Marek Martyna [*]

Jan Zwolak **








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

Słowa kluczowe:

koła zębate, naprężenia kontaktowe, poślizg międzyzębny, optymalizacja wielokryterialna


The paper presents issues concerning geometrical and durability calculations of power shift gearboxes with multi-criterion optimization, using a proprietary computer software. Among many parameters influencing the value of slippage and contact stresses, the flank ratio at P-0 correction and P correction, the value of the flank angle and the module value were analyzed. The slippage was analyzed at point B1, which belongs to the beginning of the single tooth engagement area, as well as in point B2, where the single tooth engagement area ends. Contact stresses were calculated at point C, which is the pitch point. Their values ​​were referenced to experimentally determined fatigue contact durability σHlim.




In the structure of power transmission systems of mobile engineering machines, there is a power shift gearbox [1,3,4] which characteristic feature is that the toothed gears remain in continuous engagement with each other. Change of gear ratio takes place through multidisc friction clutches (the so-called wet clutches), integrated with toothed gears mounted on a common shaft. These clutches consist of an appropriate number of friction and steel discs (depending on the transmitted torque). Friction discs have an internal spline connecting them to the toothed gear, while steel discs have an external spline that connects them to the shaft via a clutch basket. The clutch basket constitutes a permanent spline connection with the shaft, independent of the toothed gear. In the event that there is clearance between the friction discs and steel discs (there is no pressure), the clutch does not transfer the torque from the shaft to the toothed gears. The clutch switches on when the friction coupling between the friction discs and the steel discs occurs as a result of pressing them against each other through a hydraulic cylinder, which is an integral part of the clutch.

Contact stresses and slippage on individual gear ratios differ from each other quite significantly. It is possible to reduce these differences by the appropriate selection of toothed gear’s design parameters, which can be obtained by multi-criterion optimization. In the gearbox under consideration, there are toothed gears in which engagement frequency (i.e. with how many toothed gears a given gear forms a toothed pair) equals one, two and three at the largest. For example, tooth engagement equal to one is typical of toothed gear z1, which forms a toothed pair only with the z3 gear (Fig.1 and 2). Tooth engagement equal to two is assigned to gear z8, forming a toothed pair with gear z6 and gear z11. The largest tooth engagement in this gearbox equaling to three has the toothed gear z3, which engages with gears: z1, z5 and z10. An example of a toothed pair that forms a circle with gears: z1, z5, z10, has been presented in figure 1.

a)                                            b)                                       c)


Fig. 1. Toothed pairs with slip vectors: a) z3:z1, b) z3:z5, c) z3:z10

Rys. 1. Pary zębate z wektorami poślizgu: a) z3:z1, b) z3:z5, c) z3:z10

The presented engagement frequency shows that at the specified time of gearbox operation the toothed gear z3 will be subjected to the largest number of load cycles, which means that on the active surface of the gear it is most likely to experience surface wear by pitting. In the real gearbox within the toothed pair z3:z1 (Fig.1a) the driver gear is the gear z1, in which the slippage vectors are directed from the pitch point towards the head and root of the tooth. On the other hand, the z3 gear is the follower gear with slippage directed towards the pitch point. In gear z1 during operation one side of the tooth remains in continuous contact with tooth of the gear z3. In the toothed pair z3:z5 (Fig.1b) and in the toothed pair z3:z10 (Fig.1c) the driver gear is the gear z3, and the gears z5 and z10 are follower gears. Thus, in gear z3, one of the side surfaces of the teeth experiences a load from gears z5 and z10, and the other side from gear z1. At designing and constructing stages, the designer should aim to minimize the slippage, while making sure that the contact durability σHlim is fully utilized. The original computer software [7, 15] uses values ​​of fatigue contact durability σHlim and fatigue durability of the tooth root σFlim, determined experimentally [8] on a test bench within a circulating power system, for five steel grades and two finishing technologies of the gears under investigation.



The subject of numerical research in the area slippage and contact stresses was the power shift gearbox, whose kinematic scheme in axial and radial sections are shown in figure 2.


a)                                                                      b)    


Fig. 2. Kinematic scheme of the power shift gearbox: a) axial alignment, b) radial alignment

Rys. 2. Schemat kinematyczny przekładni zębatej power shift: a) układ osiowy, b) układ promieniowy


The tested gearbox has 8 gear ratios (4 forward and 4 reverse). Its main components are: 14 toothed gears, 7 shafts and 6 multidisc clutches. Clutches P and W on shaft I are integrated with toothed gears z1 and z2 responsible for forward and reverse motion, respectively. The S1 and S3 clutches are integrated with toothed gears z7 and z8 and allow connecting these wheels to shaft III. Subsequent clutches S2 and S4 integrated with gears z11 and z12 connect the gears with the IV shaft. The z8 and z9 toothed gears constitute a fixed connection with shaft V through a spline. The toothed gear z13 also constitutes a fixed spline connection with the output shaft VI. The toothed gear z14 on shaft VII plays the function of an intermediate gear in the kinematic chain of ratios i5 to i8 during reverse motion. Clutches S1 to S4 are responsible for implementing gears in the full range from i1 to i8 being under load of any toothed pair.

     Using figure 2, kinematic chains from relevant toothed pairs will be created, starting from the input shaft I to the output shaft VI, for individual gear ratios.

=                             =

=                         =

=                            =

=                          =

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

     The toothed gear z3 can be separated from the toothed pairs in figure 2 and recordings of the gear ratios. The gear simultaneously creates kinematic pairs by engaging with the gears: z1, z5, z10, which denotes that the toothed gear z3 has an engagement frequency of 3. Figure 1 shows an example of engagement frequency for the gear z3, which is a driven gear with respect to gear z1 and at the same time a follower gear with respect to gears z5 and z10.




Contact stresses being a measure of the material effort used in durability calculations [5,6,9,10,11,12,13] were determined at point C being the pitch point located in the area of single-tooth engagement. The slippage were determined in three characteristic contact points: E1 - the beginning of the active outline in area of double-tooth engagement, C - the pitch point, also called the rolling point of contact, E2 - the end of the active outline in area of double-tooth engagement. Point E2 can be placed on the tooth tip if the tooth has no bevel and in this case the diameter dE2 is equal to the diameter of the toothed gear heads, as shown in figure 3. 

At the point E1 determined by the diameter dE1, there is the beginning of the active outline of the tooth, in which area of double-tooth engagement also starts, ending at point B1 determined by diameter dB1. At the same time, point B1 is the starting point of the single tooth engagement that ends at point B2, determined by the diameter dB2. Between points B1 and B2 there is a point C called the rolling contact point or the pitch point, determined by the rolling diameter dC. Point E2 is the end point of the involute profile, which is also the end of the double-tooth engagement area within the tooth tip.

Fig. 3. Characteristic contact points on the tooth of the toothed gear

Rys. 3. Charakterystyczne punkty przyporu na zębie koła zębatego


     Contact stresses as well as slippage in toothed pairs: z1:z3, z3:z5, z3:z10 at torque load M = 2100 Nm and rotational speed n = 1500 min-1 of the input shaft, have been determined at actual parameters of the toothed gears forming the gearbox structure shown in figure 2.

     The calculated contact stresses were referred to fatigue contact durability σHlim determined experimentally [7], the value of which for AISI 8620 steel depending on the finishing technology is: chipped teeth σHlim = 1492 MPa, ground teeth σHlim = 1459 MPa. In the presented diagrams of figures 4, 5, 6, in the correct order for the toothed pairs: z1:z3, z3:z5, z3:z10, the results of calculated contact stresses were marked. Based on these figures it can be stated that the fatigue contact durability is not fully utilized in the toothed pairs under consideration.

Figure 4 shows the location of contact stresses sigmaH1 of the toothed gear z1 = 44 and sigmaH3 of the gear z3 = 44 with P-0 correction of correction factors x1 = -0.25, x3 = +0.25, with respect to fatigue contact durability shlim_1 and shlim_2. The calculated contact stresses caused by the torque M = 2100 Nm transferred by the toothed pair z1:z3 do not use all the material capacity in terms of fatigue contact durability.

Contact stresses within the toothed pair z3:z5, in which the gear z3 is the driving gear for the gear z5 are shown in figure 5.

Fig. 4. Toothed pair z1:z3 with correction P-0 (x1 = -0.25, x3 = 0.25), sigmaH1 = 708 MPa, sigmaH3 = 777 MPa

Rys. 4. Para zębata z1:z3 z korekcją P-0 (x1 = -0.25, x3 = 0.25), sigmaH1 = 708 MPa, sigmaH3 = 777 MPa

Fig. 5. Toothed pair z3:z5 with correction P-0 (x3 = 0.25, x5 = -0.25), sigmaH3 = 777 MPa, sigmaH5 = 678 MPa

Rys. 5. Para zębata z3:z5 z korekcją P-0 (x3 = 0.25, x5 = -0.25), sigmaH3 = 777 MPa, sigmaH5 = 678 MPa


     The gear z3 is also the driving gear for gear z10, for which the contact stresses are shown in figure 6. 

Fig. 6. Toothed pair z3:z10 with correction P-0 (x3 = 0.25, x10 = -0.25), sigmaH3 = 777 MPa, sigmaH10 = 817 MPa

Rys. 6. Para zębata z3:z10 z korekcją P-0 (x3 = 0.25, x10 = -0.25), sigmaH3 = 777 MPa, sigmaH10 = 817 MPa


The value of contact stresses in the toothed pair z3:z5 is respectively: sigmaH3 = 777 MPa for gear z3, and sigmaH5 = 678 MPa for gear z5. These values ​​are small compared to the fatigue contact durability σHlim = 1492 MPa or 1459 MPa. In the toothed pair z3:z10 gear z3 is subjected to contact stress of sigmaH3 = 777 MPa, and the gear z10 to stresses of sigmaH10 = 817 MPa. In all considered toothed pairs: z1:z3, z3:z5, z3:z10, contact stresses are much smaller than the fatigue contact durability σHlim, which means that the considered toothed pairs do not have properly selected geometrical parameters for the given load.

     In order to improve the use of fatigue contact durability, while maintaining the number of teeth of all gears in the tested gearbox, a multi-criterion optimization procedure was used in geometric and durability calculations [15, 16]. The software uses 11 partial criteria. The varying values in multi-criterion optimization were: modules, flank angles and correction factors. Results from calculations of contact stresses for toothed pairs: z1: z3, z3:z5, z3:z10 after optimization are shown in figures 7, 8 and 9.

Fig. 7. Toothed pair z1:z3 with correction P after optimization

Rys. 7. Para zębata z1:z3 z korekcją P po optymalizacji


Fig. 8. Toothed pair z3:z5 with correction P after optimization

Rys. 8. Para zębata z3:z5 z korekcją P po optymalizacji

Fig. 9. Toothed pair z3:z10 with correction P after optimization

Rys. 9. Para zębata z3:z10 z korekcją P po optymalizacji


The multi-criterion optimization has reduced the difference between the calculated contact stresses and the stresses corresponding to fatigue contact durability. In spite of the optimization, maintaining gear ratios unchanged, the fatigue contact durability is still largely unused.

     For toothed pairs: z1:z3, z3:z5, z5:z10 with correction P-0, slippage was determined at three contact points: E1, C, E2, which values have been graphically illustrated in figure 10.

     At point E1, which is the beginning of the active tooth outline and at point E2 that ends the active tooth outline, the slippage values ​​reach maximum (at E1 in the tooth root there is a negative value, at the E2 point in the tooth tip there is a positive value). This is due to length of the involute curvature radius. In each case, at the point C defined as the central contact point (or the pitch point), the slippage velocity equals zero.

     By introducing positive correction P in individual toothed pairs, slippage values ​​as well as their differences are reduced, which is visible in figure 11.



Fig. 10. Slippage in toothed pairs: z1:z3, z3:z5, z3:z10 with correction P-0 (x1 = -0.25, x3 = 0.25, x5 = -0.25, x10 = -0.25)

Rys. 10. Poślizgi w parach zębatych: z1:z3, z3:z5, z3:z10 z korekcją P-0 (x1 = -0.25, x3 = 0.25, x5 = -0.25, x10 = -0.25)



Fig. 11. Slippage in toothed pairs: z1:z3, z3:z5, z3:z10 with correction P (x1 = 0.0804, x3 = 0.2059, x5 = 0.2104, x10 = 0.6212)

Rys. 11. Poślizgi w parach zębatych: z1:z3, z3:z5, z3:z10 z korekcją P (x1 = 0.0804, x3 = 0.2059, x5 = 0.2104, x10 = 0.6212)


The comparison of slippage from figures 10 and 11 indicates how beneficial it is to use positive P correction. In addition to reducing the slippage value, the reduction of scatter is also visible in figure 11.




In the conducted numerical tests with optimization on the 8-ratio power shift gearbox, the criterion of the assessment was: contact stresses and slippage. The calculated values ​​of contact stresses were referred to fatigue contact durability σHlim for AISI 8620 steel, which is 1459 MPa for ground teeth and 1492 MPa for chipped teeth. Toothed pairs with P-0 correction display the greatest value of unutilized fatigue contact durability, this especially concerns the toothed pair z3:z5, in which the smallest contact stresses occur: sigmaH3 = 777 MPa, sigmaH5 = 678 MPa.

     After applying the P correction, there was an improvement within use of fatigue contact durability. On the example of a toothed pair z3:z5, after the optimization process with the correction factor x3 = 0.2059 and x5 = 0.2104 and with the module m = 4.7588 and the flank angle αo = 25o, contact stresses sigmaH3 = 929 MPa, sigmaH5 = 883 MPa. At the gear z3 there was an increase in fatigue contact durability use by 19.5%, and at gear z5 it increased by 30.2%. This article does not analyze the remaining toothed pairs that form a complete gearbox, but in each of them there are similar relationships in the use of fatigue contact durability.

     Slippage for the same toothed pairs with P-0 correction has the highest value in pair z3:z10 and amounts to: Vs3E1 = 6.647 m×s-1, Vs3E2 = 4.268 m×s-1. After applying the P correction with coefficients x3 = 0.2059 and x10 = 0.6212 and after optimization the slippage decreases, reaching the value: Vs3E1 = 2.320 m×s-1, Vs3E2 = 4.143 m×s-1. By introducing P correction and multi-criterion optimization, slippage values ​​in each toothed pair decrease.




Numerical studies with multi-criterion optimization enable selecting geometric properties of toothed gears in such a way to ensure the expected kinematic characteristics of the complete gearbox. In the analyzed gearbox, the toothed gear z3 due to tooth engagement equal to three, will be subject to the earliest wear by pitting. Hence, in this gear a greater contact durability is expected than it is in case the other gears. The use of P correction and multi-criterion optimization allows to reduce differences in contact stresses and slippage in individual toothed pairs. Therefore, gearbox engineers should use positive correction P and multi-criterion optimization to utilize the known contact fatigue durability of a particular material to the maximum extent and to obtain the smallest possible value of slippage.  




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W pracy przedstawiono zagadnienia dotyczące obliczeń geometrycznych i wytrzymałościowych przekładni zębatych power shift, z optymalizacją wielokryterialną, przy stosowaniu autorskiego programu komputerowego. Wśród wielu parametrów mających wpływ na wartość poślizgu międzyzębnego i naprężeń kontaktowych analizowano współczynnik przesunięcia zarysu przy korekcji P-0 i korekcji P, wartość kąta zarysu i wartość modułu. Poślizgi analizowano w punkcie B1, nalężącym do początku strefy jednoparowego zazębienia oraz w punkcie B2, w którym kończy się strefa jednoparowego zazębienia. Naprężenia kontaktowe obliczano w punkcie C, będącym biegunem zazębienia, a ich wartości odnoszono do wyznaczonej doświadczalnie zmęczeniowej wytrzymałości kontaktowej σHlim.


[*] Liugong Dressta Machinery, ul. Kwiatkowskiego 1, 37-450 Stalowa Wola,

   tel.15 8136284

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