1. Chapter Outline
Shigley’s Mechanical Engineering Design

2. Types of Gears Figs. 13–1 to 13–4 Spur Helical Worm Bevel
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3. Nomenclature of Spur-Gear Teeth
Fig. 13–5
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4. Tooth Size
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5. Tooth Sizes in General Use
Table 13–2
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6. Standardized Tooth Systems (Spur Gears)
Table 13–1
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7. Standardized Tooth Systems
Common pressure angle f : 20º and 25º
Old pressure angle: 14 ½º
Common face width:
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8. 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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9. 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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10. 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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11. Involute Profile Producing Conjugate Action
Fig. 13–7
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12. Circles of a Gear Layout
Fig. 13–9
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13. 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 / P
Fig. 13–9
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14. First point of contact at a where flank of pinion touches tip of gear
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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15. 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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16. Internal Gear
Fig. 13–14
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17. The contact ratio is the average number of pairs of teeth in contact.
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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18. 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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19. 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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20. 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
16.45 16
1.23 14
26.12 26
1.86 15
45.49 45 3
101.07
101
6.31 17
1309 77
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21. Increasing the pressure angle to 25º allows smaller numbers of teeth 9
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
13.33 13
1.44 10
32.39 32
3.2 11
249.23
249
22.64
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22. 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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23. Common ways of forming gear teeth Sand casting Shell molding
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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24. Common ways of cutting gear teeth Milling Shaping Hobbing
Cutting of Gear Teeth
Common ways of cutting gear teeth
Milling
Shaping
Hobbing
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25. Shaping with Pinion Cutter
Fig. 13–17
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26. Shaping with a Rack
Fig. 13–18
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27. Hobbing a Worm Gear
Fig. 13–19
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28. 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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29. 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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30. Example P=10 Kw N=3000 rpm T=? N=? Fig. 13–19
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31. Example Calculating Torque Fig. 13–19
1) P= T1*N1/ 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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32. Calculating Revolution
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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33. Example Pratic Solving Fig. 13–19 i=i1*i2 i=(Z2/Z1)*(Z4/Z3)
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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34. You must estimate some values.These are Pinion thooth number(Zp)
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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35. Slay Makers Equation(interferance control)
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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36. 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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37. Calculation of Force Fy =FS *CS
SERVICE FACTOR Cs
GEAR WORKING TIME
LOADING CASE
3 Hours/Day
8-10 Hour/Day
24 Hours/Day
Continous working
0.8 1
1.25
to light shock
1.5
to moderate shock
1.8
 to severe shock
2.0
Maximum Force From Load (FY)
Fy =FS *CS
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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38. Calculation of Force
Lewis Equation(Dişin dayanabileceği teğetsel kuvvet)
Ft =σs *b*Y*M
Number of theeth
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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39. Calculation of Force Lewis Equation Ft =σs *b*Y*M
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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40. Speed Coefficient (CV) Determination
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41. We must find module under this condition
         
Iteration
Must be
We must find module under this condition
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42. 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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43. 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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44. 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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45. 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 M
Commercial Gears
Elaborated Gears
Precision machining
Grinding Gears 4
0.05
0.025
0.0125 5
0.056 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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46. 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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47. 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 times the surface hardness.
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48. System works every day between 8 or 10 hour
Example
Coupling
Main gear
Die rolls
N=310 rpm
Electrical motor
P=20KW
Pinion gear
System works every day between 8 or 10 hour
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49. 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:
3453.6<7167 No İnterferance
SERVICE FACTOR Cs
GEAR WORKING TIME
LOADING CASE
3 Hours/Day
8-10 Hour/Day
24 Hours/Day
Continous working
0.8 1
1.25
to light shock
1.5
to moderate shock
1.8
 to severe shock
2.0
Force of Turning the Gears (FY)
b=10*M
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50. Tangential Force in Lewis Equation (Ft)
Ft  =σs *b*Y*M 
σs  : 138.3 N/mm2           (Table 1)
b= 10*M
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
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
Manganese Bronze
Aluminous Bronze
152.0
180
0.3% Carboniferous wrought steel
172.6
150
0.3% Carboniferous wrought steel (heat treated)
220.0
C30 Steel (heat treated)
220.6
C40 Steel
207.0
C45 Steel
233.4
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
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
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51. Tangent velocity of pinion and gear are same
Tangential velocity (Vt)
Tangent velocity of pinion and gear are same
Np=1000 rpm ZP=31 gear
Cv=Hız constant
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52. Fatique Calculation Barth Equation Ft max =Ft*Cv Barth Equation
For iteration:
  M=6 
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53. Pinion section circle diameter: (DP) Dp=M*ZP = 6*31 => Dp=186 mm.
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=( )/2=393 mm.
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54. CONTROL OF CORROSION STRENGTH
SERVICE FACTOR Cs
GEAR WORKING TIME
LOADING CASE
3 Hours/Day
8-10 Hour/Day
24 Hours/Day
Continous working
0.8 1
1.25
to light shock
1.5
to moderate shock
1.8
 to severe shock
2.0
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55. 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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56. Affect of Dynamic Forces From Tooth Buckingham Equation
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57. 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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58. In Tooth Corrosion Strength
İf Ep=EG
Same Material
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
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
Manganese Bronze
Aluminous Bronze
152.0
180
0.3% Carboniferous wrought steel
172.6
150
0.3% Carboniferous wrought steel (heat treated)
220.0
C30 Steel (heat treated)
220.6
C40 Steel
207.0
C45 Steel
233.4
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
The surface abrasion resistance is directly related to the surface hardness σ and BHN is times the surface hardness.
Selected material is 0,2 C Steel
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59. 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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60. We must select a harder material For 300BHN
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
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
Manganese Bronze
Aluminous Bronze
152.0
180
0.3% Carboniferous wrought steel
172.6
150
0.3% Carboniferous wrought steel (heat treated)
220.0
C30 Steel (heat treated)
220.6
C40 Steel
207.0
C45 Steel
233.4
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
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  
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