Jan Zwolak ^*

Marek Martyna ^**

THE ANALYSIS OF THE SLIPPAGE AND CONTACT STRESS IN THE MESHING OF THE POWER-SHIFT TYPE GEAR

Keywords:

gear meshing, slippage, contact stress, numerical tests, optimization

Abstract

This work is an analysis of gear slippage and contact stresses in toothed gears of a six-shaft power shift gearing. Gear meshing contains 5 characteristic contact points located within the active surface of a tooth. The contact points are as follows: A - beginning of a tooth involute profile located within double-tooth engagement area; B - the end-point of double-tooth engagement constituting the beginning of single-tooth engagement area; C - pitch point, referred to also as the central contact point; D - the last point of the single-tooth engagement being at the same time the starting point of the double-tooth engagement area, which is a part of the tooth tip; E - point at the tooth tip that closes the double-tooth engagement area.

Location of individual contact points and the resulting slippage and contact stress values depend on the geometrical parameters of cooperating gear wheels. The inter-relationship suggests that in power shift gearings the contact points have as many positions within the active surface as many cooperating gear wheels there is.

INTRODUCTION

Power shift gearings can be found in power transmission systems for contemporary mobile engineering machines [1,2,3,4]. Toothed gears in such a gearing remain in constant engagement, and a change of a gear is done with special clutches, integrated with appropriate toothed gears. The special clutches, referred to also as the wet plate clutches [2,5] are comprised of alternately positioned friction and steel disks. The friction disks with internal spline are connected with the toothed gear, while the steel disks with external spline connect the shaft through a clutch basket equipped with an internal spline.

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

[*][*] Liugong Dressta Machinery, ul. Kwiatkowskiego 1, 37-450 Stalowa Wola, tel. 15 8136284

Without load, there is a gap between friction and steel disks which disallows transferring the torque from shaft to the toothed gear. The clutch becomes activated (i.e. turned on) when the friction and steel disks are pressed against each other by the hydraulic cylinder located within the clutch.

Discussion over the contact stresses and slippage within individual stages of a gearing requires determination of appropriate pairs of toothed gears that form a kinematic chain, which accomplishes a given gear ratio. An engineer has to select such geometrical parameters, which guarantee minimal contact stresses and slippage during the torque transfer. This problem can be solved by using computer calculation methods with multi-criterion optimisation.

GENERAL CHARACTERISTICS OF THE RESEARCH SUBJECT

Analysis of contact stresses and slippage was done on a six-shaft power shift gearing [2,11,12]. Location of toothed gears within shafts provides 7 toothed pairs of the kinematic chain starting from the input shaft I to the output on shaft V. Gear ratios are secured by clutches S_p, S_w, S₁, S₂, S₃ which are integrated with toothed gears z₁, z₂, z₆, z₈, z₁₀. Figure 1 presents the kinematic scheme of the power shift gearing in axial alignment.

Fig.1 Kinematic scheme of a gearing in axial alignment: z₁ ÷ z₁₂ – toothed gears, S_p, S_w, S₁, S₂, S₃ – clutches, I ÷ V – shafts

Shaft axes from I to V are positioned in 4 vertical planes. Therefore, in order to illustrate all engaged gearings, they were presented in a radial alignment in figure 2.

Fig.2 Radial alignment of the researched gearing

Figures 1 and 2 illustrate that it is possible to create kinematic chains which secure forward motion of a vehicle by way of activating clutch S_p integrated with toothed gear z₁, and the backward motion by way of activating clutch S_w integrated with toothed gear z₂. Another possible gear ratios (gears) can be reached by creating kinematic chains in line with the following:

where: i₁,…,i₆ are the gear ratios for appropriate gears. It can be noticed that toothed gear z₅ forms a toothed pair with gear z₁ as well as with gear z₇ and gear z₃. Therefore, the toothed gear z₅ suffers the highest amount of load cycles in the given time of operation. Characteristic contact points: E₁, E₂, B₁, B₂, C, which are present in all toothed pairs [6,7], are presented in figures 3 and 4.

Fig.3 Toothed pair with the following tooth engagement area: a) B₁E₁, b) B₂E₂

Figure 3a illustrates how the operation of one toothed pair starts in point B₁ while operation of the other toothed pair finishes in point E₁. Figure 3b illustrates the point B₂ where one toothed pair finishes its operation and the point E₂ where other toothed pair begins its operation. It was assumed that the drive gear with centre O₁ rotates clockwise in order to determine the characteristic points.

The remaining geometrical parameters illustrated in figures 3a and 3b, which impact contact stress and slippage values, adopt the following meaning: ρ_1B1 – curvature radius of involute profile for tooth in gear 1 within point B₁, ρ_2B1 – curvature radius of involute profile for tooth in gear 2 within point B₁, p_b – principle scale, ρ_1C – curvature radius of involute profile for tooth in gear 1 within point C, ρ_2C – curvature radius of involute profile for tooth in gear 2 within point C, ρ_1B2 – curvature radius of involute profile for tooth in gear 1 within point B₂, ρ_2B2 – curvature radius of involute profile for tooth in gear 2 within point B₂, a_w – actual distance between axis of toothed gear 1 and toothed gear 2, α_w – rolling pressure angle. Location of curvature radii for the involute profile of a toothed pair within various points [6,7,13] was presented in figure 4.

Fig.4 The contact line and curvature radii of characteristic contact points in a toothed pair

Point N₁ is the point of contact between the contact line and the principle circle of the toothed gear with centre O₁, which means that the section O₁N₁ is the principle radius of this gear. Similar definition applies to point N₂, which refers to the toothed gear with center O₂.

Taking advantage of figures 3a, 3b and 4, it is possible to specify the characteristic points and then contact stress and slippage values of the gearing using the original computer software [8].

NUMERICAL TESTS AND RESEARCH RESULTS

Numerical tests consisted in specifying contact stresses and slippage values in characteristic points within the active surface of power shift gearing. Location of characteristic points within the involute profile of the teeth forming the toothed pair are presented in figure 5.

Fig. 5. Location of characteristic points: a) on the tooth of the drive gear, b) on the tooth of the driven gear

Definition of points located over the engagement area, referred to as the characteristic points, are as follows: E₂ – beginning of the active profile within the propelling tooth root, B₁ – internal point of single-tooth engagement, C – central point of engagement (the pitch point), B₂ – external point of single-tooth engagement, E₁ – the end of active profile within the tip of the propelling tooth.

It is worth noticing that engagement between gear z₁ (fig.5a) and gear z₂ (fig.5b) cause the following points to overlap: E₁=E₂, B₂=B₁, C=C, B₁=B₂, E₂=E₁. As shown in figures 4 i 5, part of the involute profile concerning teeth E₁B₂ and E₂B₁ at their tip in the toothed pair is located in double-tooth engagement. Part of the B₁E₂ and B₂E₁ profiles near the base of the toothed pair are located in the double-tooth engagement as well. The middle part of B₂B₁ and B₁B₂ tooth profile is located within single-tooth engagement.

Geometrical and operation parameters were defined by using an original computer software [8]. The software was developed on the basis of an algorithm, which employs formulas contained within the international standard ISO – 6336 [9] and available literature [10,14,15,16]. Thus, this work will not include any detailed formulas.

Hence, after multi-criterion optimisation, there are presented only results for contact stress and slippage values [17,18] in characteristic points within the active surface. The software conducts optimisation with 11 criteria which include: maximum number of contact points, minimal tooth shape coefficient, minimal thickness at the tooth tip, total weight of toothed gears, total mass inertial moment of toothed gears, maximal durability of tooth root and tooth edge, material effort uniformity within toothed gears, minimal relative thickness of the oil film within the area between teeth, gearing efficiency and minimal slippage value.

Due to a great amount of results obtained for 5 characteristic contact points in all toothed pairs that constitute the tested gearing, the current author will focus on providing slippage values only for the extreme points E₁ and E₂. In those points, the slippage values take maximum levels at the engagement area, which was illustrated in table 1.

Table 1. Slippage velocity values within the active surface following the 1^st calculations

Gear	Toothed pair							Contact point
	z1/z5	z6/z9	z10/z12	z5/z7	z8/z11	z2/z4	z3/z5
I	4.317	3.185	2.320					E1
II	4.317		3.972	3.759
III	4.317			3.759	5.613
IV		3.185	2.320			5.260	4.340
V			3.972	3.759		5.260	4.340
VI				3.759	5.613	5.260	4.340
I	4.295	3.182	2.081					E2
II	4.295		3.563	4.298
III	4.295			4.298	5.702
IV		3.182	2.081			5.260	4.272
V			3.563	4.298		5.260	4.272
VI				4.298	5.702	5.260	4.272

The slippage values in table 1 [m × s^-1] are the results obtained in the 1st step of calculations before optimisation. The empty fields in table 1 denote that the given toothed pair does not take part in the torque transfer.

The conducted optimisation calculations on the basis of previously mentioned 11 criteria illustrate slippage values for contact points E₁ and E₂, as shown in table 2.

Table 2. Slippage velocity values within the active surface following optimisation

Gear	Toothed pair							Contact point
	z1/z5	z6/z9	z10/z12	z5/z7	z8/z11	z2/z4	z3/z5
I	4.187	1.726	2.507					E1
II	4.187		3.030	2.579
III	4.187			2.579	5.242
IV		1.380	2.004			3.406	3.480
V			2.422	2.062		3.406	3.480
VI				2.062	4.191	3.406	3.480
I	3.202	3.751	2.197					E2
II	3.202		2.656	4.076
III	3.202			4.076	3.073
IV		2.999	1.756			3.866	2.315
V			2.123	3.258		3.866	2.315
VI				3.258	2.456	3.866	2.315

As far as durability against fatigue within surfaces of the toothed gears is concerned, the primary attention should be paid to contact stresses that result from torque transfer. Similarly as in the case of slippage values, the contact points E₁ and E₂ along with corresponding contact stress values were selected after the first step of calculations (table 3).

Table 3. Contact stress values [MPa] following 1st step of calculations

Gear	Toothed pair							Contact point
	z1/z5	z6/z9	z10/z12	z5/z7	z8/z11	z2/z4	z3/z5
I	713.2	1161.0	761.8					E1
II	713.2		582.5	856.3
III	713.2			856.3	1146.4
IV		1161.0	761.8			833.3	901.6
V			582.5	856.3		833.3	901.6
VI				856.3	1146.4	833.3	901.6
I	793.9	1292.4	848.0					E2
II	793.9		648.4	953.2
III	793.9			953.2	1276.2
IV		1292.4	848.0			927.7	1003.7
V			648.4	953.2		927.7	1003.7
VI				953.2	1276.2	927.7	1003.7

Table 4 illustrates the values of contact stresses obtained after multi-criterion optimisation conducted by way of specifying 11 criteria.

Table 4. Contact stress values [MPa] following optimisation

Gear	Toothed pair							Contact point
	z1/z5	z6/z9	z10/z12	z5/z7	z8/z11	z2/z4	z3/z5
I	928.6	1205.2	880.6					E1
II	928.6		800.8	1196.3
III	928.6			1196.3	1192.4
IV		1347.5	984.6			907.8	980.2
V			895.3	1337.5		907.8	980.2
VI				1337.5	1333.1	907.8	980.2
I	1027.3	1333.3	974.2					E2
II	1027.3		885.9	1323.4
III	1027.3			1323.4	1319.1
IV		1490.7	1089.2			1004.2	1084.4
V			990.5	1479.6		1004.2	1084.4
VI				1479.6	1474.8	1004.2	1084.4

From among 11 criteria which served as a basis for optimisation described in this work, the current author paid special attention to slippage and contact stress values. However, the global criterion refers to all the 11 criteria. Figure 6 presents the slippage chart compared with the global criterion.

Fig. 6. The example of decreasing slippage over subsequent optimisation steps

In multi-criterion optimisation the main objective consists in achieving the lowest global criterion K_G, at which it is assumed that the investigated problem has been optimally solved according to all developed criteria. The chart presents changes in slippage values from about 12 m × s^-1 to about 8 m × s^-1 at global criterion below 1 (obtained after almost 4000 optimisation steps).

Among many other criteria being calculated, the figure 7 will focus on illustrating the changing value of contact stresses.

Fig. 7. Increasing contact stresses over subsequent optimisation steps

The red curve defines contact stresses of the z₁+z₅ toothed pair on gear 2 within the contact point E₁,. In case of slippage, the lowest value was obtained after about 4000 steps of optimisation, while at stress their value increases. It is possible through the value of fatigue contact durability σ_Hlim [19] that has been entered into the PRZEKŁADNIA software [8]. In order to obtain actual contact stress values, the values presented in the chart should be multiplied by 100 MPa.

The contact stress line in 5 characteristic points within the active surface is presented in figure 8.

Fig. 8. Contact stresses in 5 characteristic points

Contact stress lines in the contact points: E₁, B₂, C, B₁, E₂ suggest that increase in their value takes place after about 4000 optimisation steps.

ANALYSIS OF THE RESULTS

In the conducted optimisation research of a 6-shaft power shift gearing that took place according to 11 criteria, the analysis focused on two: slippage and contact stress values. Those criteria, except for other conditions, should be taken into account by the engineer already at the gearing design stage. For selected slippage and contact stress values the software calculates all geometrical and durability parameters of individual toothed gears which meet conditions specified by the engineer at the stage of creating the optimisation sequence.

In case of this toothed pair (z₁+z₅) the optimal solution will be reached after about 4000 steps of optimisation. Figures 6, 7 and 8 illustrate that such number of calculations leads to a lowered slippage and elevated contact stresses. This is proved by decrease in the global criterion value, which falls below 1.0. In actual gearings, the main objective is to guarantee possibly the lowest slippage values, while the operating contact stress should correspond to fatigue contact durability σ_Hlim, which for SAE 8620 steel equals 1492 MPa. Figure 8 presents toothed pair z₁+z₅ which suffers contact stresses from 700 to 1100 MPa. Therefore, there is a great room for applying various materials within the scope of fatigue contact durability.

SUMMARY AND CONCLUSIONS

The conducted analysis with multi-criterion optimisation of the power shift gearing showed that the first stage being a toothed pair z₁+z₅ is characterized by geometrical parameters, at which the contact stresses are too low. This means that durability of the materials is not fully utilised.

The calculated contact stress values within contact point E₂ at the tooth root reach the highest levels, thus around that area the pitting will occur in the first place. This is confirmed by experimental tests.

Slippage between teeth in given toothed pairs should be kept minimized due to friction processes and the resulting high amounts of heat. Such calculations with or without optimisation are possible with the help of PRZEKŁADNIA [in English: GEARBOX] computer software.

In the central contact point C, the slippage value equals zero because the tangential velocities in this contact point for gears z₁ and z₅ are equal.

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