Page 208 - Analysis and Design of Machine Elements

Analysis and Design of Machine Elements
186
Letzonefactoras Z = √ 2 , which considers the effect of profile curvature at
H sin cos
∘
pitch point on the contact stress. For a standard spur gear, the pressure angle is 20 ,so
Z = 2.5.
H
√
The elastic coefficient Z = [ 1 ] considers the mechanical properties of
E 1− 2 1− 2
1 + 2
E 1 E 2
pinion and gear. The elastic modulus values of steels, cast steel, nodular cast iron and
grey cast iron are 206, 202, 173 and 118 GPa [9], respectively. Poisson’s ratio is usually
assumed 0.3. The elastic coefficient of matching materials can thus be calculated. For
1/2
the common match of steel pinion and steel gear, the elastic coefficient is 189.8 MPa .
With these simplifications, the contact stress can be expressed as
√ √
2KT 1 u ± 1 KF t u ± 1
= Z Z ⋅ = Z Z ⋅ (8.24)
H H E 3 H E
d u bd 1 u
d 1
8.3.2.3 Contact Strength Analysis
Excessive contact stress is the real reason for surface fatigue pitting. To prevent pitting,
contactstressesontooth surfaceshouldnot exceed theallowablevalues, that is
√
KF t u ± 1
= Z Z E ⋅ ≤ [ ] (8.25)
H
H
H
bd 1 u
The design formula can be derived from Eq. (8.25) for the determination of pitch diam-

eter of the pinion of a spur gear for surface durability, expressed as
√
( ) 2
3 2KT 1 u ± 1 Z Z E
H
d ≥ ⋅ ⋅ (8.26)
1
d u [ ]
H
In Eq. (8.26), the unit for torque is N⋅mm, the unit for face width and diameter is mm
and the unit for allowable contact stress [ ] is MPa. The determination of allowable
H
contact stresses will be introduced in Section 8.6.4.
Contact stresses, , are the same for both meshing teeth of pinion and gear, that
H
is, = . However, the ratio of [ ]/ is different for the meshing gears. This is
H1 H2 H H
because the pinion and gear are usually of different materials, heat treatments and hard-
ness, resulting in different allowable stresses [ ]. When using Eq. (8.26) in design, the
H
smaller value of allowable contact stress should be used.
It is observed from Eq. (8.24) that contact stress is affected by pinion diameter d .
H 1
The contact stress has no relation with module m. It can be interpreted that, with
H
the increase of pinion diameter d , the diameter of base circle also increases, leading to
1
a decreased curvature of involuted tooth profile, thus reducing the contact stress.
8.3.3 Tooth Bending Strength Analysis
Tooth breakage is a bending fatigue phenomenon caused by the bending action of a load.
Tooth breakage usually happens near the root fillet in a spur gear where high cyclic
bending stresses occur. To prevent tooth breakage, tooth bending strength analysis is
required. The bending stress on the meshing teeth near the root fillet is calculated by
the Lewis formula, modified by a list of adjustment factors to account for the influence