Jan Zwolak[1]
Marek Martyna[2]
Dominik Kozik[3]
DISTRIBUTION
OF CONTACT STRESSES AND INTERTOOTH SLIP IN BILATERAL AND UNILATERAL ENGAGEMENT
ROZKŁAD NAPRĘŻEŃ KONTAKTOWYCH I POŚLIZGU
MIĘDZYZĘBNEGO W ZAZĘBIENIU DWUSTRONNYM I JEDNOSTRONNYM
Key words: toothed
transmission, correction factor, unilateral and bilateral engagement.
ABSTRACT
The paper
presents the issues concerning contact stresses and intertooth slip of power
shift gear used in the powertrain of a wheel loader. The kinematic diagrams on
individual gear ratios of the transmission are presented. Using appropriate
numerical values of contour shift coefficients (correction coefficients),
normal bilateral gearing, prepitch unilateral gearing and postpitch
unilateral gearing were considered. In each of the three types of gearing,
contact stresses and intertooth slip were calculated for each gear pair and at
each gear ratio level, using an author's computer program with multicriteria
optimization.
Słowa kluczowe: przekładnia zębata,
współczynnik przesunięcia zarysu, zazębienie dwustronne i jednostronne.
STRESZCZENIE
W pracy przedstawiono zagadnienia
dotyczące naprężeń kontaktowych i poślizgu międzyzębnego
przekładni zębatej power shift
stosowanej w układzie napędowym ładowarki kołowej. Na przedmiotowej przekładni
zaprezentowano schematy kinematyczne na poszczególnych stopniach przełożenia.
Stosując odpowiednie wartości liczbowe współczynników przesunięcia zarysu
(współczynników korekcji) rozważano zazębienie dwustronne normalne, zazębienie
jednostronne przedbiegunowe oraz zazębienie
jednostronne pozabiegunowe. W każdym z trzech
rodzajów zazębienia dokonano obliczeń naprężeń kontaktowych oraz poślizgów międzyzębnych dla każdej pary zębatej i na każdym stopniu
przełożenia, stosując autorski program komputerowy z optymalizacją
wielokryterialną.
INTRODUCTION
In the design
and fabrication of gears assembled into the appropriate gear pairs to create
more or less complex gear trains, the addendum modification coefficients
(correction factors) are used in a fairly wide range. The numerical values of
these coefficients have an impact on the characteristics of engagement, which
may be defined as: normal bilateral engagement, unilateral prepitch point
engagement, unilateral postpitch point engagement.
The normal bilateral engagement is
generated by two gears, whose respective correction factors satisfy the
equation: x_{1} ˂ 1 for the driving gear and x_{2 }˃ –1
for the driven gear. The correction factors, thus defined, cause the total
length of the line of action to extend on either side of the pitch point, that
is, on the side of the driving gear and the driven gear. In this case, the intertooth
slip vectors’ senses at the working depth of the gear are pointing from the
pitch point towards the top land and the dedendum. Meanwhile, in the driven
gear, the slip vectors’ senses are pointing from the top land and dedendum
towards the pitch point. Slippage directions along the outline of the working
surface of the driving gear and the driven gear that are aligned with the
movement direction of their contacting surface are termed positive, whereas the
slippage directions opposite to the movement direction of the contacting tooth
surfaces are termed negative. In order to facilitate the determination of the
slip vector sense, it was established to state that on the faces of the
cooperating teeth occur the positive slippages, whereas on the flanks occur the
negative.
On the lateral surfaces of the teeth of
the cooperating gears, there is also a friction force, whose vector changes its
sense as well in relation to the pitch point. The constant change of the senses
of friction forces under high cyclic loading causes an accelerated wear of the
tooth top layer within the working depth. The results of our original studies,
as well as of other authors, point out a pittinginduced cumulative damage buildup
in the top layer on the dedendum, particularly in the area of unilateral
engagement.
In the unilateral prepitch point engagement, as well as in the postpitch
point one, the line of action extends on one side of the pitch point and the
sense of the intertooth slip vector does not change there. In the unilateral postpitch point engagement
the slip vectors are oriented from the top land towards the dedendum both in
the case of driving and driven gear. On the other hand, in unilateral
postpitch point engagement intertooth slippages in the driving and driven
gear are directed from the dedendum towards the top land.
POWER SHIFT GEAR
TRAIN AS A TEST OBJECT
A large number
of numerical tests were conducted on the actual power shift gear train [1, 2, 3,
5, 8] with a transmission ratio of 6 degrees, used in the wheel loader
propulsion system. The kinematic diagram of the gear train in the axial system
is shown in Figure 1.
Fig.
1. Kinematic diagram of the gearbox axial alignment
Rys. 1. Schemat kinematyczny
przekładni w układzie osiowym
The typical feature of the power shift gear train is that all
gears are continuously engage with each other and the transmission is enabled
under full load by means of the clutches integrated with the gears and
corresponding shafts. The gear z_{1} is integrated with the clutch S_{p} and the input shaft I, while the gear z_{2}
with clutch S_{W} and also with the shaft I. Using the clutch S_{p} and S_{W} enables a vehicle to move
forwards and backwards, respectively, and these two clutches are termed directional
clutches. The other clutches, such as: S_{1} integrated with the gear z_{6},
S_{2} with the gear z_{8} and S_{3} with the gear z_{10},
are counted among the group of gear clutches. Gears z_{3}, z_{4},
z_{5}, z_{7}, z_{9}, z_{11}, z_{12} are
connected with the respective shafts by means of a spline in such a way that no
axial shifting is possible. Configuration
of gears, shafts and clutches shown in Figure 1 allows for realization of six
transmission ratios
(6 forwards and 3 backwards). A complete gear train is composed of 12 gears that
form 7 gear pairs while engagement. The respective gear pairs connected
with each other through shafts and clutches, starting at the driving gears z_{1}
and z_{2, }remain in the kinematic chain on the transmission ratios
from 1 to 6, causing vehicle to move forwards and backwards. The gear pairs in the kinematic chain of the ratios
from 1 to 6 are shown in Figures 2 to 3.
a) 
b) 
c) 
Fig.
2. Gears in realization of the transmission: a) 1 ratio, b) 2 ratio, c) 3 ratio
Rys.2. Koła zębate w realizacji przełożenia: a) 1
stopnia, b) 2 stopnia, c) 3 stopnia
a) 
b) 
c) 
Fig.
3. Gears in realization of the transmission: a) 4 ratio, b) 5 ratio, c) 6 ratio
Rys. 3. Koła zębate w realizacji
przełożenia: a) 4 stopnia, b) 5 stopnia, c) 6 stopnia
In the gear pairs shown in Figure 2 and 3, which form
kinematic chains on the sequential transmission ratios, it is observed that
most gears (namely 8) take part in realization of the 4th transmission ratio.
It is also observed that a gear in a given operation period is subjected to the
biggest number of cycling loadings, because it is engage
with the gears z_{1}, z_{3} and z_{7}. Accordingly,
there is a probable danger of damaging the working surface of this gear through
pitting in the first instance. Among the clutches with the longest lifespan is
the clutch S_{3}, executing the transmission on stages 1 and 2, as well
as on 4 and 5. Meanwhile, the
smallest share (only at the 3rd and 6th transmission ratio) in the load
transfer refers to the clutch S_{2 }, which
is integrated with the gear z_{8} and
forms a gear pair with the gear z_{11} connected by means of a spline
with the output shaft V.
NORMAL BILATERAL ENGAGAMENT
In general, any
gear pair consisting of gears z_{1} and
z_{2 }is in the normal bilateral engagement
when [4, 6] the addendum modification coefficients meet the equations: x_{1} ˂ 1 and x_{2 }˃ –1.
The analysis will be carried out on the gear pair z_{1}:z_{5 }of the gear train from Figure 1, where
the gear z_{1} is a driving gear and z_{2} a driven one. Limitations of the addendum
modification coefficients, given this way, cause the total length of the action
line E_{1}E_{5} to be located on both sides of the pitch point
C. A model of engagement, as well as the intertooth forces acting between the
teeth are shown in Figure 4.
engagement line
Fig.4. Normal bilateral engagement: a) model of
engagement of gear z_{1} with gear z_{5}, b) intertooth forces
in pitch point C
Rys.4. Zazębienie
dwustronne normalne: a)model zazębienia kola z_{1} z kołem z_{5},
b)siły międzyzębne w biegunie zazębienia C
Let the purpose
of the considered gear pair be to transfer the torque M_{1} as soon as the point of engagement of the
cooperating involute teeth outlines has been located near the pitch point, and
at the same time it is a part of the addendum of the driving gear z_{1}. In
this point of engagement the normal force F_{N} appears and so does the
friction force T_{+} acting in the direction tangential to the teeth
outline. The normal force F_{N} and the friction force T_{+}
can be replaced with the resultant force W_{T+}, whose purpose is to transfer
the torque M_{1}.
By using the resultant force W_{T+} and its arm with a length a_{T+}, the
following equation for the torque M_{1 }can be proposed:
M_{1}
= W_{T+ }× a_{T}_{+}
(1)
A similar
analysis can be carried out for the point of engagement of the cooperating
teeth located near the pitch point and lying on the dedendum of the driving
tooth. In this case, the normal force F_{N} and the friction force T_{− }which was replaced by the resultant
force W_{T},
will have the action arms with a length a_{T}. Therefore, an expression for
the torque M_{1} may be written in the following form:
M_{1}
= W_{T} × a_{T}_{} (2)
The expressions
(1) and (2) may be written after equating as follows:
M_{1}
= W_{T+ }× a_{T}_{+} = W_{T}
× a_{T}_{} (3)
Based upon the
expression (3), one can determine a relationship between the resultant force W_{T+}_{ }linked
to the addendum of the tooth, and the resultant force W_{T} linked to the dedendum of the tooth:
W_{T+}
= W_{T˗} × (a_{T}_{˗}
/ a_{T}_{+}) (4)
Based on the
equation (4) and Figure 4, it can be concluded that the quotient a_{T}_{˗} / a_{T}_{+}_{ }will
always be less than one, thus the value of the force W_{T˗} acting at the dedendum will always be more than the value
of the force W_{T+ }acting at addendum with the same torque M_{1}.
The greater force value also generates higher contact stresses, and thus they
accelerate the fatigue wear of the surface layer due to pitting.
Many experimental studies of the authors
in the scope of contact fatigue strength of gears [7, 8, 9, 10] confirm that
the first traces of pitting wear emerge on the dedendum of a gear. In the
intertooth spaces of a gear pair, no matter whether the model experimental
research or realobject research is dealt with, there is oil as a lubricant
[11, 12]. Apart from its lubricating role, the oil has also a detrimental
effect consisting in penetration into microcracking gaps, which in turn results
in eroding the surface layer. In Figure 5 there is shown a mechanism of this
destructive phenomenon by using the example of the gear pair z_{1}:z_{5} taken from the considered gear train
(Figure 1).
Fig.
5. Destruction of the top layer of gear pair in normal bilateral engagement
Rys. 5. Destrukcja warstwy
wierzchniej pary zębatej zazębienia dwustronnego normalnego
Gear
pair in Figure 5 is in the normal bilateral engagement, where the gear z_{1}
is a driving gear with an angular velocity ω_{1},
and the gear z_{5} is a driven gear, and this makes it possible to
determine the senses of the friction forces T_{1} and T_{5}. Senses
of the friction forces on the addendum and dedendum are different, so the
cracking directions will also be different, because the cracks propagate into
the depth of the top layer in the opposite direction to the friction forces.
By using Figure 5 it is possible to do
an analysis of destructive action of the oil penetrating into the microcracking
gaps. Oil penetrating into gap A during a contact of the cooperating gears z_{1}
and z_{5} is encased by a tooth of the gear z_{5}. Subsequently,
the tooth of the gear z_{5 }presses on the portion of material above
the gap A and makes it bend, thereby elevating the oil pressure in the crack. Periodic
occurrence of such bending during the gear train operation leads to fatigue chipping
in this portion of material. On a dedendum there are usually more than one gaps
described, but an erosion mechanism is the same as in the case of gap A.
On the way of displacement of the
cooperating teeth’s contact point, the gap B may come up on the addendum of the
driving gear. An erosive action of the oil in this gap is of a different
character than in the gap A. Essential is a position where the lower edge of
the gap comes into contact with the tooth of the gear z_{5} and is bent
before the gap B has been closed. This bent will reduce the gap volume and
squeeze the oil out of it.
While considering the gap C on the
addendum of a tooth of the driven gear z_{5}, it is observed that the
oil is squeezed out of it in the same way as from the gap B of the driving gear
z_{1}. Further displacement of the point of engagement of the
cooperating teeth of the gears z_{1} and z_{5} reaches the gap
D located on the dedendum of the driven gear tooth z_{5}. The oil encased
in the gap D will stimulate erosion in the top layer of the gear dedendum z_{5},
like in the gap A on the dedendum of the gear z_{1}.
UNILATERAL
PREPITCH POINT ENGAGEMENT
Unilateral
prepitch point engagement is characterized in that the total line of action is
located on one side of the pitch point C [4, 6]. For the gear pair z_{1}:z_{5}
the origin of the line of action always lies at the point E_{5},
whereas its end may be at the point C or before the point C, depending on the
correction factor x_{1}. Position of the line of action for the gear
pair z_{1}:z_{5 }with the correction factors x_{1} = −1
and x_{5} = +1 was shown in Figure 6.
sliding direction cracking direction
Fig.
6. Position of line of action in unilateral prepitch point engagement
Rys. 6. Położenie odcinka przyporu w zazębieniu jednostronnym przedbiegunowym
If
the correction factor x_{1} < −1, then the line of action ends
before the point C. The correction factor x_{1} = −1 of the gear
z_{1} causes the total depth as if to comprise only a dedendum. Meanwhile,
the total depth of the tooth of the gear z_{5} with the correction
factor x_{5} = +1 is only an addendum. Accordingly, the kinematics of the
engagement shown in Figure 6 is characterized by the fact that the slip
vector’s sense (in contrast to the standard bilateral engagement) is still the
same. Slippage direction and friction forces determine a cracking direction in
the top layer on the effective surface of the gears z_{1} and z_{5}
shown in Figure 7.
Fig. 7. Destruction
of the top layer of the gear pair z_{1}:z_{5} in the unilateral
prepitch point engagement
Rys. 7. Destrukcja warstwy wierzchniej pary zębatej z_{1}/z_{5}
zazębienia jednostronnego przedbiegunowego
The oil penetrating into the gaps of the top layer of
the driving gear z_{1} acts expansively and an erosive process takes
place as was depicted in Figure 5. On the other hand, a mechanism that proceeds
in the gaps of a driven gear z_{5} is the same as in the gap C of the
normal bilateral engagement.
UNILATERAL
POSTPITCH POINT ENGAGEMENT
Unilateral postpitch
point engagement is characterized in that the total line of action is located
on one side of the pitch point C [4, 6]. The line of action is located on the
side of a driven gear and is limited by the origin point C and the end point E_{1},
as shown in Figure 8.
cracking direction sliding direction
Fig. 8. Position
of the line of action in unilateral postpitch point engagement
Rys. 8. Położenie odcinka przyporu
w zazębieniu jednostronnym pozabiegunowym
Position
of the action line in Figure 8 relates to the gear pair z_{1}:z_{5 }with
the correction factors x_{1} = +1 and x_{5} = −1,
respectively. Comparing the engagement from Figures 6 and 8, one can observe
the opposite senses of the slip vector, opposite senses of the friction force
and opposite cracking directions on the working surfaces of the cooperating
teeth. The friction force T_{+} in the unilateral postpitch point
engagement is directed constantly towards the exterior of the gear, which
creates an advantageous setup for the tooth load by extending the arm a_{T}_{+}, on which acts the resultant W_{T+}
(this is proved by the equation 4).
The resultant W_{T+} in the
unilateral postpitch point engagement is less than the resultant W_{T}
in the unilateral prepitch point engagement, so its destructive effect on the
top layer within the working depth of the cooperating teeth is smaller. In this
kind of engagement there is also observed a destructive impact of the oil as a
lubricant penetrating into microcracks in the top layer, which was shown in
Figure 9.
Fig.
9. Destruction of the top layer of the gear pair z_{1}:z_{5} in
the unilateral postpitch point engagement
Rys. 9. Destrukcja warstwy
wierzchniej pary zębatej z_{1}:z_{5} zazębienia jednostronnego pozabiegunowego
Comparing Figures 7 and 9, one
can observe a sort of „antisymmetry” in the similarity of the mechanism of the
top layer destruction in the unilateral pre and postpitch point engagement. The
similarity is that a tooth of
a driving gear in the case of the unilateral prepitch point engagement is
affected by the same destructive mechanism as a driven gear tooth in the unilateral
postpitch point engagement. A similar „antisymmetry” can be applied with
respect to the tooth of a driven gear in the unilateral prepitch point
engagement and to the tooth of a driving gear in the unilateral postpitch
engagement.
It is necessary to remember, however,
that the selection of a unilateral engagement, independently of whether it is
to be a pre or postpitch point one, is limited to some extent by the number
of teeth that the gears should have. It
depends upon the quantity of the gear teeth what value of a correction factor may
be used. The commonly known criterion for using the positive correction is
tooth easing. Meanwhile, the criterion for using a negative correction is undercutting
the dedendum.
NUMERICAL TESTS OF CONTACT STRESSES AND INTERTOOTH SLIDING
Numerical tests were carried out on a complete power shift transmission
with six gear ratios, calculating contact stress and interdental slip at
characteristic contact points, using a proprietary computer program [7]. Characteristic
concurrent contact on the active surface of the tooth profile reflecting the
cooperation of the toothed pair z_{1}/z_{5}, measured in
diameters: d_{E1} – beginning of actual profile, d_{B1} – end
of twopair engagement zone, beginning of twopair engagement zone, d_{C} – central point of engagement or pitch point,
d_{B5} – end of onepair engagement zone, a beginning of twopair engagement
zone, d_{E5} – end of actual tooth profile, end of twopair engagement
zone, are shown in Figure 10.
Fig. 10. Characteristic points of engagement gear pair z_{1}:z_{5}
Rys. 10.
Charakterystyczne punkty przyporu pary zębatej z_{1}:z_{5}
The characteristic points visible in Figure 10 and their position on the
engagement line with the measure of the radius of curvature of the involute
(the active surface of the gear work together has the involute profile) are
shown in Figure 11.
Fig. 11. Position of the characteristic points on the
engagement line
Rys. 11. Położenie
charakterystycznych punktów na linii przyporu
Contact stresses being a measure of the resistance of the surface layer
to tribological wear were determined using a computer program [10] for all gear
pairs in the pre, postpitch and normal engagement forming the investigated gearbox.
Stress results are presented in Table 1.
Table 1. Contact stresses σ_{H }[MPa] in gear pairs
Tabela 1. Naprężenia
kontaktowe σ_{H}_{ }[MPa] w parach zębatych
Gear pair 
Type of
gearing 

prepitch_{} 
postpitch 
normal 

z_{1}/z_{5} 
1148
/ 1049 
1176
/ 1176 
1152
/ 1112 
z_{6}/z_{9} 
1182
/ 1067 
1122
/ 1134 
1145
/ 1122 
z_{10}/z_{12} 
1260
/ 1067 
1160
/ 1160 
1163
/ 1090 
z_{5}/z_{7} 
1049
/ 1116 
1176
/ 1122 
1112
/ 1095 
z_{8}/z_{11} 
1050
/ 1050 
1071
/ 1199 
1143
/ 1183 
z_{2}/z_{4} 
1183
/ 1078 
1070
/ 1170 
1182
/ 1182 
z_{3}/z_{5} 
1186
/ 1049 
1157
/ 1176 
1140
/ 1112 
At the same characteristic point of engagement (except point C, where
the slip speed is always zero), the slip value was calculated for each gear
pair in the appropriate gear ratio. Gear pairs in a normal engagement achieve a
slip speed (at an input speed of n = 2000 min^{1}) at characteristic
contact points, as shown in Table 2.
Table 2. Slip values [m*s^{1}] in normal gear
engagement
Tabela 2. Wartości
poślizgu [m*s^{1}] w parach zębatych o zazębieniu normalnym
Gear ratio 
Gear pair 
Contact
point 

z_{1}/z_{5} 
z_{6}/z_{9} 
z_{10}/z_{12} 
z_{5}/z_{7} 
z_{8}/z_{11} 
z_{2}/z_{4} 
Z_{3}/z_{5} 

1 
2.759 
1.999 
1.476 




E_{1} 
1.783 
1.231 
0.720 




B_{1} 

1.663 
1.217 
0.594 




B_{5} 

2.759 
2.013 
1.601 




E_{5} 

2_{} 
2.759 

2.526 
2.695 



E_{1} 
1.783 

1.232 
1.556 



B_{1} 

1.663 

1.018 
1.663 



B_{5} 

2.759 

2.741 
2.617 



E_{5} 

3 
2.759 


2.695 
4.191 


E_{1} 
1.783 


1.556 
1.750 


B_{1} 

1.663 


1.663 
1.807 


B_{5} 

2.759 


2.617 
4.134 


E_{5} 

4_{} 

1.999 
1.476 


4.029 
2.759 
E_{1} 

1.231 
0.720 


1.465 
1.783 
B_{1} 


1.217 
0.594 


1.465 
2.880 
B_{5} 


2.013 
1.601 


4.029 
2.880 
E_{5} 

5 


2.526 
2.695 

4.029 
2.759 
E_{1} 


1.232 
1.556 

1.465 
1.783 
B_{1} 



1.018 
1.633 

1.465 
2.880 
B_{5} 



2.741 
2.617 

4.029 
2.880 
E_{5} 

6 



2.695 
4.191 
4.029 
2.759 
E_{1} 



1.556 
1.750 
1.465 
1.783 
B_{1} 




1.633 
1.807 
1.465 
2.880 
B_{5} 




2.617 
4.134 
4.029 
2.880 
E_{5} 
The slip speed in each case was calculated according to the equations
[6]:
where:
w_{1}  angular velocity of the gear z_{1},
w_{5}  angular velocity of the gear z_{5},
r_{1 } radius of curvature of the
outline of an involute gear of gear 1 at points respectively: E_{1}, B_{1},
C, B_{5}, E_{5},
r_{5 } radius of curvature of the
involute gear outline of gear 5 respectively at points: E_{1}, B_{1},
C, B_{5}, E_{5}.
The same gear pairs as in the normal bilateral gearing, but with a
correction for a unilateral prepitch gearing (driving gear with correction
factor ˗1, driven gear with correction factor +1) at the same input speed
value of n = 2000 min^{1} achieve the slip speeds shown in Table 3.
Table 3. Slip values [m*s^{1}] in prepitch
gear engagement
Tabela 3. Wartości
poślizgu [m*s^{1}] w parach zębatych o zazębieniu przedbiegunowym
Gear ratio 
Gear pairs 
Contact
point 

z_{1}/z_{5} 
z_{6}/z_{9} 
z_{10}/z_{12} 
z_{5}/z_{7} 
z_{8}/z_{11} 
z_{2}/z_{4} 
Z_{3}/z_{5} 

1 
0.729 
0.491 
0.399 




E_{1} 
5.271 
3.721 
1.797 




B_{1} 

1.059 
0.640 
0.566 




B_{5} 

5.602 
3.870 
2.761 




E_{5} 

2_{} 
0.729 

0.682 
5.241 



E_{1} 
5.271 

3.076 
0.991 



B_{1} 

1.059 

0.968 
4.637 



B_{5} 

5.602 

4.726 
0.387 



E_{5} 

3 
0.729 


5.241 
0.183 


E_{1} 
5.271 


0.991 
5.757 


B_{1} 

1.059 


4.637 
1.641 


B_{5} 

5.602 


0.387 
7.582 


E_{5} 

4_{} 

0.491 
0.399 


0.730 
0.090 
E_{1} 

3.721 
1.797 


4.764 
4.633 
B_{1} 


0.640 
0.566 


1.514 
1.059 
B_{5} 


3.870 
2.761 


7.007 
5.602 
E_{5} 

5 


0.682 
5.241 

0.730 
0.090 
E_{1} 


3.076 
0.991 

4.764 
4.633 
B_{1} 



0.968 
4.637 

1.514 
1.059 
B_{5} 



4.726 
0.387 

7.007 
5.602 
E_{5} 

6 



5.241 
0.183 
0.730 
0.090 
E_{1} 



0.991 
5.757 
4.764 
4.633 
B_{1} 




4.637 
1.641 
1.514 
1.059 
B_{5} 




0.387 
7.582 
7.007 
5.602 
E_{5} 
The application of the correction factor +1 for the driving gear and −1
for the driven gear results in a unilateral postpitch gearing. For such a
gearing, the slip values for all gear pairs at the characteristic points of
engagement are shown in Table 4.
Table 4. Slip values [m*s^{1}] in postpitch
gear engagement
Tabela 4. Wartości
poślizgu [m*s^{1}] w parach zębatych o zazębieniu pozabiegunowym
Gear ratio 
Gear pair 
Contact
point 

z_{1}/z_{5} 
z_{6}/z_{9} 
z_{10}/z_{12} 
z_{5}/z_{7} 
z_{8}/z_{11} 
z_{2}/z_{4} 
Z_{3}/z_{5} 

1 
5.645 
4.153 
2.966 




E_{1} 
1.103 
0.923 
0.771 




B_{1} 

4.319 
3.158 
1.960 




B_{5} 

0.223 
0.073 
0.236 




E_{5} 

2_{} 
5.645 

5.077 
0.209 



E_{1} 
1.103 

1.319 
4.041 



B_{1} 

4.319 

3.354 
1.083 



B_{5} 

0.223 

0.404 
5.333 



E_{5} 

3 
5.645 


0.209 
8.052 


E_{1} 
1.103 


4.041 
2.112 


B_{1} 

4.319 


1.083 
5.331 


B_{5} 

0.223 


5.333 
0.609 


E_{5} 

4_{} 

4.153 
2.966 


8.006 
5.649 
E_{1} 

0.923 
0.771 


2.513 
1.107 
B_{1} 


3.158 
1.960 


4.837 
4.319 
B_{5} 


0.073 
0.236 


0.656 
0.223 
E_{5} 

5 


5.077 
0.209 

8.006 
5.649 
E_{1} 


1.319 
4.041 

2.513 
1.107 
B_{1} 



3.354 
1.083 

4.837 
4.319 
B_{5} 



0.404 
5.333 

0.656 
0.223 
E_{5} 

6 



0.209 
8.052 
8.006 
5.649 
E_{1} 



4.041 
2.112 
2.513 
1.107 
B_{1} 




1.083 
5.331 
4.837 
4.319 
B_{5} 




5.333 
0.609 
0.656 
0.223 
E_{5} 
In tables 2 to 4, blank positions mean that a given gear pair is not
involved in the transmission of the rotational movement at a given gear ratio,
even though all gears remain in the gear engagement.
DISCUSSION OF NUMERICAL SURVEY RESULTS
During analysing the values of contact
stresses in Table 1, it is noted that the gear z_{5}, which is work
together with the gear: z_{1}, z_{3}, z_{7}, has the
lowest value of σ_{H} = 1049 MPa in the
prepitch gearing. On the other hand, the z_{8}/z_{11} gear
pair in the prepitch engagement is subjected to contact stress σ_{H} = 1050 MPa, both for z_{8} and z_{11}
gears. In the case of out pitch engagement, the lowest values of contact
stresses σ_{H} = 1070 MPa are
found in the gear z_{2} which work together with the gear z_{4}.
In a bilateral normal engagement in a gear z_{12} gear pair z_{10}/z_{12},
the lowest contact stress σ_{H} = 1090
MPa is greater than the lowest stress occurring in a prepitch or postpitch engagement.
Out of all 12 gears present in the investigated gearbox, the highest number of
load cycles within the specified service life will be performed by the gear z_{5},
because it work together with the gear pairs: z_{1}/z_{5}, z_{3}/z_{5},
z_{5}/z_{7}. Therefore, the lowest value of contact stress σ_{H} = 1049 MPa in the unilateral prepitch engagement is
important for this gear. The values of the intertooth slip shown in tables 2,
3 and 4 for gear pair z_{1}/z_{5} for the respective gearing (normal
bilateral engagement, prepitch unilateral engagement, postpitch unilateral engagement)
are the same at each gear ratio level (1 to 3) at the respective points of engagement.
This is due to the constant speed of n = 1800 rpm of the shaft I input, on
which the driving gear z_{1} is placed, forming a gear pair with the
gear z_{5}. Also on the shaft I input there is a z_{2} driving gear
work together with a gear z_{4}, which realizes gear ratios from 4 to
6. In next pairs of teeth, the numerical values of the slip already result from
the respective gear ratios. It is noted in Tables 2, 3 and 4 that the numerical
values of slip in the respective toothed pairs and at the corresponding points
of engagement in gear ratio 1 to 3 correspond to the numerical values of slip
of these pairs involved in gear ratio 4 to 6. This is shown in Figures 2 and 3,
where the number of teeth in the gear z_{2} equals the number of teeth
in the gear z_{4}. The highest slip value in both bilateral normal and
unilateral prepitch and postpitch engagement is found in the z_{8}/z_{11}
and z_{2}/z_{4} gear pairs at the extreme point of engagement E_{1}
and E_{2}, where the top of the driving gear tooth is in contact with
the beginning of the active outline of the driven gear. On the active surfaces
of the gear z_{5} in three gears pairs (z_{1}/z_{5}, z_{5}/z_{7},
z_{3}/z_{5}) an intertooth slide with the lowest value of V_{s}
= 0.09 m*s^{1} at the E_{1} prepitch point of engagement (Table
3) of the gear pair z_{3}/z_{5} to the highest value of V_{s}
= 5.649 m*s^{1} of the same gear in the postpitch engagement (Table
4) at the E_{1} outpitch point of engagement. The minimum and maximum interteeth
slip range for normal bilateral engagement are much smaller and for z_{3}/z_{5}
gear pair are in the range of 1.783 to 2.88 m*s^{1} (Table 2).
SUMMARY
Conducting numerical research with multicriteria optimization enables
the implementation of multivariant design solutions and then selecting the
best solution based on the adopted criteria. The complete gearbox shows that
the gear z_{5} in combination with the z_{1}, z_{3} and
z_{7} gears is subjected to the lowest contact stresses in the case of unilateral
prepitch engagement. This is a beneficial engagement case for the z_{5}
gear due to its number of load cycles in service, which is the largest of all
in twelve gears. Unilateral prepitch engagement is obtained by applying a
correction factor x = −1 for the driving gear and x = +1 for the driven
gear. The correction factor x = +1 in the driving gear and x = −1 in the driven
gear provides a unilateral out of pitch engagement.
The use of unilateral engagement
is also beneficial because the sense of the slip vector on the active surfaces
of the work together tooth sides is always the same. No change in the sense of
a slip vector also ensures a constant sense of friction forces, which in the
case of a bilateral normal engagement change when passing through the central
point of the engagement (the gearing pole). The constant sense of a slip vector
and friction forces has a positive effect on the lubrication quality and
stability of the gearbox work, which together reduces the vibroacoustic
activity of the entire system associated with the gearbox.
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[1] Uczelnia Państwowa
im. Jana Grodka w Sanoku, 38500 Sanok, ul. Mickiewicza 21, email:
jazwol@ur.edu.pl, ORCID: 0000000292316306
[2] Liugong Dressta Machinery Sp. z
o. o., email: marek.martyna@dressta.com, 37450 Stalowa Wola, ul.
Kwiatkowskiego 1, ORCID: 0000000306228375
[3] Centrum Nowych Technologii Dominik Kozik w Rzeszowie, email: mechatron1@wp.pl, ORCID: 0000000181343408.