Spur Gear Strength Calculation With CSN 01 4686, ISO 6336 and DIN 3990

Based on the fixed-end beam calculation. Contains the majority of effects.

Safety factors

Contact fatigue

where:

	σ Hlim	contact fatigue limit (material property)
	F t	tangential force acting at teeth
	b w	operating face width
	d 1	pitch diameter of pinion

Contact during one-time loading

where:

	σ HPmax	permissible contact stress
	K AS	one-time overloading factor

Bending fatigue

where:

	σ Flim	bending fatigue limit (material property)
	b wF1,2 = min (b 1,2 , b w + 2m)	tooth width for bending

Bending during one-time loading

where:

σ FPmax

permissible bending stress

Factor calculations

Z _N ... life factor (for contact)

1 ≤ Z _N ≤ 1.3 nitridated steels

1 ≤ Z _N ≤ 1.6 other steels

	N Hlim	base number of load cycles for contact (material property)
	N K1,2 = 60 L h n 1,2	required number of load cycles (speed)

Y _N ... life factor (for bending)

1 ≤ Y _N ≤ 1.6 nitridated steels

1 ≤ Y _N ≤ 2.5 other steels

	N Flim	base number of load cycles for bending (material property)
	N K1,2 = 60 L h n 1,2	required number of load cycles (speed)

Z _L ... lubricant factor

	DIN and ISO:
		Z L = C ZL + 4 (1 - C ZL ) 0.158
		C ZL = σ Hlim / 4375 + 0.6357
		for σ Hlim < 850 Mpa C ZL = 0.83
		for σ Hlim > 1200 Mpa C ZL = 0.91

Z _R ... roughness factor

Z _V ... speed factor

	CSN:
		Z v = 0.95 + 0.08 log v
	ISO and DIN:
		C ZV = C ZL + 0.02

Z _E ... elasticity factor

where:

	μ	Poisson's ratio (material property)
	E	modulus of elasticity (material property)

Z _H ... zone factor

Z _B ... single pair tooth contact factor

for ε β ≥ 1 or internal gearing:
	Z B1,2 = 1
for ε β = 0:
for ε β < 1:
	Z B1,2 = Z B0 - ε b (Z B0 - 1) where:
	Z B0 = Z B1,2 calculated for ε β = 0

Z ε ... contact ratio factor

for ε β = 0:
for ε β < 1:
for ε β ≥ 1:

Y _ε ... contact ration factor (for bending)

CSN:

	for ε β < 1:
	for ε β ≥ 1:

ISO and DIN:

Z _β ... helix angle factor (for contact)

CSN:

Z β = 1

ISO and DIN:

Y _β ... helix angle factor (for bending)

CSN:

Y βmin = 1 - 0.25 ε β ≥ 0.75

ISO and DIN:

	for ε β > 1 the ε β = 1 is used
	for β > 30° the β = 30° is used

Z _x ... size factor (for contact)

Y _x ... size factor (for bending)

Z _W ... work hardening factor

Y _Fa ... form factor

where:

	h Fa	bending arm of a force acting on the tooth end
	s Fn	thickness of dedendum dangerous section of alternate gear
	α Fan	bending angle at the end of straight tooth of alternate gear

Y _Sa ... stress correction factor

Y _Sa = (1.2 + 0.13 L _a ) q _s ^exp

Y _Sag ... teeth with grinding notches factor

Y _δ ... notch sensitivity factor (depends on the material and curvature radius of dedendum transition)

Y _R ... tooth root surface factor

K _H ... additional loads factor (for contact)

K _H = K _A K _Hv K _Hb K _Ha

K _F ... additional loads factor (for bending)

K _F = K _A K _Fv K _Fb K _Fa

K _A ... application factor (external dynamic forces)

K _Hv ... dynamic factor (internal dynamic forces) for contact

K _Fv ... dynamic factor (internal dynamic forces) for bending

for CSN:

at K A F t / b w < 150 considering K A F t / b w = 150

for ISO and DIN:

	at K A F t / b w < 100 considering K A F t / b w = 100
	where: K P , K Q ... table values

K _Hβ ... face load factor (for contact)

for CSN:

where:

	c = 0.4	gears with hardened tooth sides
	c = 0.3	non-hardened gears
	f ky = | f sh1 + f sh2 | + f kZ - y β
	f b , f x , f y ... teeth tolerance y β ... table value

for ISO and DIN:

	for
	otherwise ( < 1):
	F βy = F βx χ β
	for gears with hardened tooth sides χ β = 0.85
	for others
	F βx = 1.33 f sh + f ma

	q' = 0.04723 + 0.15551/z v1 + 0.25791/z v2 - 0.00635 x 1 - 0.11654 x 1 /z v1 - 0.00193 x 2 - 0.24188 x 2 /z v2 + 0.00529 x 1 2 + 0.00182 x 2 2
	for F t K A / b w < 100 the values are interpolated
	for ISO c' = c' [(F t K A / b w ) / 100] 0.25
	for DIN c' = c' (F t K A / b w ) / 100
	C M = 0.8
	C R = 1 for solid gears
	C B = [1 + 0.5 (1.2 - h f /m)] [1 - 0.02 (20° - α)]
	E steel = 206 000
	c γ = c' (0.75 ε α + 0.25)
	A, B ... table values depend on the arrangement of teeth gears, shafts, and bearings

K _Fβ ... face load factor (for bending)

K _Fβ = (K _Hβ ) ^NF

where:

	h = 2 m/ε α	spur gears
	h = 2 m	helical gears

K _Fa ... transverse load factor (for bending)

for ε γ < 2:
for ε γ > 2:
	at K A F t / b w < 100 considering K A F t / b w = 100
limit values:
	for CSN: 1 ≤ K Fα ≤ε γ

K _Hα ... transverse load factor (for contact)

for CSN:
	K Hα = 1 for straight teeth
	K Hα = K Fα for helical teeth
DIN and ISO:
	K Hα = K Fα
for limit values: