Spur Gears Equations
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Transcript of Spur Gears Equations
Equations for Spur Gears
by
Dr. Robert Rizza Associate Professor
Department of Mechanical Engineering Milwaukee School of Engineering
1025 N. Broadway Milwaukee, WI 53202
(414) 277-7377 Fax:(414) 277-2222
Email: [email protected] http://people.msoe.edu/~rizza
The equations are believed to be correct. But, if you find any errors please let me know by writing to [email protected].
1
Nomenclature Symbol* Meaning
a Addendum length b Dedendum length c Clearance C Gear and Pinion center distance
Cf (or ZR) Surface condition factor CH Hardness factor
CP (or ZE) Elastic coefficient d Pitch diameter E Young’s modulus hu Working depth
ht=a+b Whole depth F Face width Fn Transmitted load Fr Radial load Ft Tangential load I Surface Geometry factor φ Pressure angle m Module J Bending Strength Geometry Factor
KB Rim thickness factor KM (CM) Load distribution factor
Ko Overload factor KR Reliability factor KS Size factor KT Temperature factor KV Dynamic Factor MG Gear ratio (never less than 1) mB Backup ratio Mp Contact ratio Mv Angular velocity ratio n Number of revolutions per minute N Number of teeth ν Poisson ratio ω Angular velocity
ωarm Angular velocity of the arm Pb Base Pitch
Pc Circular Pitch Pd Diametral Pitch Qv Quality index r Pitch circle radii t Tooth thickness ρ Radii of curvature TF Oil film temperature Sb Allowable bending stress Sc Allowable surface contact stress Sfb Bending fatigue strength Sfc Surface fatigue contact strength T Torque tR Rim thickness Vt Velocity along the pitch circle YN Stress cycle factor for bending z Length of action
ZN Stress cycle for pitting resistance * Subscripts p and g are used to indicate the pinion and g for some of these parameters.
2
KINEMATICS
Fundamental Relationships
out in inV
in out out
r Nmr N
ω= = =
ω out
Ain V
T 1mT m
= =
cdP
Nπ
= dNPd
= b cP P cos= φ
dmN
=
( ) ( ) ( ) ( )2 2 2 2p p p g g gz r a r cos r a r cos Csin= + − φ + + − φ − φ
dP
P zMcos
=π φ
MG= Mv or MA depending on which is greater than 1
3
Standard Gears Tables 1, 2, 3 (Norton tables 11-1, 11-2 and 11-3, or Shigley 13-1 to 13-3) Table 1 (Norton Table 11-1, Shigley Table 13-1): Standard Gear Parameters
Parameter Coarse Pitch (Pd < 20) Fine Pitch (Pd > 20) φ 20 or 25 degrees 20 degrees a 1.000/ Pd 1.000/ Pdb 1.250/ Pd 1.250/ Pdhu 2.000/ Pd 2.000/ Pdht 2.250/ Pd 2.200/ Pd +0.002 in t 1.571/ Pd 1.571/ Pd
Fillet radius 0.3000/ Pd Not standardized Minimum basic clearance 0.250/ Pd 0.200/ Pd +0.002 in
Minimum width of top land
0.250/ Pd Not standardized
c 0.350/ Pd 0.350/ Pd +0.002 in Table 2 (Norton Table 11-2, Shigley Table 13-2): Standard Diametral Pitches
Coarse Pitch (Pd < 20) Fine Pitch (Pd > 20) 1.25 20 1.5 24 1.75 32
2 48 2.5 64 3 72 4 80 5 96 6 120 8 10 12 14 16 18
4
Table 3 (Norton Table 11-3, Shigley 13-3): Standard Metric Modules m (mm) Equivalent Pd (1/in)
0.3 84.67 0.4 63.50 0.5 50.80 0.8 31.75 1 25.40
1.25 20.32 1.5 16.93 2 12.70 3 8.47 4 6.35 5 5.08 6 4.23 8 3.18 10 2.54 12 2.12 16 1.59 20 1.27 25 1.02
Table 4 Minimum Number of Teeth Required for Avoiding Interference in a 20° Full-Depth Pinion and Full-Depth Gear (Norton Table 11-5). Minimum Pinion Teeth Maximum Gear Teeth
17 1309 16 101 15 45 14 26 13 16
Table 5 Minimum Number of Teeth Required for Avoiding Interference in a Full-Depth Pinion and a Full-Depth Rack (Norton Table 11-4).
Pressure Angle (f) Minimum Number of Teeth. 14.5 32 20 18 25 12
5
Gear Trains 1. Simple fixed Gears
out
in
product of the number of teeth of input gearsproduct of the number of teeth of output gears
ω=
ω
2. Planetary Gears
L arm
F arm
product of the number of teeth of input gears product of the number of teeth of output gears
ω − ω=
ω − ω
STRESSES Transmitted Loads
t n
r n t
F F cosF F sin F tan
= φ= φ = φ
Velocity of the Pitch Circle
(( )
d
t
dn Nn (in FPM) 12 12P
V dn in in/s n in rpsmNn in m/s
π π⎧ =⎪⎪⎪= π⎨⎪π⎪⎪⎩
) (See also Norton Table 11-7)
Bending Stress
t db o m v S B
F PK K K K K
FJσ = English Units
tb a m v S B
FK K K K K
FmJσ = SI units
Bending Strength Geometry Factor, J For values of J see Norton Tables 11-8 to 11-15 or Shigley Figure 14-6 below.
6
Shigley Figure 14-6. Dynamic Factor
tv t
v tt
v t
50 VK (V in ft/min)
50ForQ 5 (V 2,500 ft/min (13m/s))50 200V
K (V in m/s)50
⎧ +=⎪
⎪≤ ≤ ⎨+⎪
=⎪⎩
( )
( )2
3
Bt
v t
Bt
v tv
v
A VK (V in ft/min)
A
A 200VK (V in m/s)For 6 Q 11 A
A=50+56 1-B
12 QB
4
⎧ ⎛ ⎞+⎪ = ⎜ ⎟⎪ ⎜ ⎟⎝ ⎠⎪
⎪ ⎛ ⎞+⎪⎪ = ⎜ ⎟≤ ≤ ⎨ ⎜ ⎟⎝ ⎠⎪
⎪⎪⎪ −⎪ =⎪⎩
7
Experimental results are limited by
( ) (
( )
)
( )
2t max v
2v
t max
V A Q 3 in ft/min
A Q 3V
200
⎡ ⎤= + −⎣ ⎦
⎡ ⎤+ −⎣ ⎦= in m/s
Shigley Figure 14-9 Kv versus Vt. Load distribution factor, KM In general keep 8/Pd < F < 16/Pd. (Shigley equation 14-30) ( )m mc pf pm maK 1 C C C C C= + + e
mc1 for uncrowned teeth
C0.8 for crowned teeth
⎧= ⎨
⎩
pf
2
F 0.025 F 1 in10d
FC 0.0375 0.0125F 1 <F 17 in10d
F 0.1109 0.0207F 0.000228F 17 <F 40 in10d
⎧ − ≤⎪⎪⎪= − + ≤⎨⎪⎪ − + − ≤⎪⎩
Note for values F/(10d) < 0.05, F/(10d)=0.05 is used.
8
1pm
1
1 for straddle mounted pinion with S /S 0.175C
1.1 for straddle mounted pinion with S /S 0.175<⎧
= ⎨ ≥⎩
Shigley Figure 14-10. Definitions of S1 and S.
(Shigley Table 14-9 or Table 5 for values of A, B and C) 2
maC A BF C= + + F (Shigley Figure 14-11 may also be used) Table 5 (Shigley Table 14-9) Values of A, B and C Condition A B C Open gearing 0.247 0.0167 -0.765(10-4) Commercial, enclosed units
0.127 0.0158 -0.093(10-4)
Precision, enclosed units
0.0675 0.0128 -0.926(10-4)
Extraprecision enclosed gear units
0.0030 0.0102 -0.822(10-4)
e0.8 for gearing adjusted at assembly or compatability is improved by lapping or both
C1 for all other conditions
⎧= ⎨
⎩
9
Shigley Figure 14-11. Variation of Cma with F. Size correction factor, KS Most applications KS=1, for gears with large teeth use KS=1.25 to 1.5. Rim thickness factor, KB
RB
t
B B
BB
tm
h2.2421.6 ln m 1.2m
K1.0 m 1.21.0 solid gear
=
⎧ ≤⎪⎪= ⎨ >⎪⎪⎩
See also Shigley Figure 14-16
10
Shigley Figure 14-11. KB as a function of backup ratio. Overload factor Ko Table 6 (Ko Norton Table 11-17 not in Shigley).
Driven Machine Driving Machine Uniform Moderate Shock Heavy Shock
Uniform (electric motor, turbine)
1.00 1.25 1.75 or higher
Light Shock (Multicylinder
engine)
1.25 1.50 2.00 or higher
Medium shock (Single-cylinder
engine)
1.50 1.75 2.25 or higher
Surface Stress
m fP t o v s
cH R
E t o v sI
K CC F K K K US units
dF IK Z
Z F K K K SI unitsdF Z
⎧⎪⎪σ = ⎨⎪⎪⎩
For the value of d use the pitch diameter of the smaller of the two gears in contact.
11
Surface Geometry factor, I (or ZI)
G
G
G
G
mcos sin external gears2 m 1
Imcos sin internal gears
2 m 1
φ φ⎧⎪ +⎪= ⎨
φ φ⎪⎪ +⎩
Elastic Coefficient, Cp (or ZE) (in (lbf/in2)½ or (N/mm2)½)
p 2 2p g
p g
1C1 1
E E
=⎡ ⎤⎛ ⎞ ⎛ ⎞− ν − ν⎢ ⎥⎜ ⎟ ⎜ ⎟π +⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦
See also Norton Table 11-18 (Shigley Table 14-8) for ν=0.3. Surface condition factor, Cf (or ZR) Standard values have yet to be established by AGMA.
Cf=1 for conventional manufactured gears, higher for rougher finishes. Fatigue Strength
Nfb t
T R
YS S
K K=
For values of St, see Tables 7 and 8 (Shigley tables 14-3 and 14-4).
12
Table 7 (Shigley Table 14-3) St at 107 cycles and 0.99 Reliability for Steel Gears
Allowable Bending Stress Number St Psi
Material Designation
Heat Treatment Minimum Surface
Hardness Grade 1 Grade 2 Grade 3 Through-hardened
Fig 14-2 Fig 14-2 Fig 14-2 -
Flame or induction
hardened with type A pattern
AGMA Table 8
45,000 55,000 -
Flame or induction
hardened with type B pattern
AGMA Table 8
22,000 22,000 -
Carburized and hardened
AGAM Table 9
55,000 65,000 or 70,000 (bainite
and cracks limited to
grade 3 levels)
75,000
Steel
Nitrided (through hardened
steels)
83.5 HR15N Fig 14-3 Fig 14-3 -
Nitralloy 135M Nitralloy N,
and 2.5% chrome (no aluminum)
Nitrided 87.5 HR15N Fig 14-4 Fig 14-3 Fig 14-4
13
Shigley Figure 14-2
Shigley Figure 14-3
14
Shigley Figure 14-4 Table 8 (Shigley Table 14-4) Applied Bending Strength St for Iron and Bronze Gears at 107 Cycles and 0.99 Reliability
Material Material Designation
Heat Treatment Typical Minimum
Surface Hardness
Allowable Bending Stress
Number, St (psi) Class 20 As cast - 5000 Class 30 As cast 174 HB 8500
ASTM A48 gray
Class 40 As cast 201 HB 13,000 Grade 60-40-18 Annealed 140 HB 22,000-33,000 Grade 80-55-06 Quenched and
Tempered 179 HB 22,000-33,000
Grade 100-70-03 Quenched and Tempered
229 HB 27,000-40,000
ASTM A536 ductile (modular)
Iron
Grade 120-90-02 Quenched and Tempered
269 HB 31,000-44,000
Bronze Sand cast Min. Tensile strength 40,000
psi
5700
ASTM B-148 Alloy 954
Heat treated Min. Tensile strength 90,000
psi
23,600
15
Stress cycle factor for bending, YN
Shigley Figure 14-14 Values of YN. Temperature factor, KT
o
FT oF
F
1 T 250 FK 460 T T 250 F
620
⎧ ≤⎪= ⎨ +>⎪
⎩
From Norton not Shigley
Reliability factor, KR
R0.658 0.0759ln(1 R) 0.5 < R < 0.99
K0.50 - 0.109ln(1-R) 0.99 R 0.9999
− −⎧= ⎨ ≤ ≤⎩
Surface Fatigue Strength
N H cfc
T R H
Z C SS
K K S=
16
Hardness factor, CH
a. Through hardness
( )H GC 1 A m 1= + −
p
g
p p
g g
p
g
HB0 1.2
HB
HB HBA 0.00898 0.00829 1.2 1.7
HB HB
HB0.00698 1.7
HB
⎧<⎪
⎪⎪⎪= − ≤⎨⎪⎪⎪ >⎪⎩
≤
b. Surface hardened Pinion (> 48 HRC)
( )q
q
H g
0.0112R
0.052R
C 1 B 450 HB
0.00075e (english)B
0.00075e (SI)
−
−
= + −
⎧⎪= ⎨⎪⎩
Stress cycle for pitting resistance, ZN
See Shigley Figure 14-15
Shigley Figure 14-15.
17
Allowable Surface Contact Stress Sc See Table 9 and 10 (Shigley Tables 14-6 and 14-7) Table 9 (Shigley Table 14-6) Repeatedly Applied Contact Strength Sc at 107 and 0.99 Reliability for Steel Gears
Allowable Contact Stres Number Sc, Psi Material Designation
Heat Treatment
Minimum Surface
Hardness Grade 1 Grade 2 Grade 3
Through hardened
Fig 14-5 Fig 14-5 Fig 14-5 -
50 HRC 170,000 190,000 - Flame or Induction hardened
54 HRC 175,000 195,000 -
Carburized and
hardened
AGMA Table 9
180,000 225,000 275,000
83.5 HR1 5N 150,000 163,000 175,000
Steel
Nitrided (through hardened
steels)
84.5 HR1 5N 155,000 168,000 180,000
2.5% Chrome (no aluminum)
Nitrided 87.5 HR1 5N 155,000 172,000 189,000
Nitralloy 135M Nitrided 90 HR1 5N 170,000 183,000 195,000 Nitralloy N Nitrided 90 HR1 5N 172,000 188,000 205,000
2.5% Chrome (no aluminum)
Nitrided 90 HR1 5N 176,000 196,000 216,000
18
Table 10 (Shigley Table 14-7) Repeatedly Applied Contact Strength Sc at 107 and 0.99 Reliability for Iron and Bronze Gears
Material Material Designation
Heat Treatment Typical Minimum
Surface Hardness
Allowable Bending Stress
Number, St (psi) Class 20 As cast - 50,000-60,000 Class 30 As cast 174 HB 65,000-75,000
ASTM A48 gray
Class 40 As cast 201 HB 75,000-85,000 Grade 60-40-18 Annealed 140 HB 77,000-92,000 Grade 80-55-06 Quenched and
Tempered 179 HB 77,000-92,000
Grade 100-70-03 Quenched and Tempered
229 HB 92,000-112,000
ASTM A536 ductile (modular)
Iron
Grade 120-90-02 Quenched and Tempered
269 HB 103,000-126,000
Bronze Sand cast Min. Tensile strength 40,000
psi
30,000
ASTM B-148 Alloy 954
Heat treated Min. Tensile strength 90,000
psi
65,000
Shigley Figure 14-5
19