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
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Equations for Spur Gears
by
Dr. Robert Rizza Associate Professor
Department of Mechanical Engineering Milwaukee School of Engineering
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.
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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
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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
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
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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.
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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
⎧ ⎛ ⎞+⎪ = ⎜ ⎟⎪ ⎜ ⎟⎝ ⎠⎪
⎪ ⎛ ⎞+⎪⎪ = ⎜ ⎟≤ ≤ ⎨ ⎜ ⎟⎝ ⎠⎪
⎪⎪⎪ −⎪ =⎪⎩
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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.
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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
⎧= ⎨
⎩
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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
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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.
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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)