Jan Zwolak *

Marek Martyna **

 

 

ANALYSIS OF CONTACT AND BENDING STRESSES IN GEARBOXES SWITCHING UNDER LOAD

 

ANALIZA NAPRĘŻEŃ KONTAKTOWYCH I NAPRĘŻEŃ ZGINAJĄCYCH W PRZEKŁADNI ZĘBATEJ PRZEŁĄCZANEJ POD OBCIĄŻENIEM

 

Keywords:

toothed gears, contact stress, bending stress, multi-criterion optimization

Słowa kluczowe:

koła zębate, naprężenia kontaktowe, naprężenia zginające, optymalizacja wielokryterialna

ABSTRACTS

 

The work analyses contact stresses that occur within the active surface of toothed gears as well as bending stresses that take place at the tooth root. Contact stresses have been designated at the beginning of the single-tooth engagement area, within the pitch point and in the end of single-tooth engagement area. Designation of bending stresses at the tooth root has been made by applying the interteeth force to the external point of single-tooth engagement.

     The calculated numerical values of contact and bending stresses were compared to fatigue contact durability σH lim and fatigue bending strength σF lim which were obtained experimentally. Calculation of contact stresses and bending stresses was done with multi-criterion optimisation, which makes it possible to select such geometrical parameters of toothed gears that allow utilizing fatigue durability σH lim and σF lim in reference to a given material and technology of manufacturing toothed gears.

 

[*]  Uniwersytet Rzeszowski, al. Rejtana 16c, 35-959 Rzeszów, tel.17 8518582

[*][*] LiuGong Dressta Machinery Sp. z.o.o, ul. Kwiatkowskiego 1, 37-450 Stalowa Wola, tel. 15 8136284

INTRODUCTION

 

Analysis of damage within toothed gears forming power transmission systems in engineering machines shows that the most common type of damage is tooth fracture at the tooth root and pitting wear within top areas of the tooth active surface. Hence, the most intense are the numerical tests and experimental test within the scope of bending stresses σF and contact stresses σH. Designation of contact stresses or bending stresses without knowing the values of tooth root fatigue durability σF lim and tooth edge fatigue durability σH lim does not constitute a comprehensive or sufficient analysis of the problem with operation of gearboxes.

     Hence, calculations of bending stress σF and contact stress σH with the use of multi-criterion optimisation show that known values of σF lim and σH lim [7,21] allow the designer to take advantage of the properties of a material, which was used to manufacture the toothed gears. The material is also connected with the technology of manufacturing the toothed gears, which affects values σFlim and σHlim. The conducted analysis of contact stress and bending stress will take into consideration the total impact of material and technology that are manifested in fatigue durability of the tooth edge σH lim and fatigue durability of the tooth root σF lim.

   

PROPERTIES OF THE ANALYZED GEARBOX

 

Researches into contact stress have been conducted on toothed pairs that create an internal structure of a 8-ratio gearbox referred to as the power shift gearbox [1,2,3,4] which is used in power transmission systems of wheel loaders and agricultural tractors. This gearbox includes 7 shafts, 14 toothed gears and 6 multidisc clutches. On shaft I, the clutches P and W integrated with toothed gears z1 and z2 allow forward or backward movement, respectively. Clutches S1 and S3 integrated with gears z7 and z6 connected to shaft III as well as clutches S4 and S2 integrated respectively with toothed gears z11 and z12 on shaft IV are used to accomplish gear ratios from 1 to 8. Kinematic scheme in axial alignment within the analysed gearbox have been illustrated in figure 1.

 

 

Fig. 1. Kinematic scheme of a gearing in axial alignment

Rys. 1. Schemat kinematyczny przekładni zębatej w układzie osiowym

 

In axial alignment it is not possible to present all toothed gears in their engagement. Thus, there is a requirement to create a radial alignment scheme as shown in figure 2.

 

Fig. 2. Radial alignment of the researched gearbox

Rys. 2. Układ promieniowy badanej przekładni zębatej

 

On the basis of figures 1 and 2 it is possible to create kinematic chains for individual gear ratios, starting with input shaft I to output shaft VI.

=                             =

=                         =

=                            =

=                          =

The recorded ratios from i1 to i4 allow operation with gears from 1 to 4 in forward direction.  On the other hand, ratios from i5 to i8 allow operation with gears from 5 to 8 and the reverse gear.

     Using the toothed pairs in figure 2 and the records of ratios it is possible to separate toothed gear z3, which at the same time creates kinematic pairs by way of engaging with gears: z1, z5, z10. Hence, it may be stated that toothed gear z3 has three engagement cycles. Examples of teeth engagement cycles for z1 gear were illustrated in figure 3.

Fig. 3. Tooth engagement cycles of z3 gear with gears: z1, z5, z10

Rys. 3. Krotność zazębienia koła z3 z kołami: z1, z5, z10

 

Analysis of z3 gear meshing shows that in toothed pair z1: z3, the gear z3 is the follower gear of z1 being the driver gear. In this pair, the same surface is active all the time. In the toothed pair z3:z5 and z3:z10, gear z3 is a driver gear, while gears z5 and z10 are the follower gears. In this case, the working surface are the opposite surfaces of toothed gears z3 rather than surfaces in toothed pair z1:z3.

     Hence, during the set time of operation of the gearbox, one side of a tooth in toothed gear z3 will be subjected to a lesser number of load cycles than the other side. Here we also observe variations in contact stresses, which is important in terms of creation and development of the pitting wear.

 

NUMERICAL TEST RESULTS

 

Numerical tests were applied to designate contact stresses and bending stresses within all toothed gears from z1 to z14 that form the gearbox being analysed. However, research results were presented only for toothed pairs:  z1 : z3, z5 : z3, z10 : z3 because the toothed gear z3 which has 3 engagement cycles is the most exposed to damage that arises from pitting wear and fatigue durability against bending (fracture of tooth at its root).

     We can distinguish 5 characteristic points within the active surface of toothed gears remaining in engagement with cooperating gear wheels. Those points on a tooth of the driver gear are named as follows: E2 – beginning of the tooth active surface, B1 – beginning of the single-tooth engagement area and at the same time the end of double-tooth engagement area and the internal point of single-tooth engagement, C – pitch point known also and the central point of engagement, B2 – beginning of the double-tooth engagement area and at the same time the end of single-tooth engagement area, referred to also as the outer point of single-tooth engagement, E1 – end of the tooth active surface. Location of characteristic points within the profile of the tooth is presented in figure 4.

      

 

Fig. 4. Characteristic points within the tooth active surface: a) on the tooth of the driver gear; b) on the tooth of the follower gear

Rys. 4. Charakterystyczne punkty na wysokości czynnej zęba: a) na zębie koła napędzającego, b) na zębie koła napędzanego

 

Nominal contact stresses are calculated at parameters of engaged toothed gears that designate the central contact point C [5, 6, 8, 9]. On the other hand, bending stress at the tooth root are designated for the event of highest bending torque, which occurs when interteeth force applies to the external point of a single-tooth engagement of B2 [8, 15, 18, 19]. Contact stress and bending stress are stresses which have the highest impact on operational durability of gearboxes [10, 11, 12, 13, 14, 16, 17, 20].

     Thus, it is very important to pay attention to the designing stage, in which the stresses are being designated. Even more important are the values of fatigue contact durability σHlim and fatigue resistance to fracture σFlim of the teeth in toothed gears [7, 22]. The calculated stress values at loading the input shaft I with M = 2200 Nm and engine speed n = 2000 min-1 for toothed pairs: z1:z3, z3:z5, z3:z10 have been presented in table 1.

 

Table 1. Contact stress values [MPa] for B2 , C and B1 points before optimisation

Tabela 1. Naprężenia kontaktowe [MPa] w punktach B2 , C i B1, przed optymalizacją

Gear ratio

Toothed pair

Contact point

z1/z3

z3/z5

z3/z10

B2

i1

1002.7 / 1010.3

954.8 / 950.5

 

i2

1002.7 / 1010.3

 

1008.7 / 1004.5

i3

1002.7 / 1010.3

954.8 / 950.5

 

i4

1002.7 / 1010.3

 

1008.7 / 1004.5

i5

 

954.8 / 950.5

 

i6

 

 

1008.7 / 1004.5

i7

 

954.8 / 950.5

 

i8

 

 

1008.7 / 1004.5

 

 

 

 

 

i1

921.7 / 928.6

884.5 / 880.5

 

C

i2

921.7 / 928.6

 

933.3 / 929.4

i3

921.7 / 928.6

884.5 / 880.5

 

i4

921.7 / 928.6

 

933.3 / 929.4

i5

 

884.5 / 880.5

 

i6

 

 

933.3 / 929.4

i7

 

884.5 / 880.5

 

i8

 

 

933.3 / 929.4

 

 

 

 

 

i1

1002.7 / 1010.3

954.8 / 950.5

 

B1

i2

1002.7 / 1010.3

 

1008.7 / 1004.5

i3

1002.7 / 1010.3

954.8 / 950.5

 

i4

1002.7 / 1010.3

 

1008.7 / 1004.5

i5

 

954.8 / 950.5

 

i6

 

 

1008.7 / 1004.5

i7

 

954.8 / 950.5

 

i8

 

 

1008.7 / 1004.5

 

Results in table 1 present the calculated values of contact stress for points: B2, C, B1 resulting from single calculations (without optimization) and operational load M = 2200 Nm on shaft I. The places without stress values mean that toothed pairs: z1:z3, z3:z5, z3:z10, do not participate in transferring load on respective gear ratios.

     Stress values in table 1 are too low in relation to fatigue contact durability σHlim = 1528 MPa calculated for steal 18H2N4MA [22] as well as in relation to acceptable contact stresses σHP = 1428 MPa. At such combination of the results, there are significant resources of unutilized fatigue contact durability.

In such a case, it is necessary to perform a multi-criterion optimisation [7] which decreases its geometrical parameters and makes the stress values be closer to the σHlim value. Results of contact stresses after optimization have been presented in table 2.

 

Table 2. Contact stress values [MPa] for B2 , C and B1 points after optimisation

Tabela 2. Naprężenia kontaktowe [MPa] w punktach B2 , C i B1, po optymalizacji

Gear ratio

Toothed pair

Contact point

z1/z3

z3/z5

z3/z10

B2

i1

1218.7 / 1223.9

1191.2 / 1191.2

 

i2

1218.7 / 1223.9

 

1210.4 / 1210.4

i3

1218.7 / 1223.9

1191.2 / 1191.2

 

i4

1218.7 / 1223.9

 

1210.4 / 1210.4

i5

 

1191.2 / 1191.2

 

i6

 

 

1210.4 / 1210.4

i7

 

1191.2 / 1191.2

 

i8

 

 

1210.4 / 1210.4

 

 

 

 

 

i1

1116.6 / 1121.3

1111.4 / 1111.4

 

C

i2

1116.6 / 1121.3

 

1118.7 / 1118.7

i3

1116.6 / 1121.3

1111.4 / 1111.4

 

i4

1116.6 / 1121.3

 

1118.7 / 1118.7

i5

 

1111.4 / 1111.4

 

i6

 

 

1118.7 / 1118.7

i7

 

1111.4 / 1111.4

 

i8

 

 

1118.7 / 1118.7

 

 

 

 

 

i1

1218.7 / 1223.9

1191.2 / 1191.2

 

B1

i2

1218.7 / 1223.9

 

1210.4 / 1210.4

i3

1218.7 / 1223.9

1191.2 / 1191.2

 

i4

1218.7 / 1223.9

 

1210.4 / 1210.4

i5

 

1191.2 / 1191.2

 

i6

 

 

1210.4 / 1210.4

i7

 

1191.2 / 1191.2

 

i8

 

 

1210.4 / 1210.4

 

Optimization calculations have been conducted using 11 criteria: minimal tooth face contact ratio, maximum tooth form factor, minimal tooth thickness on pitch line, total weight of toothed gears, total mass inertial moment of toothed gears, maximal durability of tooth root, maximum contact durability available, effort uniformity of the material, minimal relative thickness of the oil film, gearing efficiency, maximum slippage value. Actual accomplishment of optimization calculations for each criterion is done through attribution of appropriate weighting factors that allow reaching compromise solutions for a set of adopted criteria. It is worth mentioning that there is no need to apply all 11 criteria at the same time, but it is possible to use e.g. 4 criteria and set the remaining 7 to zero value. In the analysed case, the adopted optimization model applies 141 variables and 632 non-linear limits. Decision-making factors are values which describe toothed gears (module, profile angle, number of teeth, correction factor, addendum factor, width of toothed wheel rim, thickness and width of wheel hub, diameter of the wheel rim hub, outer diameter of the wheel rim hub, inner diameter of the wheel rim hub).

Apart from the contact stress, the analysis of gearbox durability designates also the bending stresses at the tooth root for toothed pairs: z1:z3, z3:z5, z3:z10 at input shaft load M = 2200 Nm and engine speed n = 2000 min-1. Results of stresses after one calculation stage without optimization and after optimization have been presented in table 3.

 

Table 3. Bending stress values for a tooth [MPa] before and after optimization

Tabela 3. Naprężenia zginające zęba [MPa] przed i po optymalizacji

 

Toothed pair

z1/z3

z3/z5

z3/z10

before  optimization

376.5 / 381.6

381.6 / 418.9

381.6 / 390.1

 

 

 

 

after optimization

572.4 / 600.4

600.4 / 598.4

600.4 / 578.3

 

Value of acceptable bending stresses for toothed gears made of steel 18H2N4MA equals σFP = 782 MPa [7]. Hence, multi-criterion optimisation allows selecting such geometrical parameters of a gearbox, in which toothed gears will have their resources of fatigue durability utilized to a greater extent [7].

The course of increasing values of contact stresses in toothed pair z1: z3 within characteristic contact points: E2, B1, C, B2, E1 referring to toothed gear z1 during accomplishment of optimization procedure have been presented in figure 5.

 

Fig. 5. Contact stresses within z1 gear over subsequent optimisation steps

Rys. 5. Naprężenia kontaktowe koła z1 w kolejnych krokach optymalizacji

 

For the same toothed pair z1:z3 and the same characteristic contact points: E2, B1, C, B2, E1 referring to toothed gear z3, the course of variability in contact stress have been presented in figure 6.

 

Fig. 6. Contact stresses within z3 gear over subsequent optimisation steps

Rys. 6. Naprężenia kontaktowe koła z3 w kolejnych krokach optymalizacji

 

Figure 7 presents the outline of changes within bending stress in toothed gears z1 and z3 that constitute toothed pair transferring torque M = 2200 Nm through input shaft I at engine speed n =  2000 min-1.

 

Fig. 7. Bending stresses within z1 and z3 gears over subsequent optimisation steps

Rys. 7. Naprężenia zginające koła z1 i z3 w kolejnych krokach optymalizacji

 

The red curve refers to toothed gear z1, while the green curve refers to gear z3. The outlines of contact stresses presented in figures 5 and 6 as well as the bending stress from figure 7 reach their final values after 14000 optimization steps with 11 optimization criteria.

 

ANALYSIS AND DISCUSSION OVER THE RESEARCH RESULTS

 

Numerical tests on optimization calculations have been conducted on a 8-ratio power shift gearbox with application of 11 criteria. Two of them have been presented herein. They include: contact stresses and bending stresses at the tooth root of toothed gears that create kinematic pairs: z1: z3, z3: z5, z3: z10.

     Contact stresses are the source of wear over active surface of the tooth edge. The most common form of damage is the pitting wear that infiltrates within the top surface. Another type of stress, which more dangerous than contact stress, are the bending stress at tooth root that initiate cracks leading to its fracture.

     Stress outlines from figure 5 concern points: E2, B1, C, B2, E1 in toothed gear z1 have similar properties. Local maximum is visible around the 1500th optimization step, then a local minimum around the 2000th optimization calculation. Then we observe a slow increase in stresses up to 14,000th calculation step at low load that corresponds to over 6000 calculations.

     The curves of stresses illustrated in figure 6 referring to toothed gear z3 have a similar course. They refer to contact points: E2, B1, C, B2, E1. Differences in numerical values of contact stresses within toothed gears z1 and z3 shown in tables 1 and 2 are low and they result from the difference between curvature radii of involute profile within engaged teeth in toothed gears z1 and z3.

     The course of bending stress curves in figure 7 indicates that up to 2500 optimization steps the stresses remain virtually at the same level. Further on, bending stresses increase together with increase in number of the optimization steps.

 

SUMMARY AND CONCLUSIONS

 

Numerical tests and analysis of the power shift gearbox with multi-criterion optimization have shown that its application allows more comprehensive utilization of fatigue contact durability σHlim and fatigue durability of the tooth root σFlim. This is proven by contact stresses and bending stresses presented in tables 1, 2 and 3 as well as the curves of those stresses illustrated in figures 5, 6 and 7.

     The toothed gear z3 , which is present in the kinematic chain within all gear ratios, has 3 engagement cycles and it is exposed to the highest number of load cycles. This, in turn, is connected with the possibility to observe the earliest damage in the form of pitting wear or fracture. Application of multi-criterion optimization can help to avoid damaging impact of natural processes that occur within the gearbox and at the same time to maintain control over resources of fatigue contact durability of the tooth edge and fatigue durability of the tooth root.

 

 

 

LITERATURE

 

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3.     Mara Tanelli, Giulio Panzaui, Sergio M. Savaresi, Carlo Pirola: Transmission control for power shift agricultural tractors. Mechatronics, vol. 21, February 2011.

4.     Molari G., Sedoni E.: Experimental evaluation of power losses in a power shift agricultural tractor transmission. Biosystems Engineering 100 (2008).

5.     Li Ting, Pan Cunyun: On grinding manufacture technique and tooth contact and stress analysis of ring involute spherical gears. Mechanism and Machine Theory 44 (2009).

6.     Jose I. Pedrero, Miguel Pleguezuelos, Mariano Artes, Juan A. Antona: Load   distribution model along the line of contact for involute external gears. Mechanism and Machine Theory, vol. 45, 2010.

7.     Martyna  M., Zwolak  J.: Program  komputerowy  z  optymalizacją    wielokryterialną PRZEKŁADNIA. www.gearbox.com.pl.

8.     ISO Standard 6336, 1996: Calculation of load capacity of spur and helical gears.

9.     Hui Ma, Xu Pang, Ranjiao Feng, Rongze Song, Bangchun wen: Fault features analysis of cracked gear considering the effects of the extended tooth contact. Engineering Failure Analysis 48 (2015).

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15.  Miryam B. Sanchez, Jose I. Pedrero, Miguel Pleguezuelos: Critical stress and load conditions for bending calculations of involute spur and helical gears. International Journal of Fatigue 48 (2013).

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21.  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 Uniwersity, vol. 23, issue 1, January 2016.

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STRESZCZENIE

W pracy dokonano analizy naprężeń kontaktowych występujących na wysokości czynnej zębów kół zębatych oraz naprężeń zginających u podstawy zęba. Naprężenia kontaktowe wyznaczano na początku strefy jednoparowego zazębienia, w biegunie zazębienia oraz w końcu strefy jednoparowego zazębienia. Wyznaczenia naprężeń zginających u podstawy zęba dokonano przykładając siłę międzyzębną w zewnętrznym punkcie jednoparowego zazębienia.

     Obliczone wartości liczbowe naprężeń kontaktowych i naprężeń zginających odnoszono do wyznaczonych doświadczalnie wartości zmęczeniowej wytrzymałości kontaktowej σH lim i zmęczeniowej wytrzymałości na zginanie σF lim. Przy obliczaniu naprężeń kontaktowych jak i naprężeń zginających stosowano optymalizację wielokryterialną, dającą możliwości doboru takich parametrów geometrycznych kół zębatych, które umożliwiają wykorzystanie zdolności wytrzymałości zmęczeniowej σH lim i σF lim w odniesieniu do określonego materiału i określonej technologii wytwarzania kół zębatych.