Section 3 Design
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Clasifications Register Rules and Regulations - Rules and Regulations for the Classification of Inland Waterways Ships, July 2022 - Part 5 Main and Auxiliary Machinery - Chapter 3 Gearing - Section 3 Design

Section 3 Design

Parent topic: Chapter 3 Gearing

3.1 Symbols

3.1.1 For the purposes of this Chapter, the following symbols apply:

a = centre distance, in mm
b = facewidth, in mm
d = reference diameter, in mm
d a = tip diameter, in mm
d an = virtual tip diameter, in mm
d b = base diameter, in mm
d bn = virtual base diameter, in mm
d en = virtual diameter to the highest point of single tooth pair contact, in mm
d f = root diameter, in mm
d fn = virtual root diameter, in mm
d n = virtual reference diameter, in mm
d s = shrink diameter, in mm
d w = pitch circle diameter, in mm
f ma = tooth flank misalignment due to manufacturing errors, in μm
f pb = maximum base pitch deviation of wheel, in μm
f Sh = tooth flank misalignment due to wheel and pinion deflections, in μm
f Sho = intermediary factor for the determination of fSh
g α = length of line of action for external gears, in mm:
=
= for internal gears:
=
h = total depth of tooth, in mm
h ao = basic rack addendum of tool, in mm
h F = bending moment arm for root stress, in mm
h W = sum of actual tooth addenda of pinion and wheel, in mm
m n = normal module, in mm
n = rev/min of pinion
q = machining allowances, in mm
q s = notch parameter
q ' = intermediary factor for the determination of Cγ
u = gear ratio
v = linear speed at pitch circle, in m/s
x = addendum modification coefficient
y α = running in allowance, in μm
y β = running in allowance, in μm
z = number of teeth
z n = virtual number of teeth
=
C γ = tooth mesh stiffness (mean total mesh stiffness per unit facewidth), in N/mm μm
F t = nominal tangential tooth load, in N
=
F β = total tooth alignment deviation (maximum value specified), in μm
F βx = actual longitudinal tooth flank deviation before running in, in μm
F βy = actual longitudinal tooth flank deviation after running in, in μm
HV = Vickers hardness number
K A = application factor
K Fα = transverse load distribution factor
K Fβ = longitudinal load distribution factor
K Hα = transverse load distribution factor
K Hβ = longitudinal load distribution factor
K v = dynamic factor
K vα = dynamic factor for spur gears
K vβ = dynamic factor for helical gears
K γ = load sharing factor
P = transmitted power, in kW
P r = radial pressure at shrinkage surface, in N/mm2
P ro = protuberance of tool, in mm
Q = accuracy grade derived from ISO 1328 Cylindrical gears – ISO system of accuracy
R a = surface roughness – arithmetical mean deviation (C.L.A.) as determined by an instrument having a minimum wavelength cut-off of 0,8 mm and for a sampling length of 2,5 mm, in μm
S pr = residual undercut left by protuberance in mm
S F min = minimum factor of safety for bending stress
S Fn = tooth root chord in the critical section, in mm
S H min = minimum factor of safety for Hertzian contact stress
Y D = design factor
Y F = tooth form factor
Y R rel T = relative surface finish factor
Y S = stress concentration factor
Y ST = stress correction factor
Y x = size factor
Y β = helix angle factor
Y δ rel T = relative notch sensitivity factor
Z E = material elasticity factor
Z H = zone factor
Z R = surface finish factor
Z V = velocity factor
Z X = size factor
Z β = helix angle factor
Z ∊ = contact ratio factor
αen = pressure angle at the highest point of single tooth contact, in degrees
αn = normal pressure angle at reference diameter, in degrees
αt = transverse pressure angle at reference diameter, in degrees
αtw = transverse pressure angle at pitch circle diameter, in degrees
αF en = angle for application of load at the highest point of single tooth contact, in degrees
β = helix angle at reference diameter, in degrees
βb = helix angle at base diameter, in degrees
γ = intermediary factor for the determination of f Sh
α = transverse contact ratio
=
αn = virtual transverse contact ratio
β = overlap ratio
=
γ = total contact ratio
ρao = tip radius of tool, in mm
ρc = relative radius of curvature at pitch point, in mm
=
ρF = tooth root fillet radius at the contact of the 30° tangent, in mm
σy = yield or 0,2 per cent proof stress, in N/mm2
σB = ultimate tensile strength, in N/mm2
σF = bending stress at tooth root, N/mm2
σF lim = endurance limit for bending stress in N/mm2
σFP = allowable bending stress at the tooth root, in N/mm2
σH = Hertzian contact stress at the pitch circle, in N/mm2
σH lim = endurance limit for Hertzian contact stress, in N/mm2
σHP = allowable Hertzian contact stress, in N/mm2
Subscript:  
  1 = pinion
  2 = wheel
  0 = tool.

3.2 Tooth form

3.2.1 The tooth profile in the transverse section is to be of involute shape, and the roots of the teeth are to be formed with smooth fillets of radii not less than 0,25 m n.

3.2.2 All sharp edges left on the tips and ends of pinion and wheel teeth after hobbing and finishing are to be removed.

3.3 Tooth loading factors

3.3.1 For values of application factor, K A, see Table 3.3.1 Values of K A .

Table 3.3.1 Values of K A

Main and auxiliary gears K A
Main propulsion engine reduction gears:  
  Hydraulic coupling or equivalent on input 1,10
  High elastic coupling on input 1,30
  Other coupling 1,50
Auxiliary gears:  
  Electric and engine drives with hydraulic coupling or equivalent on input 1,0
Engine drives with high elastic coupling on input 1,20
  Engine drives with other couplings 1,40

3.3.2 Load sharing factor, K γ. The value for K γ is to be taken as 1,15 for multi-engine drives or split torque arrangements. Otherwise K γ is to be taken as 1. Alternatively, where measured data exists, a derived value will be considered.

3.3.3 Dynamic factor, K v:

For helical gears with ∊β ≥ 1:
  K v = 1 + Q 2vz1 10–5 = K vβ
For helical gears with ∊β < 1:
  K v = K vα – ∊β (K vα – K vβ )
For spur gears:
  K v = 1 + 1,8Q 2 vz 110–5 = K vα
where for helical gears, and
where for spur gears, the value of K v will be
  specially considered.

3.3.4 Longitudinal load distribution factors, K Hβ and K Fβ:

K Hβ =

Calculated values of KHβ > 2 are to be reduced by improved accuracy and helix correction as necessary:

where
F βy = F βx – y β and
F βx = 1,33 f Sh + f ma
f ma = F β at the design stage, or
f ma = F β where helix correction has been applied
f Sh = f Sho
where
fSho = 23γ10–3 μm mm/N for gears without helix correction and without end relief, or
= 16γ10–3 μm mm/N for gears without helix correction but with end relief

γ = for single helical and spur gears
= for double helical gears

The following minimum values are applicable, these also being the values where helix correction has been applied:

fSho = 10 x 10–3 μm mm/N for helical gears, or
= 5 x 10–3 μm mm/N for spur gears

For through-hardened steels and surface hardened steels running on through-hardened steels:

y β = F βx when
y β =

For surface hardened steels, when

y β = 0,15F βx
y β = ≤ 6 μm
K Fβ = K Hβ n

where

n =

Note 1. is to be taken as the smaller of or

Note 2. For double helical gears is to be substituted for b in the equation for n.

3.3.5 Transverse load distribution factors, K Hα and K Fα

K Hα = K Fα ≥ 1,00
where

∊γ ≤ 2

K Hα =
where

∊γ> 2

K Hα = 0,9 + 0,4 , but
K Hα = and
K Fα = and

When tip relief is applied, f pb is to be half of the maximum specified value:

y α = f pb for through-hardened steels, when
y α = μm and
y α = 0,075f pb for surface hardened steels, when
y α = ≤ 3 μm

When pinion and wheel are manufactured from different materials:

y α =

3.3.6 Tooth mesh stiffness, C γ :

C γ = cos β (0,75∊α + 0,25) N/mm μm
where
q ' = 0,04723 0,00635x 1— – 0,00193x 2 – + 0,00529x 1 2 + 0,00182x 2 2

For internal gears z n2 = ∞

Other calculation methods for C γ will be specially considered.

3.4 Tooth loading for surface stress

3.4.1 The Hertzian contact stress, σH, at the pitch circle is not to exceed the allowable Hertzian contact stress, σHP.

σH = Z H Z E Z ∊ Z β K A K γ K v K Hβ K Hα and
σHP = for the pinion/wheel combination.
where
Z H =
Z E = 189,8 for steel
Z ∊ = for ∊β < 1 and
Z ∊ = for ∊β ≥ 1
Z β =
Z R = but Z R ≤ 1,14
where

R a is the surface roughness value of the tooth flanks. When pinion and wheel tooth flanks differ, then the larger value of R a is to be taken.

Z v = 0,88 + 0,23
For values of Z x, see Table 3.3.2 Values of Z x
For values of σH lim, see Table 3.3.3 Values of endurance limit for Hertzian contact stress, σ H lim
For values of S H min, see Table 3.3.4 Factors of safety.

Table 3.3.2 Values of Z x

Pinion heat treatment Z x
Carburized and induction-hardened m n ≤ 10 1,0
10 < m n < 30 1,05 – 0,005m n
30 ≤ m n 0,9
Nitrided m n < 7,5 1,0
7,5 < m n < 30 1,08 – 0,011m n
30 ≤ m n 0,75
  All modules 1,0

Table 3.3.3 Values of endurance limit for Hertzian contact stress, σ H lim

Heat Treatment
Pinion Wheel σH lim N/mm2
Through-hardened Through-hardened 0,46σB2 + 255
Surface-hardened Through-hardened 0,42σB2 + 415
Carburised, nitrided or induction-hardened Soft bath nitrided
(Tufftrided) 		1000
Carburised, nitrided or induction-hardened Induction-hardened 0,88 Hv2 + 675
Carburised or nitrided Nitrided 1300
Carburised Carburised 1500

Table 3.3.4 Factors of safety

  S H min S F min
Main propulsion gears 1,25 1,50
Main propulsion gears for multiple screw 1,20 1,45
Auxiliary gears 1,15 1,40

3.5 Tooth loading for bending stress

3.5.1 The bending stress at the tooth root, σF is not to exceed the allowable tooth root bending stress σFP

σF = N/mm2
σFP = N/mm2
For values of S F min, see Table 3.3.4 Factors of safety.
For values of σF lim, see Table 3.3.5 Values of endurance limit for bending stress, σF lim .
Stress correction factor Y ST = 2.

Table 3.3.5 Values of endurance limit for bending stress, σF lim

Heat treatment σF lim N/mm2
Through-hardened carbon steel 0,09σB + 150
Through-hardened alloy steel 0,1σB + 185
Soft bath nitrided (Tufftrided) 330
Induction hardened 0,35 Hv + 125
Gas nitrided 390
Carburized A 450
Carburized B 410

Note 1. A is applicable for Cr Ni Mo carburising steels.

Note 2. B is applicable for other carburising steels.

3.5.2 Tooth form factor, Y F:

Y F =
where

h F, αF en and S Fn are shown in Figure 3.3.1 Normal tooth section.

= z n sin
where
ν = tan ν – H
G =
H =
E =

E, h ao, αn, S pr and ρao are shown in Figure 3.3.2 External tooth forms

=
d en =
where
d an = d n + d a – d
d n =
d bn = d n cos αn
∊αn =
γe =
where
αen = arc cos
=
where
αF en = αen – γe

Figure 3.3.1 Normal tooth section

Figure 3.3.2 External tooth forms

3.5.3 For internal tooth forms, the form factor is calculated, as an approximation, for a substitute gear rack with the form of the basic rack in the normal section, but having the same tooth depth as the internal gear.

= , and
=
where

αF en is taken as being equal to an

ρF2 =

3.5.4 Stress concentration factor, Y s

Y s =
where
L =
q s =
when

q s < 1, the value of Y s is to be specially considered.

The formula for Y s is applicable to external gears with αn = 20° but may be used as an approximation for other pressure angles and internal gears.

3.5.5 Helix angle factor, Y β

Y β = , if ∊β > 1 let ∊β = 1
but
Y b = ≥ 1 – 0,25∊β ≥ 0,75

3.5.6 Relative notch sensitivity factor, Yδ rel T

Yδ rel T = 1 + 0,036 (q s – 2,5) for through-hardened steels
Yδ rel T = 1 + 0,008 (q s – 2,5) for carburized and induction-hardened steels, and
Yδ rel T = 1 + 0,04 (q s – 2,5) for nitrided steels

3.5.7 Relative surface finish factor, YR rel T

YR rel T = 1,674 – 0,529 (6R a + 1)0,1 for through-hardened, carburized and induction hardened steels, and
YR rel T = 4,299 – 3,259 (6R a + 1)0,005 for nitrided steels

3.5.8 Size factor, Y x

Y x = 1,0, when m n ≤ 5
Y x = 1,03 – 0,006m n for through hardened steels
Y x = 0,85, when m n ≥ 30
Y x = 1,05 – 0,01 m n for surface-hardened steels
Y x = 0,80, when m n ≥ 25

3.5.9 Design factor, Y D

Y D = 0,83 for gears treated with a controlled shot peening process
Y D = 1,5 for idler gears
Y D = 1,25 for shrunk on gears, or
Y D = , otherwise
Y D = 1,0.

3.6 Factors of safety

3.6.1 Factors of safety are shown in Table 3.3.4 Factors of safety.

3.7 Design of enclosed gear shearing

3.7.1 The following symbols apply:

P in kW and R in rpm, see Pt 5, Ch 1, 3.3 Power ratings 3.3.1

L = span between shaft bearing centres, in mm
αn = normal pressure angle at the gear reference diameter, in degrees
β = helix angle at the gear reference diameter, in degrees
d w = pitch circle diameter of the gear teeth, in mm
σu = specified minimum tensile strength of the shaft material, in N/mm2

Note Numerical value used for σu is not to exceed 800 N/mm2 for gear and thrust shafts and 1100 N/mm2 for quill shafts.

3.7.2 This sub-Section is applicable to the main and ancillary transmission shafting, enclosed within the gearcase.

3.7.3 The diameter of the enclosed gear shafting adjacent to the pinion or wheel is to be not less than the greater of d b or d t, where:

d b =
d t =
where
s b = 45 + 0,24(σu – 400)
and
s s = 42 + 0,09(σu – 400)

3.7.4 For the purposes of the above, it is assumed that the pinion or wheel is mounted symmetrically spaced between bearings.

3.7.5 Outside a length equal to the required diameter at the pinion or wheel, the diameter may be reduced, if applicable, to that required for d t.

3.7.6 For bevel gear shafts, where a bearing is located adjacent to the gear section, the diameter of the shaft is to be not less than d t. Where a bearing is not located adjacent to the gear the diameter of the shaft will be specially considered.

3.7.7 The diameter of quill shaft (not axially constrained and subject only to external torsional loading) is to be not less than given by the following formula:

Diameter of quill shaft:

dq =

3.7.8 Where a shaft, located within the gearcase, is subject to the main propulsion thrust, the diameter at the collars of the shaft transmitting torque, or in way of the axial bearing where a roller bearing is used as a thrust bearing, is to be not less than 1,1d t. For thrust bearings located outside the gearcase, see Pt 5, Ch 4 Main Propulsion Shafting.

3.8 External shafting and components

3.8.1 For shafting external to the gearbox and other components ancillaries, see Pt 5, Ch 4 Main Propulsion Shafting.

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