BEVEL GEARS
BEVEL GEARS: CLASSIFICATION
Straight Bevel Gear Spiral Bevel Gear
BEVEL GEARS: CLASSIFICATION
Miter Gears
External bevel gear Crown gear Internal bevel gear
(pitch angle is less than 90°) (pitch angle equal to 90°) (pitch angle is greater than 90°)
BEVEL GEARS: NOMENCLATURE
Pitch Cone
Cone Centre (O)
Cone Distance (A0)
Pitch Angle (ɣ)
Addendum Angle (α)
Dedendum Angle (δ)
Face Angle (ɣ +α)
Root Angle (ɣ- δ)
Back Cone
Back Cone Distance/Radius (rb)
Addendum (ha)
Dedendum (hf)
BEVEL GEARS: FORMATIVE SPUR GEAR
z : Actual Number of teeth
D : Pitch circle diameter
𝑟𝑏 : pitch circle radius of formative
spur gear
z’ : Virtual number of teeth
𝑫 𝟐𝒓𝒃
𝒛= 𝒛′ =
𝒎 𝒎
Triangle ABC:
𝑫/𝟐
𝒔𝒊𝒏 (𝟗𝟎 − 𝞬) = ;
𝒓𝒃
𝑫 𝒛
𝒓𝒃 = ; 𝒛′ =
𝟐𝒄𝒐𝒔 𝞬 𝒄𝒐𝒔 𝞬
BEVEL GEARS: PITCH ANGLE AND CENTER DISTANCE
Triangle OBA:
𝑫𝒑/𝟐 𝒎𝒛𝒑 𝒛𝒑
𝒕𝒂𝒏 𝞬 = = = ;
𝑫𝒈/𝟐 𝒎𝒛𝒈 𝒛𝒈
𝒛𝒈
𝒕𝒂𝒏 𝜞 = ;
𝒛𝒑
π
𝜞+𝞬=
𝟐
Cone distance A0
𝟏 𝟎. 𝟓
A0 = 𝑫𝒑𝟐 + 𝑫𝒈𝟐
𝟐
BEVEL GEARS: FORCE ANALYSIS
P: Normal Force b/w meshed tooth (along BD)
Pt = Tangential component (N)
Pr = Radial component (N)
Pa = Axial or thrust component (N)
2𝜋𝑛𝑝 2𝑀𝑡
Pt = 𝑷𝒄𝒐𝒔 α Power =
60
𝑀𝑡 𝑃𝑡 =
𝑟𝑚
rm = mean radius [0.5(ra + rf )]
𝑫𝒑 𝒃
rm = - 𝒔𝒊𝒏 𝞬
𝟐 𝟐
Ps = 𝑷𝒔𝒊𝒏 α = Pt 𝒕𝒂𝒏 α
Pr = Ps 𝒄𝒐𝒔 𝞬 = Pt 𝒕𝒂𝒏 α 𝒄𝒐𝒔 𝞬
Pa = Ps 𝒔𝒊𝒏 𝞬 = Pt 𝒕𝒂𝒏 α 𝒔𝒊𝒏 𝞬
BEVEL GEARS: FORCE DIRECTIONS
Pt : Tangential component
Pt =(N)
𝑷𝒄𝒐𝒔 α
• The direction of tangential component for the driving gear is opposite to the direction
of rotation. (Dot coming out from the plane of screen)
• The direction of tangential component for the driven gear is same as the direction of
rotation. (Cross going inside the plane of screen)
Pr : Radial component (N)
The radial component on the pinion acts towards the centre of the pinion as well as gear.
Pa : Axial or thrust component (N)
• The thrust component on the pinion is equal and opposite of the radial component on
the gear.
• The thrust component on the gear is equal and opposite of the radial component on the
pinion. The tendency of thrust components is to separate
BEVEL GEARS: FORCE DIRECTIONS
A pair of bevel gears transmitting 7.5 kW at 300 rpm is shown below. The pressure
angle is 20°. Determine the components of the resultant gear tooth force and draw a
free-body diagram of forces acting on the pinion and the gear.
𝑫𝒑/𝟐 𝒎𝒛𝒑 𝒛𝒑
𝒕𝒂𝒏 𝞬 = = = ;
𝑫𝒈/𝟐 𝒎𝒛𝒈 𝒛𝒈
𝑫𝒑 𝒃
rm = - 𝒔𝒊𝒏 𝞬
𝟐 𝟐
[Answer Key: Pt =3789.40 N; Pr = 1103.38 N; Pa = 827.54 N]
BEVEL GEARS: FORCE DIRECTIONS
A pair of bevel gears transmitting 7.5 kW at 300 rpm is shown below. The pressure
angle is 20°. Determine the components of the resultant gear tooth force and draw a
free-body diagram of forces acting on the pinion and the gear.
[Answer Key: Pt =3789.40 N; Pr = 1103.38 N; Pa = 827.54 N]
Bevel Gears: Beam Strength
Beam Strength of Spur Gears: Sb = 𝒎𝒃σb𝒀
Applying the Lewis equation to a formative spur
gear at a distance x from the apex:
δ(Sb) = 𝑚𝑥𝑏𝑥𝜎b𝑌
where,
𝑟𝑥 𝑥 𝑥
= → 𝑟𝑥= R Sb = Beam strength of the tooth (N)
𝑅 𝐴0 𝐴0 m = Module at the large end of the tooth (mm)
2𝑟𝑥 2𝑥𝑅 𝑚𝑥 b = Face width (mm)
𝑚𝑥 = = = σb = Permissible bending stress (Sut/3) (N/mm2)
𝑧 𝑧𝐴0 𝐴0
Y = Lewis form factor based on formative number of teeth
𝜎b𝑌𝑚𝑥𝑑𝑥
δ(Sb) = Ao = cone distance (mm)
𝑏
𝐴0 1− 𝐴 =Bevel factor.
0
𝜎b𝑌𝑚𝑥𝑑𝑥 𝜎b𝑌𝑚𝑅𝑥2𝑑𝑥
𝑟𝑥 δ(Sb) = 𝑟𝑥 = M
𝐴0 𝐴02 Pt = 𝑹 t
𝑏 𝑏2 𝑏 𝑏2 𝒃
Mt= 𝑚𝑏𝜎b𝑌𝑅 [1- + 2] → Sb = 𝑚𝑏𝜎b𝑌 [1− + 2 ] → Sb = 𝒎𝒃𝝈b𝒀 [1− ] :at large end
𝐴0 3𝐴0 𝐴0 3𝐴0 𝑨𝟎
Bevel Gears: Effective Load on tooth
Rated Power, Rated speed and torque relation:
2πN
P= M
60 t
Tangential component of force:
Mt
Pt =
R
Service factor:
Cs=maximum torque/rated torqued
(Mt)max (Pt)max
Cs= =
Mt Pt
(Pt)max =Cs*Pt
Service factor
Effective Load on Gear Tooth
Dynamic Factor (Gears operating at high speeds):
inaccuracies of the tooth profile;
errors in tooth spacing;
misalignment between bearings;
elasticity of parts; and
inertia of rotating disks
Preliminary design
π𝑑𝑁
𝑣= ; Pitch line velocity
60
6
𝐶𝑣 = ; 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑓𝑎𝑐𝑡𝑜𝑟 𝑓𝑜𝑟 𝑐𝑢𝑡 𝑡𝑒𝑒𝑡ℎ
6+𝑣
5.6
𝐶𝑣 = 0.5 ; 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑓𝑎𝑐𝑡𝑜𝑟 𝑓𝑜𝑟 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑒𝑑 𝑡𝑒𝑒𝑡ℎ
5.6 + 𝑣
𝑪 𝒔 𝑷𝒕
𝑬𝒇𝒇𝒆𝒄𝒕𝒊𝒗𝒆 𝑳𝒐𝒂𝒅 𝒐𝒏 𝒕𝒐𝒐𝒕𝒉; 𝑷𝒆𝒇𝒇 =
𝑪𝒗
Effective Load on Gear Tooth
Buckingham’s equation for dynamic loads (high operating speeds)
𝑬𝒇𝒇𝒆𝒄𝒕𝒊𝒗𝒆 𝑳𝒐𝒂𝒅 𝒐𝒏 𝒕𝒐𝒐𝒕𝒉; 𝑷𝒆𝒇𝒇 = 𝑪𝒔 𝑷𝒕 +𝑷𝒅
Pd is dynamic load/additional load coming under
dynamic conditions
𝟐𝟏𝒗 (𝑪𝒆𝒃 + 𝑷𝒕 )
Pd = .
𝟐𝟏𝒗 + 𝑪𝒆𝒃 + 𝑷𝒕 𝟎 𝟓
Maximum expected error (e) between two
Pd = dynamic load or incremental dynamic load (N)
meshing teeth (mm)
v = pitch line velocity (m/s)
C = deformation factor (N/mm2)
e = sum of errors between two meshing teeth (mm)
Class-1 : Well cut commercial gear teeth
b = face width of tooth (mm)
Class-2 : Gear teeth cut with great care
Pt = tangential force due to rated torque (N)
Class-3 : Ground and lapped precision gear teeth
Lewis Bending Equation: Effective Load on Gear Tooth
k = constant depending upon the form of tooth
Ep = modulus of elasticity of pinion material (N/mm2 )
𝑘 Eg = modulus of elasticity of gear material (N/mm2 )
C=
1 1
+
Ep Eg
k = 0.107 (for 14.5° full depth teeth)
= 0.111 (for 20° full depth teeth)
= 0.115 (for 20° stub teeth)
Values of deformation factor C (N/mm2)
Estimating Bevel Gear Module Based on Beam Strength
Condition to avoid failure due to bending: Sb > Peff
Adding factor of safety: Sb = 𝑓𝑜𝑠 ∗ Peff ; [recommended fos =1.5-2]
2Mt 2Mt 2 60∗1006(Power in 𝑘𝑊)
Pt = = =
D mz mz 2π𝒏
𝑪𝒔 𝑷𝒕
Peff = = or 𝑪𝒔 𝑷𝒕 +𝑷𝒅
𝑪𝒗
𝒃
Sb = 𝒎𝒃𝝈b𝒀 [1− ]
𝑨𝟎
𝐴0
Assumption for face width: 𝑏 = or b=10m whichever is smaller
3
Beam strength (Sb)indicates the maximum value of the tangential force at the large end of the tooth that the
tooth can transmit without bending failure.
Wear Strength : Bevel Gears
Wear Strength of spur gears: 𝑺𝒘 = 𝒃𝑸𝒅𝒑 𝑲
Wear Strength of Bevel gears:
Sw = 𝑏𝑄𝑑𝑝 𝐾
𝐷𝑝
dp= 2rb =
𝑐𝑜𝑠 𝞬
𝒃𝑸𝑫𝒑𝑲 𝒃𝑸𝑫𝑲
Sw = =
𝒄𝒐𝒔 𝞬 𝒄𝒐𝒔 𝞬
𝟎.𝟕𝟓 𝒃𝑸𝑫𝑲
Sw = ; for overhanging bevel gears
𝒄𝒐𝒔 𝞬
𝑧𝑔′ 𝑧𝑝 𝑧𝑔 𝑧𝑔 2𝑧𝑔
𝑄=2 𝑧𝑝 ′ = ; 𝑧𝑔 ′ = = 𝑄=2
𝑧𝑔′ + 𝑧𝑝′ 𝑐𝑜𝑠 𝞬 𝑐𝑜𝑠 𝛤 𝑐𝑜𝑠(90−𝞬) 𝑧𝑔 + 𝑧𝑝 𝑡𝑎𝑛 𝞬
Wear Strength : Bevel gears
2
1 1
2𝑧𝑔 σc 𝒔𝒊𝒏 𝜶 𝒄𝒐𝒔 𝜶 (𝑬 + 𝑬 )
𝟎. 𝟕𝟓 𝒃𝑸𝑫𝑲 𝑄=2 𝒑 𝒈
Sw = 𝑧𝑔 + 𝑧𝑝 𝑡𝑎𝑛 𝞬 K=
𝒄𝒐𝒔 𝞬 𝟏. 𝟒
Special case: The expression for the load-stress factor K can be simplified when both the gears
are made of steel with a 20° pressure angle. 𝐸1 = 𝐸2 = 206 000 N/mm2 ; 𝛼 = 20°
According to G. Niemann,
σc= 0.27 (BHN) kgf/mm2 = 0.27 (9.81)(BHN) N/mm2
where BHN is the Brinell Hardness Number.
𝑩𝑯𝑵 𝟐
K=0.156
𝟏𝟎𝟎
The wear strength (Sw) indicates the maximum value of the tangential force at the large end of the tooth that the
tooth can transmit without pitting failure.
Estimating Bevel Gear Module Based on Wear Strength
Condition to avoid failure from pitting: Sw > Peff
Adding factor of safety: Sw = 𝑓𝑜𝑠 ∗ Peff ; [recommended fos =1.5-2]
2Mt 2Mt 2 60∗1006(Power in 𝑘𝑊)
Pt = = =
d mz mz 2π𝒏
𝑪𝒔 𝑷𝒕
Peff = or 𝑪𝒔 𝑷𝒕 +𝑷𝒅
𝑪𝒗
𝟎. 𝟕𝟓 𝒃𝑸𝑫𝑲
Sw =
𝒄𝒐𝒔 𝞬
Bevel Gear
A pair of straight bevel gears is mounted on shafts, which are intersecting at right angles. The
number of teeth on the pinion and gear are 30 and 45 respectively. The pressure angle is 20°. The
pinion shaft is connected to be an electric motor developing 16.5 kW rated power at 500 rpm.
The service factor can be taken as 1.5. The pinion and the gear are made of steel (Sut = 570
N/mm2) and heat treated to a surface hardness of 350 BHN. The gears are manufactured in such a
way that the error between two meshing teeth is limited to 20 mm. The module and face width
are 6 mm and 50 mm respectively. Determine the factor of safety against bending as well as
pitting.
[ Answer Key: 1.25 and 1.85 ]
Bevel Gear
A pair of straight bevel gears consists of a 24-teeth pinion meshing with a 48 teeth gear. The
module at the outside diameter is 6 mm, while the face width is 50 mm. The gears are made of
grey cast iron FG 220 (Sut = 220 N/mm2). The pressure angle is 20°. The teeth are generated and
assume that velocity factor accounts for the dynamic load. The pinion rotates at 300 rpm and the
service factor is 1.5. Calculate
(i) the beam strength of the tooth;
(ii) the static load that the gears can transmit with a factor of safety of 2 for bending
consideration
(iii) the rated power that the gears can transmit.
[ Answer Key: (i) 5267.74 N (ii) 1384.19 N (iii) 3.13]
