Creation of a general mathematical model of Torsen T-2 differential units for kinematical analysis, geometric design and CAD modelling

Sándor Bodzás; Zsolt Tiba

doi:10.1556/1848.2025.00955

International Review of Applied Sciences and Engineering

Volume/Issue:: Accepted Manuscript / Online First

Creation of a general mathematical model of Torsen T-2 differential units for kinematical analysis, geometric design and CAD modelling

Authors:

Sándor Bodzás

Sándor Bodzás Department of Mechanical Engineering, Faculty of Engineering, University of Debrecen, Debrecen, Ótemető str. 2-4. 4028, Hungary

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bodzassandor@eng.unideb.hu https://orcid.org/0000-0001-8900-2800

and

Zsolt Tiba

Zsolt Tiba Department of Vehicles Engineering, Faculty of Engineering, University of Debrecen, Debrecen, Ótemető str. 2-4. 4028, Hungary

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tiba@eng.unideb.hu

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Online Publication Date:: 21 Feb 2025

Publication Date:: 21 Feb 2025

Article Category:: Research Article

DOI:: https://doi.org/10.1556/1848.2025.00955

Keywords:: Torsen; gear; model; surface

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Abstract

The power can be taken from the ICE crankshaft is a function of the continuously changing adhesion coefficient between the tire and the road surface and the normal force of the wheel. In order to maximize vehicle dynamic performance, Torsen differentials were developed change the power transmission ratio between the wheels or axles depending on the tractive force can be transmitted. The Torsen T-2 differential having internal kinematic ratio i = −1, can be used both as a front and rear as well as a central differential. The torque ratio between the axles connected via Torsen T-2 differential is ensured by the high internal mechanical friction, can be derived from the axial tooth force component of the helical gear drive applied.

The aim of this study is to create a general mathematical model of the Torsen T-2 standard construction. Using this model enables to perform a detailed kinematic analysis of the operation of the entire mechanism. This model will be created by the motion of the gears since coordinate systems are ordered to each moving gears. Based on the Connection I statement the conjugated gear profiles of the gear pairs can be determined by mathematical and computational ways. After that, the CAD models of the gears can be created using 3D software for further finite element analysis. These CAD models are also required for computer-aided manufacturing (CAM) and CNC programming. We prove the usefulness of the model in the case of creating a concrete geometric facility produced by 3D printing.

Abstract

1 Introduction

1.1 Function of differential

The task of the differentials is to transmit and distribute the engine torque in an appropriate proportion to the two axles it connects; enable the rpm difference of the connected axles and thus allow the vehicle cornering. The differential is a two-degree-of-freedom gearbox that operates in power splitting mode in vehicle drive and in power summing mode in coast drive situation. Gear differentials can be derived from the KB type planetary gear set. The kinematic ratio depends on the teeth number of the sun and ring gear. A symmetrical (i = −1) differential may be used between the two wheels of the given axle, or as a central differential; a non-symmetrical (i ≠ −1) differential can be used only as a central one [1].

The power taken from the ICE's crankshaft is equal to the power transmitted to the road (disregarding the mechanical loss of the drive train), which is a function of the constantly changing friction coefficient between the tire and the road surface, and the normal force on the wheel. In order to achieve the greatest possible power transmission, differentials have been developed that change the power transmission ratio depending on the transmittable traction force of the axles. In the following, we analyze what affects the transmittable traction force in the different drive modes [2].

1.2 Vehicle cornering

When cornering, the loads of the inner wheels and outer wheels are also relocated. The outer wheels are loaded up, the inner wheels are loaded down (they may even lose touch with the road). Accordingly, the outer wheels are capable of transmitting higher power to the road, provided that the power transmission capability of the inner wheels does not limit it.

A common feature of differential gears is that they provide a torque distribution between the connected axles according to their gear ratio, and the frictional torque of the differential gear is added to the torque of the slower rotating axle (wheel) [3].

1.3 Vehicle accelerating

When accelerating the vehicle, the loads on the axles change due to the load relocation, which causes the front axle to be loaded down and the rear axle to be loaded up. When accelerating, the driving torque that provides tractive force on the driven wheels also causes the load relocation of the axles. The measure of the normal force on the wheel basically determines the transmittable tractive force and thus the performance [4].

The different constructions of four-wheel and all-wheel drive ensure in different ways that the rear wheels take as much power from the ICE as possible, if the front wheels are on the verge of slipping. Figure 1 shows the classification of the vehicle drivetrain systems. The highest power can be transmitted by the rear wheels in the case of the four-wheel drive, where the rear wheels are driven via the transmission with a positive connection and the front wheels are driven via a frictional clutch (e.g. BMW Xdrive). Although this drive layout provides the highest drive dynamic operation, its disadvantage is the limited lifetime of the multi disc clutch actuating the front wheel drive and the high maintenance cost. The clutch constantly slips due to the rpm difference between the front and rear wheels; and is therefore more exposed to the different tire pressure and wear.

Fig. 1.

Classification of the vehicle drivetrain systems

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00955

The all-wheel drive having a central differential instead of the multi disc clutch, distributes the power between the axles without slip. The different types of central differential (open, LSD, Torsen T-1, Torsen T-2, Torsen T-3) share the power in different way and ratio between the front and rear axles. The kinetic ratio of Torsen T-3 is 40:60 (applied in Audi Q7) with its TBR can transmit the highest ratio of the engine power to the rear axle. Car manufacturers can apply either the layout of the four-wheel drive or all-wheel drive with different types of central differential depending on the required vehicle operation style. In this paper we are focusing on the Torsen T-2 differential.

1.4 Bevel gear open differentials

Its construction can be derived from the KB type planetary gear, see Fig. 2 [5]. The input takes place through the arm of the planetary gears.

Fig. 2.

Differential gear = bevel gear basic planetary gear set

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00955

The two-degree-of-freedom system has two outputs, one is the sun gear, and the other sun gear derived from the ring gear of the original construction. The two sun gears are kinematically connected via the planetary gears. If the two sun gears have the same teeth number, the ratio of torque distribution is 50–50%, kinematic ratio: i = −1.

When cornering, the wheel running on the inner curve (the wheel that rotates slowly) is affected by the torque of the wheel running on the outer curve and the internal frictional torque of the differential. It applies to all operating modes that if one of the wheels slips, the torque of the slipping wheel is applied to the non-slipping wheel, to which the internal friction torque of the planetary gear is added (negligible for an open differential). The construction of the bevel gear differential is shown in Fig. 3.

Fig. 3.

A bevel gear differential

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00955

The internal friction of the differential can be increased by installing a frictional clutch (LSD, Limited Slip Differential). If the magnitude of the internal friction torque is not sufficient in a given operating mode, a differential lock can be used. This case, by locking any two elements of the planetary gear set, the connected axles form a rigid unit.

1.5 Bevel gear limited slip differentials

For increasing the internal friction of the differential, the wet multi-plate clutch became widespread. In the construction shown in Fig. 4, the axles of the planetary gears sit in a wedge-shaped seat of a split cylindrical insert. The insert axially guided can move in the housing. When the sun gears rotate relative to each other, the planetary gear axles push the split inserts apart, thereby prestressing the multi-plate clutches [6].

Fig. 4.

Limited slip differential

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00955

1.6 Torsen (torque sensing) differentials

The common feature of Torsen differentials is the Torque Bias Ratio (TBR), which is the torque transmission/distribution ratio. The high internal friction of Torsen differentials is caused by the large gear width and helix angle screw gear (Torsen T-1), or is due to the axial tooth force component of the helical gear drive (Torsen T-2, Torsen T-3), which pushes the gears axially against the planetary arm. The sliding friction that occurs between the tooth surfaces, as well as between the side surface of the gears and, in the case of some constructions, the outer surface of the gears, causes the high internal friction of the differential [7]. In order to further increase the internal friction, a multi-plate clutch can be installed as well.

The Torsen differential has four drive modes, depending on which of the axles it connects slips either in drive or coast operation. If the traction on the two sides is different, the wheel with the smaller traction will not spin until the ratio of the traction on the two sides exceeds the TBR. If it is exceeded, the wheel with less traction slips, and the wheel with better traction get the torque of the spinning side times the TBR. The torque ratio between the two sides is maintained as long as the wheel with better traction can transmit the increased torque (if provided by the ICE) to the road in the form of tractive force [8].

The advantage of Torsen differentials over LSDs is that their internal friction is proportional to the transmitted driving torque, thus avoiding undue tire wear when cornering under light load [9]. Another advantage of Torsen differentials is that they operate as a pure mechanical mechanism without electronic control or delay [10]. However, if the vehicle is equipped with a stability control system, the application of the differential braking makes its operation even more effective [11, 12].

1.7 Torsen T-2 differential

The kinematic ratio of the Torsen T-2 differential is also I = −1. Planetary gears are helical gears with a large tooth width that engage simultaneously with each other and with the sun gear of the drives, see Fig. 5 [13].

Fig. 5.

The Torsen T-2 differential

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00955

The TBR varies between 2 and 2.5 and can be used both as a front and rear differential as well as a central differential. In the version developed for rally sport known as “Torsen T-2R”, a preloaded frictional clutch ensures that there is a frictional torque on the traction wheel even when there is no traction on the other wheel.

The reason for the high internal friction of Torsen differentials is the axial tooth force component of the meshing screw or helical gears. In the case of the same geometric dimensions, the TBR can be modified simply by choosing the appropriate tooth helix angle. Increasing this angle the load capacity and the tooth connection can be modified. During operation, keeping the fluctuation of the TBR within narrow range is ensured by the adequate stiffness of the large-width helical gears meshing along the long connecting line, which can be realized with FEM design and the high precision manufacturing technology [14].

2 Construction of the general mathematical model

The compilation of the general mathematical model can be seen in Fig. 6.

Fig. 6.

The general mathematical model of the Torsen T-2 differential unit system

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00955

The differential is driven by the transaxle or transmission depending on the actual application via a final ratio that is provided in our study by a spiral bevel gear drive (Figs 6 and 7). It has higher efficiency and load bearing capacity due to the uniform tooth connection comparing to other type of bevel gear drives.

Fig. 7.

Coordinate system arrangement between the spiral bevel gears

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00955

The translation matrices between the C_bevel1R and C_bevel2R coordinate systems (Fig. 7):

M_{b e v e l 2 R, b e v e l 1 R} = M_{b e v e l 2 R, b e v e l 2 S} ∙ M_{b e v e l 2 S, b e v e l 1 S} ∙ M_{b e v e l 1 S, b e v e l 1 R}

(1)

M_{b e v e l 1 R, b e v e l 2 R} = M_{b e v e l 1 R, b e v e l 1 S} ∙ M_{b e v e l 1 S, b e v e l 2 S} ∙ M_{b e v e l 2 S, b e v e l 2 R}

(2)

Knowing the translation matrices between the coordinate systems, the position vector in any other coordinate system related to any given point in the C_bevel1R coordinate system can be defined in two parametric forms, as a surface point as well, in the following way:

\vec{r_{b e v e l 1 R}} = \vec{r_{b e v e l 1 R}} (η, ϑ)

(3)

In case the surface which is connected to C_bevel2R coordinate system - which is connected to the given

\vec{r_{b e v e l 1 R}} = \vec{r_{b e v e l 1 R}} (η, ϑ)

surface in C_bevel1R coordinate system – is sought, we can use the fact that the two surfaces during their movements overlap each other, and considering the transmission ratio between the elements [17–19]:

φ_{2} = i_{b e v e l} ∙ φ_{1}

(4)

In derivative geometry, it is proved that for the independence of the parameters

η

and

ϑ

, the following expression is necessary [16–20]:

\frac{\partial \vec{r_{b e v e l 1 R}}}{\partial η} \times \frac{\partial \vec{r_{b e v e l 1 R}}}{\partial ϑ} \neq 0

(5)

The plane defined by tangents of the parameter lines

\frac{\partial \vec{r_{b e v e l 1 R}}}{\partial η}

and

\frac{\partial \vec{r_{b e v e l 1 R}}}{\partial ϑ}

is the tangent plane in the given point of the surface. The

\vec{n_{b e v e l 1 R}}

surface normal is perpendicular to the tangent plane and can be defined by the following expression [15–19]:

\vec{n_{b e v e l 1 R}} = \frac{\partial \vec{r_{b e v e l 1 R}}}{\partial η} \times \frac{\partial \vec{r_{b e v e l 1 R}}}{\partial ϑ} = [\begin{array}{c} \vec{i} & \vec{j} & \vec{k} \\ \frac{\partial x_{b e v e l 1 R}}{\partial η} & \frac{\partial y_{b e v e l 1 R}}{\partial η} & \frac{\partial z_{b e v e l 1 R}}{\partial η} \\ \frac{\partial x_{b e v e l 1 R}}{\partial ϑ} & \frac{\partial y_{b e v e l 1 R}}{\partial ϑ} & \frac{\partial z_{b e v e l 1 R}}{\partial ϑ} \end{array}]

(6)

The relative velocity between the two surfaces can be defined by the transformation between the rotating C_bevel1R coordinate system of the Bevel gear 1 and the rotating C_bevel2R coordinate system of the Bevel gear 2 in the C_bevel2R system:

\vec{v_{b e v e l 1 R}^{(1, 2)}} = M_{b e v e l 1 R, b e v e l 2 R} ∙ \vec{v_{b e v e l 2 R}^{(1, 2)}} = = M_{b e v e l 1 R, b e v e l 2 R} ∙ \frac{d}{d t} (M_{b e v e l 2 R, b e v e l 1 R}) ∙ \vec{r_{b e v e l 1 R}} = P_{1} ∙ \vec{r_{b e v e l 1 R}}

(7)

where

P_{1} = M_{b e v e l 1 R, b e v e l 2 R} ∙ \frac{d}{d t} (M_{b e v e l 2 R, b e v e l 1 R})

(8)

is the matrix of the kinematical mapping that belongs to the bevel gear connection.

On the surface of the teeth of the connecting members, as contact lines mutually overlapping each other, the Connection I statement which expresses the connecting equation [15–19].

\vec{n_{b e v e l 1 R}} ∙ \vec{v_{b e v e l 1 R}^{(1, 2)}} = \vec{n_{b e v e l 2 R}} ∙ \vec{v_{b e v e l 2 R}^{(1, 2)}} = 0

(9)

and vector – scalar function defining the surface of the teeth should be solved at the same time [16–20]:

\begin{matrix} \vec{n_{b e v e l 1 R}} ∙ \vec{v_{b e v e l 1 R}^{(1, 2)}} = 0 \\ \vec{r_{b e v e l 1 R}} = \vec{r_{b e v e l 1 R}} (η, ϑ) \\ \vec{r_{b e v e l 2 R}} = M_{b e v e l 2 R, b e v e l 1 R} ∙ \vec{r_{b e v e l 1 R}} \end{matrix}}

(10)

The correlation between the Bevel gear 2 and Sun gear 1 can be seen in Fig. 8.

Fig. 8.

Coordinate system arrangement between the Bevel gear 2 and the Sun gear 1

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00955

The translation matrices between the C_bevel2R and the C_sun1R coordinate systems (Fig. 8) since the Sun gear 1 can rotate together with the Bevel gear 2:

M_{s u n 1 R, b e v e l 2 R} = M_{s u n 1 R, s u n 1 S} ∙ M_{s u n 1 S, b e v e l 2 R}

(11)

M_{b e v e l 2 R, s u n 1 R} = M_{b e v e l 2 R, s u n 1 S} ∙ M_{s u n 1 S, s u n 1 R}

(12)

The correlation between the Sun gear 1 and the Planet gear 1 can be seen in Fig. 4. The Planet gear 1 is an a₁ centre distance far from the axis of the Sun gear 1. As you know more Planet gear 1 gears are around the Sun gear 1. They can do two rotation motions at the same time: rotation around their own axis and rotation around the axis of the Sun gear 1.

The translation matrices between the C_sun1R and the C_planet1R coordinate systems (Fig. 9):

M_{p l a n e t 1 R, s u n 1 R} = M_{p l a n e t 1 R, p l a n e t 1 S} ∙ M_{p l a n e t 1 S, s u n 1 S} ∙ M_{s u n 1 S, s u n 1 R}

(13)

M_{s u n 1 R, p l a n e t 1 R} = M_{s u n 1 R, s u n 1 S} ∙ M_{s u n 1 S, p l a n e t 1 S} ∙ M_{p l a n e t 1 S, p l a n e t 1 R}

(14)

Fig. 9.

Coordinate system arrangement between the Sun gear 1 and the Planet gear 1

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00955

Two parametric vector – scalar function of the tooth surface in the C_sun1R coordinate system is given:

\vec{r_{s u n 1 R}} = \vec{r_{s u n 1 R}} (η, ϑ)

(15)

As two surfaces during their movements overlap each other (Sun gear 1 and Planet gear 1 connection), we can consider the transmission ratio between the elements:

φ_{4} = i_{s u n p l a n e t} ∙ φ_{3}

(16)

The

\vec{n_{s u n 1 R}}

surface normal is similar to (6) equation [15–19]:

\vec{n_{s u n 1 R}} = \frac{\partial \vec{r_{s u n 1 R}}}{\partial η} \times \frac{\partial \vec{r_{s u n 1 R}}}{\partial ϑ} = [\begin{array}{c} \vec{i} & \vec{j} & \vec{k} \\ \frac{\partial x_{s u n 1 R}}{\partial η} & \frac{\partial y_{s u n 1 R}}{\partial η} & \frac{\partial z_{s u n 1 R}}{\partial η} \\ \frac{\partial x_{s u n 1 R}}{\partial ϑ} & \frac{\partial y_{s u n 1 R}}{\partial ϑ} & \frac{\partial z_{s u n 1 R}}{\partial ϑ} \end{array}]

(17)

The relative velocity between the two surfaces similarly to (7) and (8) equations

\vec{v_{s u n 1 R}^{(1, 2)}} = M_{s u n 1 R, p l a n e t 1 R} ∙ \vec{v_{p l a n e t 1 R}^{(1, 2)}} = = M_{s u n 1 R, p l a n e t 1 R} ∙ \frac{d}{d t} (M_{p l a n e t 1 R, s u n 1 R}) ∙ \vec{r_{s u n 1 R}} = P_{2} ∙ \vec{r_{s u n 1 R}}

(18)

where

P_{2} = M_{s u n 1 R, p l a n e t 1 R} ∙ \frac{d}{d t} (M_{p l a n e t 1 R, s u n 1 R})

(19)

Considering the Connection I statement and the equation of the tooth surface of the Sun gear I the surface of the connecting gear (Planet gear 1) can be determinable [15, 16]:

\begin{matrix} \vec{n_{s u n 1 R}} ∙ \vec{v_{s u n 1 R}^{(1, 2)}} = 0 \\ \vec{r_{s u n 1 R}} = \vec{r_{s u n 1 R}} (η, ϑ) \\ \vec{r_{p l a n e t 1 R}} = M_{p l a n e t 1 R, s u n 1 R} ∙ \vec{r_{s u n 1 R}} \end{matrix}}

(20)

The correlation between the Planet gear 1 and Planet gear 2 can be seen in Fig. 10.

Fig. 10.

Coordinate system arrangement between the Planet gear 1 and the Planet gear 2

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00955

The translation matrices between the C_planet1R and the C_planet2R coordinate systems (Fig. 10) are

M_{p l a n e t 2 R, p l a n e t 1 R} = M_{p l a n e t 2 R, p l a n e t 2 S} ∙ M_{p l a n e t 2 S, p l a n e t 1 S} ∙ M_{p l a n e t 1 S, p l a n e t 1 R}

(21)

M_{p l a n e t 1 R, p l a n e t 2 R} = M_{p l a n e t 1 R, p l a n e t 1 S} ∙ M_{p l a n e t 1 S, p l a n e t 2 S} ∙ M_{p l a n e t 2 S, p l a n e t 2 R}

(22)

Since the two tooth surfaces overlap each other the two parametric vector – scalar function of the tooth surface in the C_planet1R coordinate system is given based on (20):

\vec{r_{p l a n e t 1 R}} = \vec{r_{p l a n e t 1 R}} (η, ϑ)

(23)

We can consider the transmission ratio between the elements:

φ_{5} = i_{p l a n e t p l a n e t} ∙ φ_{4}

(24)

The

\vec{n_{p l a n e t 1 R}}

surface normal is similar to (6) equation:

\vec{n_{p l a n e t 1 R}} = \frac{\partial \vec{r_{p l a n e t 1 R}}}{\partial η} \times \frac{\partial \vec{r_{p l a n e t 1 R}}}{\partial ϑ} = [\begin{array}{c} \vec{i} & \vec{j} & \vec{k} \\ \frac{\partial x_{p l a n e t 1 R}}{\partial η} & \frac{\partial y_{p l a n e t 1 R}}{\partial η} & \frac{\partial z_{p l a n e t 1 R}}{\partial η} \\ \frac{\partial x_{p l a n e t 1 R}}{\partial ϑ} & \frac{\partial y_{p l a n e t 1 R}}{\partial ϑ} & \frac{\partial z_{p l a n e t 1 R}}{\partial ϑ} \end{array}]

(25)

The relative velocity between the two surfaces similarly to (7) and (8) equations:

\vec{v_{p l a n e t 1 R}^{(1, 2)}} = M_{p l a n e t 1 R, p l a n e t 2 R} ∙ \vec{v_{p l a n e t 2 R}^{(1, 2)}} = = M_{p l a n e t 1 R, p l a n e t 2 R} ∙ \frac{d}{d t} (M_{p l a n e t 2 R, p l a n e t 1 R}) ∙ \vec{r_{p l a n e t 1 R}} = P_{3} ∙ \vec{r_{p l a n e t 1 R}}

(26)

where:

P_{3} = M_{p l a n e t 1 R, p l a n e t 2 R} ∙ \frac{d}{d t} (M_{p l a n e t 2 R, p l a n e t 1 R})

(27)

Considering the Connection I statement and the equation of the tooth surface of the Planet gear 1 the surface of the connecting gear (Planet gear 2) can be determinable [16–20]:

\begin{array}{c} \vec{n_{p l a n e t 1 R}} ∙ \vec{v_{p l a n e t 1 R}^{(1, 2)}} = 0 \\ \vec{r_{p l a n e t 1 R}} = \vec{r_{p l a n e t 1 R}} (η, ϑ) \\ \vec{r_{p l a n e t 2 R}} = M_{p l a n e t 2 R, p l a n e t 1 R} ∙ \vec{r_{p l a n e t 1 R}} \end{array}

(28)

The translation matrices between the C_planet2R and the C_sungear2R coordinate systems (Fig. 11) are:

M_{s u n 2 R, p l a n e t 2 R} = M_{s u n 2 R, s u n 2 S} ∙ M_{s u n 2 S, p l a n e t 2 S} ∙ M_{p l a n e t 2 S, p l a n e t 2 R}

(29)

M_{p l a n e t 2 R, s u n 2 R} = M_{p l a n e t 2 R, p l a n e t 2 S} ∙ M_{p l a n e t 2 S, s u n 2 S} ∙ M_{s u n 2 S, s u n 2 R}

(30)

Fig. 11.

Coordinate system arrangement between the Planet gear 2 and the Sun gear 2

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00955

Previously the two parametric vector – scalar function of the tooth surface on the C_planet2R coordinate system has been calculated based on (28):

\vec{r_{p l a n e t 2 R}} = \vec{r_{p l a n e t 2 R}} (η, ϑ)

(31)

We can consider the transmission ratio between the elements:

φ_{6} = i_{s u n p l a n e t I I} ∙ φ_{5}

(32)

The

\vec{n_{p l a n e t 2 R}}

surface normal is similar to (6) equation:

\vec{n_{p l a n e t 2 R}} = \frac{\partial \vec{r_{p l a n e t 2 R}}}{\partial η} \times \frac{\partial \vec{r_{p l a n e t 2 R}}}{\partial ϑ} = [\begin{array}{c} \vec{i} & \vec{j} & \vec{k} \\ \frac{\partial x_{p l a n e t 2 R}}{\partial η} & \frac{\partial y_{p l a n e t 2 R}}{\partial η} & \frac{\partial z_{p l a n e t 2 R}}{\partial η} \\ \frac{\partial x_{p l a n e t 2 R}}{\partial ϑ} & \frac{\partial y_{p l a n e t 2 R}}{\partial ϑ} & \frac{\partial z_{p l a n e t 2 R}}{\partial ϑ} \end{array}]

(33)

The relative velocity between the two surfaces similarly to (7) and (8) equations:

\vec{v_{p l a n e t 2 R}^{(1, 2)}} = M_{p l a n e t 2 R, s u n 2 R} ∙ \vec{v_{s u n 2 R}^{(1, 2)}} = = M_{p l a n e t 2 R, s u n 2 R} ∙ \frac{d}{d t} (M_{s u n 2 R, p l a n e t 2 R}) ∙ \vec{r_{p l a n e t 2 R}} = P_{4} ∙ \vec{r_{p l a n e t 2 R}}

(34)

where:

P_{4} = M_{p l a n e t 2 R, s u n 2 R} ∙ \frac{d}{d t} (M_{s u n 2 R, p l a n e t 2 R})

(35)

Considering the Connection I statement and the equation of the tooth surface of the Planet gear 2 the surface of the connecting gear (Sun gear 2) can be determinable [16–20]:

\begin{array}{c} \vec{n_{p l a n e t 2 R}} ∙ \vec{v_{p l a n e t 2 R}^{(1, 2)}} = 0 \\ \vec{r_{p l a n e t 2 R}} = \vec{r_{p l a n e t 2 R}} (η, ϑ) \\ \vec{r_{s u n 2 R}} = M_{s u n 2 R, p l a n e t 2 R} ∙ \vec{r_{p l a n e t 2 R}} \end{array}

(36)

The developed mathematical model is usable for the tooth contact analysis of the connecting elements, the mathematical generation of the tooth surface of the conjugated tooth pairs and further geometric development of the Torsen T-2 differential unit. Furthermore, it is also usable to do kinematical motion analysis or follow the load distribution and spreading along the overall gear transmission.

3 CAD modelling of a Torsen T-2 differential unit

Considering the created general mathematical model, knowing the sizes and the geometric establishment of a Torsen T-2 differential unit we could create the CAD model of the gear system. The exploded view can be seen in Fig. 12.

Fig. 12.

The exploded view of the gear system

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00955

The tooth connection between the planet gears and the sun gears can be seen in Fig. 13.

Fig. 13.

The tooth connection between the sun gears and the planet gears

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00955

4 3D printing of the Torsen T-2 differential unit

The application of the 3D printing is reasonable due to the complex geometric shape of the gear unit. The main advantage of the printing process is the fast creation of the prototype and saving manufacturing cost. After the printing a real body model system is received that is touchable that is why the incidental shape errors can be detected easier. If any mistake on the 3D model is detected, the geometric design can be easily modified.

The 3D printing was made by Bambulab P1P printer (Fig. 14). The specification of this printer can be seen on Table 1.

Fig. 14.

The Bambulab P1P 3D printer

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00955

Table 1.

The specifications of the Bambulab P1P printer [20]

Body

Build Volume: 256 × 256 × 256 mm³, Chassis: Welded Steel, Shell: Open frame (Printable Modplates Available)

Speed

Max Speed of Toolhead: 500 mm s⁻¹, Max Acceleration of Toolhead: 20 m s⁻²

Toolhead

Hot End: All-Metal, Nozzle: Stainless Steel, Max Hot End Temperature: 300 °C

Toolhead Cable: Standard toolhead cable

Cooling & Filtration

Control Board Fan: Optional, Chamber Temperature Regulator Fan: Optional

Auxiliary Part Cooling Fan: Optional, Air Filter: Optional

Supported Filaments

PLA, PETG, TPU, PVA, PET: Ideal, ABS, ASA: Capable, PA, PC: Capable

The applied printing parameters can be seen on Table 2.

Table 2.

The applied printing parameters

The layer of thickness	in case of toothed gears: 0.12 mm
The layer of thickness	in case of the gear house: 0.16 mm
The type of the applied material	ecoPLA
The printing speed	initial layer: 50 mm s⁻¹
	filling of the initial layer: 105 mm s⁻¹
	external wall: 100 mm s⁻¹
	internal wall: 150 mm s⁻¹
	filling of the other layers: 200 mm s⁻¹
The acceleration of the printing head	10,000 mm s⁻²
Temperature	65 °C
Total printing time	15–20 h

The printed parts can be seen in Fig. 15. After the 3D printing the assemblage is compiled (Fig. 16).

Fig. 15.

The printed elements of the gear unit

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00955

Fig. 16.

The compilation of the printed elements

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00955

Analysing the geometry of the compilation we may design a Torsen T-2 differential unit using the general coordination system and the modelling process. The following step would be the finite element analysis where the tooth contact analysis can be executable on different loads. After that the forthcoming huge task is the manufacturing design of the elements that need a lot of engineering tasks and organizations.

5 Conclusion

A functional analysis of differential units was carried out that is usable in different application fields of vehicles. The differential is a two-degree-of-freedom gear mechanism that ensures the distribution of the engine's power between the axles it connects, and in the given drive mode allows the wheels to rotate with the desired speed difference. In addition, Torsen differentials are able to compensate the effect of the reduced transmittable traction force of one of the wheels on the wheel that still has traction within the limits of the TBR.

A general mathematical model was developed due to the enhancement of the whole geometric design, the detailed functional analysis (kinematics analysis) and the tooth connection analysis between the teeth. The tooth design is happening with double motion wrapping that means starting from the geometric shape of the input gear tooth the tooth surface of the connecting gear tooth can be determinable thus the CAD models of each gear can be determinable consequently. The necessary kinematical equations and matrixes are also determined. These equations are created in general way thus knowing of concrete input parameters the general equations are usable. This complex mathematical model was created for the Torsen T-2 type differential unit. The practical significance of this study is the improvement possibility of the Torsen T-2 type differential unit based on geometric design and manufacturing way. In further research we intend to develop a universal complex mathematical model that includes the property of most of the differential units in one complex system.

Because of the validation of the usability of the model, a concrete Torsen T-2 differential unit was designed. Importing the received contact points between the connecting teeth of the gears and the geometric parameters the CAD models have to be generated.

The unification of the designed elements and the motion simulation is happening in the field of assembly.

Receiving acceptable results, the 3D printing process is designable where we have to give solutions for the following problems: arrangement of the elements on the machine table, the type of the support and the printing material, the printing structure, the printing strategy, the optimization of the printing time, etc.

Continuing the design process, the FEM analysis and after that the manufacturing design are the following huge tasks.

In the following paper we clarify the TBR calculation of different Torsen differentials.

Conflict of interest

The Authors are members of the Editorial Board of the journal, therefore they did not take part in the review process in any capacity and the submission was handled by a different member of the editorial board. The submission was subject to the same process as any other manuscript and editorial board membership had no influence on editorial consideration and the final decision.

Acknowledgement

We would like to thank for the help of Bálint Vértessy and András Törő B.Sc. Vehicle Engineering students concerning the CAD modelling and the 3D printing.

References

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Bambu Lab website: https://bambulab.com/hu/p1?product=p1p (download: 24. 01. 2024).
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Nomenclature

Symbol	Unit	Name
φ₁, φ₂, φ₃, φ₄, φ₅, φ₆	(°)	angular displacements of the gears
ΔN	rpm	rpm difference of the two shafts connected
ΔT	Nm	torque difference of the two shafts connected
$\vec{n_{b e v e l 1 R}}$		Unit normal vector of Bevel gear 1 surface in C_bevel1R coordinate system
$\vec{n_{p l a n e t 1 R}}$		Unit normal vector of Planet gear 1 surface in C_planet1R coordinate system
$\vec{n_{p l a n e t 2 R}}$		Unit normal vector of Planet gear 2 surface in C_planet2R coordinate system
$\vec{n_{s u n 1 R}}$		Unit normal vector of Sun gear 1 surface in C_sun1R coordinate system
$\vec{r_{b e v e l 1 R}}$		Position vector of moving point on the Bevel gear 1
$\vec{r_{b e v e l 2 R}}$		Position vector of moving point on the Bevel gear 2
$\vec{r_{p l a n e t 1 R}}$		Position vector of moving point on the Planet gear 1
$\vec{r_{p l a n e t 2 R}}$		Position vector of moving point on the Planet gear 2
$\vec{r_{s u n 1 R}}$		Position vector of moving point on the Sun gear 1
$\vec{v_{b e v e l 1 R}^{(1, 2)}}$	m min⁻¹	Velocity vector of Bevel gear 1 and 2 surfaces in the C_bevel1R coordinate system
$\vec{v_{b e v e l 2 R}^{(1, 2)}}$	m min⁻¹	Velocity vector of Bevel gear 1 and 2 surfaces in the C_bevel2R coordinate system
$\vec{v_{p l a n e t 1 R}^{(1, 2)}}$	m min⁻¹	Velocity vector of Sun gear 1 and Planet gear 1 surfaces in the C_planet1R coordinate system
$\vec{v_{p l a n e t 2 R}^{(1, 2)}}$	m min⁻¹	Velocity vector of Planet gear 1 and Planet gear 2 surfaces in the C_planet2R coordinate system
$\vec{v_{s u n 1 R}^{(1, 2)}}$	m min⁻¹	Velocity vector of Sun gear 1 and Planet gear 1 surfaces in the C_sun1R coordinate system
$\vec{v_{s u n 2 R}^{(1, 2)}}$	m min⁻¹	Velocity vector of Sun gear 2 and Planet gear 2 surfaces in the C_sun2R coordinate system
$M_{b e v e l 1 R, b e v e l 1 S}$		Coordinate transformation matrix (transforms C_bevel1S to C_bevel1R)
$M_{b e v e l 1 R, b e v e l 2 R}$		Coordinate transformation matrix (transforms C_bevel2R to C_bevel1R)
$M_{b e v e l 1 S, b e v e l 1 R}$		Coordinate transformation matrix (transforms C_bevel1R to C_bevel1S)
$M_{b e v e l 1 S, b e v e l 2 S}$		Coordinate transformation matrix (transforms C_bevel2S to C_bevel1S)
$M_{b e v e l 2 R, b e v e l 1 R}$		Coordinate transformation matrix (transforms C_bevel1R to C_bevel2R)
$M_{b e v e l 2 R, b e v e l 2 S}$		Coordinate transformation matrix (transforms C_bevel2S to C_bevel2R)
$M_{b e v e l 2 R, s u n 1 R}$		Coordinate transformation matrix (transforms C_sun1R to C_bevel2R)
$M_{b e v e l 2 R, s u n 1 S}$		Coordinate transformation matrix (transforms C_sun1S to C_bevel2R)
$M_{b e v e l 2 S, b e v e l 1 S}$		Coordinate transformation matrix (transforms C_bevel1S to C_bevel2S)
$M_{b e v e l 2 S, b e v e l 2 R}$		Coordinate transformation matrix (transforms C_bevel2R to C_bevel2S)
$M_{p l a n e t 1 R, p l a n e t 1 S}$		Coordinate transformation matrix (transforms C_planet1S to C_planet1R)
$M_{p l a n e t 1 R, p l a n e t 2 R}$		Coordinate transformation matrix (transforms C_planet2R to C_planet1R)
$M_{p l a n e t 1 R, s u n 1 R}$		Coordinate transformation matrix (transforms C_sun1R to C_planet1R)
$M_{p l a n e t 1 S, p l a n e t 1 R}$		Coordinate transformation matrix (transforms C_planet1R to C_planet1S)
$M_{p l a n e t 1 S, p l a n e t 2 S}$		Coordinate transformation matrix (transforms C_planet2S to C_planet1S)
$M_{p l a n e t 1 S, s u n 1 S}$		Coordinate transformation matrix (transforms C_sun1S to C_planet1S)
$M_{p l a n e t 2 R, p l a n e t 1 R}$		Coordinate transformation matrix (transforms C_planet1R to C_planet2R)
$M_{p l a n e t 2 R, p l a n e t 2 S}$		Coordinate transformation matrix (transforms C_planet2S to C_planet2R)
$M_{p l a n e t 2 R, s u n 2 R}$		Coordinate transformation matrix (transforms C_sun2R to C_planet2R)
$M_{p l a n e t 2 S, p l a n e t 1 S}$		Coordinate transformation matrix (transforms C_planet1S to C_planet2S)
$M_{p l a n e t 2 S, p l a n e t 2 R}$		Coordinate transformation matrix (transforms C_planet2R to C_planet2S)
$M_{p l a n e t 2 S, s u n 2 S}$		Coordinate transformation matrix (transforms C_sun2S to C_planet2S)
$M_{s u n 1 R, p l a n e t 1 R}$		Coordinate transformation matrix (transforms C_planet1R to C_sun1R)
$M_{s u n 1 R, b e v e l 2 R}$		Coordinate transformation matrix (transforms C_bevel2R to C_sun1R)
$M_{s u n 1 R, s u n 1 S}$		Coordinate transformation matrix (transforms C_sun1S to C_sun1R)
$M_{s u n 1 S, b e v e l 2 R}$		Coordinate transformation matrix (transforms C_bevel2R to C_sun1S)
$M_{s u n 1 S, p l a n e t 1 S}$		Coordinate transformation matrix (transforms C_planet1S to C_sun1S)
$M_{s u n 1 S, s u n 1 R}$		Coordinate transformation matrix (transforms C_sun1R to C_sun1S)
$M_{s u n 2 R, p l a n e t 2 R}$		Coordinate transformation matrix (transforms C_planet2R to C_sun2R)
$M_{s u n 2 R, s u n 2 S}$		Coordinate transformation matrix (transforms C_sun2S to C_sun2R)
$M_{s u n 2 S, p l a n e t 2 S}$		Coordinate transformation matrix (transforms C_planet2S to C_sun2S)
$M_{s u n 2 S, s u n 2 R}$		Coordinate transformation matrix (transforms C_sun2R to C_sun2S)
$\vec{i}, \vec{j}, \vec{k}$		unit vectors
$i_{b e v e l}$		Transmission ratio between the Bevel gear 1 and Bevel gear 2
$i_{p l a n e t p l a n e t}$		Transmission ratio between the Planet gear 1 and Planet gear 2
$i_{s u n p l a n e t}$		Transmission ratio between the Sun gear 1 and Planet gear 1
$i_{s u n p l a n e t I I}$		Transmission ratio between the Planet gear 2 and Sun gear 2
A, B, C, D, E	mm	linear distances between the coordinate systems
a₁	mm	centre distance between the planet gears and the sun gear
C_bevel1R (x_bevel1R, y_bevel1R, z_bevel1R)		Rotated coordinate system fixed to the Bevel gear 1
C_bevel1S (x_bevel1S, y_bevel1S, z_bevel1S)		Stationary coordinate system fixed to the Bevel gear 1
C_bevel2R (x_bevel2R, y_bevel2R, z_bevel2R)		Rotated coordinate system fixed to the Bevel gear 2
C_bevel2S (x_bevel2S, y_bevel2S, z_bevel2S)		Stationary coordinate system fixed to the Bevel gear 2
C_planet1R (x_planet1R, y_planet1R, z_planet1R)		Rotated coordinate system fixed to the Planet gear 1
C_planet1S (x_planet1S, y_planet1S, z_planet1S)		Stationary coordinate system fixed to the Planet gear 1
C_planet2R (x_planet2R, y_planet2R, z_planet2R)		Rotated coordinate system fixed to the Planet gear 2
C_planet2S (x_planet2S, y_planet2S, z_planet2S)		Stationary coordinate system fixed to the Planet gear 2
C_sun1R (x_sun1R, y_sun1R, z_sun1R)		Rotated coordinate system fixed to the Sun gear 1
C_sun1S (x_sun1S, y_sun1S, z_sun1S)		Stationary coordinate system fixed to the Sun gear 1
C_sun2R (x_sun2R, y_sun2R, z_sun2R)		Rotated coordinate system fixed to the Sun gear 2
C_sun2S (x_sun2S, y_sun2S, z_sun2S)		Stationary coordinate system fixed to the Sun gear 2
n_out1, n_out2	1/min	output number of revolutions on the rims
O₁, O₂, O₃, O₄, O₅, O₆		Origins of the coordinate systems
P, Q, R, T		Contact points between the gear surfaces
P₁, P₂, P₂, P_3, P₄		Kinematic projection matrix for direct method
S₁, S₂, S₃, S₄, S₅, S₆		connecting surfaces of the gears
$η, ϑ$		Internal parameters of the tooth surfaces

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International Review of Applied Sciences and Engineering

Print ISSN:: 2062-0810

Online ISSN:: 2063-4269

Editorial Board

Mohammad Nazir AHMAD, Institute of Visual Informatics, Universiti Kebangsaan Malaysia, Malaysia

Murat BAKIROV, Center for Materials and Lifetime Management Ltd., Moscow, Russia

Nicolae BALC, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Umberto BERARDI, Toronto Metropolitan University, Toronto, Canada

Ildikó BODNÁR, University of Debrecen, Debrecen, Hungary

Sándor BODZÁS, University of Debrecen, Debrecen, Hungary

Fatih Mehmet BOTSALI, Selçuk University, Konya, Turkey

Samuel BRUNNER, Empa Swiss Federal Laboratories for Materials Science and Technology, Dübendorf, Switzerland

István BUDAI, University of Debrecen, Debrecen, Hungary

Constantin BUNGAU, University of Oradea, Oradea, Romania

Shanshan CAI, Huazhong University of Science and Technology, Wuhan, China

Michele De CARLI, University of Padua, Padua, Italy

Robert CERNY, Czech Technical University in Prague, Prague, Czech Republic

Erdem CUCE, Recep Tayyip Erdogan University, Rize, Turkey

György CSOMÓS, University of Debrecen, Debrecen, Hungary

Tamás CSOKNYAI, Budapest University of Technology and Economics, Budapest, Hungary

Anna FORMICA, IASI National Research Council, Rome, Italy

Alexandru GACSADI, University of Oradea, Oradea, Romania

Eugen Ioan GERGELY, University of Oradea, Oradea, Romania

Janez GRUM, University of Ljubljana, Ljubljana, Slovenia

Géza HUSI, University of Debrecen, Debrecen, Hungary

Ghaleb A. HUSSEINI, American University of Sharjah, Sharjah, United Arab Emirates

Nikolay IVANOV, Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia

Antal JÁRAI, Eötvös Loránd University, Budapest, Hungary

Gudni JÓHANNESSON, The National Energy Authority of Iceland, Reykjavik, Iceland

László KAJTÁR, Budapest University of Technology and Economics, Budapest, Hungary

Ferenc KALMÁR, University of Debrecen, Debrecen, Hungary

Tünde KALMÁR, University of Debrecen, Debrecen, Hungary

Milos KALOUSEK, Brno University of Technology, Brno, Czech Republik

Jan KOCI, Czech Technical University in Prague, Prague, Czech Republic

Vaclav KOCI, Czech Technical University in Prague, Prague, Czech Republic

Imre KOCSIS, University of Debrecen, Debrecen, Hungary

Imre KOVÁCS, University of Debrecen, Debrecen, Hungary

Angela Daniela LA ROSA, Norwegian University of Science and Technology, Trondheim, Norway

Éva LOVRA, Univeqrsity of Debrecen, Debrecen, Hungary

Elena LUCCHI, Eurac Research, Institute for Renewable Energy, Bolzano, Italy

Tamás MANKOVITS, University of Debrecen, Debrecen, Hungary

Igor MEDVED, Slovak Technical University in Bratislava, Bratislava, Slovakia

Ligia MOGA, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Marco MOLINARI, Royal Institute of Technology, Stockholm, Sweden

Henrieta MORAVCIKOVA, Slovak Academy of Sciences, Bratislava, Slovakia

Phalguni MUKHOPHADYAYA, University of Victoria, Victoria, Canada

Balázs NAGY, Budapest University of Technology and Economics, Budapest, Hungary

Husam S. NAJM, Rutgers University, New Brunswick, USA

Jozsef NYERS, Subotica Tech College of Applied Sciences, Subotica, Serbia

Bjarne W. OLESEN, Technical University of Denmark, Lyngby, Denmark

Stefan ONIGA, North University of Baia Mare, Baia Mare, Romania

Joaquim Norberto PIRES, Universidade de Coimbra, Coimbra, Portugal

László POKORÁDI, Óbuda University, Budapest, Hungary

Roman RABENSEIFER, Slovak University of Technology in Bratislava, Bratislava, Slovak Republik

Mohammad H. A. SALAH, Hashemite University, Zarqua, Jordan

Dietrich SCHMIDT, Fraunhofer Institute for Wind Energy and Energy System Technology IWES, Kassel, Germany

Lorand SZABÓ, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Csaba SZÁSZ, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Ioan SZÁVA, Transylvania University of Brasov, Brasov, Romania

Péter SZEMES, University of Debrecen, Debrecen, Hungary

Edit SZŰCS, University of Debrecen, Debrecen, Hungary

Radu TARCA, University of Oradea, Oradea, Romania

Zsolt TIBA, University of Debrecen, Debrecen, Hungary

László TÓTH, University of Debrecen, Debrecen, Hungary

László TÖRÖK, University of Debrecen, Debrecen, Hungary

Anton TRNIK, Constantine the Philosopher University in Nitra, Nitra, Slovakia

Ibrahim UZMAY, Erciyes University, Kayseri, Turkey

Andrea VALLATI, Sapienza University, Rome, Italy

Tibor VESSELÉNYI, University of Oradea, Oradea, Romania

Nalinaksh S. VYAS, Indian Institute of Technology, Kanpur, India

Deborah WHITE, The University of Adelaide, Adelaide, Australia

International Review of Applied Sciences and Engineering
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International Review of Applied Sciences and Engineering
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