BS ISO 6336-2 Calculation of load capacity of spur and helical gears - Part 2 : Calculation of surface durability (pitting PDF

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BRITISH STANDARD BS ISO 6336-2:2006 Incorporating corrigendum June 2008 Calculation of load capacity of spur and helical gears — Part 2 : Calculation of surface durability (pitting) ICS 21.200 12&23<,1*:,7+287%6,3(50,66,21(;&(37$63(50,77('%<&23<5,*+7/$: BS ISO 6336...

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BRITISH STANDARD

BS ISO 6336-2:2006 Incorporating corrigendum June 2008

Calculation of load capacity of spur and helical gears — Part 2 : Calculation of surface durability (pitting)

ICS 21.200

12&23 2). See ISO 6336-1:2006, 4.2, for the definition of Ft and comments on particular characteristics of double-helical gearing. The value b of mating gears is the smaller of the facewidths at the root circles of pinion and wheel ignoring any intentional transverse chamfers or tooth-end rounding. Neither unhardened portions of surface-hardened gear tooth flanks nor the transition zones shall be included.

5.4

Permissible contact stress, σHP

The limit values of contact stresses (see Clause 10) should preferably be derived from material tests using meshing gears as test pieces (see Introduction). The more closely test gears and test conditions resemble the service gears and service conditions, the more relevant to the calculations the derived values will be. 5.4.1

Determination of permissible contact stress σHP — Principles, assumptions and application

Several procedures for the determination of permissible contact stresses are acceptable. The method adopted shall be validated by carrying out careful comparative studies of well-documented service histories of a number of gears. 5.4.1.1

Method A

In Method A the permissible contact stress σHP (or the pitting stress limit, σHG) for reference stress, long and limited life and static stresses is calculated using Equation (4) or (5) from the S-N curve or damage curve derived from tests of actual gear pair duplicates under appropriate service conditions. The cost required for this method is in general only justifiable for the development of new products, failure of which would have serious consequences (e.g. for manned space flight). Similarly, the permissible stress values may be derived from consideration of dimensions, service conditions and performance of carefully monitored reference gears. The more closely the dimensions and service conditions of the actual gears resemble those of the reference gears, the more effective will be the application of such values for purposes of design ratings or calculation checks. 5.4.1.2

Method B

Damage curves, characterized by the allowable stress number values, σH lim, and the limited life factors, ZNT, have been determined for a number of common gear materials and heat treatments from the results of gear loading tests with standard reference test gears. These test gear values are converted to suit the dimensions and service conditions of the actual gear pair using the (relative) influence factors for lubricant ZL, pitch line velocity Zv, flank surface roughness ZR, work hardening ZW and size ZX. Method B is recommended for reasonably accurate calculation whenever pitting resistance values are available from gear tests, from special tests or, if the material is similar, from ISO 6336-5 (see Introduction). 5.4.1.3

Method BR

Material characteristic values are determined by rolling pairs of disks in loaded contact. The magnitude and direction of the sliding speed in these tests should be adjusted to represent the in-service slide and roll conditions of the tooth flanks in the areas at risk from pitting. Method BR may be used when stress values derived from gear tests are not available. The method is particularly suitable for the determination of the surface durability of various materials relative to one another.

5

BS ISO 6336-2:2006

5.4.2

Permissible contact stress, σHP: Method B

The permissible contact stress is calculated from

σHP =

σ H lim Z NT S H min

ZL Z v ZR Z W Z X =

σ HG S Hmin

(6)

where

σH lim

is the allowable stress number (contact) (see Clause 10 and ISO 6336-5), which accounts for the influence of material, heat treatment and surface roughness of the standard reference test gears;

ZNT

is the life factor for test gears for contact stress (see Clause 11), which accounts for higher load capacity for a limited number of load cycles;

σHG

is the pitting stress limit (= σHP SH min);

SH min

is the minimum required safety factor for surface durability.

ZL, ZR, Zv

are factors that, together, cover the influence of the oil film on tooth contact stress;

ZL

is the lubricant factor (see Clause 12), which accounts for the influence of the lubricant viscosity;

ZR

is the roughness factor (see Clause 12), which accounts for the influence of surface roughness;

Zv

is the velocity factor (see Clause 12), which accounts for the influence of pitch line velocity;

ZW

is the work hardening factor (see Clause 13), which accounts for the effect of meshing with a surface hardened or similarly hard mating gear.

ZX

is the size factor for contact stress (see Clause 14), which accounts for the influence of the tooth dimensions for the permissible contact stress.

a)

Permissible contact stress (reference), σHP ref, is derived from Equation (6), with ZNT = 1 and the influence factors σH lim, ZL, Zv, ZR, ZW , ZR, ZX and SH min calculated using Method B.

b)

Permissible contact stress (static), σHP stat, is determined in accordance with Equation (6), with all influence factors (for static stress) following Method B.

5.4.3

Permissible contact stress for limited and long life: Method B

In Method B, provision is made for determination of σHP by graphical or computed linear interpolation on a log-log scale between the value obtained for reference in accordance with 5.4.2 a) and the value obtained for static stress in accordance with 5.4.2 b). Values appropriate to the relevant number of load cycles, NL, are indicated by the S-N curve. See Clause 11. 5.4.3.1

Graphical values

Calculate σHP for reference stress and static stress in accordance with 5.4.2 and plot the S-N curve corresponding to the life factor ZNT. See Figure 1 for principle. σHP for the relevant number of load cycles NL may be read from this graph.

6

BS ISO 6336-2:2006

Key X

number of load cycles, NL (log)

Y

permissible contact stress, σHP (log)

1

static

2 3

limited life long life

a

Example: permissible contact stress, σHP for a given number of load cycles.

Figure 1 — Graphic determination of permissible contact stress for limited life — Method B

5.4.3.2

Determination by calculation

Calculate σHP ref for reference and σHP stat for static strength in accordance with 5.4.2 and, using these results, determine σHP, in accordance with Method B for limited life and the number of load cycles NL in the range as follows (see ISO 6336-1:2006, Table 2, for an explanation of the abbreviations used). a)

St, V, GGG(perl., bain.), GTS(perl.), Eh, IF, if a certain number of pits is permissible: ⎯

For the limited life stress range, 6 × 105 < NL u 107 in accordance with Figure 6: ⎛ 3 × 10 8 σHP = σHP ref ZN = σHP ref ⎜ ⎜ NL ⎝

⎞ ⎟ ⎟ ⎠

exp

(7)

7

BS ISO 6336-2:2006

where exp = 0,370 5 log ⎯

σ HP stat σ HP ref

(8)

For the limited life stress range, 107 < NL u 109 in accordance with Figure 6: ⎛ 10 9 σHP = σHP ref ZN = σHP ref ⎜ ⎜ NL ⎝

⎞ ⎟ ⎟ ⎠

exp

(9)

where exp = 0,279 1 log b)

σ HP stat σ HP ref

(10)

St, V, GGG(perl., bain.), GTS(perl.), Eh, IF, when no pits are permissible: ⎯

For the limited life stress range, 105 < NL u 5 × 107 in accordance with Figure 6: ⎛ 5 × 10 7 σHP = σHP ref ZN = σHP ref ⎜ ⎜ NL ⎝

⎞ ⎟ ⎟ ⎠

exp

(11)

where exp is as in Equation (8). c)

GG, GGG(ferr.), NT(nitr.), NV(nitr.) ⎯

For the limited life stress range, 105 < NL u 2 × 106 in accordance with Figure 6: ⎛ 2 × 10 6 σHP = σHP ref ZN = σHP ref ⎜ ⎜ NL ⎝

⎞ ⎟ ⎟ ⎠

exp

(12)

where exp = 0,768 6 log d)

σ H P stat σ HP ref

(13)

NV(nitrocar.) ⎯

For the limited life stress range, 105 < NL u 2 × 106 in accordance with Figure 6:

⎛ 2 × 10 6 ⎞ ⎟⎟ σ HP = σ HP ref Z N = σ HP ref ⎜⎜ ⎝ NL ⎠

exp

(14)

where ˆ exp = 0,768 6 log

σ HP stat ‰ σ HP ref

Corresponding calculations may be determined for the range of long life.

8

(15)

BS ISO 6336-2:2006

6

Zone factor, ZH, and single pair tooth contact factors, ZB and ZD

These factors account for the influence of tooth flank curvature on contact stress.

6.1

Zone factor, ZH

The zone factor, ZH, accounts for the influence on Hertzian pressure of tooth flank curvature at the pitch point and transforms the tangential load at the reference cylinder to normal load at the pitch cylinder. 6.1.1

Graphical values

ZH can be taken from Figure 2 as a function of (x1 + x2) / (z1 + z2) and β for external and internal gears having normal pressure angles αn = 20°, 22,5° or 25°.

Key X

zone factor, ZH

Y

helix angle at reference circle β

Figure 2 — Zone factor, ZH 6.1.2

Determination by calculation

The zone factor is calculated by: ZH =

2 cosβ b cosα wt cos 2 α t sinα wt

(16)

9

BS ISO 6336-2:2006

Single pair tooth contact factors, ZB and ZD, for εα u 2

6.2

The single pair tooth contact factors, ZB and ZD, are used to transform the contact stress at the pitch point of spur gears to the contact stress at the inner point B of single pair tooth contact of the pinion or at the inner point D of single pair tooth contact of the wheel if ZB > 1 or ZD > 1. See Figure 3 and 5.1.

External gearing

Internal gearing

Key 1 2

pinion wheel

Figure 3 — Radii of curvature at pitch point C and single pair tooth contact point B of pinion and D of wheel for determination of pinion single pair tooth contact factor ZB in accordance with Equation (17) and wheel single pair tooth contact factor ZD in accordance with Equation (18) (only for external spur gears) In general, ZD should only be determined for gears when u < 1,5. When u > 1,5, M2 is usually less than 1,0 in which case ZD is made equal to 1,0 in Equation (17). For internal gears, ZD shall be taken as equal to 1,0. Determination by calculation: ˆ

M1 =

ρ C1 ρ C2 = ρ B1 ρ B2 ⎛ 2 ⎜ d a1

⎜ d2 ⎝ b1

10

tan α wt ⎞⎛ ⎞ 2 2π ⎟⎜ d a2 2π ⎟ −1− − 1 − (ε α − 1) 2 z1 ⎟⎜ d b2 z2 ⎟ ⎠⎝ ⎠

‰

(17)

BS ISO 6336-2:2006

ˆ M2 =

ρ C1 ρ C2 = ρ D1 ρ D2 ⎛ d2 ⎜ a2 ⎜ d2 ⎝ b2

tan α wt −1 −

⎞⎛ ⎞ 2 2π ⎟ ⎜ d a1 2π ⎟ ‰ − 1 − (ε α − 1) 2 z 2 ⎟ ⎜ d b1 z1 ⎟ ⎠⎝ ⎠

(18)

Equation (17) and (18) are not valid, if undercut shortens the path of contact. See 8.2.1 for calculation of the profile contact ratio εα. a)

b)

Spur gears with εα > 1: if M1 u 1 then ZB = 1;

if M2 u 1 then ZD = 1;

if M1 > 1 then ZB = M1;

if M2 > 1 then ZD = M2.

Helical gears with εα > 1 and εβ W 1: ZB = ZD = 1

c)

Helical gears with εα > 1 and εβ < 1: ZB and ZD are determined by linear interpolation between the values for spur and helical gearing with

εβ W 1:

ZB = M1 − εβ (M1 − 1) and ZB W 1 ZD = M2 − εβ (M2 − 1) and ZD W 1

If ZB or ZD are made equal to 1, the contact stresses calculated using Equation (4) or (5) are the values for the contact stress at the pitch cylinder. d)

Helical gears with εα u 1 and with εγ > 1: not covered by ISO 6336 — a careful analysis of the decisive contact stress along the path of contact is necessary.

Methods a), b) and c) apply to the calculation of contact stress when the pitch point lies in the path of contact. If the pitch point C is determinant and lies outside the path of contact, then ZB and/or ZD are determined for contact at the adjacent tip circle. For helical gearing when εβ is less than 1,0, ZB and ZD are determined by linear interpolation between the values (determined at the pitch point or at the adjacent tip circle as appropriate) for spur gears and those helical gears with εβ W 1.

6.3

Single pair tooth contact factors, ZB and ZD, for εα > 2

In the case of meshing gear pairs of high precision with 2 < εα u 2,5, the entire tangential load in any transverse plane is supported by two pairs, or three pairs, of teeth in continued succession. For such gears, the calculation of contact stress is based on the inner point of two pair tooth contact of the pinion.

7

Elasticity factor, ZE

The elasticity factor, ZE, takes into account the influences of the material properties E (modulus of elasticity) and ν (Poisson's ratio) on the contact stress. ZE =

1 ⎛ 1−ν 2 1 π⎜ ⎜ E1 ⎝

1 − ν 22 + E2

⎞ ⎟ ⎟ ⎠

(19)

11

BS ISO 6336-2:2006

When E1 = E2 = E and ν1 = ν2 = ν : E

ZE =

(20)

2π(1 − ν 2 )

For steel and aluminium ν = 0,3 and therefore:

Z E = 0,175 E

(21)

For mating gears in material having different moduli of elasticity E1 and E2, the equivalent modulus E=

2 E1 E 2 E1 + E 2

(22)

may be used. For some material combinations ZE can be taken from Table 1. Table 1 — Elasticity factor, ZE, for some material combinations Wheel 1 Material a

St, V, Eh, IF, NT, NV

Modulus of elasticity, E N/mm2

Wheel 2 Poisson’s ratio, ν

206 000

0,3 St(cast)

a

8

202 000

GGG, GTS

173 000

GG

126 000 to 118 000

Material

Modulus of elasticity, E N/mm2

St, V, Eh, IF, NT, NV

206 000

189,8

St(cast)

202 000

188,9

GGG, GTS

173 000

181,4

GG

126 000 to 118 000

165,4 to 162,0

St(cast)

202 000

GGG, GTS

173 000

180,5

GG

118 000

161,4

GGG, GTS

173 000

173,9

GG

118 000

156,6

118 000

146,0 to 143,7

GG

Poisson’s ratio, ν

0,3

ZE

N/mm 2

188,0

See ISO 6336-1:2006, Table 2, for explanation of abbreviations used.

Contact ratio factor, Zε

The contact ratio factor, Zε, accounts for the influence of the transverse contact and overlap ratios on the surface load capacity of cylindrical gears. Calculation of the contact stress is based on a virtual facewidth bvir instead of the actual facewidth b:

bvir 1 = b z ε2

(23)

The average length of the line of contact calculated on a simplified basis is used as the appropriate value for helical gearing with εβ > 1.

12

BS ISO 6336-2:2006

8.1

Determination of contact ratio factor, Zε

8.1.1

Graphical values

Zε for known contact and overlap ratio factors may be read from Figure 4.

Key X

transverse contact ratio, εα

Y

contact ratio factor, Zε

Figure 4 — Contact ratio factor, Zε 8.1.2 a)

Determination by calculation

Spur gears: Zε =

4 − εα 3

(24)

The conservative value of Zε = 1,0 may be chosen for spur gears having a contact ratio less than 2,0. b)

Helical gears: Zε =

Zε =

εβ 4 − εα (1 − ε β ) + εα 3 1

εα

for εβ W 1

for εβ < 1

(25)

(26)

13

BS ISO 6336-2:2006

Calculation of transverse contact ratio, εα, and overlap ratio, εβ

8.2 8.2.1

Transverse contact ratio, εα

The calculation is based on the roll angle ξ and the angular pitch τ, both expressed in radians in the following equations.

εα =

ξ fw 1 + ξ aw 1 τ1

=

ξ fw 2 + ξ aw 2

(27)

τ2

where

ξfw1,2

are the roll angles from the root form diameters to the working pitch point, taken as the least value of ⎯

limited by the base diameters:

ξ fw1,2 = tan α wt ⎯

⎯

limited by the root form diameters:

ξ fw 1 = tan α wt − tan arccos

d b1 d soi1

(29)

ξ fw 2 = tan α wt − tan arccos

d b2 d soi2

(30)

limited by the tip diameters of the wheel/pinion (start of active profile): ⎛

ξ fw1 = ⎜ tan arccos ⎝

⎛

ξ fw 2 = ⎜ tan arccos ⎝

ξaw1,2

⎞ db2 − tan α wt ⎟ z 2 d a2 ⎠ z1

(31)

⎞ d b1 − tan α wt ⎟ z1 d a1 ⎠ z2

(32)

are the roll angles from the working pitch point to the tip diameter

ξ aw 1 = ξ fw 2 τ1,2

(28)

z2 z , ξ aw 2 = ξ fw 1 1 z1 z2

(33)

is the pinion/wheel angular pitch:

τ 1=

2π z1

,τ 2 =

2π

(34)

z2

Equations (28) to (34) do not take into account undercut (see Annex A). 8.2.2

Overlap ratio, εβ

This is calculated by εβ =

b sin β π mn

See Equation (3) for the definition of facewidth.

14

(35)

BS ISO 6336-2:2006

9

Helix angle factor, Zβ

Independent of the influence of the helix angle on the length of path of contact, the helix angle factor, Zβ, accounts for the influence of the helix angle on surface load capacity, allowing for such variables as the distribution of load along the lines of contact. Zβ is dependent only on the helix angle, β. For most purposes, the following empirical relationship is in sufficiently good agreement with experimental and service experience, but that agreement is only achieved when high accuracy and optimum modifications are employed: ˆ Zβ =

1 cos β

‰

(36)