slay maker’s equation:
Force of Turning the Gears (FY)
3453.6<7167 No İnterferance
SERVICE FACTOR Cs
b=10*M
GEAR WORKING
TIME
LOADING
CASE
3 Hours/Day
8-10
Hour/Day
24 Hours/Day
Continous
working
to light shock
0.8
1
1.25
1
1.25
1.5
to moderate
shock
1.25
1.5
1.8
1.8
2.0
to severe shock
1.5
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Tangential Force in Lewis Equation (Ft)
Ft =σs *b*Y*M
σs : 138.3 N/mm2
b= 10*M
ZP =31, Ф=200
To get a full gear;
Y= 0.358 Table 3
Ft =138.3*10*M*0.36*M
Ft =498*M2
(Table 1)
STRENGTH (σs) AND HARDNESS OF GEAR MATERIAL WHICH
WILL USED IN LEWIS FORMULA
MATERIAL
STRENGT
H OF
GEAR
σ(N/mm^2)
HARDNES
S (BHN)
20 Grade Iron casting
47.1
200
25 Grade Iron casting
56.4
220
35 Grade Iron casting
56.4
225
35 Grade Iron casting (heat treated)
78.5
300
0.2% Carboniferous steel
138.3
100
0.2% Carboniferous steel (heat treated)
193.2
250
Bronze
68.7
80
Phosphorous Bronze
82.4
100
Manganese Bronze
138.3
100
Aluminous Bronze
152.0
180
0.3% Carboniferous wrought steel
172.6
150
0.3% Carboniferous wrought steel (heat
treated)
220.0
200
C30 Steel (heat treated)
220.6
300
C40 Steel
207.0
150
C45 Steel
233.4
200
Alloy Steel-surface hardeness
345.2
650
Cr-Ni alloy 0.45% carboniferous steel
462.0
400
Cr-Va alloy 0.45% carboniferous steel
516.8
450
Plastic
58.8
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Tangential velocity (Vt)
Tangent velocity of pinion and
gear are same
Np=1000 rpm
ZP=31 gear
Cv=Hız
constant
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Barth Equation
Ft max =Ft*Cv
Barth Equation
Fatique Calculation
For iteration:
M=6
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Gear width (b):
b= 10*M=10*6=> b=60mm.
Pinion section circle diameter: (DP)
Dp=M*ZP = 6*31 => Dp=186 mm.
The main gear section circle diameter: (DG)
DG=M*ZG = 6*100 => Dp=600 mm.
Gear distance between axes: C(mm)
C(mm)= (Dp +DG )/2
C=(186+600)/2=393 mm.
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CONTROL OF CORROSION STRENGTH
SERVICE FACTOR Cs
GEAR WORKING TIME
LOADING
CASE
3 Hours/Day
8-10
Hour/Day
24 Hours/Day
Continous
working
to light shock
0.8
1
1.25
1
1.25
1.5
to moderate
shock
1.25
1.5
1.8
1.5
1.8
2.0
to severe shock
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Ф=200 For full-sized gears: k=0.111*e
c=11655*e
e: for error factor Graphics1.
V=9.7 m/ sn => e=0.04mm.
=> c=11655*0.04=466 N/mm
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Affect of Dynamic Forces From Tooth
Buckingham Equation
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In Tooth Corrosion Strength
FW: Abrasion force (N)
DP: pinion gear section circle diameter
(mm)
b: Gear width (mm)
Q: Ratio factor
K: load stress factor (N / mm2)
σ A: Surface strength resistance (N / mm2)
Ф: Gear pressure angle
E: Material elastic modulus (N / mm2)
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In Tooth Corrosion Strength
İf Ep=EG
Same Material
The surface abrasion resistance is directly
related to the surface hardness σ and BHN
is 2.6487 times the surface hardness.
Selected material is 0,2 C Steel
STRENGTH (σs) AND HARDNESS OF GEAR MATERIAL WHICH WILL USED IN LEWIS
FORMULA
MATERIAL
STRENGTH
OF GEAR
σ(N/mm^2)
HARDNESS
(BHN)
20 Grade Iron casting
47.1
200
25 Grade Iron casting
56.4
220
35 Grade Iron casting
56.4
225
35 Grade Iron casting (heat treated)
78.5
300
0.2% Carboniferous steel
138.3
100
0.2% Carboniferous steel (heat treated)
193.2
250
Bronze
68.7
80
Phosphorous Bronze
82.4
100
Manganese Bronze
138.3
100
Aluminous Bronze
152.0
180
0.3% Carboniferous wrought steel
172.6
150
0.3% Carboniferous wrought steel (heat
treated)
220.0
200
C30 Steel (heat treated)
220.6
300
C40 Steel
207.0
150
C45 Steel
233.4
200
Alloy Steel-surface hardeness
345.2
650
Cr-Ni alloy 0.45% carboniferous steel
462.0
400
Cr-Va alloy 0.45% carboniferous steel
516.8
450
Plastic
58.8
Shigley’s Mechanical Engineering Design
In Tooth Corrosion Strength
FW=DP*b*Q*K
Dp=186 mm.
B=60 mm.
Q=1.52
K=0.5
FW=186*60*1.52*0.5
FW=8,481 N
Must be required to corrosion:
FD =19,719 N > FW=8,481 N
Corrosion Strength is not enough
We must change the metarial
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STRENGTH (σs) AND HARDNESS OF GEAR MATERIAL WHICH WILL USED IN
LEWIS FORMULA
We select new material
We must select a harder material
For 300BHN
FW=DP*b*Q*K
→ FW=186*60*1.52*1.4
FW=23,816 N
FD =19,719 N < FW=23,816 N
Corrosion Strengt is Suitable
MATERIAL
STRENGTH
OF GEAR
σ(N/mm^2)
HARDNESS
(BHN)
20 Grade Iron casting
47.1
200
25 Grade Iron casting
56.4
220
35 Grade Iron casting
56.4
225
35 Grade Iron casting (heat treated)
78.5
300
0.2% Carboniferous steel
138.3
100
0.2% Carboniferous steel (heat treated)
193.2
250
Bronze
68.7
80
Phosphorous Bronze
82.4
100
Manganese Bronze
138.3
100
Aluminous Bronze
152.0
180
0.3% Carboniferous wrought steel
172.6
150
0.3% Carboniferous wrought steel (heat
treated)
220.0
200
C30 Steel (heat treated)
220.6
300
C40 Steel
207.0
150
C45 Steel
233.4
200
Alloy Steel-surface hardeness
345.2
650
Cr-Ni alloy 0.45% carboniferous steel
462.0
400
Cr-Va alloy 0.45% carboniferous steel
516.8
450
Plastic
58.8
Shigley’s Mechanical Engineering Design