Jan Zwolak[1]

Marek Martyna[2]

Dominik Kozik[3]               

 

DISTRIBUTION OF CONTACT STRESSES AND INTER-TOOTH 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 inter-tooth 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, pre-pitch unilateral gearing and post-pitch unilateral gearing were considered. In each of the three types of gearing, contact stresses and inter-tooth slip were calculated for each gear pair and at each gear ratio level, using an author's computer program with multi-criteria 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 pre-pitch point engagement, unilateral post-pitch point engagement.

        The normal bilateral engagement is generated by two gears, whose respective correction factors satisfy the equation: x1 ˂ 1 for the driving gear and x2 ˃ 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 inter-tooth 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 pitting-induced cumulative damage build-up in the top layer on the dedendum, particularly in the area of unilateral engagement.

        In the unilateral pre-pitch point engagement, as well as in the post-pitch point one, the line of action extends on one side of the pitch point and the sense of the inter-tooth slip vector does not change there.  In the unilateral post-pitch 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 post-pitch point engagement inter-tooth 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 z1 is integrated with the clutch Sp and the input shaft I, while the gear z2 with clutch SW and also with the shaft I. Using the clutch Sp and SW enables a vehicle to move forwards and backwards, respectively, and these two clutches are termed directional clutches. The other clutches, such as: S1 integrated with the gear z6, S2 with the gear z8 and S3 with the gear z10, are counted among the group of gear clutches. Gears z3, z4, z5, z7, z9, z11, z12 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 z1 and z2, 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 z1, z3 and z7. 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 S3, 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 S2 , which is integrated with the gear z8 and forms a gear pair with the gear z11 connected by means of a spline with the output shaft V.

 

NORMAL BILATERAL ENGAGAMENT

 

In general, any gear pair consisting of gears z1 and z2 is in the normal bilateral engagement when [4, 6] the addendum modification coefficients meet the equations: x1 ˂ 1 and x2 ˃ 1. The analysis will be carried out on the gear pair z1:z5 of the gear train from Figure 1, where the gear z1 is a driving gear and z2 a driven one. Limitations of the addendum modification coefficients, given this way, cause the total length of the action line E1E5 to be located on both sides of the pitch point C. A model of engagement, as well as the inter-tooth forces acting between the teeth are shown in Figure 4.

engagement line

Fig.4. Normal bilateral engagement: a) model of engagement of gear z1 with gear z5, b) inter-tooth forces in pitch point C

Rys.4. Zazębienie dwustronne normalne: a)model zazębienia kola z1 z kołem z5, b)siły międzyzębne w biegunie zazębienia C

 

Let the purpose of the considered gear pair be to transfer the torque M1 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 z1. In this point of engagement the normal force FN appears and so does the friction force T+ acting in the direction tangential to the teeth outline. The normal force FN and the friction force T+ can be replaced with the resultant force WT+, whose purpose is to transfer the torque M1. By using the resultant force WT+ and its arm with a length aT+, the following equation for the torque Mcan be proposed:

 

M1 = WT+ × aT+                                                    (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 FN and the friction force T which was replaced by the resultant force WT-, will have the action arms with a length aT-. Therefore, an expression for the torque M1 may be written in the following form:

 

M1 = WT- × aT-                                                   (2)

 

The expressions (1) and (2) may be written after equating as follows:

 

M1 = WT+ × aT+ = WT- × aT-                               (3)

 

Based upon the expression (3), one can determine a relationship between the resultant force WT+ linked to the addendum of the tooth, and the resultant force WT- linked to the dedendum of the tooth:

 

WT+ = W × (aT˗ / aT+)                                      (4)

 

Based on the equation (4) and Figure 4, it can be concluded that the quotient aT˗ / aT+ will always be less than one, thus the value of the force W acting at the dedendum will always be more than the value of the force WT+ acting at addendum with the same torque M1. 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 inter-tooth spaces of a gear pair, no matter whether the model experimental research or real-object 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 z1:z5 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 z1 is a driving gear with an angular velocity ω1, and the gear z5 is a driven gear, and this makes it possible to determine the senses of the friction forces T1 and T5. 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 z1 and z5 is encased by a tooth of the gear z5. Subsequently, the tooth of the gear z5 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 z5 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 z5, it is observed that the oil is squeezed out of it in the same way as from the gap B of the driving gear z1. Further displacement of the point of engagement of the cooperating teeth of the gears z1 and z5 reaches the gap D located on the dedendum of the driven gear tooth z5. The oil encased in the gap D will stimulate erosion in the top layer of the gear dedendum z5, like in the gap A on the dedendum of the gear z1.

 

UNILATERAL PRE-PITCH POINT ENGAGEMENT

 

Unilateral pre-pitch 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 z1:z5 the origin of the line of action always lies at the point E5, whereas its end may be at the point C or before the point C, depending on the correction factor x1. Position of the line of action for the gear pair z1:z5 with the correction factors x1 = −1 and x5 = +1 was shown in Figure 6.

 

sliding direction

 

cracking direction

 

Fig. 6. Position of line of action in unilateral pre-pitch point engagement

Rys. 6. Położenie odcinka przyporu w zazębieniu jednostronnym przedbiegunowym

 

If the correction factor x1 < −1, then the line of action ends before the point C. The correction factor x1 = −1 of the gear z1 causes the total depth as if to comprise only a dedendum. Meanwhile, the total depth of the tooth of the gear z5 with the correction factor x5 = +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 z1 and z5 shown in Figure 7.

 

Fig. 7. Destruction of the top layer of the gear pair z1:z5 in the unilateral pre-pitch point engagement

Rys. 7. Destrukcja warstwy wierzchniej pary zębatej z1/z5 zazębienia jednostronnego przedbiegunowego

 

The oil penetrating into the gaps of the top layer of the driving gear z1 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 z5 is the same as in the gap C of the normal bilateral engagement.

UNILATERAL POST-PITCH POINT ENGAGEMENT

 

Unilateral post-pitch 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 E1, as shown in Figure 8.

 

cracking direction

 

sliding direction

 

Fig. 8. Position of the line of action in unilateral post-pitch point engagement

Rys. 8. Położenie odcinka przyporu w zazębieniu jednostronnym pozabiegunowym

 

Position of the action line in Figure 8 relates to the gear pair z1:z5 with the correction factors x1 = +1 and x5 = −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 post-pitch point engagement is directed constantly towards the exterior of the gear, which creates an advantageous set-up for the tooth load by extending the arm aT+, on which acts the resultant WT+ (this is proved by the equation 4).

        The resultant WT+ in the unilateral post-pitch point engagement is less than the resultant WT- in the unilateral pre-pitch 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 z1:z5 in the unilateral post-pitch point engagement

Rys. 9. Destrukcja warstwy wierzchniej pary zębatej z1:z5 zazębienia jednostronnego pozabiegunowego

 

Comparing Figures 7 and 9, one can observe a sort of „anti-symmetry” in the similarity of the mechanism of the top layer destruction in the unilateral pre- and post-pitch point engagement. The similarity is that a tooth of
a driving gear in the case of the unilateral pre-pitch point engagement is affected by the same destructive mechanism as a driven gear tooth in the unilateral post-pitch point engagement. A similar „anti-symmetry” can be applied with respect to the tooth of a driven gear in the unilateral pre-pitch point engagement and to the tooth of a driving gear in the unilateral post-pitch engagement.

        It is necessary to remember, however, that the selection of a unilateral engagement, independently of whether it is to be a pre- or post-pitch 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 INTER-TOOTH 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 z1/z5, measured in diameters: dE1 – beginning of actual profile, dB1 – end of two-pair engagement zone, beginning of two-pair engagement zone, dC – central point of engagement or pitch point, dB5 – end of one-pair engagement zone, a beginning of two-pair engagement zone, dE5 – end of actual tooth profile, end of two-pair engagement zone, are shown in Figure 10.

 

Fig. 10. Characteristic points of engagement gear pair z1:z5

Rys. 10. Charakterystyczne punkty przyporu pary zębatej z1:z5

 

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-, post-pitch and normal engagement forming the investigated gearbox. Stress results are presented in Table 1.

                                                                      

 

 

Table 1. Contact stresses σH [MPa] in gear pairs

Tabela 1. Naprężenia kontaktowe σH [MPa] w parach zębatych

Gear pair

Type of gearing

pre-pitch

post-pitch

normal

z1/z5

1148 / 1049

1176 / 1176

1152 / 1112

z6/z9

1182 / 1067

1122 / 1134

1145 / 1122

z10/z12

1260 / 1067

1160 / 1160

1163 / 1090

z5/z7

1049 / 1116

1176 / 1122

1112 / 1095

z8/z11

1050 / 1050

1071 / 1199

1143 / 1183

z2/z4

1183 / 1078

1070 / 1170

1182 / 1182

z3/z5

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

z1/z5

z6/z9

z10/z12

z5/z7

z8/z11

z2/z4

Z3/z5

1

2.759

1.999

1.476

 

 

 

 

E1

1.783

1.231

0.720

 

 

 

 

B1

1.663

1.217

0.594

 

 

 

 

B5

2.759

2.013

1.601

 

 

 

 

E5

2

2.759

 

2.526

2.695

 

 

 

E1

1.783

 

1.232

1.556

 

 

 

B1

1.663

 

1.018

1.663

 

 

 

B5

2.759

 

2.741

2.617

 

 

 

E5

3

2.759

 

 

2.695

4.191

 

 

E1

1.783

 

 

1.556

1.750

 

 

B1

1.663

 

 

1.663

1.807

 

 

B5

2.759

 

 

2.617

4.134

 

 

E5

4

 

1.999

1.476

 

 

4.029

2.759

E1

 

1.231

0.720

 

 

1.465

1.783

B1

 

1.217

0.594

 

 

1.465

2.880

B5

 

2.013

1.601

 

 

4.029

2.880

E5

5

 

 

2.526

2.695

 

4.029

2.759

E1

 

 

1.232

1.556

 

1.465

1.783

B1

 

 

1.018

1.633

 

1.465

2.880

B5

 

 

2.741

2.617

 

4.029

2.880

E5

6

 

 

 

2.695

4.191

4.029

2.759

E1

 

 

 

1.556

1.750

1.465

1.783

B1

 

 

 

1.633

1.807

1.465

2.880

B5

 

 

 

2.617

4.134

4.029

2.880

E5

 

The slip speed in each case was calculated according to the equations [6]:

 

 

 

                                   (5)

 

 

 

where:

w1 - angular velocity of the gear z1,

w5 - angular velocity of the gear z5,

r1 - radius of curvature of the outline of an involute gear of gear 1 at points respectively: E1, B1, C, B5, E5,

r5 - radius of curvature of the involute gear outline of gear 5 respectively at points: E1, B1, C, B5, E5.

 

The same gear pairs as in the normal bilateral gearing, but with a correction for a unilateral pre-pitch 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 pre-pitch gear engagement

Tabela 3. Wartości poślizgu [m*s-1] w parach zębatych o zazębieniu przedbiegunowym

Gear ratio

Gear pairs

Contact point

z1/z5

z6/z9

z10/z12

z5/z7

z8/z11

z2/z4

Z3/z5

1

0.729

0.491

0.399

 

 

 

 

E1

5.271

3.721

1.797

 

 

 

 

B1

1.059

0.640

0.566

 

 

 

 

B5

5.602

3.870

2.761

 

 

 

 

E5

2

0.729

 

0.682

5.241

 

 

 

E1

5.271

 

3.076

0.991

 

 

 

B1

1.059

 

0.968

4.637

 

 

 

B5

5.602

 

4.726

0.387

 

 

 

E5

3

0.729

 

 

5.241

0.183

 

 

E1

5.271

 

 

0.991

5.757

 

 

B1

1.059

 

 

4.637

1.641

 

 

B5

5.602

 

 

0.387

7.582

 

 

E5

4

 

0.491

0.399

 

 

0.730

0.090

E1

 

3.721

1.797

 

 

4.764

4.633

B1

 

0.640

0.566

 

 

1.514

1.059

B5

 

3.870

2.761

 

 

7.007

5.602

E5

5

 

 

0.682

5.241

 

0.730

0.090

E1

 

 

3.076

0.991

 

4.764

4.633

B1

 

 

0.968

4.637

 

1.514

1.059

B5

 

 

4.726

0.387

 

7.007

5.602

E5

6

 

 

 

5.241

0.183

0.730

0.090

E1

 

 

 

0.991

5.757

4.764

4.633

B1

 

 

 

4.637

1.641

1.514

1.059

B5

 

 

 

0.387

7.582

7.007

5.602

E5

 

The application of the correction factor +1 for the driving gear and −1 for the driven gear results in a unilateral post-pitch 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 post-pitch gear engagement

Tabela 4. Wartości poślizgu [m*s-1] w parach zębatych o zazębieniu pozabiegunowym

Gear ratio

Gear pair

Contact point

z1/z5

z6/z9

z10/z12

z5/z7

z8/z11

z2/z4

Z3/z5

1

5.645

4.153

2.966

 

 

 

 

E1

1.103

0.923

0.771

 

 

 

 

B1

4.319

3.158

1.960

 

 

 

 

B5

0.223

0.073

0.236

 

 

 

 

E5

2

5.645

 

5.077

0.209

 

 

 

E1

1.103

 

1.319

4.041

 

 

 

B1

4.319

 

3.354

1.083

 

 

 

B5

0.223

 

0.404

5.333

 

 

 

E5

3

5.645

 

 

0.209

8.052

 

 

E1

1.103

 

 

4.041

2.112

 

 

B1

4.319

 

 

1.083

5.331

 

 

B5

0.223

 

 

5.333

0.609

 

 

E5

4

 

4.153

2.966

 

 

8.006

5.649

E1

 

0.923

0.771

 

 

2.513

1.107

B1

 

3.158

1.960

 

 

4.837

4.319

B5

 

0.073

0.236

 

 

0.656

0.223

E5

5

 

 

5.077

0.209

 

8.006

5.649

E1

 

 

1.319

4.041

 

2.513

1.107

B1

 

 

3.354

1.083

 

4.837

4.319

B5

 

 

0.404

5.333

 

0.656

0.223

E5

6

 

 

 

0.209

8.052

8.006

5.649

E1

 

 

 

4.041

2.112

2.513

1.107

B1

 

 

 

1.083

5.331

4.837

4.319

B5

 

 

 

5.333

0.609

0.656

0.223

E5

 

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 z5, which is work together with the gear: z1, z3, z7, has the lowest value of σH = 1049 MPa in the pre-pitch gearing. On the other hand, the z8/z11 gear pair in the pre-pitch engagement is subjected to contact stress σH = 1050 MPa, both for z8 and z11 gears. In the case of out pitch engagement, the lowest values of contact stresses σH = 1070 MPa are found in the gear z2 which work together with the gear z4. In a bilateral normal engagement in a gear z12 gear pair z10/z12, the lowest contact stress σH = 1090 MPa is greater than the lowest stress occurring in a pre-pitch or post-pitch 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 z5, because it work together with the gear pairs: z1/z5, z3/z5, z5/z7. Therefore, the lowest value of contact stress σH = 1049 MPa in the unilateral pre-pitch engagement is important for this gear. The values of the inter-tooth slip shown in tables 2, 3 and 4 for gear pair z1/z5 for the respective gearing (normal bilateral engagement, pre-pitch unilateral engagement, post-pitch 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 z1 is placed, forming a gear pair with the gear z5. Also on the shaft I input there is a z2 driving gear work together with a gear z4, 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 z2 equals the number of teeth in the gear z4. The highest slip value in both bilateral normal and unilateral pre-pitch and post-pitch engagement is found in the z8/z11 and z2/z4 gear pairs at the extreme point of engagement E1 and E2, 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 z5 in three gears pairs (z1/z5, z5/z7, z3/z5) an inter-tooth slide with the lowest value of Vs = 0.09 m*s-1 at the E1 pre-pitch point of engagement (Table 3) of the gear pair z3/z5 to the highest value of Vs = 5.649 m*s-1 of the same gear in the post-pitch engagement (Table 4) at the E1 out-pitch point of engagement. The minimum and maximum inter-teeth slip range for normal bilateral engagement are much smaller and for z3/z5 gear pair are in the range of 1.783 to 2.88 m*s-1 (Table 2).

 

SUMMARY

 

Conducting numerical research with multi-criteria optimization enables the implementation of multi-variant design solutions and then selecting the best solution based on the adopted criteria. The complete gearbox shows that the gear z5 in combination with the z1, z3 and z7 gears is subjected to the lowest contact stresses in the case of unilateral pre-pitch engagement. This is a beneficial engagement case for the z5 gear due to its number of load cycles in service, which is the largest of all in twelve gears. Unilateral pre-pitch 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.

 

REFERENCES

 

1.      Li Baogang, Sun Dongye, Hu Minghui, ZhouXingyu, Liu Junlog, Wang Dongyang: Coordinated control of gear shifting proces with multiple clutches for power shift transmission. Mechanism and Machine Theory, vol. 140, October 2019, pages 274 – 291.

2.      Molari G., Sedoni E.: Experimental evaluation of power losses in a power shift agricultural tractor transmission. Biosystems Engineering, vol. 100, issue 2, June 2008, pages 177 – 183.

3.      Mara Tanelli, Giulio Panzani, Sergio M. Savaresi, Carlo Pirola: Transmission control for power shift agricultural tractors: Design and end – of – line automatic tuning. Mechatronics, vol. 21, issue 1, February 2011, pages 285 – 297.

4.      Rohonyi W.: Untersuchung verschiedener Profilverschiebungsysteme aufgrund elastohydrodynamischer Erkenntnisse. Konstruktion 26, nr 3, 1974.

5.      Baogang Li, Dongye Sun, Munghui Hu, Junlog Liu: Automatic starting control of tractor with a novel power shift transmission. Mechanism and Machine Theory, vol. 131, January 2019, pages 75 – 91.

6.      Muller L.: Przekładnie zębate projektowanie. WNT, Warszawa 1996.

7.      Martyna M., Zwolak J.: Software with multi-criteria optimization PRZEKŁADNIA. www.gearbox.com.pl.

8.      Zwolak J., Martyna M.: The analysis of the slippage and contact stress in the meshing of the power shift type gear. Tribologia, nr 5, 2016, pages 229 – 241.

9.      Zwolak J., Palczak A.: The effect of the gear teeth finishing method on the properties of the teeth surface layer and its resistance to the pitting wear creation. Journal of Central South University, January 2016, vol. 23, issue 1, pages 68 – 76.

10.   Zwolak J., Martyna M.: Analysis of contact and bending stresses in gearbox switching under load. Tribologia, nr 4, 2017, pages 133 – 138.

11.   Amarnath M., Sujatha C., Swarnamani S.: Experimental studies on the effects of reduction in gear tooth stiffness and lubricant film thickness in a spur geared system. Tribology International, vol. 42, issue 2, February 2009, pages 340 – 352.

12.   Yiyao Jiang, Xiaozhou Hu, Shunjun Hong, Pingping Li, Minggui Wu: Influences of an oil guide device on splash lubrication performance in a spiral bevel gearbox. Tribology International, vol. 136, August 2019, pages 155 – 164.

 

 



[1] Uczelnia Państwowa im. Jana Grodka w Sanoku, 38-500 Sanok, ul. Mickiewicza 21, e-mail: jazwol@ur.edu.pl, ORCID: 0000-0002-9231-6306

[2] Liugong Dressta Machinery Sp. z o. o., e-mail: marek.martyna@dressta.com, 37-450 Stalowa Wola, ul. Kwiatkowskiego 1, ORCID: 0000-0003-0622-8375

[3] Centrum Nowych Technologii Dominik Kozik w Rzeszowie, e-mail: mechatron1@wp.pl, ORCID: 0000-0001-8134-3408.