Basics of Involute Gears
2.1
Force and Power
Gears transmit force from tooth of the driving gear on to the
meshing driven tooth as shown in Fig- 2.1. The tooth of the
driving gear 1 pushes the meshing tooth on gear 2 along the
line of action. The force is normal to the teeth profiles and has
been designated as Fn. At the pitch point, normal force Fn may
be resolved into two components, the tangential component
Ft and the radial component Fr.
(2.1)
where α is the pressure angle
The power P transmitted is
(2.2)
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where
v = pitch line velocity, m/s
d1 = pitch diameter of pinion, mm
n1 = speed of pinion, rpm
Fig. 2.1 Force diagram of
spur gears
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2.2
Gear Tooth Failures
Gear teeth fail due to fatigue. Failure is exhibited either as a
facture of one or more teeth at the root or failure due to
excessive wear of contact surfaces due to pitting, scoring and
wear.
Fatigue Fracture due to Bending
Fatigue fracture due to bending starts with a small crack at
the fillet in the tensile stress zone at the root of the teeth. Fig
2.2a.
The crack slowly progresses inward and downward, Fig. 2.2b,
and finally (c) when the tooth becomes too weak because of
the reduced cross-section, It breaks suddenly as at Fig. 2.2c.
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Fig. 2.2 Fatigue fracture of a gear tooth due to bending
Tangential component of the normal force is used to calculate
the tensile stress at the root. Compressive and shear stresses
are neglected. Wilfred Lewis (1892) was the first to analyse
the bending stresses and his analysis is used in calculating the
beam strength of a gear tooth.
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Fatigue Failure of Tooth Surface
Surface failures of gear teeth are generally due to excessive
loading and lubrication breakdown, resulting in various
combinations of abrasion, scoring and pitting.
Abrasive wear is caused by the ingress of foreign particles due
to improper sealing of gear box or due to inadequate filtration
of oil supply or both. Change of contaminated lubricant at
requisite intervals help in increasing the life.
Scratching and scoring is generally caused by a hard particle in
the lubricant under high speed and inadequate lubrication.
High speed and inadequate lubrication results in high local
temperature at which an ordinary iron particle converts into
ferrous carbide and causes surface failure.
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Pitting is a surface fatigue due to high complex stress contact
zone. Pitting occurs in the vicinity of pitch line where high
stresses and zero sliding velocity are likely to breakdown the
oil film. Metal contact leads to local welding and breaking of
high points leading to pitting.
It may be concluded that a suitable design value of contact
stress and proper lubrication system are needed for
preventing surface failure. Surface finish of the order of 0.5
micron is desirable to reduce pitting and to improve surface
strength
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2.3
Lewis Tooth Bending Stress
Lewis considered tooth as a cantilever subjected to a static
normal force F applied at the tip. Force F is resolved in two
components, tangential Ft and radial Fr. The bending stress at
the point where tooth force meets the middle section is zero;
and then it increase parabolically as shown in Fig 2.3. The
constant strength parabola is inscribed in the tooth profile.
Since the parabola is inside the tooth profile except for the
section at ‘a’ where parabola and tooth profile are tangential
to each other. Hence section as ‘a’ critical and the bending
stress is:
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Fig. 2.3 Gear tooth as a cantilever
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(a)
From two similar triangles formed at point ‘a’
(b)
Substituting (b) into (a)
(c)
is called the Lewis form factor
Since
(2.3)
, we may rewrite the above equation as
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Since,
, we may rewrite the above equation as
(2.4)
(2.5)
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1.
2.
3.
4.
5.
6.
7.
8.
It may be noticed that we considered stresses due to tangential
component of the force on the tooth and made following
assumptions:
The full load is applied to the tip of a single tooth. In practice full
load does not act at the tip because there would be another pair
of teeth in mesh
The load is applied statically, the load is dynamic.
The compressive stresses due to radial component Fr are
neglected; it is on safe side.
Shear stresses due to Ft are neglected; shear stresses are rater
low.
Only the tensile stresses due to tangential load Ft are considered;
negative stresses are not considered, and it is on the safe side.
It is assumed that the load is distributed uniformly across the full
face width
Forces due to tooth sliding friction are neglected. The forces are
generally small.
Stress concentration at the tooth fillet is neglected
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In-spite of all these assumptions, Lewis equation gives a good
estimate of static bending stress.
The numerical values of form factor y or modified form factor
Y depend on the number of teeth on a gear for a particular
standard tooth-system and these values are available in the
published literature in the form of empirical formulae such as
Lewis form factor for 14.50 involute system
(2.6)
Lewis form factor for 200 full depth involute
(2.7)
Lewis form factor for 200 stub tooth involute
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(2.8)
However, the stress at the root depends on the geometry of
both the driving and driven gears and the same are indicated
by geometry factors J. AGMA Standard 908-B89 provides
tables of J factors as well as the algorithm to calculate them.
Several text books and data book also give these values.
In the derivation of Lewis equations; only one tooth carries
the whole load at the tip, which is not true in practice. As the
contact ratio is generally of the order of 1.5, the greatest load
is not exerted at the tip but it is much below the tip where a
single pair is in contact. However, we shall use full Ft at the tip
to be on the safe side.
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2.5 Tooth-Bending Dynamic Stress or Design Stress
Gear teeth are subjected to several loads due to the dynamic
characteristics of driving power source and driven source,
surface finish of teeth, inertia etc. Static stress is multiplied by
several factors the numerical values of which is base on
experience. The resulting dynamic stress at the root of a tooth
is written as
(2.9)
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where,
σd = dynamic stress at the root of the tooth (or design stress)
J = spur geometry factor
Ko = overload factor ( or load application factor)
Kv = velocity factor ( a dynamic factor)
Km = load distribution factor
If the data for J is available, the same should be used in
calculating σd, the design stress at the root. Else we shall use Y
in place of J, i.e,
(2.10)
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Overload Factor or Load Application Factor
Overloads are due to external sources such as characteristics
of driving and driven machines, ratio of inertias and
characteristics of couplings. As there is no analytical method
to evaluate overload factors, empirical values based on
experience given in Table 2.1 are used.
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Table 2.1 Overload Factor, Ko (or Load Application Factor)
Source of Power
Driven Machinery
Uniform1
Moderate Shock2 Heavy Shock3
Uniform (Electric Motor,
Turbine)
1.00
1.25
1.75
Light Shock (MultiCylinder I.C. Engine)
1.25
1.50
2.00
Heavy Shock (singlecylinder I.C. Engine)
1.50
1.75
2.25
1.
2.
3.
Generator, belt conveyor, platform conveyor, light elevator, electric hoist, feed gears of
machine tools, ventilators, turbo blower, mixer for constant density material.
Main drive to machine tool, heavy elevator, turning gears of crane, mine ventilator,
mixer for variable density material, multi-cylinder piston pump, feed pump.
Press, shear, rubber dough mill, rolling mill drive, power shovel, heavy centrifuge, heavy
feed pump, rotary drilling apparatus, briquette press, pug mill.
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Dynamic Load Factor or velocity factor, Kv
The velocity factor accounts for:
The severity of impact as successive pairs of teeth come into
engagement.
Factors such as pitch line velocity, manufacturing and
assembly accuracies.
Polar mass moments of inertia of pinion and gear mesh, shaft
and bearing stiff-nesses.
The following empirical equations based on experience may
be used to estimate the numerical value of Kv:
(2.11)
(2.12)
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(2.13)
(2.14)
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Load Distribution Factor
Beacause of unavoidable Inaccuracies in mounting of gear
shats and in the manufacture of gears, the contact across the
full face of teeth may not take place. Separate longitudinal
and transverse load distribution factors are recommended in
standards. However, a combined load distribution factor will
be used as given in the Table 2.2
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Table 2.2 Load Distribution Factor, Km
Characteristics of
support
Accurate mountings,
small bearing clearance,
small deflections,
precision gears
Less rigid mountings,
less accurate gears,
contact across the full
face
Accuracy and mounting
such that less than full
face contact exists
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0-50
Face width (mm)
50-150 150-225 225-up
1.3
1.4
1.5
1.8
1.6
1.7
1.8
2.2
More the 2.2
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2.6 Endurance Limit of Gear Material in Bending, S’e,bending
The endurance limit is defined as the stress which can be
endured at the root for 3x106 cycles for constant load
direction. A higher value of stress may be used if the desired
life is less than three million cycles, e.g. for a life of 10,000
cycles or less, bending stress of 2.5 S’e,bending may be used. In
case of Idler gear, 0.7 times of S’e,bending is used as endurance
limit.
S’e,bending can be found by plusating tests for any material for
any desired state for use in design. However, values of
endurance limit of several materials is given in IS4460. The
data given is for 99% reliability, 2.0 stress concentration and
for constant load
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2.7 Endurance strength of Gear in Bending, Se,bending
Endurance strength in bending is influenced by size,
roughness, notch sensitivity, desired life and desired reliability
of the gear.
Thus the endurance strength Se,bending of a pinion or gear in
bending can be calculated as follows:
Se,bending = S’e,bending (surface finish factor) (notch sensitivity
factor) (size factor) (life factor) (reliability factor)
Some of these factors will be considered, but in majority of
cases to keep the calculations simple, we shall assume that
Se,bending = 0.33 Sut for CI and Steel
(2.15)
= 0.40 Sut for bronze
(for idlers, Se,bending should be 70% of the above)
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For a factor of safety of (fs), the permissible stress [σ] is given
as
(2.16)
Hence the design equation from bending consideration is
(2.17)
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2.8
Factor of Safety is Bending
(2.16)
Hence
(2.17)
Thus the maximum tangential force Ft transmitted by a pair of
gears is minimum of the following two:
(2.18 a)
(2.18 b)
If the gear and pinion are manufactured from the same
material and to the same accuracies, gear teeth are stronger;
hence design the pinion only.
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2.9 Buckingham’s Dynamic Equation For Bending
Buckingham considered the dynamic load composed of two
parts Ft and Fi as follows:
FDB = Ft + Fi where
(2.19)
Ft = Tangential Force, N
Fi = Incremental Load, N
FDB = Design load for bending as per Buckingham’s method, N
According to him, incremental load is caused by
Small machining errors
Deflection of teeth due to load
Machining errors and deflections cause impacts and inertia
forces leading to incremental dynamic loads.
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Incremental Dynamic Load is given by the empirical equation
N
(2.20)
V = pitch line velocity, m/s
C = deformation factor, N/mm2
e = Sum of errors between meshing teeth, mm
b = face width of tooth, mm
Deformation factor, C is a function of modulii of elasticity of materials
of pinion and gears, and pressure angle. Values of C are given in Table
2.4
Sum of errors between meshing teeth, e depend on many factors, the
important one being tolerance on module, pitch circle diameter and
gear quality or grade. Values of error e are given in Table 2.5
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Table 2.4 Values of deformation factor C (N/mm2)
Materials
Pinion
Gear
14.5o
full depth teeth
20o
full depth teeth
20o
stub teeth
Grey CI
Grey CI
5600
5800
6000
Steel
Grey CI
7800
8000
8200
Steel
Steel
11000
11400
12000
Table 2.5 Expected Sum of Errors Between Two Meshing Teeth
(mm)
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2.10
Factor of Safety in Bending as per Buckingham’s Method
As before, the permissible stress [σ] is given as
(2.16)
(2.21)
where
(2.22)
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