INFLUENCE OF A NON-STANDARD GEOMETRY ... - Dunarea de Jos

14
THE ANNALS OF UNIVERSITY “DUNĂREA DE JOS “ OF GALAŢI
FASCICLE VIII, 2004, ISSN 1221-4590
TRIBOLOGY
4.1. Influence of Hertzian deflection on
sliding velocity
The Hertzian compression, as an effect of
tooth’s load, modifies the flank geometry, changing
the standard involute into a straight line [10].
Following Weber’s analysis [13], the real contact
point extends down to the pressure line by an amount
2δ H :
M t
2δ H =
(5)
2 ⋅ Rb
⋅ k H
where M t is the transmitted torque and k H – the
stiffness of Hertzian contact. According to Hertzian
law and assuming that the gear tooth surfaces are two
cylinder of diameters d 1 ’ and d 2 ’ respectively, the
stiffness is given by:
π⋅ E ⋅b
k H =
(6)
' '
2 2 2d1
2d 2
4(1 − ν ) ⋅(
+ ln + ln )
3 s s
where E and ν are the Young’s modulus and
Poisson’s ratio, s – the width of rectangular contact
area. From the results derived by Yang [14] the
stiffness k H is practically a constant along the entire
line of action and it can be calculated by the relation:
π⋅ E ⋅b
k H =
2
(6’)
4(1 − ν )
So, the deflection due to Hertzian contact is
constant and independent to the position of contact
point. It increases the sliding velocity by an amount:
H
∆ vs = ( ω1
+ ω2
) ⋅ 2δ
(7)
H
4.2. Influence of bending deflection on
sliding velocity
Bending deflection is calculated assuming that
the tooth is a beam based on a rigid foundation. For
the standard involute flank geometry, the contact
force, exerted during gear meshing, is along the line
of action. In the case of plastic gears, the variation of
tooth profile determines contacts in points that do not
belong to the theoretical line of action and, therefore,
the actions of the contact force are in unknown
directions. But, since the contact points are in the
vicinity of pressure line and, furthermore, they have
the same sliding velocity as the point being on the
same flank and on the line of action, the study on
bending deflection can be developed, with small
errors, considering standard conditions.
The displacement of the point where the load is
applied has a bending and a shear component. Using
Walton’s assumptions and calculation [9], the
deflection due to both bending and shear is expressed
as follow:
- bending deflection:
32M t 3
b =
⋅ h
3 3 j
π ⋅b
⋅ E ⋅ m ⋅ Rb
δ (8)
where h j is the current tooth height, measured from
the deddendum circle to the contact point. The
increase in sliding velocity, due to bending
deflection, varies along the gear tooth flank with:
b
∆ vs = ( ω1
+ ω2
) ⋅δb
(9)
- shear deflection:
2M t
δ sh =
π⋅b
⋅G
⋅ m ⋅ R
⋅ h j
(10)
where G is the shear modulus. The increase in sliding
velocity, due to shear deflection, varies along the gear
tooth flank with:
sh
∆ v = ( ω + ω ⋅δ
(11)
s 1 2 )
5. THE REAL SLIDING VELOCITY
Taking into account the total deflection of the
plastic gear tooth, the real sliding velocity should be
calculated as:
r
H b sh
vs
= vs
+ ∆vs
+ ∆vs
+ ∆v
(12)
s
To analyse both the influence of the nonstandard
geometry and the influence of the low
stiffness of the plastic gear tooth on the sliding
velocity, the curved face width gear with the
geometrical parameters mentioned in paragraph 3, is
treated as a series of n standard spur gears. It is
assumed that the transmitted torque is equally
distributed on the elementary spur gears, with
specific geometry. The gear is made by ERTALON
66SA with E = 3450MPa, G = 1320 MPa and ν = 0.3.
Using equations 5 -11 it was found that:
- the Hertzian deflection is not significant and it can
be ignored. Along the gear face width, the ideal
sliding velocity is increased by 0.12 ÷ 0.1%; the
maximum influence is recorded at the gear’s centre,
where the gear tooth flank has the standard involute,
and is decreasing towards the gear face width ends
due to the increase in base circle radius;
- the shear deflection is also not important on the
sliding velocity. The maximum increase of 0.86% is
recorded at the gear centre and is decreasing along
the gear face width to a minimum value of 0.73% at
the gear end sections;
- the bending deflection is one of the components of
gear tooth deflection that also decreases along the
gear face width, a consequence of the higher gear
tooth rigidity. The sliding velocity varies by 2.5÷
1.8% from its ideal value due to bending deflection.
Figure 7 shows the variation of the maximum
sliding velocity (at speed of 1000 revs/min) at points
the gears come out of mesh, where the bending
deflection is maximum. It can be seen that the real
sliding velocity, compared to its ideal value, varies
from 3.5% at the gear half face width to 2.67% at the
gear ends.
b
sh