BS gear Stranded
Short Description
Descripción: BS gear Stranded...
Description
BRITISH STANDARD 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
BS 436-3:1986 (Reprinted, incorporating Amendment No. 1)
Spur and helical gears — Part 3: Method for calculation of contact and root bending stress limitations for metallic involute gears
UDC 621.833.1
BS 436436-3:1 3:198 986 6 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Committees responsible for this British Standard The preparation of this British Standard was entrusted by the M achinery and Components Standards Committee (MCE/-) to Technical Committee MCE/5 upon which the following bodies were represented: Association of Consulting Engineers British Clock and Watch Manufacturers’ Association British Gear Manufacturers Association British Horological Institute British Railways Board Engineering Equipment and Materials Users’ Association Federation of Manufacturers of Construction Equipment and Cranes Gauge and Tool Makers’ Association Institution of Mechanical Engineers Institution of Production Engineers Lloyds Register of Shipping Machine Tool Industry Research Association Milling Cutter Association Ministry of Defence National Coal Board Society of Motor Manufacturers and Traders Limited
This British Standard, having been prepared under the direction direction of the Machinery Machinery and Components Components Standards Standards Committee was published under the authority authority of the Board Board of BSI and and comes comes into into effec effectt on 30 September 1986 © BSI 06-1999
The following BSI references relate to the work on this standard: Committee reference MCE/5 Draft for comment comment 84/73219 84/73219 DC
ISBN 0 580 15227 8
Amendments issued since publication Amd. No.
Date of issue
Comments
5797
May 1988
Indicated by a sideline in the margin
BS 436436-3:1 3:198 986 6 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Committees responsible for this British Standard The preparation of this British Standard was entrusted by the M achinery and Components Standards Committee (MCE/-) to Technical Committee MCE/5 upon which the following bodies were represented: Association of Consulting Engineers British Clock and Watch Manufacturers’ Association British Gear Manufacturers Association British Horological Institute British Railways Board Engineering Equipment and Materials Users’ Association Federation of Manufacturers of Construction Equipment and Cranes Gauge and Tool Makers’ Association Institution of Mechanical Engineers Institution of Production Engineers Lloyds Register of Shipping Machine Tool Industry Research Association Milling Cutter Association Ministry of Defence National Coal Board Society of Motor Manufacturers and Traders Limited
This British Standard, having been prepared under the direction direction of the Machinery Machinery and Components Components Standards Standards Committee was published under the authority authority of the Board Board of BSI and and comes comes into into effec effectt on 30 September 1986 © BSI 06-1999
The following BSI references relate to the work on this standard: Committee reference MCE/5 Draft for comment comment 84/73219 84/73219 DC
ISBN 0 580 15227 8
Amendments issued since publication Amd. No.
Date of issue
Comments
5797
May 1988
Indicated by a sideline in the margin
BS 436-3 436-3:19 :1986 86 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Contents Page Inside front cover ii
Committees responsible Foreword Section 1. General 1 Scope and field of application 2 Definitions and symbols 3 Guide to using the calculation procedure
Section 2. Contact stress calculations 4 Basic equations for contact stress calculations 5 Nomi No mina nall tan tange gent ntia iall for force ce for for con conta tact ct stre stress ss,, F Ht 6 Zone Zone fac factor tor for for cont contac actt stre stress ss,, Z H 7 Cont Contac actt rat ratio io fact factor or for for cont contac actt str stres ess, s, Z & 8 Elas Elasti tici city ty fact factor or for for cont contac actt stre stress ss,, Z E 9 Basi Basicc end endur uran ance ce limi limitt for for cont contac actt str stres ess, s, BH lim 10 Mater Materia iall qual quality ity for for conta contact ct stre stress ss,, Z M 11 Lubrica Lubricant nt influe influence nce,, rough roughnes nesss and speed speed factor factorss for for conta contact ct stress, Z L, Z R and Z v 12 Work Work harden hardenin ing g fact factor or for for cont contac actt stre stress, ss, Z W 13 Size Size fac facto torr for for cont contac actt stre stress ss,, Z X 14 Life Life fact factor or for for con conta tact ct stre stress ss,, Z N 15 Appl pplication fac factor tor, K A 16 Dynamic fa factor, K v 17 Load Load dist distri ribu buti tion on fact factor ors, s, K K H! and K and K H ¶ 18 Minimum Minimum demande demanded d and and actual actual safety safety fact factors ors on cont contact act stress stress,, S H min and S H Section 3. Bending stress calculations 19 Basic equations for tooth root bending stress 20 Nomi No minal nal tang tangen entia tiall forc force e for for bendi bending ng str stres ess, s, F Ft 21 Geom Ge omet etry ry facto factors rs for for bendi bending ng stres stress, s, Y F, Y S, Y ¶ 22 Basi Basicc endu endura ranc nce e lim limit it for for ben bendin ding g str stres esss BF0 23 Mater Materia iall qual qualit ity y fact factor or for for ben bendi ding ng str stres ess, s, Y M 24 Sensi Sensiti tivi vity ty fact factor or for for ben bendi ding ng stre stress ss,, Y B 25 Surf Surfac ace e cond conditi ition on fac facto torr for for bend bendin ing g stres stress, s, Y R 26 Size Size fac facto torr for for bend bendin ing g str stres ess, s, Y X 27 Life Life fact factor or for for ben bendi ding ng stre stress ss,, Y N 28 Load Load fac facto tors rs for for bend bendin ing g stre stress ss,, K F! and K and K F ¶ 29 Minimum Minimum demande demanded d and and actual actual safety safety factor factorss on on tooth tooth root root stress, S F min and S F
8 8 9 9 10 10 12 13 13 13 13 16 18 22 26 27 28 28 31 32 33 34 34 36 36 36
Appendix A Variable duty calculations Appendix B Gearing equations Appendix C Design guidance on tooth modifications Appendix D Typical residual stresses Appendix E Tooth and mesh stiffness c and c * Appendix F Definition of material quality Appendix G Examples of calculations Appendix H Equations of graphs
39 41 42 44 45 46 47 52
Figure 1 — Yield strength for contact stress, BHY Figure 2 — Values of BHD Figure 3 — Values of Z G2
10 11 12
9
© BSI 06-1999
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BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Page 12 14 14 15 16 17 21 21 24 25 29 32 35 35 37
Figure 4 — Casedepth factor, Z c Figure 5 — Combined speed and lubricant factors, Z LZ v Figure 6 — Roughness factor, Z R Figure 7 — Values of Z W Figure 8 — Life factor for contact stress, Z N Figure 9 — Constituent parts of typical gear load Figure 10 — K v350 ¶ for helical gears, & ¶ W1 Figure 11 — K v350! for spur gears Figure 12 — Constant K for calculation of f sh Figure 13 — Values of qy Figure 14 — Dimensions of the basic rack of the gearing Figure 15 — Values of BF0 Figure 16 — Values of Y R Figure 17 — Values of Y X Figure 18 — Y N for through hardened steel Figure 19 — Y N for thick case surface hardened steel and cast iron Figure 20 — Y N for thin case surface hardened steel, grey cast iron and bronze Figure 21 — Typical S /N curve Figure 22 — Height and length of end relief Figure 23 — Height of crowning
37 38 39 42 43
Table 1 — Value of Z E for some material combinations Table 2 — Limiting casedepth Table 3 — Values of Z M Table 4 — Values of application factor, K A Table 5 — Examples of prime mover with different working characteristics Table 6 — Examples of driven machines with different working characteristics Table 7 — Values of X Table 8 — Auxiliary value, A Table 9 — Minimum and maximum values of K H! Table 10 — Value of Y M Table 11 — Values of Ô Table 12 — Default values for contact stress S /N curve parameters Table 13 — Typical values of residual stress, BR Table 14 — Change in residual stress due to post-hardening operations Table 15 — Variable duty calculation example Table 16 — Values of K v350 ¶ at discontinuities Table 17 — K v350 ¶ termination points Table 18 — Values of K v350! at discontinuities Table 19 — K v350! termination points 9
Publication referred to
ii
10 11 13 16 16 17 19 23 26 33 33 40 44 45 52 55 55 55 56
Inside back cover
© BSI 06-1999
BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Foreword This Part of BS 436 has been prepared under the direction of the Machinery and Components Standards Committee. It is a general application standard for spur and helical external and internal gears. The standard follows the principles developed by the International Organization for Standardization (ISO) in that the stress levels in the tooth flank and in the tooth root are calculated and compared with basic permissible stress levels derived from tests on simple test specimens. Modifying factors used to calculate stress levels are based on the ISO proposals but have been adjusted to avoid the step functions which occur in these proposals. This Part of BS 436 together with BS 436-1 “ Basic rack form pitches and accuracy (diametrical pitch series)” and BS 436-2 “Basic rack form, modules and accuracy (1 to 50 metric module)” supersede BS 436:1940 which is therefore withdrawn. BS 436-1 is retained solely for the purpose of supplying replacement gears designed in accordance with the imperial system of units. To assist in the data processing of the calculations given in this standard, FORTRAN sub-routines can be obtained from the British Gear Association 1). Procedures for some factors are extracted or derived from Draft International Standard ISO/DIS/6336/1, 2 and 3, “Calculation of load capacity of spur and helical gears”. A British Standard does not purport to include all the necessary provisions of a contract. Users of British Standards are responsible for their correct application. Compliance with a British Standard does not of itself confer immunity from legal obligations.
Summary of pages This document comprises a front cover, an inside front cover, pages i to iv, pages 1 to 58, an inside back cover and a back cover. This standard has been updated (see copyright date) and may have had amendments incorporated. This will be indicated in the amendment table on the inside front cover. 1)
British Gear Association, c/o Institution of mechanical Engineers, Birdcage Walk, London SW1 H9JJ.
© BSI 06-1999
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2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
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BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Section 1. General 1 Scope and field of application 1.1 This Part of BS 436 is a general application standard for spur and helical external and internal gears of accuracy grades 3 to 10 operating at any pitch line speed. The calculations given in this standard are not applicable to the prediction of gear damage caused by scuffing, wear, welding or fracture of the gear rims, web, or hub. The standard covers methods for determining the actual and permissible contact stresses and bending stresses in a pair of involute gears. Stress levels on the tooth flan k and in the tooth root are calculated and compared with basic permissible stress levels derived from simple test specimens. Modification factors are given to take account of: a) the effects of dimensional variations arising from manufacture and assembly; b) vibrations arising from sources internal and external to the gears; c) the effect of the lubricant film and gear flank roughness; d) stress concentration effects in the tooth root; e) the effect of different depths of case or surface hardened gears; f) the effect of bending stress on the tooth flank stress cycle; g) the effect of residual stress in the tooth root. Procedures are included for calculating the peak load capacity and for taking account of variable duty. 1.2 The gear type and qualifications in respect of the gear design are as follows: a) Types of gears: internal and external spur, helical and double helical gears. b) Range of speed: no restriction but note that at pitch line speeds less than 1 m/s the load capacity is often limited by wear. c) Gear accuracy: grade 3 to 10 of BS 436-2. The calculation of load modifying factors are based on the largest deviation allowed for the particular manufacturing grade. d) Range of transverse contact ratio: 1.2 e) Range of helix angle: ¶ u 45°.
u
&! u 1.9.
f) Basic racks: no restriction. g) Pinion and pinion shaft: solid or hollow pinion with dil/dfl u 0.52). h) Gear blank and rim: solid gear blanks and fabricated or cast wheels with rim thickness under the root greater than 3.5m n 2). j) Material: 1) through hardened steel; 2) surface hardened steel; 3) cast iron; 4) bronze. Three grades of material and material production quality are specified (see Appendix F). The permissible stresses are reduced by a factor Z M on contact stress and Y M on bending stress for lower quality materials (see clauses 10 and 23). The effect of residual stress at the tooth root is included in this standard. Surface hardening processes, e.g. carburizing, nitriding and induction hardening, induce beneficial compressive residual stress at the surface balanced by tensile residual stress in the region of the case/core junction. Grinding the tooth surface after hardening can reduce the compressive stress and may leave a tensile stress at the surface. A compressive stress can be introduced (or re-introduced after grinding) by means of controlled shot peening. Typical values of residual stresses resulting from good heat treatment practice are included in Appendix D. Appendix H gives the equations and data from which the graphs in t he appropriate figures are derived.
2) If
the gears have dimensions outside these limitations then additional calculations are necessary to check the stress levels at the root of the teeth.
© BSI 06-1999
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BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Where no in-house data are available for the calculation of the endurance limit for contact stress, suita ble fatigue factors can be determined from the information contained in clause 9. NOTE The titles of the publications referred to in this standard are listed on the inside back cover.
2 Definitions and symbols 2.1 Definitions For the purposes of this Part of BS 436 the definitions given in BS 2519-1 apply together with the following. 2.1.1 effective casedepth, ceff the depth at which the hardness falls below 500 HV for carburized and nitrided cases or below 450 HV for induction hardening 2.1.2 endurance limit for contact stress, Ö H lim the maximum contact stress that may be sust ained for an infinite number of cycles without the occurrence of progressive fatigue damage (pitting) 2.1.3 limiting casedepth, clim that effective casedepth beyond which a further increase in casedepth does not produce a further increase in failure load 2.1.4 nominal tangential force for bending stress, F Ft the force tangential to the reference cylinder and perpendicular to its straight generators 2.1.5 nominal tangential force for contact stress, F Ht the force tangential to the reference cylinder and perpendicular to its straight generators 2.1.6 peak torque capacity for bending stress, T F max that torque which may be transmitted for up to 1 000 tooth cycles during the design life of the gears without causing failure due to bending stress 2.1.7 peak torque capacity for contact stress, T H max that torque which may be transmitted for up to 1 000 tooth cycles during the design life of the gears without causing failure due to contact stress 2.1.8 tooth stiffness constant, c and c * that force which will deform one or several meshing gear teeth having a facewidth of 1 mm by an amount of 1 4m 9
2.2 Symbols For the purposes of this British Standard the following symbols apply. NOTE This subclause is based on the symbols of BS 2519-2:1976.
2
© BSI 06-1999
BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Symbol
Designation
Units
a
centre distancea facewidth
mm
effective casedepth
mm
b ceff c
9
c *
maximum tooth stiffness of one tooth pair in normal section mean value of total tooth stiffness (or mesh stiffness) per unit facewidth
mm
N/(mm·4m) N/(mm·4 m)
d d1
reference diameter reference diameter of pinion
mm mm
d2
reference diameter of wheel
mm
da
tip diameter
mm
dan
virtual tip diameter
mm
db
base diameter
mm
dbn
virtual base diameter
mm
den df
virtual diameter to highest point of single tooth pair contact root diameter
mm mm
df2
root diameter of internal gear
mm
dfn2
virtual root diameter of internal gear
mm
di
internal diameter
mm
dm
mean diameter [= (da + df )/2]
mm
dn
virtual reference diameter
mm
dw1
pitch diameter of pinion
mm
dw2
pitch diameter of wheel
mm
f
individual deviation
f f
profile tolerance (maximum of pinion and wheel)
f ma
mesh misalignment due to manufacturing tolerance
f p
permissible single pitch deviation
4m 4m 4m 4m
f pe
tolerance on pitch
4m
f sh
mesh misalignment due to shaft deflections
4m
g g !
grinding allowance length of path of contact
mm mm
h ha0
tooth depth addendum of the basic rack of the tool
mm mm
hf2
dedendum of internal gears
mm
hfp
dedendum of the basic rack of the gearing
mm
hpr
height of protuberance
mm
hF
bending moment arm
mm
l lc
bearing span length of end relief per flank
mm mm
m
module
—
mn
normal module
mm
a
a, u and z2 are negative for internal gears.
© BSI 06-1999
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BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Symbol
Designation
Units
n p
rotational speed pitch
r/min mm
pbn
normal base pitch
mm
pbt
transverse base pitch
mm
pr
protuberance of tool
mm
q qs
combined flexibility of a pair of teeth notch parameter
mm·4 m/N —
qy
reduction factor on misalignment due to running-in
—
sFn
thickness of virtual tooth at critical section
mm
spr
residual undercut left by the protuberance
mm
u
—
v
gear ratio a pitch line velocity
w
specific loading F t/b
N/mm
wm
mean load intensity
N/mm
x y y!
addendum modification coefficient running-in allowance (only with subscript ! or ¶) running-in allowance for K H!
— 4m 4m
z
number of teetha
—
zv
virtual number of teeth
—
B C a
parameter for effect of load height of tip or root relief
— 4m
C b
height of end relief
C c
height of crowning
4m 4m
D E
tool diameter
mm
Young’s modulus of elasticity
F m
mean tangential force
MN/m2 N
F t
nominal tangential force at reference circle
N
F Ft
nominal tangential force for bending stress
N
F Ht
nominal tangential force for contact stress
N
F ¶x
mesh misalignment prior to running-in
4m
F ¶y
effective mesh misalignment
4m
(HV ) I
Vickers hardness value polar moment of inertia
—
K v
dynamic factor
—
K v350
dynamic factor for F t K A /b = 350
—
K v!
dynamic factor for spur gears
—
K v ¶
dynamic factor for helical gears (& ¶W 1)
m/s
kg·m 2
K A
application factor
— —
K F!
transverse load factor for bending stress
—
K F ¶
face load factor for bending stress
—
K H!
transverse load factor for contact stress
—
a
4
a, u and z2 are negative for internal gears.
© BSI 06-1999
BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Symbol
Designation
Units
K H ¶
face load factor for contact stress (Hertzian pressure)
—
M red P F
mass per unit facewidth of the gear pair referred to the line of action actual power capacity based on bending stress
kg/mm kW
P FP
permissible power capacity based on bending stress
kW
P H
actual power capacity based on contact stress
kW
P HP
permissible power capacity based on contact stress
kW
R Ra
roughness arithmetic average roughness (CLA value)
Rz
mean roughness
4m 4m 4m
S
S H
safety factor actual safety factor for bending stress (against breakage) minimum demanded safety factor for bending stress (against breakage) actual safety factor for contact stress
S H min
minimum demanded safety factor for contact stress
—
T
torque
N·m
T 1
pinion torque
N·m
T 2
wheel torque
N·m
T F
actual torque based on bending stress
N·m
T F max
peak torque capacity for bending stress
N·m
T FP
permissible torque based on bending stress
N·m
T H
actual torque based on contact stress
N·m
T H max
peak torque capacity for contact stress
N·m
T HP
permissible torque based on contact stress
N·m
Y
factor for bending stress
—
Y F
tooth form factor for bending stress
—
Y M
material quality factor for bending stress
—
Y N
life factor for bending stress
—
Y R
surface condition factor for bending stress
—
Y S
stress correction factor for bending stress
—
Y X
size factor for bending stress
—
Y ¶
helix angle factor for bending stress
—
Y ¸
sensitivity factor for bending stress
—
Z
factor for contact stress
—
Z c
casedepth factor for contact stress
—
Z v
speed factor for contact stress
—
Z E
elasticity factor for contact stress
—
Z G
disc/gear correlation factor for contact stress
—
Z H
zone factor for Hertzian pressure at pitch point for contact stress lubricant factor for contact stress
— —
S F S F min
Z L
© BSI 06-1999
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— — —
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BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Symbol
Designation
Units
Z M
material quality factor for contact stress
—
Z N
life factor for contact stress
—
Z R
roughness factor for contact stress
—
Z W
work hardening factor for contact stress
—
Z X
size factor for contact stress
—
Z &
contact ratio factor for contact stress
—
! en
!n
pressure angle at highest point of single tooth pair contact normal pressure angle at reference cylinder
radians radians
! t
transverse pressure angle at reference cylinder
radians
! tw
transverse pressure angle at pitch cylinder
radians
! Fen
angle for application of load at highest point of single tooth pair contact helix angle (without subscript: at reference cylinder)
radians radians
base helix angle
radians
º º!
contact ratio transverse contact ratio
— —
º! n º ¶
virtual transverse contact ratio
—
overlap ratio
—
º *
total contact ratio
—
v
Ô2
Poisson’s ratio root radius of internal gear
— mm
Ôa0
tip radius of the basic rack of the tool
mm
Ôf Ôfp
tooth root fillet radius
mm
root fillet radius of the basic rack of the gearing
mm
Ôrel ÔF
radius of relative curvature
mm
root fillet radius at critical section
mm
Ö B
ultimate tensile strength
MN/m2
Ö F
actual tooth root bending stress
MN/m2
Ö F0
MN/m2
Ö FP
basic endurance limit of a polished specimen under a reversing bending load permissible tooth root bending stress
Ö FY
yield strength for bending stress
MN/m2
Ö H
actual contact stress (Hertzian pressure)
MN/m2
Ö H lim
endurance limit for contact stress for gears
MN/m2
Ö HD
endurance limit for contact stress for discs
MN/m2
Ö HP
MN/m2
Ö HY
permissible contact stress (permissible Hertzian pressure) yield strength for contact stress
Ö R
residual stress
MN/m2
¶ ¶b
6
MN/m2
MN/m2
© BSI 06-1999
BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Subscripts 1
pinion
2
wheel
core
core material properties of surface hardened steels
eff
effective values, real stress
est
estimated value
lim
value of endurance limit
min
minimum
max
maximum
red
reduced
sh
shaft
stat
static load
3 Guide to using the calculation procedure 3.1 Calculation procedure The calculation procedure is used to calculate gear load carrying capacity either: a) in terms of stresses; or b) in terms of power. In the case of a) the permissible stress is calculated from the material endurance limits modified by the stress modifying factors. This has to exceed the actual stress which is calculated from the nominal tangential force, modified by the load modifying factors and the gear geometry. In the case of b) the power capacity of the gear pair is calculated from the permissible stress, the load modifying factors and the gear geometry. This has to exceed the power required of the gear pair. In either case the calculation is performed separately for four cases: 1) pinion contact stress (see section 2); 2) wheel contact stress (see section 2); 3) pinion bending stress (see section 3); 4) wheel bending stress (see section 3). Illustrative examples are given in Appendix G. NOTE The values of some factors (K v, K H! , K H ¶ , K F! , K F ¶ ) depend on the value of the nominal tangential force, F t. When calculating the maximum rating of a pair of gears the value of F t has to be estimated in order to calculate these factors. It is recommended that if the final rated value of F t differs from the estimated value by more than 10 % then the load dependant factors are re-calculated using the rated value of F t.
3.2 Lubrication The procedure is valid for gears having adequate lubrication. NOTE At slow speed, particular care is required to ensure an adequate supply of lubricant at the mesh. It should also be ensured that the lubricant will not cause corrosion of the gears or any other parts of the gear unit. Corrosion is not co vered by this procedure.
3.3 System dynamics The values of application factor provided for design purposes do not apply if the driving or driven machinery causes an excitation at a frequency at or close to one of the system’s natural frequencies. NOTE In such cases it is recommended that the designer of the system supplies a value of the application factor, based on calculation or measurements on similar systems.
3.4 Safety factors The use of this procedure requires a realistic appraisal of the influencing factors. When experience has been gained by running other similar gears in similar environments such experience can be used in choosing appropriate values of the safety factors.
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7
BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Section 2. Contact stress calculations 4 Basic equations for contact stress calculations 4.1 Permissible contact stress Ö HP The permissible contact stress, Ö HP , is calculated separately for pinion and wheel from the equation: (1)
4.2 Actual calculated contact stress Ö H The actual calculated contact stress, Ö H , is calculated separately for pinion and wheel from the equation: (2) where the value of F Ht is the nominal tangential force at t he reference circles appropriate to pinion or wheel calculations respectively from equation (6) or (7). 4.3 Permissible power capacity and torque based on contact stress P HP and T HP The permissible power capacity, P HP, based on contact stress is calculated separately for pinion and wheel from the equation:
(3) The permissible torque, T HP, based on contact stress is calculated separately for pinion and wheel from the equation: (4) 4.4 Peak torque capacity for contact stress, T H max The peak torque capacity for contact stress expressed as a multiple of T HP is given by the equation: (5) where Ö HY is obtained from Figure 1. line corresponding to NOTE The lines for surface hardened steels in Figure 1 are to be extended downwards as far as the Ö B the condition of the core.
5 Nominal tangential force for contact stress, F Ht The nominal force for contact stress, F Ht, is calculated from either equation (6) or (7): (6)
(7) For information: (8)
8
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BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
If either a) the gear pair is subject to variable duty; or b) the gear pair is subject to intermittent high loads (for instance at start-up) greater than the nominal running torque then the pinion torque, T H1, in equation (6) is calculated in accordance with the variable duty procedure in Appendix A for T H1. Z
6 Zone factor for contact stress, Z H 6.1 Purpose of Z H The zone factor, Z H, accounts for the influence of tooth flank curvature at the pitch point on Hertz ian stress and converts the tangential force at the reference cylinder to a normal force at the pitch cylinder. 6.2 Calculation of Z H The zone factor, Z H, is calculated from the equation: (9)
For gears at standard centres this simplifies to: (10)
7 Contact ratio factor for contact stress, Z & 7.1 Purpose of Z & The contact ratio factor accounts for the load sharing influence of the transverse contact ratio and the overlap ratio on the specific loading. 7.2 Calculation of Z & 7.2.1 For spur gears, the contact ratio factor, Z &, is calculated from the equation: (11) 7.2.2 For helical gears with & ¶ < 1, the contact ratio factor, Z (, is calculated from the equation: (12) 7.2.3 For helical gears with & ¶
W
1, the contact ratio factor, Z º, is calculated from the equation: (13)
For information, equations for
© BSI 06-1999
º! and º ¶ are given in B.8 and B.9, respectively.
9
BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Figure 1 — Yield strength for contact stress, Ö HY
8 Elasticity factor for contact stress, Z E 8.1 Purpose of Z E The elasticity factor accounts for the influence of the specific material properties E (modulus of elasticity) and v (Poisson’s ratio) on the Hertzian stress. 8.2 Calculation of Z E The elasticity factor Z E is calculated from the equation: (14)
This is tabulated for some gear materials in Table 1. For properties of bronzes see BS 1400. Table 1 — Value of Z E for some material combinations Gear materials
Z E
Steel/steel
189
Steel/SG cast iron
181
SG cast iron/SG cast iron
174
Grey iron/grey iron
146
9 Basic endurance limit for contact stress, Ö Hlim lim, is calculated separately for pinion and wheel from the The basic endurance limit for contact stress, Ö H equation: (15)
10
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BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
where is obtained from Figure 2; Ö HD Z c
for through hardened steels = 1.0;
Z c
for surface hardened steels is obtained from Figure 4 and the limiting casedepth from Table 2;
Z G2
is obtained from Figure 3;
Z G1
is the greater of: (16) (17) Table 2 — Limiting casedepth
Hardening process
Limiting casedepth
clim
Carburizing and hardening
0.16 mn
Nitriding
0.16 mn
Induction hardening
0.32 mn
in this figure are derived from disc tests performed under the auspices of the Admiralty Vickers Gear NOTE Values of Ö HD Research Association and the Navy and Vickers Gear Research Association
Figure 2 — Values of Ö HD
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11
BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
NOTE For internal gears the value of Ôrelfor an equivalent external gear should be used, i.e. the value of Ôrel obtained from B.11 should be multiplied by (|u| – 1)/(|u| + 1).
Figure 3 — Values of Z G2
Figure 4 — Casedepth factor, Z c
10 Material quality for contact stress, Z M 10.1 Purpose of Z M Better quality control exercised in the manufacture of a material results in less scatter on the mechanical properties of the finished material. Hence, for a given confidence level, better quality materials have a higher permissible stress and, conversely, lower quality materials a lower permissible stress. Material qualities are defined in Appendix F. 10.2 Values of Z M The value of Z M is obtained from Table 3.
12
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BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Table 3 — Values of Z M Z M
Material Quality A
Quality B
Quality C
Surface hardened steels
1.0
0.9
0.8
Through hardened or normalized steels
1.0
0.9
0.8
Through hardened or normalized cast steels or bronze. 0.9
0.8
0.7
Nodular cast iron
0.9
0.8
0.7
Other cast irons
0.7
0.5
0.5
11 Lubricant influence, roughness and speed factors for contact stress, Z L, Z R and Z v 11.1 Purpose of Z L, Z R, Z v The lubricant viscosity, surface roughness and pitch line speed affect the lubricant film thickness which in turn affects the Hertzian component of the total stress at the pitch cylinder. 11.2 Calculation of Z L, Z R and Z v The value of the product (Z LZ v) is obtained from Figure 5. The roughness factor Z R is obtained from Figure 6. If the pinion and wheel roughnesses are different, then: Ra = (Ra1+Ra2)/2 If the roughness is measured in terms of Rz, then the value of Ra is calculated from the equation: Ra = Rz/6
(18)
(19)
NOTE The values of Ra1 and Ra2 relate to the flank roughness in the finished condition after completing any running-in treatment or other manufacturing process which may improve the roughness of the flanks. This includes running-in during commissioning, when it is specified.
12 Work hardening factor for contact stress, Z W 12.1 Purpose of Z W The work hardening factor accounts for the increase of surface durability due to meshing a through hardened steel wheel with surface hardened pinion. In all other cases, Z W = 1.0. 12.2 Calculation of Z W For pinions, Z W = 1.0. For wheels of hardness less than 400 HV, the value of Z W is obtained from Figure 7. If the roughness is measured in terms of Rz then calculate Ra from equation (19). For wheels of hardness greater or equal to 400 HV, Z W = 1.0.
13 Size factor for contact stress, Z x The size factor is included to take into account possible influences of size on material quality and its response to heat treatment and other manufacturing processes. The value is taken as Z x = 1.0.
14 Life factor for contact stress, Z N 14.1 Purpose of Z N The life factor for contact stress takes account of the increase in permissible stress if the number of stress cycles is less than the endurance life.
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BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Figure 5 — Combined speed and lubricant factors, Z LZv
Figure 6 — Roughness factor, Z R
14
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BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
14.2 Calculation of Z N If the gear pair is subject to variable duty (and the design torque has therefore been calculated using the variable duty procedure in Appendix A), then Z N = 1.0. Otherwise, the value of Z N is derived from the S–N curve of the material if it is available, failing which it is t aken from Figure 8. The number of cycles of tooth loading, N , are those appropriate to pinion and wheel, respectively, taking into account the gear ratio and the number of pinions and wheels in mesh. Material type 1 applies to through hardened steels, surface hardened steels with casedepth greater than or equal to the limiting casedepth 3) and cast irons other than grey cast iron when some pitting is permissible (see curve 1 in Figure 8). Material type 2 applies to through hardened steels and cast irons other than grey cast iron when pitting is not permissible (see curve 2 in Figure 8). Material type 3 applies to surface hardened steels with casedepth greater than or equal to the limiting casedepth3) when pitting is not permissible (see curve 3 in Figure 8). Material type 4 applies to surface hardened steels with casedepth less than the limiting casedepth3), bronze and grey cast iron (see curve 4 in Figure 8). Material type 5 applies to bath nitrided steels (see curve 5 in Figure 8).
Figure 7 — Values of Z W
3)
See Table 2 for values of limiting casedepth.
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BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Figure 8 — Life factor for contact stress, Z N
15 Application factor, K A 15.1 Purpose of K A The application factor accounts for load fluctuations from the mean load or loads in the load histogram caused by sources external to the gearing. The fluctuations depend on the characteristics of the prime mover, the driven machinery and the system vibration response to the working conditions. A typical total gear load is shown broken down into individual components including the application factor in Figure 9. 15.2 Determination of K A The application factor is assessed from measurements on similar existing systems or, if such information is not available, from the empirical information given in Table 4,Table 5 and Table 6. Table 4 — Values of application factor, K A Load characteristic of prime mover
Load on driven machine Uniform
Moderate shock
Medium shock
Heavy shock
Uniform
1.0
1.25
1.50
1.75
Light shock
1.10
1.35
1.60
1.85
Moderate shock
1.25
1.50
1.75
2.0
Heavy shock
1.50
1.75
2.0
2.25
Table 5 — Examples of prime mover with different working characteristics Character of operation
Prime mover
Uniform
Electric motor
Light shock
Steam turbine, gas turbine
Moderate shock
Multi-cylinder combustion engine
Heavy shock
Single cylinder combustion engine
16
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BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Table 6 — Examples of driven machines with different working characteristics Character
Uniform
Driven machine
Generator, uniformly loaded belt or platform conveyors, worm conveyors, light elevators, packaging machines, feed gears for machine tools, ventilators, light centrifuges, centrifugal pumps, mixer for light fluids or constant density material, shearing, pressing, punching, turning gears, moving gears
Moderate shock Non-uniformly (e.g. mixed cargo) loaded belt or platform conveyors, main drives of machine tools, heavy elevators, turning gears of cranes, industrial and mine ventilators, heavy centrifuges, centrifugal pumps, mixer for high viscosity or variable density material, multi-cylinder piston pumps, feed pumps, extruders (general), calenders, rotary furnaces, rolling mills (continuous zinc strip, aluminium strip as well as wire and bar rolling mills) Medium shock
Extruders for rubber, mixers with interrupted operation for rubber and plastics, ball mills (light), wood working (mills, saws, lathes), billet rolling mills, lifting gear, single cylinder piston pump
Heavy shock
Excavators (bucket wheel gears, multi-bucket gears, sieve gears, power shovels), ball mills (heavy), rubber dough mills, breaker (stone, ore) metallurgical machines, heavy feed pumps, rotary drilling apparatus, brick moulding press, braking drums, peeling machines, cold strip rolling mills, briquette press, pug mills
Figure 9 — Constituent parts of typical gear load
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17
BS 436436-3:1 3:1986 986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
16 Dynamic factor, K factor, K v 16.1 Purpose of K of K v The dynamic factor accounts for load fluctuations arising from contact conditions at the gear mesh. The main influences are: a) gear tooth accuracy; b) the tooth contact frequency divided by the natural frequency of torsional oscillations due to pinion and wheel inertias acting against the mesh stiffness. The portion of a typical gear load accounted for by the dynamic factor is illustrated in Figure 9. The values included in this standard are appropriate. NOTE At lowNOTE low-loa load d the val value ue of K of K v may be higher than given by the standard, but the stress will not exceed the stress at the maximum rating of the gear on which the values of K of K v are based. If a gear pair is operating at or near resonance speed or at multiples or sub-multiples of resonance speed (particularly the second and third harmonics and sub-harmonics, respectively) then a thorough dynamic analysis is recommended. This is beyond the scope of this standard.
16.2 Calculation of K of K v The value of K of K v is calculated from the equation: (20) For helical gears of overlap ratios greater than or equal to unity, K v350 = K v350 ¶ where K where K v350 ¶ is obtained from Figure 10. For spur gears, K v350 = K v350 ! where K where K v350 ! is obtained from Figure 11. For helical gears of overlap ratio less than unity: (21) where K v350 ¶ is obtained from Figure 10; K v350 ! is obtained from Figure 11. The value of B of B is calculated from the equation: (22) where X where X is is obtained from Table 7. If the value of F t K 100. A /b is less than 100, then use F t K A /b = 100 If the procedure is being used to calculate a maximum rating, then estimate the value of F t K A /b from the equation: (23)
where Ö HP and Ö HP2 . is the lesser of Ö HP1 It is advisable to check the accuracy of this estim ate when the rating has been calculated an d, if necessary, re-calculate K re-calculate K v using the new value of F t.
18
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BS 436-3 436-3:19 :1986 86 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Table 7 — Values of X of X Grade
F t K /b A
#
350
F t K A /b > 350
3
0.4736
0.2931
4
0.6110
0.4211
5
0.7153
0.5402
6
0.8017
0.6361
7
0.8635
0.7303
8
0.9005
0.7954
9
0.9334
0.8687
10
0.9530
0.8954
16.3 Values of auxiliary parameter Q v 16.3.1 Q v takes account of the shift in the resonance frequency of the gear pair when both or either of the pinion and wheel are not solid gears. The general equation for Q v is: (24)
where M red 0 is the value of M red for the particular example of a solid pinion matin g with a solid wheel and: (25) where M 1 is the moment of inertia of the pinion and (26) M 2 is the moment of inertia of the wheel, and (27) 16.3.2 Q v and M red for some common gear arrangements when the pinion and wheel material are of the same density are given in a) to d) below. If the density of the pinion and wheel materials are different, Q v is calculated from equations (24) to (27). a) For a solid pinion meshing with a solid wheel: Q v = 1.0
(28)
b) For a solid pinion meshing with a wheel with a fabricated rim:
(29)
where dm2 = (da2 + df2)/2
© BSI 06-1999
(30)
19
BS 436436-3:1 3:1986 986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
c) Planetary gears 1) for sun pinion with planet gear wheel:
(31)
where np la is the number of planets meshing with the sun; M pla is the moment of inertia of the planet gear and is calculated from equation (27); M sun is the moment of inertia of the sun pinion and is calculated from equation (26). 2) planet gear with annulus rigidly connected to the gear case. In this and other cases where the mass of the stationary annulus is sufficiently large to be assumed infinite:
(32)
3) planet gear with rotating annulus:
(33)
where do2 is the outside diameter of the annulus. (d) Idler gears (34)
where M l, M idl and M 2 are the reduced masses of the small gear (pinion), idler gear and large gear (wheel) respectively.
20
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BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Figure 10 — K v350 ¶ for helical gears, & ¶ W 1
Figure 11 — K v350 ! for spur gears
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21
BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
17 Load distribution factors, K H! and K H ¶ 17.1 Purpose of the face load factor for contact stress K H ¶ K H ¶ is the maximum specific load divided by the mean specific load. It accounts for the increase in local load due to mal-distribution of load across the face of the gear caused by deflections, alignment tolerances and helix modifications. These include: pinionshaft bending deflections pinionshaft torsional deflections misalignment because of manufacturing tolerances end relief 4) helix correction 4) crowning4) 17.2 Calculation of K H ¶ 17.2.1 The calculation of the face load factor involves the following. a) determination of the mean load intensity wm; b) determination of the mesh misalignment due to deflections and manufacturing tolerance modified by the effect of running-in and helix modifications F ¶y; c) determination of mesh stiffness, c * d) calculation of the face load factor, K H ¶ 17.2.2 Calculate the value of beff from the equation: beff = b – lc/2
(35)
17.2.3 Calculate the value of wm· If the tangential load on the gears is known and the procedure is being used to calculate a safety factor, then wm is calculated from the equation: wm = F t K K v/beff A
(36)
If the calculated value of wm is less than 100 N/mm then use wm = 100 N/mm. If the procedure is being used to determine the maximum rating of the gear pair, then an estimated value of F m is calculated from the equation: (37) where Ö HP and Ö HP2 then: is the minimum of Ö HP1 wm est = F m est/beff
(38)
wm est is then used in place of wm in the following analysis. If higher accuracy is required, then the procedure can be used iteratively by calculating a value of K H ¶ using wm est, then re-calculating using wm est/ K H ¶. Three such iterations will normally converge to give a constant value of K H ¶. 17.2.4 Calculate the value of F ¶y . If: a) the gears are helix corrected, or b) the gear layout does not conform to Figure 12, or c) substantial forces other than pure shaft torque are to be applied (e.g. pulley loads), or d) wheel shaft deflections are significant, then F ¶y is calculated by a thorough analysis of all contritutions to the mesh misalignment (bearing clearances, case and shaft deflections, manufacturing tolerances, etc.). 4)
Recommendations on the design of end relief, helix correction and crowning are given in Appendix C.
22
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BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Otherwise F ¶y is calculated as follows. 1) Calculate the value of f sh. Values of K , l and s are defined in Figure 12 and A is taken from Table 8. If the value of s is greater than 0.3l then use s = 0.3 l
(39)
Then for spur and single helical gears:
(40)
or for double helical gears
(41)
The value of s for double helical gears is the distance to the centre of the helix which is nearer to the torqued end of the gear. Table 8 — Auxiliary value, A Gear pairs without crowning or end relief
Gears with suitably chosen crowning
Gear pairs with suitably chosen end relief
mm· 4m/N
mm·4m/N
mm·4m/N
0.023
0.012
0.016
2) Calculate f ma. The value of fma depends on the manufacturing tolerance of the gears, the case and bearings and the bearing clearances. For gears without helix modifications and without any adjustment on assembly, use: f ma = f H ¶
(42)
where f H ¶ is the larger of the tooth alignment tolerances of the pinion and wheel given in Table 5 of BS 436-2:1970. For gears with crowning or gears where the contact is adjusted on assembly f ma = 0.5 f H ¶
(43)
provided that this assumption is verified by inspection of the contact marking under light load. For gears with suitable end relief f ma = 0.7 f H ¶
(44)
3) Calculate F ¶x from the equation: F ¶x = |1.33 f sh ± f ma|
(45)
The negative sign is to be used only if the gears are adjusted on assembly and if the contact pattern is inspected to justify the assumption. The value of F ¶x is to be the maximum of i) the value from equation (45); ii) 0.005 wm; iii) half the actual manufacturing tolerance (higher value of pinion and wheel)
© BSI 06-1999
23
BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
4) The value of qy is obtained from Figure 13. If the pinion and wheel material are different then: qy = (qy1 + qy2)/2
(46)
5) Calculate the value of F ¶y from the equation: F ¶y = qyF ¶ x
(47)
Factor K Shrink fit
Key fit
a)
0.0 (s = 0)
0.0 (s = 0)
b)
0.0 (s = 0)
0.0 (s = 0)
c)
0.48
0.8
d)
– 0.48
– 0.8
e)
1.33
f)
– 0.36
– 0.6
g)
– 0.6
– 1.0
1.33
Figure 12 — Constant K for calculation of f sh 17.2.5 For gears conforming to the basic rack profile specified in BS 436-1 or BS 436-2 and with 1.2 u (! u 1.9 average values of c * are: c * = 20 N/(mm· 4m) for a steel/steel gear pair; c * = 18.2 N/(mm· 4m) for a steel/SG cast iron gear pair; c * = 16.8 N/(mm· 4m) for an SG cast iron/SG cast iron gear pair; c * = 14.8 N/(mm· 4m) for a steel/grey cast iron gear pair; c * = 11 N/(mm· 4m) for a grey cast iron/grey cast iron gear pair. 24
© BSI 06-1999
BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
If greater accuracy is required use the procedure in Appendix E. 17.2.6 Calculate the value of K H ¶ from either:
(48)
(49)
Figure 13 — Values of qy 17.3 Purpose of the transverse load factor for contact stress K H! The transverse load factor for contact stress accounts for the mal-distribution of load down the tooth f lank due to profile and pitch deviations and tooth modifications. 17.4 Calculation of K H! For gears with º * < 2 (50)
where f pe is the single pitch tolerance given in Table 3 of BS 436-2:1970. For gears with º *
W 2
(51)
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BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Table 9 — Minimum and maximum values of K H! Gear type
Minimum value of K H !
Maximum value K H !
Spur gears
1.0
1/Z "2
Helical gears
1.15
º!/cos2 ¶b
17.5 Running-in allowance y! For through hardened steels and cast steels: (52) when v u 5: no limit 5 < v u 10: y! u 12 800/BH lim v > 10: y! u 6 400/BH lim For cast iron and bronze y! = 0.275 f pe
(53)
when v u 5: no limit 5 < v u 10: y! u 22 v >10: y! u 11 For surface hardened steels y! = 0.075 f pe
(54)
subject to y! u 3 at any speed. If the pinion and wheel are of different materials then y! = ( y!1 + y!2)/2
(55)
18 Minimum demanded and actual safety factors on contact stress, S Hmin and S H 18.1 Minimum demanded safety factor S Hmin The choice of the minimum demanded safety factor is to be agreed between t he gear manufacturer and the purchaser. NOTE The value should reflect the confidence in the actual operating conditions and material properties being truly reflected in this standard. When the load histogram is not surely known or where high tooth loads are likely to occur due to circumstances outside the scope of this standard an appropriately high value of the minimum demanded safety factor should be used. The recommended ranges of S Hmin are a) for normal industrial applications S Hmin = 1.0 to 1.2; b) for high reliability and critical applications (high consequential damage, loss of life etc.) S Hmin = 1.3 to 1.6.
18.2 Calculation of the actual safety factor S H The value of S H is calculated from the equation: S H = S H min Ö HP /Ö H
26
(56)
© BSI 06-1999
BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Section 3. Bending stress calculations 19 Basic equations for tooth root bending stress 19.1 Permissible bending stress Ö FP The permissible bending stress is calculated separately for pinion and wheel from the equation: (57) NOTE The residual stress, Ö R, is viewed algebraically, i.e. Ö R is negative if compressive or positive if tensile. Typical values of Ö R resulting from good heat treatment practice are given in Appendix D. When requested by the user, the designer should make known, with justification, the residual stresses assumed for calculation purposes.
19.2 Actual calculated bending stress, Ö F The actual calculated bending stress is calculated separately for pinion and wheel from the equation: (58) 19.3 Permissible power capacity and torque based on bending stress P FP and T FP The permissible power capacity based on ben ding stress is calculated separately for pin ion and wheel from the equation:
(59) The permissible torque based on bending stress is calculated separately for pinion and wheel from the equation: (60) 19.4 Sub-surface bending in surface hardened gears Because of the possibility of a sub-surface bending failure, especially when the hardened case is thin, a check is made at the case/core junction. In this calculation the permissible stresses are calculated from equation (61) in which the values of material properties relate to the core material. (61) To allow for the reduced bending stress below the surface the actual bending stress is calculated from equation (62), in which Ö F is obtained from equation (58). (62) where either (63) or 1, whichever is the greater. The power capacity based on core bending stress is calculated from the equation: (64)
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BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
19.5 Idler duty For frequently reversing duties or any drive where the direction of loading on the teeth is reversed each cycle (for example, idler gears or planets in epicyclic gearing) the permissible tooth root stress is calculated from the equation: (65) and Ö FP W Ö F · For surface hardened gears the permissible core root stress is calculated from the equation: (66)
19.6 Peak torque capacity for bending stress T F max The peak torque capacity for bending stress expressed as a multiple of T FP is given by the equation: (67) For surface hardened gears, a further requirement is a calculation of the peak torque capacity of the core material. In this calculation, the value of Ö FY in equation (67) is replaced by Ö FY core where (68)
20 Nominal tangential force for bending stress, F Ft The nominal tangential force for bending stress, F Ft, is calculated from either equation (69) or (70): (69)
(70)
(71) NOTE If either: a) the gear pair is subject to variable duty, or b) the gear pair is subject to intermittent high loads for instance at start-up) greater than the nominal running torque, then the pinion torque T F1 in equation (69) is calculated in accordance with the variable duty procedure in Appendix A.
21 Geometry factors for bending stress, Y F, Y s and Y " 21.1 Purpose of geometry factors 21.1.1 Form factor, Y F The form factor takes into account the influence of the tooth form on the nominal bending stress for application of load at the highest point of single tooth pair contact. 21.1.2 Stress correction factor, Y S The stress correction factor takes into account the stress increasing effect of the fillet and the proximity of the bending moment arm on the nominal bendin g stress for application of load at the highest point of sin gle tooth pair contact.
28
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BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
21.1.3 Helix angle factor for bending stress, Y ¶ The helix angle factor for bending stress accounts for the fact that the conditions for tooth root stress in helical gears are more favourable as a result of the inclined line of contact than for the virtual spur gears on which the calculations are based. 21.2 Calculation of geometry factors 21.2.1 Geometrical equations The parameters used for calculating the auxiliary parameters in 21.2.2 and 21.2.3 are calculated from equations (72) to (79). (72) (73) (74) (75) (76) (77)
a
dana = dn + da – d
(78)
dana = mnzv + (da – d)
(79)
dfn is calculated by substituting dan and da by dfn and df , respectively.
21.2.2 Auxiliary parameters Ú, G, H and q s for Y F, Y S and Y ¶ of external gears. Auxiliary parameters used for calculating Y F, Y S and Y ¶for external gears are calculated from equations (80) to (90).
NOTE For the purposes of calculation it may be assumed that hfp = ha0 and Ô fp = Ôao·
Figure 14 — Dimensions of the basic rack of the gearing
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BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
For the calculation of these parameters the dimensions of the basic rack of the gearing, hfp and Ôfp are introduced, these are defined in Figure 14. The nominal tooth form is taken as a basis, i.e. for this purpose the influence of backlash and grinding allowance is ignored. (80)
Ú = ;/6 radians can be used as a starting point for the solution of equation (80). Where (81) (82)
(83) For tools without protuberance use S pr = 0 (84) (85)
(86) (87)
(88) (89) (90)
* Inv is the involute form of ! which is tan ! – ! .
30
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BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
21.2.3 Auxiliary parameters for Y F, Y S and Y ¶ of internal gears (approximated by a rack). Auxiliary parameters used for calculating Y F,Y S and Y ¶ for internal gears are calculated from equations (91) to (94). (91) (92)
@
@
F2 fp1 As a first approximation for calculation purposes --------- = ---------mn 2
(92a)
(93) (94)
with den2 according to equation (86) and dfn2 analogous to dan according to equation (78) or (79). 21.2.4 Tooth form factor Y F The value of Y F is calculated from the equation:
(95)
with parameters according to 21.2.2 for external gears and 21.2.3 for internal gears. 21.2.5 Stress correction factor Y S The value of Y S is calculated from the equation: (96) where (97) qs is the notch parameter and (98) with parameters according to 21.2.2 for external gears and 21.2.3 for internal gears. 21.2.6 Helix angle factor Y ¶ The value of Y ¶is calculated from the equation: (99) When & ¶ > 1, use & ¶ = 1, when ¶ > 30°, use ¶ = 30°.
22 Basic endurance limit for bending stress, Ö Fo The basic endurance limit for bending stress used in this standard is based on the fully reversed bending endurance limit of a 0.3 in (7.62 mm) diameter polished specimen, at a 99 % confidence level. © BSI 06-1999
31
BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
This stress has to be modified for the effects of loading condit ion, size, surface finish, quality and life before it can be used as a permissible stress for gear tooth root bending calculations. These factors are included in equation (57). The basic endurance limit is taken from the graph in Figure 15. If the value of (HV ) is known, this shall be used in preference to Ö B when using Figure 15. For values of (HV ) not covered by Figure 15, use the equations in H.12. 9
Figure 15 — Values of Ö F0
23 Material quality factor for bending stress, Y M 23.1 Purpose of Y M Better quality control exercised in the manufacture of a material results in less scatter on the mechanical properties of the finished material. Hence for a given confidence level, better quality materials have a higher permissible stress and, conversely, lower quality materials a lower permissible stress. Material qualities are defined in Appendix F. 23.2 Values of Y M The value of Y M is taken from Table 10.
32
© BSI 06-1999
BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Table 10 — Values of Y M Y M
Material Quality A
Quality B
Quality C
Surface hardened steels
1.0
0.9
0.6
Through hardened or normalized steels
1.0
0.9
0.6
Through hardened or normalized cast steels or bronze
0.9
0.8
0.5
Nodular cast iron
0.7
0.6
0.5
Other cast irons
0.3
0.2
0.2
24 Sensitivity factor for bending stress, Y ¸ 24.1 Purpose of Y ¸ The sensitivity factor accounts for the sensitivity of the gear material to the presence of notches (i.e. the root fillet). 24.2 Calculation of Y ¸ 24.2.1 Y ¸ at the endurance limit The value of Y ¸ at the endurance limit is calculated from the equation: (100) where Ô! is obtained from Table 11 and qs is obtained from equation (98). Linear interpolation can be used between the values given in Table 11. 24.2.2 Y ¸ at static strength (peak torque capacity) The value of Y ¸ is calculated as follows: a) for grey cast iron: Y ¸ stat = 1.0
(101) Table 11 — Values of Ô
9
Material
Grey cast iron Nitrided steel Soft steel and bronze Through hardened
Ö FY = 300 MN/m2 Ö FY = 400 MN/m2 Ö 0.2 a = 500 MN/m2
Ô (in mm) 9
0.310 0.100 0.083 0.045 0.028
steel cast steel
Ö 0.2 = 600 MN/m2
0.019
and S.G. cast iron
Ö 0.2 = 800 MN/m2
0.006
B0.2 = 1 000 MN/m2
0.001
Carburized, induction and flame hardened steels
0.003
a
Ö 0.2 is the 0.2 % proof stress.
b) for nitrided steels, from the equation: Y $ stat = 0.27 Y s + 0.72
© BSI 06-1999
(102)
33
BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
c) for soft steels and bronze, from the equation: (103) d) for through hardened steel, from the equation: (104) where Ö 0.2 is the 0.2 % proof stress e) for carburized, induction and flame hardened steels, from the equation: Y $ stat = 0.77 Y s + 0.22
(105) (106)
Equation 106 deleted.
25 Surface condition factor for bending stress, Y R 25.1 Purpose of Y R The surface condition factor accounts for the reduction of endurance limit due to flaws in the material and the surface roughness of the tooth root fillets. 25.2 Calculation of Y R The surface condition factor is taken from Figure 16. Material type 1 applies to through hardened steels, bronze and cast iron other than grey cast iron (see curve 1 in Figure 16). Material type 2 applies to surface hardened steels with casedepth greater than or equal to the limiting casedepth (see Table 2 for values of limiting casedepth) (see curve 2 in Figure 16). Material type 3 applies to surface hardened steels with casedepth less than the limiting casedepth (see Table 2 for values of limiting casedepth) and grey cast iron (see curve 3 in Figure 16).
26 Size factor for bending stress, Y x The size factor is included to take into account possible influences of size on material quality and its response to heat treatment and other manufacturing processes. Its value is taken from Figure 17. Material type 1 applies to through hardened steels and cast iron other than grey cast iron (see curve 1 in Figure 17). Material type 2 applies to surface hardened steels (see curve 2 in Figure 17). Material type 3 applies to grey cast iron and bronze (see curve 3 in Figure 17).
34
© BSI 06-1999
BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
NOTE
Ra is the roughness (CLA value) of the root.
Figure 16 — Values of Y R
Figure 17 — Values of Y x
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BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
27 Life factor for bending stress, Y N 27.1 Purpose of Y N The life factor for bending stress accounts for the increase in permissible stress if the number of stress cycles is less than the endurance life. 27.2 Calculation of Y N If the gear is subject to variable duty (and the design torque has therefore been calculated using the variable duty procedure in Appendix A), then the value of Y N is 1.0. If the S/N curve of the material is known then this is used as the basis for the life factor. Otherwise, Y N is obtained from Figure 18, Figure 19 or Figure 20, as appropriate. Figure 18 applies to through hardened steels. Figure 19 applies to surface hardened steels with cased epth greater than or equal to the limiting casedepth (see Table 2 for values of limiting casedepth) and cast iron. Figure 20 applies to surface hardened steels with casedepth less than the limiting casedepth (see Table 2 for values of limiting casedepth), grey cast iron and bronze. For bath nitrided steel Y N = 1.0 for all values of N , the number of tooth cycles.
28 Load factors for bending stress, K F! and K F ¶ 28.1 Purpose of K f ! and K F ¶ The load factors for bending stress account for uneven distribution of bending moment across the facewidth caused by uneven loading across the face. 28.2 Calculation of K F! and K F ¶ K F! = K H!
(107)
where K H! is obtained from 17.4. (108) where K H ¶ is obtained from 17.2 and h/b is the maximum of h1/b1 and h2/b2.
29 Minimum demanded and actual safety factors on tooth root stress, S Fmin and S F 29.1 Minimum demanded safety factor S F min The choice of the minimum demanded safety factor is to be agreed between t he gear manufacturer and the purchaser. NOTE The value should reflect the confidence in the actual operating conditions and material properties being truly reflected in this standard. When the load histogram is not surely known or where high tooth loads are likely to occur due to circumstances outside the scope of this standard an appropriately high value of the minimum demanded safety factor should be used. The recommended ranges of S F min are a) for normal industrial applications S F min = 1.4 to 1.5; b) for high reliability and critical applications (high consequential damage, loss of life, etc.) S F min = 1.6 to 3.0.
29.2 Calculation of the actual safety factor S F The value of S F is calculated from the equation: S F = S Fmin Ö FP /Ö F
36
(109)
© BSI 06-1999
BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Figure 18 — Y N for through hardened steel
Figure 19 — Y N for thick case surface hardened steel and cast iron
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BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Figure 20 — Y N for thin case surface hardened steel, grey cast iron and bronze
38
© BSI 06-1999
BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Appendix A Variable duty calculations A.1 Symbols, terms and units Symbol
Designation
N i
number of tooth cycles endured at torque T i
—
N Ö m ax
number of cycles at upper knee of S/N curve (see Figure 21) a
—
N
number of cycles at lower knee of S/N curve (see Figure 21) a
—
Q
exponent (see equation (110) or (111) a
—
T i
applied pinion torques
T
design torque for
Ö lim
stress at lower knee of S/N curve (see Figure 21) a
MN/m2
Ö max
stress at upper knee of S/N curve (see Figure 21) a
MN/m2
torque number (i = 1 for largest torque)
—
Z
Z
Units
W
N·m
N ¸ cycles equivalent to the specified dutya
N·m
Subscripts i a
With additional subscript H for contact stress or subscript F for bending stress calculations.
A.2 Preliminary calculations (applicable to both contact and bending stress) A.2.1 Arrange the applied pinion torques, T i, in descending order of magnitude. A.2.2 For each level of torque calculate the number of cycles of tooth engagement experienced by pinion and wheel respectively. The number of pinions and wheels in engagement and the gear ratio have to be taken into account. A.3 Contact stress calculations The value of Q H is calculated from the equation: (110) or if the S /N curve of the material is not available, use the value in Table 12.
Figure 21 — Typical S /N curve
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BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Table 12 — Default values for contact stress S/N curve parameters Material type (see 14.2) 1
2
3
4
5
N H
109
5 ! 107
5 ! 107
2 ! 106
2 ! 106
N B H max
104
5 ! 104
105
105
105
Ö H max/Ö H lim
2.0
2.0
1.6
1.3
1.1
Q H
8.305
4.983
6.611
5.709
15.72
Z
NOTE In order to facilitate the calculations for surface stress when pitting is allowed, the three sections of the life curve (see Figure 8) have been approximated to one straight line. This will result in a slightly over-estimated value of T H for low numbers of cycles. Z
A.4 Bending stress calculations The value of Q F is calculated from the equation: (111)
where N min is the minimum of the values N i, or N Ö F max calculated for pinion or wheel, and Y N(Nmin) is the value of Y N for N min cycles. If the S /N curve of the material is not available, values of N F and N Ö F max can be taken from the appropriate life curve in clause 27. Z
A.5 Variable duty calculations A.5.1 The following calculation has to be performed for each of the following conditions: a) contact stress values of Q H and N using Z
1) pinion tooth cycles; 2) wheel tooth cycles; b) bending stress values of Q F and N using Z
1) pinion tooth cycles; 2) wheel tooth cycles. A.5.2 Starting with i = 1 (i.e. including the largest torque only) and continuing with further values of T i in descending order, calculate T i for successive values of T i using the equation: Z
(112) Then: a) If any value of T i is greater than the next level of torque T (i+1) then that value of T i is used as T in the calculation of actual stress or power. Z
Z
Z
b) If T i is less than T (i+1) (i.e. if condition a) does not apply), but T (i + 1)is greater than T (i + 1), then T (i + 1) is used as T in the calculation of actual stress or power. Z
Z
Z
c) If when all torques have been included the final value of T i is less than the final T i, then T i is used as T in the calculation of actual stress or power. If the final value of T i is greater than the final value of T i then the final value of T i is used. This is equivalent to using a) and b) above with an additional imaginary infinitesimal torque T i. Z
Z
40
Z
Z
© BSI 06-1999
BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Appendix B Gearing equations B.1 Reference diameter of pinion, d1 The value of d1 is calculated from the equation:
B.2 Length of path of contact g ! The value of g ! is calculated from the following equations.
NOTE
Centre distance a is a negative quantity for internal gears.
B.3 Transverse base pitch, pbt The value of pbt is calculated from the equation:
B.4 Pitch line velocity, v The value of v is calculated from the equation:
B.5 Transverse pressure angle at reference cylinder, !t The value of ! t is calculated from the equation:
B.6 Transverse pressure angle at pitch cylinder, !tw The value of ! tw is calculated from the equation:
B.7 Base helix angle, ¶b The value of ¶b is calculated from the equation:
¶b = sin –1(cos ! n sin ¶) B.8 Transverse contact ratio, &! The value of &! is calculated from the equation:
B.9 Overlap ratio, &" The value of &" is calculated from the equation:
© BSI 06-1999
41
BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
B.10 Total contact ratio, & * The value of & * is calculated from the equation:
& * = &" + &! B.11 Radius of relative curvature, Ôrel The value of Ôrel is calculated from either of the equations:
B.12 Virtual number of teeth, Z v The value of zv is calculated from the equation:
Appendix C Design guidance on tooth modifications C.1 General This design guide is included as an aid to gear designers in choosing suitable amounts of tooth reliefs and corrections. C.2 End relief End relief is used: a) on a helical gear to allow for tooth deflections such that a tooth entering the meshing zone receives minimum shock which is important in gearing with a low noise requirement, and/or b) to prevent high loading at the ends of the teeth due to mesh misalignment. The height of end relief is related to tooth deflection and misalignment. The recommended range of length of end relief is: lc = 0.05 b
(113)
lc = 0.1 b
(114)
to The recommended range of height of end relief is: C b = F ¶ Y
(115)
to C b = F ¶ Y + 20
(116)
where F ¶y is calculated in accordance with 17.2.4.
Figure 22 — Height and length of end relief
42
© BSI 06-1999
BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Figure 23 — Height of crowning C.3 Helix correction Helix correction is determined by calculating the deflected shape of the pinion at a particular load assuming a uniform distribution of load across the facewidth (but excluding the end-relieved portion of the facewidth). The complement of this deflected shape (u sually approximated to a straight line) is then applied to the tooth in terms of metal removed. Since helix correction is fully effective only when the gear is op erating at the load for which t he correction was calculated, the choice of design load is important. The following points should be considered when choosing the load. a) When running at loads less than the design load, K H ¶ will increase and may become greater than its value without helix correction. The specific loading will however be less than at the design load. b) When running at loads greater than the design load, K H ¶ will increase but will always be less than the value without helix correction at the particular load. Unless the correction is designed after a no-load meshing test, manufacturing and location errors cannot be compensated for by helix correction. It is recommended therefore that helix correction is used only for gears of sufficient manufacturing and location accuracy that any such errors are negligible, unless corrections are designed to accommodate errors as measured on a no-load meshing test. Crowning (see C.4) is recommended in preference to helix correction if significant alignment errors are anticipated. C.4 Crowning Crowning is a crude form of helix modification used to compensate for manufacturing errors and deformations of the gear under load. It is recommended that crowning be used only if the value of K H ¶ without crowning (clause 17) is greater than 2. The recommended range of crowning height is C c = F ¶#
(117)
C c = F ¶# + 20
(118)
to
where f sh is calculated in accordance with 17.2.4. C.5 Tip and root relief On helical gears, tip and root relief is used to prevent high loading at the tip/root contact where the sliding velocity is a maximum and scuffing is likely to occur. On spur gears, tip and root relief is used to allow for tooth deflection, prevent tip loading and to reduce noise levels at tooth-contact frequency. The lengths and heights of relief depend largely on the capabilities of the gear cutting machine used to finish the gears. The following recommendations can therefore only be regarded as typical values. The length and height of tip relief should not exceed the maximum permissible amounts specified in BS 436-2.
© BSI 06-1999
43
BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
The recommended ranges of height of relief are: a) for helical and double helical gears: C a = (0.06 F t/b) + 5
(119)
to C a = (0.06 F t/b) + 25
(120)
b) for spur gears: C a = f p + (0.06 F t/b) + 5
(121)
to C a = f p + (0.06 F t/b) + 25
(122)
The value selected from these ranges should be the minimum of the tolerance band. This amount of relief may be disposed as tip and/or root relief on pinion and/or wheel in one of the following ways: 1) the relief calculated in equations (119) to (122) applied as tip relief only on both pinion and wheel; 2) the relief calculated in equations (119) to (122) applied as both tip and root relief on one gear only (usually the pinion); 3) one-half the relief calculated in equations (119) to (122) applied as both tip and root on both pinion and wheel; 4) other dispositions between pinion and wheel tip and root with the relief calculated in equations (119) to (122) disposed between pinion tip and wheel root, and between pinion root and wheel tip.
Appendix D Typical residual stresses D.1 General The residual stresses in Table 13 are typical of the values pertaining after the gear material has undergone the listed hardening processes. If the actual residual stress of the gear or a test piece is measured, then that value should be used in preference to the values in Table 13 and Table 14 and should be made known, with justification, to the user. D.2 Carburizing and hardening Incorrect heat treatment may result in compressive residual stress levels lower than those indicated. In particular, decarburization will reduce the residual stress level. The degree of decarburization can be detected by surface hardness measurement or by filing and re-checking. D.3 Gear grinding Grinding of the surface may affect the residual stress at the immediate surface and, for guidance, the values in Table 14 may be added algebraically to those in Table 13. Table 13 — Typical values of residual stress, Ö R Hardening process
Quality A or B
Quality C
Ö R
Ö Rcore
Ö R
Ö Rcore
MN/m2
MN/m2
MN/m2
MN/m2
Carburized and hardened
– 400
240
0
240
Nitride hardened
– 400
140
0
140
Induction hardened
– 140
450
0
450
D.4 Shot peening Shot peening can be used to increase the surface residual compressive stress. Correct selection of shot and of intensity is extremely important and should be controlled by Almen strips. For guidance, a typical value of the change in residual stress after shot peening with correct technique is given in Table 14.
44
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BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Table 14 — Change in residual stress due to post-hardening operations Algebraic change to Ö R
Operation
Very light grinding
MN/m2
carefully controlled
0
Light full-form grinding
+ 300 (tensile)
Heavy abrasive grinding
+ 600 (tensile)
Controlled shot peening
– 500 (compressive)
Appendix E Tooth and mesh stiffness c and c * 9
E.1 General The main factors influencing tooth stiffness are: a) tooth thickness; b) tooth form; c) tooth height; d) helix angle; e) radii of profile curvature; f) rim section design; g) gear manufacturing deviations and alignment errors; h) Youngs modulus. E.2 Calculation of c9 and c * E.2.1 Single stiffness c The maximum stiffness of one pair of teeth (single stiffness) in the normal plane can be evaluated using equations (123) and (124). 9
(123)
(124) Where the coefficients C 1 ... C 9 are: C 1 =
0.04723; C 2 =
0.15551; C 3 =
0.25791;
C 4 = – 0.00635; C 5 = – 0.11654; C 6 = – 0.00193; C 7 = – 0.24188; C 8 =
0.00529; C 9 =
0.00182.
E.2.2 Mesh stiffness c * The mean value of the total tooth stiffness constant in the transverse plane can be calculated from equation (125). This equation is valid for spur gears and for helical gears up to ¶= 45°. c * = c (0.75 &! + 0.25) 9
(125)
with c from equation (123). 9
E.2.3 Materials other than steel Equations (123) to (125) apply to steel/steel gear pairs.
© BSI 06-1999
45
BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
For other materials: (126) (127) where ß is the ratio of E for the material to E for steel, i.e. (128) and (129)
Appendix F Definition of material quality F.1 General The following three levels of material quality are recommended. a) Quality C defines the minimum requirements for materials and their heat treatments for lightly loaded gears in non-critical applications. b) Quality B defines the requirements for materials and their heat treatments for the majority of industrial gears at reasonable cost. c) Quality A defines the maximum requirements for materials and their heat treatments for special applications, e.g. high power or high reliability requirements. F.2 Quality specifications F.2.1 General In order to rate a pair of gears to a particular material quality level, a specification needs to be drawn up to define the level of inspection expected at different stages of the manufacture. Such a specification may be the subject of agreement between the gear manufacturer and the gear purchaser. Guidance and comments are given in F.2.2 to F.2.4 under headings which would normally be included at each grade level. This is not intended to form the basis of a specification, but rather to be an aid to determining the quality level of a material manufactured to an existing specification. In practice, a full specification will probably cover additional items. F.2.2 Quality C The inspection requirements for quality C are as follows. a) Mechanical properties (in final condition): HB or HV only (see BS 240-1 and BS 427-1). b) Casedepth: Etch check on a test piece carburized with the gear. For induction or flame hardened gears, etch check on the tooth end face of a sample gear. c) Weld repairs: For cast steels, permitted with an approved procedure. For other materials, not permitted in toothed areas. F.2.3 Quality B For cast iron other than nodular cast iron, the permissible stresses for quality B are the same as for quality C on the grounds of safety. The inspection requirements are therefore as for quality C. For other materials the inspection requirements are as follows. a) Chemical analysis: Supplier certification. b) Mechanical properties (in final condition): HB, HV or HRC (see BS 240-1, BS 427-1 and BS 891-1). Ö B and Charpy or Izod (random samples) (see BS 131-1 and BS 131-2). c) Crack detection: Magnetic particle inspection (100 % inspection on surface hardened, otherwise random samples) (see BS 6072). d) Weld repairs: For cast steels, permitted with an approved procedure. For other materials, not permitted in toothed areas. e) Ultrasonic inspection: 100 % checks on cast materials (see BS 4080).
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BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
f) Casedepth: Etch check on a test piece carburized with the gear. For induction or flame hardened gears, etch check on tooth end face and full tooth profile on a sample or every gear (dependent on batch size). g) Core structure (nodular irons): Sample check from batch for pearlite and ferrite content and graphite spheroidization. h) Heat treatment (carburized steel): Furnace fitted with atmosphere control instrumentation. F.2.4 Quality A All tests applicable to quality B apply also to quality A, but random sampling (when applicable to quality B) is replaced by 100 % testing. Additional inspection requirements are as follows: a) Ultrasonic inspection: 100 % check (see BS 4080, BS 4124-1 and BS 5996). b) Casedepth: Hardness traverse check on a test piece (same material cast and heat treatm ent condition) carburized with the gear. For induction or flame hardened gears, hardness traverse check at nominated positions across the facewidth on a sacrificial test gear or segment hardened un der the same conditions as the gear. c) Surface structure (carburized steel): Check on test piece: essentially fine martensite and specified retained austenite and carbide. d) Heat treatment: Furnace temperature chart records.
Appendix G Examples of calculations G.1 Example 1 G.1.1 The following pair of single helical gears are to be rated against the standard to find the maximum power rating. Gear details Pinion
Wheel
Centre distance, a
325 mm
325 mm
Normal module, mn
8 mm
8 mm
Normal pressure angle at reference cylinder, !n
20
Helix angle, ¶
18 37
Number of teeth, z1, z2
19
58
Facewidth, b
125 mm
125 mm
Reference diameters, d1, d2
160.392 mm 489.619 mm
Tip diameters, da1, da2
180.708 mm 501.292 mm
Tooth depth, h1, h2
19.2 mm
Mounting arrangement
Both central between bearings
Shaft diameters
130 mm
Number of revolutions, n1
1 450 r/min
Manufacturing accuracy
BS 436-2, grade 5
Construction
solid
solid
Helix modification
none
none
Mean roughness, Ra (flank)
0.8 4m
0.8 4m
© BSI 06-1999
°
°
20 9
°
°
18 37
9
19.2 mm
—
47
BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
3.2 4m
Mean roughness, Ra (root)
3.2 4m
Hob details Thickness at reference diameter
12.72 mm
Hob addendum, ha0
11.2 mm
Hob tip radius, Ôa0
2.40 mm
Offset of tip radius from centreline, D
0.80 mm
Material and operating conditions (pinion and wheel) Material type
steel quality B
Hardening process
carburized and hardened
Surface hardness
825 HV
Effective casedepth, ceff
1.3 mm
Ultimate tensile strength, Ö B
2 130 MN/m2
Core tensile strength, Ö B core
1 000 MN/m2
Residual stress (surface), Ö R
– 400 MN/m2
Residual stress (core), Ö R core
240 MN/m2
°
Lubricant viscosity at 40 C
303 cSt
Application factor, K A
1.0
Required life
25 000 h
Minimum demanded safety factor for contact stress, S H min
1.0
Minimum demanded safety factor for bending stress, S F min
1.4
G.1.2 From the given gear details the following geometry is calculated. dw1
= 2 a/(u + 1)
= 160.390 mm
dw2
= dw1 u
= 489.610 mm
! t
= tan –1 (tan! n/cos ¶)
= 21° 0 36
db1
= d1 cos!t
= 149.729 mm
db2
= d2 cos!t
= 457.068 mm
& ¶
= b sin ¶/(mn;)
= 1.588
g !
from B.2
= 37.015 mm
g !
used in subsequent = 34.462 mm calculations, allowing for undercut on the pinion (calculated separately from this procedure)
Ôbt
from B.3
= 24.757 mm
&!
= g ! / Ôbt
= 1.392
¶b
= sin –1 (cos ! n·sin ¶)
= 17° 27 24
48
9
9
0
0
© BSI 06-1999
BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Ôrel
from B.11
= 22.703 mm
zv1
= z1/cos3 ¶
= 22.32
zv2
= z2/cos3 ¶
= 68.15
v
= d1n1/19098
= 12.178 m/s
G.1.3 Contact stress factors are now calculated as follows. Zone factor, Z H, from equation (9)
= 2.3877
Contact ratio factor, Z &, from equation (12)
= 0.8477
Elasticity factor, Z E, from Table 1
= 189
Endurance limit for contact stress, Ö H lim
BHD from Figure 2
= 2 130 MN/m2
Z G2 from Figure 3
= 0.925
Z G1 from equation (16)
= 0.887
Limiting casedepth from Table 2
= 1.28 mm
Z c from Figure 4
= 1.0
then Ö H lim 1 from equation (15)
= 1 888 MN/m2
and Ö H lim 2 from equation (15)
= 1 969 MN/m2
Material quality factor, Z M, from Table 3
= 0.9
Lubricant influence, roughness and speed factors, Z L, Z R, Z v Z LZ v from Figure 5
= 1.076
Z R from Figure 6
= 0.943
Work hardening factor, Z w, from clause12
= 1.0
Size factor for constant stress, Z x, from clause 13= 1.0 Life factor for contact stress, Z N Number of tooth cycles N 1 = 25 000
!
60 !1450
= 2.18
!
109
= 7.13
!
108
Number of tooth cycles N 2 = N 1/u then Z N1 and Z N2 from curve 3 of Figure 8
= 1.0
Minimum demanded safety factor for contact stress, S H mi n (value specified)
= 1.0
Permissible contact stress, Ö HP1 from equation (1)
= 1 724 MN/m2
and Ö HP2 from equation (1)
= 1 798 MN/m2
Application factor, K A (uniformly loaded)
© BSI 06-1999
= 1.0
49
BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Dynamic factor, K v Q v from 16.3
= 1.0
K v350 ¶ from Figure 10
= 1.05
(F t K A /b)est from equation (23)
= 2 337 N/mm
B from equation (22)
= 0.3586
then K v
= 1.018
Load distribution factors K H!, K H ¶ beff
from equation (35)
= 125 mm
F m est
from equation (37)
= 306 700 N
wm est
from equation (38)
= 2 454 N/mm
K
from Figure 12
=0
A
from Table 8
= 0.023
f sh
from equation (40)
= 34.3 4m
f ma
from equation (42)
= 12.9 4m
F ¶X
from equation (45)
= 58.5 4m
q Y
from Figure 13 and equation (46)
= 0.83
F ¶ Y
from equation (47)
= 48.6 4m
c *
from 17.2.5
= 20 N/(mm· 4m)
K H ¶
from equation (49)
= 1.20
f pe
from Table 3 of BS 436-2:1970
= 12.2 4m
y!
from equation (54)
= 0.9 4m
K H!
from 17.4
= 1.15
Power capacity P HP1 from equation (3)
= 2 659 kW
P HP2 from equation (3)
= 2 892 kW
G.1.4 Bending stress factors are now calculated as follows: (sFn/mn) from equation (84):
pinion = 2.075 wheel = 2.159
( ÔF/mn) from equation (85):
pinion = 0.475 wheel = 0.548
! Fen from
equation (89):
pinion = 0.387 rad wheel = 0.340 rad
(hF/mn) from equation (90):
pinion = 1.188 wheel = 1.384
Then form factor Y F from equation (95):
pinion = 1.630 wheel = 1.788
L from equation (97):
pinion = 1.748 wheel = 1.559
50
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BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
qs from equation (98):
pinion = 2.187 wheel = 1.968
Then stress correction factor Y s from equation (96):
pinion = 1.945 wheel = 1.805
Helix angle factor Y ¶ from equation (99):
= 0.845
Basic endurance limit for bending stress, Ö F0 from Figure 15 (pinion and wheel) = 560 MN/m2 Surface residual stress, Ö R (value supplied)
= – 400 MN/m2
Material quality factor Y M from Table 10
= 0.9
Sensitivity factor Y ¸ from equation (100): pinion = 1.032 wheel = 1.030 Surface condition factor, Y R from Figure 16
= 0.904
Size factor for bending, Y x from clause 26
= 0.97
Life factors for bending, Y N1 and Y N2 from clause 27
= 1.0
Load factors for bending K F! from equation (107)
= 1.15
K F ¶ from equation (108)
= 1.168
Minimum demanded safety factor, S F min (value specified) Permissible bending stress, BFP from equation (57):
= 1.4
pinion = 629 MN/m2 wheel = 628 MN/m2
Permissible power capacity, P FP from equation (59):
pinion = 2 090 kW wheel = 2 051 kW
Sub-surface bending calculations: Permissible core bending stress, Ö FP core from equation (61): Actual bending stress, Ö Fcore from equation (62):
= 319 MN/m2 pinion = 218 MN/m2 wheel = 235 MN/m2
Core power capacity from equation (64):pinion = 3 056 kW wheel = 2 787 kW
© BSI 06-1999
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BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
G.2 Example 2: variable duty calculation using Appendix A The design torque of the pair of gears in example 1 is to be calculated from Table 15. From Table 12, Q H = 6.611 and N H
Z
=5
!
107
For the calculation of Q F, pinion
wheel
Y X
= 0.97
0.97
Y R
= 0.904
0.904
Ö B
= 2 130
2 130
Ö F0
= 560
560
N min
= 3 ! 105
9.8 ! 104
N F
= 3 ! 106
3 ! 106
Y N(N min)
= 1.352
1.565
Then Q F
= 9.685
9.852
Z
Then T H 1
= 9 486 N·m [see A.5.1 a)]
Z
T H 2
= 8 017 N·m [see A.5.1 a)]
T F 1
= 14 336 N·m [see A.5.1 b)]
T F 2
= 12 838 N·m [see A.5.1 b)]
Z
Z
Z
Table 15 — Variable duty calculation example Required duty i
T i
Pinion calculation
n1
Duration of applied torque
N·m
N i1
Wheel calculation
T H i1
T F i1
N·m
h
Z
T H i2
T F i2
N·m
N·m
N·m
Z
N i2
Z
Z
1
17 500
500
10
3.0
!
105 8 072
13 797
9.8 ! 104 6 815
12 366
2
15 000
500
20
6.0
!
105 8 763
14 336
2.0 ! 105 7 408
12 838
3
12 500
1 450
25
2.1
!
106 9 259
6.9 ! 105 7 827
4
10 000
1 450
60
5.2
!
106 9 486
1.7 ! 106 8 017
5
8 000
1 450
Continuous
> 5 ! 107
> 5 ! 107
Appendix H Equations of graphs H.1 Equations for graph of yield strength for contact stress, Ö HY , in Figure 1 For through hardened steels and cast irons:
Ö HY = 0.96362 Ö B + 838.755 For surface hardened steels:
Ö HY = 9.81 [(31.0 " log10 (6ceff ) – 275.5) " log10 ( Ôrel) + 130.0 " log10 (6ceff ) + 526] H.2 Equations for graph of values of Ö HD in Figure 2 If Ö B < 2 130, then Ö HD = Ö B . If Ö B W 2 130, then Ö HD = 2 130.
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© BSI 06-1999
BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
H.3 Equations for graph of values of Z G in Figure 3
if Ö B W 2 130 or the material is surface hardened and Ôrel/mn W 12.0 then Z G = 0.5667. Use linear interpolation for values of Ö B between the curves. H.4 Equations for graph of casedepth factor, Z c, in Figure 4 If ceff /clim < 1.0 then Z c = [5.0 + 3(ceff /clim)]/8. If ceff /clim > 1.0, then Z c = 1.0. H.5 Equation for graph of combined speed and lubricant factor, Z LZ v, in Figure 5
H.6 Equation for graph of roughness factor, Z R, in Figure 6
H.7 Equation for graph values of Z W in Figure 7
© BSI 06-1999
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BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
H.8 Equations for graph of life factor for contact stress, Z N, in Figure 8 H.8.1 Curve 1
H.8.2 Curve 2
H.8.3 Curve 3
H.8.4 Curve 4
H.8.5 Curve 5
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© BSI 06-1999
BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
H.9 Data for graph of K v350 for helical gears, &"
W
1, in Figure 10
Table 16 gives values of K v350 ¶ at discontinuities Table 16 — Values of K v350 ¶ at discontinuities Grade
Q vvz1 0
1 400
1 440
2 000
2 540
Values of K v350 ¶
3
1.0
1.18
1.47
1.47
1.18
4
1.0
1.23
1.54
1.54
1.23
5
1.0
1.31
1.65
1.65
1.31
6
1.0
1.45
1.80
1.80
1.45
7
1.0
1.65
—
—
—
8
1.0
1.96
—
—
—
9
1.0
2.44
—
—
—
10
1.0
3.38
—
—
—
Use linear interpolation between the above points. The lines for grades 7 to 10 are terminated at the points given in Table 17. Table 17 — K v350 ¶ termination points Grade
K v350 ¶
7
1.50
8
1.50
9
1.50
10
1.42
H.10 Data for graph of K v350 for spur gears in Figure 11 Table 18 gives values of K v350! at discontinuities Table 18 — Values of K v350! at discontinuities Grade
Q vvz1 0
1 000
1 040
1 400
1 800
Values of K v350!
3
1.0
1.21
1.77
1.77
1.21
4
1.0
1.30
1.89
1.89
1.30
5
1.0
1.42
2.06
2.06
1.42
6
1.0
1.61
2.26
2.26
1.65
7
1.0
1.92
2.60
2.60
1.98
8
1.0
2.29
—
—
—
9
1.0
2.78
—
—
—
10
1.0
3.50
—
—
—
Use linear interpolation between the above points. The lines for grades 7 to 10 are terminated at the points given in Table 19.
© BSI 06-1999
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BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Table 19 — K v350! termination points Grade
K v350!
7
2.40
8
1.96
9
1.96
10
1.82
H.11 Equation for graph of values of q# in Figure 13
H.12 Equations for graph of values Ö Fo in Figure 15
H.13 Equations for graph of values of Y R in Figure 16 H.13.1 Curve 1 Y R = 1.49 – 0.471(6Ra + 1)0.1 H.13.2 Curve 2 Y R = 4.924 – 3.9(6Ra + 1)0.01 H.13.3 Curve 3 Y R = 4.161 – 3.155(6Ra + 1)0.005 H.14 Equations for graph of values of Y X in Figure 17 H.14.1 All curves If mn u 5, then
Y x = 1.0
H.14.2 Curve 1 If 5 < mn < 30, then
Y x = 1.03 – 0.006 mn
If mn W 30, then
Y x = 0.85
H.14.3 Curve 2 If 5 < mn < 30, then
Y x = 1.05 – 0.01 mn
If mn W 30, then
Y x = 0.75
H.14.4 Curve 3 If 5 < mn < 25, then
Y x = 1.075 – 0.015 mn
If mn W 25, then
Y x = 0.70
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© BSI 06-1999
BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
H.15 Equation for graph of Y N for through hardened steel in Figure 18
H.16 Equation for graph of Y N for thick case surface hardened steel and cast iron in Figure 19
H.17 Equation for graph of Y N for thin case surface hardened steel, grey cast iron and bronze in Figure 20
© BSI 06-1999
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2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
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BS 436-3:1986 2 1 0 2 n o i t u t i t s n I s d r a d n a t S h s i t i r B e h T ) c ( , y p o C d e l l o r t n o c n U , 7 3 : 6 1 3 1 0 2 / 2 0 / 4 0 , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U e h T , y r a r b i L r e t s e h c n a M f o y t i s r e v i n U : y p o C d e s n e c i L
Publications referred to BS 131, Methods for notched bar tests. BS 131-1, The Izod impact test on metals. BS 131-2, The Charpy V-notch impact test on metals. BS 240, Method for Brinell hardness test. BS 240-1, Testing of metals. BS 427, Method for Vickers hardness test. BS 427-1, Testing of metals. BS 436, Spur and helical gears. BS 436-1, Basic rack form, pitches and accuracy (diametral pitch series). BS 436-2, Basic rack form, modules and accuracy (1 to 50 metric module). BS 891, Method for Rockwell hardness test. BS 891-1, Testing of metals. BS 1400, Specification for copper alloy ingots and copper alloy and high conductivity copper castings. BS 2519, Glossary for gears. BS 2519-1, Geometrical definitions. BS 2519-2, Notation. BS 4080, Methods for non-destructive testing of steel castings. BS 4124, Non-destructive testing of steel forgings. BS 4124-1, Ultrasonic flaw detection. BS 5996, Methods for ultrasonic testing and specifying quality grades of ferritic steel plate. BS 6072, Method for magnetic particle flaw detection. BS 6443, Method for penetrant flaw detection.
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