# Influence of Open Differential Design on the Mass Reduction Function

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}and root mean square error (RMSE), an accuracy check of the proposed mathematical model was performed. According to the proposed algorithm and mathematical model, the two mentioned design elements of the ODT were optimized. After optimization, the overall mass of the ODM was reduced by 16.5%.

## 1. Introduction

#### 1.1. Differential Transmissions Overview

_{LEFT}= n

_{RIGHT}). Satellite gears rotate around the axis of the output shafts. During the movement of the vehicle in a turn, the satellite gears rotate around the axis of the output shafts, but also around their own axis. Therefore, the rotation speeds of the satellite gears are equal, but in the opposite direction. Depending on the direction of deflection, the rotation speeds of the sun gears are different (n

_{LEFT}< n

_{RIGHT}or n

_{LEFT}> n

_{RIGHT}). By slipping one of the drive wheels, the open differential distributes most of the torque to the wheel that has less resistance to movement (the slipping wheel). Therefore, a wheel that does not slip has an angular velocity of zero, and the torque is also zero. This is one of the main disadvantages of the ODT.

#### 1.2. Mass Reduction Impact on Improving Energy Efficiency

_{Z}and nominal gear root stress were proposed. This is important from the aspect of increasing the accuracy of the gear calculation and gear optimization. Authors in [28] analyze the influence effects of center distance deviation on polymer gearbox life. Papers [29,30], using genetic algorithms, designed the minimum mass of the optimal combination of spur gears. By applying multi-objective optimization methods [31], two parameters, volume and center distance, are optimized and the Pareto frontier is obtained. Authors in this research used the teaching–learning-based optimization (TLBO) method. By applying RSM, authors in [32] analyze, through the response surface diagrams, the influence of individual design parameters on the set objective functions. TOM deals with the problem of optimal material distribution [33]. Their application is particularly significant in aerospace structure design, dynamic responses design, shape preserving design and smart structure design [34]. By applying these methods, the objective function refers to the saving of materials by reducing the mass of the design, taking care to preserve the functionality of the design.

## 2. Proposed Methodology and Research Structure

## 3. Design of the ODT

#### 3.1. Design Requirements of the ODT

_{3}= 12; teeth number of sun bevel gears z

_{4}= 16 and gearbox transmission ratios (i

_{1}= 3.5, i

_{2}= 2.1, i

_{3}= 1.32, i

_{4}= 0.97, i

_{5}= 0.76, i

_{R}= 3.55). Also, the degrees of efficiency of the elements of the kinematic chain (Figure 2) are the efficiency of the gearbox η

_{g}= 0.96, efficiency of the cardan shaft η

_{c}= 0.98 and efficiency of the open differential η

_{d}= 0.97. Tire dimensions 205/55 R16. Since the drive shaft and pinion are made of one piece and form one design element, the drive shaft is made of the same steel as the pinion. The driven shaft is made of St 70-2 steel. The design requirements of the vehicle for which it is necessary to design a differential transmission are shown in Table 1.

#### 3.2. Loads on the Input and Output Side of the Differential Transmission

_{Tmax}(Table 1) at which T

_{max}(Table 1) is achieved, the power on the motor output shaft is determined by the expression:

_{1}is presented in Table 1. Input torque to the differential transmission is determined by the expression:

_{w}) and friction force (F

_{f}) act, through the drive wheel, from the outside on the differential transmission (Figure 3). Reaction force in the wheel is determined by the expression:

_{v}is the vehicle weight force calculated as a product of the maximum vehicle mass (Table 1) and the acceleration of the earth’s gravity (g = 9.81 m/s

^{2}). The calculated amount of the reaction force is presented in Table 2.

_{w}= T′

_{L}= T′

_{R}). The torque of the drive wheels is determined by the expression:

_{L}and T′

_{R}are the torques on the left and right drive wheels, r

_{d}is the dynamic wheel radius. Since the dimension of the tire is determined by the request (205/55 R16), the dynamic wheel radius is 97% of the length of the actual wheel radius. Its calculation amount is shown in Table 1.

_{2}), which was obtained without including the losses of the kinematic chain, is equal to the sum of the torques of the left (T′

_{L}) and of the right (T′

_{R}) drive wheels. This torque is determined by the expression:

_{2}), it is necessary to include in the calculation the losses of the open differential transmission, i.e., efficiency of the open differential (η

_{d}). For the determination of T

_{2}, it is necessary to determine the output power of the differential transmission. This calculation includes the efficiency of the open differential (η

_{d}). The output power of the differential transmission is determined by the expression:

_{2}is presented in Table 2. Hence, the torque of the hypoid ring bevel gear is determined by the expression:

_{2}is the angular velocity of the hypoid ring bevel gear. The angular velocity of the hypoid ring bevel gear is determined by the expression:

_{d}is the transmission ratio of the differential transmission (Table 2). The transmission ratio of the differential transmission is determined by the expression:

_{2}is shown in Table 2. Also, the final amounts of the torque on the drive wheels are equally divided between the left and right wheels (T

_{w}= T

_{L}= T

_{R}= T

_{2}/2). Their calculated amounts are presented in Table 2.

_{TRD}), whose calculation value is presented in Table 2. In the calculation of the total resistance force, the geometry of the vehicle body is not defined, and therefore air resistance is not included in this calculation [38]. In order to achieve the movement of the vehicle, the amount of the traction force (F

_{TR}) should be greater than the amount of the total resistance force (F

_{TRD}). The values of these forces are presented in Table 2.

_{RO}is the rolling resistance factor. The amount of this factor for rolling on the asphalt is f

_{RO}= 0.02. The calculated amount of this force is presented in Table 2.

_{v}is vehicle acceleration (Table 1) and ψ is a contribution factor of rotating masses for the first degree of transmission. The amount of this factor for this calculation is ψ = 1.165. The calculated amount of this force is presented in Table 2.

#### 3.3. Hypoid Bevel Gears Calculation

#### 3.3.1. Dimensioning of the Driving and Driven Hypoid Bevel Gears

_{1}, z

_{2}) that are in mutual grip (Figure 2). The pinion (z

_{1}) is a part of the drive shaft of the differential transmission. By iterative calculation, variables needed to design the pinion and driven hypoid bevel gear (ring gear) were obtained. Iteration is carried out until the following condition is satisfied reached:

_{m1}is the mean cone distance of the pinion.

_{m1}is calculated according to the expression:

_{m1}is pinion mean pitch diameter. Diameter d

_{m1}is determined by the expression:

_{mint}is determined using the following expression:

_{3}, A

_{4}, A

_{5}, A

_{6}and A

_{7}are intermediate variables.

_{1}and δ

_{2}obtained by Equations (21) and (22) are added in the first iteration, and their new values will be determined in the calculation of the first iteration (Table 3). Other variables and their values, which were determined by the iteration calculation, are presented in Table 3. Values determined in the last iteration represent the final solutions of the iterative calculation.

#### 3.3.2. Stress Calculation of Hypoid Bevel Gears in the Tooth Root and Tooth Face Flank

_{Flim1}= σ

_{Flim2}= 500 MPa and amount of permanent dynamic strength for tooth face flank is σ

_{Hlim1}= σ

_{Hlim2}= 1630 MPa.

_{Fmin}is the minimum safety factor against root breakage, K

_{FX}is size influence factor and Y

_{S}is notch action factor.

_{mt1}is a tangential force on the dividing circle, Y

_{F}is tooth shape factor, Y

_{εV}is load share factor of auxiliary gears, Y

_{β}is impact factor of tooth locking on the stress distribution in the root, K

_{Fα}is load distribution factor and K

_{Fβ}is load distribution factor along the length of the tooth flank. Solutions obtained by expressions (29) and (30) are shown in Table 5.

_{Hmin}is minimum safety factor against tooth flank fracture, K

_{L}is influence factor of lubricating oil, K

_{HX}is factor of dimensions that influence the load capacity of the tooth flank, Z

_{V}is speed influence factor and Z

_{N}is service life factor.

_{HV}is tooth shape factor, Z

_{M}is material factor, Z

_{εV}is coverage factor, K

_{Hα}is load distribution factor and K

_{Hβ}is load distribution factor along the length of the tooth flank. Solutions obtained by expressions (31) and (32) are shown in Table 5.

#### 3.4. Sun and Planetary Bevel Gears Calculation

_{r}= 6.637 mm and m

_{f}= 8.49 mm. Between these two modules, for the further calculation of the bevel gears, was adopted a module m = m

_{f}= 8.5 mm.

#### 3.4.1. Dimensioning of the Planetary and Sun Bevel Gears

#### 3.4.2. Stress Calculation of Planetary and Sun Bevel Gears in the Tooth Root and the Tooth Face Flank

_{N}is service life factor, Y

_{R}is roughness factor, Y

_{S}and K

_{FX}are factors according to [35].

_{mt3}is tangential force on the middle pitch circle, Y

_{F}, K

_{Fα}, K

_{Fβ}and Y

_{εV}are factors according to [35].

_{Hlim}is minimum anti-pitting factor, Z

_{R}is roughness influence factor, Z

_{W}is influence factor of flank hardening, K

_{L}, K

_{HX}, Z

_{V}and Z

_{N}are factors according to [35].

_{HV}, Z

_{M}, Z

_{εV}, K

_{Fα}and K

_{Fβ}are factors according to [35].

## 4. Numerical Analysis of Differential Transmission

#### 4.1. Analysis of Stress and Displacement of Bevel Gear and Pinion Gear

_{1}(Figure 6b), the value of which is given in Table 1.

#### 4.2. Submodeling of the Pinion and Bevel Gear

#### 4.3. Analysis of Stress and Displacement of Drive Shaft

## 5. Optimization of Differential Transmission

#### 5.1. Topological Optimization and Determination of Design Parameters

#### 5.2. RSM Model

^{2}and root mean square error (RMSE) values is shown in Table 11. From the graph in Figure 16 and the data in Table 11, some scatter in the results can be seen. This is due to the complexity of the geometry of the teeth in the contact between the pinion and ring gear (radii) and the density of the finite element mesh. These data should be taken into account when interpreting the above results. It can be concluded that the accuracy of the numerical and mathematical model is acceptable considering the above data. The accuracy was additionally confirmed by the numerical control calculation of the optimized design elements, which is presented in Section 6.

#### 5.3. Analysis of Sensitivity of Parameter Change on Response Objective

## 6. Numerical Calculation of the Optimized Differential Gear

## 7. Conclusions

_{all}), mass reduction of the ring gear (m

_{1}) and mass reduction of the drive shaft (m

_{2}). From the local sensitivity analysis, the parameters of the inner radius of the drive shaft (r) and overhang height of the ring gear (h) have the greatest influence on the m

_{all}response variable. Then, we followed the parameters vertical and horizontal chamfer of the ring gear (a and b). Based on the partial objective functions of mass reduction of each optimized design element (m

_{1}and m

_{2}), a significant influence of the selected design parameters on this design element can be seen. All ring gear parameters h, b and a have influence on objective function m

_{1}where parameter h is dominant. This is visible through the obtained mathematical Equation (40). The parameter r has the largest effect on the function m

_{2}.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

a | hypoid offset |

a_{v} | vehicle acceleration |

A_{3}, A_{4}, A_{5}, A_{6}, A_{7} | intermediate variables |

b | tooth width |

b_{1} | pinion gear face width |

b_{2} | ring gear face width |

b_{3} | tooth width of the planetary gear |

b_{4} | tooth width of the sun gear |

c | clearance |

d_{a3} | planetary gear addendum circle diameter |

d_{a4} | sun gear addendum circle diameter |

d_{ia3} | planetary gear inner addendum diameter of tooth |

d_{ia4} | sun gear inner addendum diameter of tooth |

d_{m1} | pinion mean pitch diameter |

d_{m2} | ring gear mean pitch diameter |

d_{m3} | planetary gear mean pitch diameter |

d_{m4} | sun gear mean pitch diameter |

d_{v3} | pitch diameter of equivalent gear (planetary) |

d_{v4} | pitch diameter of equivalent gear (sun) |

d_{va3} | addendum diameter of equivalent gear (planetary) |

d_{va4} | addendum diameter of equivalent gear (sun) |

d_{vb3} | base diameter of equivalent gear (planetary) |

d_{vb4} | base diameter of equivalent gear (sun) |

d_{vm3} | pitch diameter of middle equivalent gear (planetary) |

d_{vm4} | pitch diameter of middle equivalent gear (sun) |

d_{3} | pitch circle diameter of the planetary gear |

d_{4} | pitch circle diameter of the sun gear |

F | hypoid dimensional factor |

F_{f} | friction force |

F_{IN} | vehicle’s inertial force |

F_{mt1} | tangential force on the pinion |

F_{mt2} | tangential force on the ring gear |

F_{mt3} | tangential force on the planetary gear |

F_{mt4} | tangential force on the sun gear |

F_{RI} | rising resistance force |

F_{RO} | rolling resistance force |

f_{RO} | rolling resistance factor |

F_{TR} | traction force |

F_{TRD} | total resistance force of driven machine |

F_{w} | reaction force in the wheel |

g | acceleration of the earth’s gravity |

G_{v} | vehicle weight force |

h | overhang height of the ring gear |

h_{3} | planetary gear whole depth |

h_{4} | sun gear whole depth |

h_{a3} | addendum depth for zero pair of bevel gears (planetary) |

h_{a4} | addendum depth for zero pair of bevel gears (sun) |

h_{f3} | dedendum depth for zero pair of bevel gears (planetary) |

h_{f4} | dedendum depth for zero pair of bevel gears (sun) |

i_{d} | transmission ratio of the differential transmission |

i_{1}, i_{2}, i_{3}, i_{4}, i_{5}, i_{R} | gearbox transmission ratios |

K_{FX} | size influence factor |

K_{Fα} | load distribution factor |

K_{Fβ} | load distribution factor along the length of the tooth flank |

K_{HX} | factor of dimensions influence on the load capacity of the tooth flank |

K_{Hα} | load distribution factor |

K_{Hβ} | load distribution factor along the length of the tooth flank |

K_{L} | influence factor of lubricating oil |

m_{all} | overall mass of optimized design elements |

m_{f} | module considering of load of tooth face flank |

m_{n} | mean normal module |

m_{r} | module considering of load of tooth root |

m_{v, max} | maximum vehicle mass |

m_{1} | mass of the ring gear |

m_{2} | mass of the drive shaft |

P_{max} | driving machine power (engine power) |

P_{o max} | power on the motor output shaft |

P_{1} | power delivered to the drive bevel gear (pinion gear) |

P_{2} | output power of the differential transmission |

R_{a} | length of the derivative of the dividing cone |

R_{mint} | intermediate variable |

R_{m1} | mean cone distance of the pinion |

R_{m2} | mean cone distance of the ring gear |

r | inner radius of the drive shaft |

S_{Fmin} | minimum safety factor against root breakage |

S_{Hlim} | minimum anti-pitting factor |

S_{Hmin} | minimum safety factor against tooth flank fracture |

T_{L} | torques on the left drive wheel |

${T}_{\mathrm{L}}^{\prime}$ | torques on the left drive wheel |

T_{max} | maximum engine torque |

T_{R} | torques on the right drive wheel |

${T}_{\mathrm{R}}^{\prime}$ | torques on the right drive wheel |

T_{W} | torque on the drive wheels |

${T}_{\mathrm{W}}^{\prime}$ | torque on the drive wheels |

t_{zm1} | crossing point to calculation point along pinion axis |

t_{zm2} | crossing point to calculation point along ring gear axis |

T_{1} | input torque to the differential transmission |

T_{2} | torque of the hypoid ring bevel gear (with included efficiency of the open differential) |

${T}_{2}^{\prime}$ | torque of the hypoid ring bevel gear (without including the losses of the kinematic chain) |

u_{v} | transmission ratio of supplementary bevel gears |

Y_{F} | tooth shape factor |

Y_{N} | is service life factor |

Y_{R} | roughness factor |

Y_{S} | notch action factor |

Y_{β} | impact factor of tooth locking on the stress distribution in the root |

Y_{εV} | load share factor of auxiliary gears |

Z_{HV} | tooth shape factor |

Z_{M} | material factor |

Z_{N} | service life factor |

Z_{R} | roughness influence factor |

Z_{V} | speed influence factor |

Z_{W} | influence factor of flank hardening |

z_{1} | teeth number of the hypoid pinion bevel gear |

z_{2} | teeth number of the hypoid ring bevel gear |

z_{3} | teeth number of the planetary bevel gear |

z_{4} | teeth number of the sun bevel gear |

Z_{εV} | coverage factor |

α_{lim} | limit pressure angle |

α_{n} | mean normal pressure angle |

β_{m1} | spiral angle of hypoid pinion bevel gear |

β_{m2} | spiral angle of hypoid ring bevel gear |

γ | rising angle (slope angle) |

δ_{a3} | face angle of planetary gear |

δ_{a4} | face angle of sun gear |

δ_{1} | pinion gear pitch angle |

δ_{2} | ring gear pitch angle |

δ_{3} | planetary gear pitch angle |

η_{c} | efficiency of the cardan shaft |

η_{d} | efficiency of the open differential |

η_{g} | efficiency of the gearbox |

κ_{a3} | planetary gear teeth face angle |

κ_{a4} | sun gear teeth face angle |

λ | tooth width factor |

µ | friction factor for asphalt surface |

ΔΣ | shaft angle departure |

Σ | shaft angle |

σ_{Flim1} | amount of permanent dynamic strength for tooth root (pinion) |

σ_{Flim2} | amount of permanent dynamic strength for tooth root (ring gear) |

σ_{Flim3} | amount of permanent dynamic strength for tooth root (planetary gear) |

σ_{Flim4} | amount of permanent dynamic strength for tooth root (sun gear) |

σ_{F1} | bending stress on pinion tooth root |

σ_{F2} | bending stress on ring gear tooth root |

σ_{F3} | bending stress on planetary tooth root |

σ_{F4} | bending stress on sun tooth root |

σ_{FP1} | allowed stress on pinion tooth root |

σ_{FP2} | allowed stress on ring gear tooth root |

σ_{FP3} | allowed stress on planetary tooth root |

σ_{FP4} | allowed stress on sun tooth root |

σ_{Hlim1} | permanent dynamic strength for tooth face flank (pinion) |

σ_{Hlim2} | permanent dynamic strength for tooth face flank (ring gear) |

σ_{Hlim3} | permanent dynamic strength for tooth face flank (planetary gear) |

σ_{Hlim4} | permanent dynamic strength for tooth face flank (sun gear) |

σ_{H1} | Hertzian stress in the kinematic pole of pinion |

σ_{H2} | Hertzian stress in the kinematic pole of ring gear |

σ_{H3} | Hertzian stress in the kinematic pole of planetary gear |

σ_{H4} | Hertzian stress in the kinematic pole of sun gear |

σ_{HP1} | allowed Hertzian stress on pinion tooth face flank |

σ_{HP2} | allowed Hertzian stress on ring gear tooth face flank |

σ_{HP3} | allowed Hertzian stress on planetary tooth face flank |

σ_{HP4} | allowed Hertzian stress on sun tooth face flank |

ζ_{m} | pinion offset angle in the axial plane |

ζ_{mp} | pinion offset angle in the pitch plane |

υ_{m} | auxiliary angle |

ψ | contribution factor of rotating masses |

ω_{max} | angular velocity at which T_{max} is reached |

ω_{1} | angular velocity of the hypoid pinion bevel gear |

ω_{2} | angular velocity of the hypoid ring bevel gear |

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**Figure 9.**Submodel of the pinion and ring gear: (

**a**) finite element mesh; (

**b**) distribution of total displacement in mm.

**Figure 12.**Stress and displacement distribution of the drive shaft: (

**a**) distribution of equivalent von Mises stress in MPa; (

**b**) total deformation in mm.

**Figure 13.**Ring gear geometry: (

**a**) geometry obtained by the topological optimization method; (

**b**) corrected geometry.

**Figure 16.**Values of the numerical model versus predicted values: (

**a**) mass of the ring gear; (

**b**) mass of the drive shaft; (

**c**) overall mass of the optimized design elements.

**Figure 17.**Sensitivity indices for the responses: overall mass, mass of the ring gear and mass of the drive shaft.

**Figure 18.**Stress and displacement distribution of the optimized drive shaft: (

**a**) distribution of equivalent stresses according to von Mises in MPa; (

**b**) total displacements in mm.

**Figure 20.**Mass comparison of the design elements of the differential transmission before and after optimization.

Design Requirements | Amount |
---|---|

Maximum vehicle speed at the highest transmission ratio (v_{max}), km/h | 220 |

Time required to reach a speed of 100 km/h (t_{0–100}), s | 8.9 |

Vehicle mass (m_{v}), kg | 1550 |

Maximum vehicle mass (m_{v, max}), kg | 2100 |

Vehicle acceleration (a_{v}), m/s^{2} | 2.5 |

Dynamic wheel radius (r_{d}), mm | 394.2 |

Technical characteristics of the driving machine | |

Max. engine torque (T_{max}), Nm | 340 |

Number of revolutions at which T_{max} is reached (n_{Tmax}), min^{−1} | 1600 |

Power delivered to the drive bevel gear (P_{1}), kW | 53.05 |

Input torque to the differential transmission (T_{1}), Nm | 1108.14 |

**Table 2.**Values of forces, transmission ratio, power and torques from the transmission’s input/output side.

Differential Transmission Forces and Torques | Amount |
---|---|

Reaction force in the wheel (F_{w}), N | 5150.25 |

Friction force (F_{f}), N | 4635.23 |

Traction force (F_{TR}), N | 9270.45 |

Transmission ratio of the differential transmission (i_{d}) | 3.286 |

Output power of the differential transmission (P_{2}), kW | 51.457 |

Rolling resistance force (F_{RO}), N | 412.02 |

Rising resistance force (F_{RI}), N | 2153.39 |

Vehicle’s inertial force (F_{IN}), N | 6116.25 |

Total resistance force of driven machine (F_{TRD}) | 8681.66 |

Torque of the hypoid ring bevel gear (T_{2}), Nm | 3546.31 |

Torque of drive wheel (T_{W}, T_{R}, T_{L}), Nm | 1773.16 |

Variables | First Iteration | Last Iteration |
---|---|---|

Ring gear mean pitch diameter (d_{m2}), mm | 304.08 | 305.59 |

Pinion offset angle in the axial plane (ζ_{m}), ° | 11.28 | 10.51 |

Pinion offset angle in the pitch plane (ζ_{mp}), ° | 11.78 | 11.34 |

Pinion gear pitch angle (δ_{1}), ° | 16.59 | 22.31 |

Ring gear pitch angle (δ_{2}), ° | 67.57 | 67.69 |

Mean normal module (m_{n}), mm | 5.41 | 5.44 |

Spiral angle of pinion (β_{m1}), ° | 46.78 | 46.34 |

Hypoid dimensional factor (F) | 1.196 | 1.187 |

Pinion mean pitch diameter (d_{m1}), mm | 110.7 | 110.36 |

Mean cone distance of the pinion (R_{m1}), mm | 190.07 | 145.35 |

Mean cone distance of the ring gear (R_{m2}), mm | 158.93 | 165.16 |

Auxiliary angle (υ_{m}), ° | 10.8 | 9.74 |

Iteration condition: ($\left|{R}_{\mathrm{mint}}-{R}_{\mathrm{m}1}\right|<0.0001\xb7{R}_{\mathrm{m}1}$) | |−47.537| < 0.019 | |−0.010| < 0.0145 |

Geometric Quantities | Amount | |
---|---|---|

Pinion | Ring Gear | |

Shaft angle departure (ΔΣ), ° | 0 | 0 |

Pinion offset angle in the axial plane (ζ_{m}), ° | 10.68 | - |

Pinion offset angle in the pitch plane (ζ_{mp}), ° | 11.55 | - |

Mean normal module (m_{n}), mm | 5.5 | 5.5 |

Limit pressure angle (α_{lim}), ° | −1.32 | −1.32 |

Mean normal pressure angle (α_{n}), ° | 20 | 20 |

Crossing point to calculation point along ring gear axis (t_{zm2}), mm | - | 55 |

Crossing point to calculation point along pinion axis (t_{zm1}), mm | 149.71 | - |

Pinion gear | Stress on tooth root | σ_{FP1} = 357.14 MPa | σ_{F1} < σ_{FP1} |

F_{mt1} = 20,082.3 N | |||

σ_{F1} = 226.79 MPa | |||

Stress on tooth face flank | σ_{HP1} = 1310.14 MPa | σ_{H1} < σ_{HP1} | |

σ_{H1} = 1286 MPa | |||

Ring gear | Stress on tooth root | σ_{FP2} = 357.14 MPa | σ_{F2} < σ_{FP2} |

F_{mt2} = 23,987 N | |||

σ_{F2} = 149 MPa | |||

Stress on tooth face flank | σ_{HP2} = 1324 MPa | σ_{H2} < σ_{HP2} | |

σ_{H2} = 852.23 MPa |

Geometric Quantities | Mark | Amount | ||
---|---|---|---|---|

Planetary | Sun | Planetary | Sun | |

Pitch circle diameter, mm | d_{3} | d_{4} | 102 | 136 |

Length of the derivative of the dividing cone, mm | R_{a} | 85 | ||

Tooth width, mm | b_{3} | b_{4} | 26 | 26 |

Middle pitch diameter of bevel gear, mm | d_{m3} | d_{m4} | 85 | 115.2 |

Clearance, mm | c | 2.125 | ||

Whole depth, mm | h_{3} | h_{4} | 19.125 | 19.125 |

Addendum depth for zero pair of bevel gears, mm | h_{a3} | h_{a4} | 8.5 | 8.5 |

Dedendum depth for zero pair of bevel gears, mm | h_{f3} | h_{f4} | 10.625 | 10.625 |

Addendum circle diameter, mm | d_{a3} | d_{a4} | 115.599 | 146.2 |

Teeth face angle, ° | κ_{a3} | κ_{a4} | 5.71 | 5.71 |

Face angle of bevel gear, ° | δ_{a3} | δ_{a4} | 42.58 | 58.84 |

Inner addendum diameter of tooth, mm | d_{ia3} | d_{ia4} | 84.24 | 101.48 |

Pitch diameter of middle equivalent gear, mm | d_{vm3} | d_{vm4} | 106.25 | 192 |

Pitch diameter of equivalent gear, mm | d_{v3} | d_{v4} | 127.5 | 226.67 |

Addendum diameter of equivalent gear, mm | d_{va3} | d_{va4} | 144.5 | 243.67 |

Base diameter of equivalent gear, mm | d_{vb3} | d_{vb4} | 119.81 | 213 |

Planetary gear | Stress on tooth root | σ_{FP3} = 512.15 MPa | σ_{F3} < σ_{FP3} |

F_{mt3} = 15.391 N | |||

σ_{F3} = 274.93 MPa | |||

Stress on tooth face flank | σ_{HP3} = 1441.1 MPa | σ_{H3} < σ_{HP3} | |

σ_{H3} = 1400.04 MPa | |||

Sun gear | Stress on tooth root | σ_{FP4} = 512.5 MPa | σ_{F4} < σ_{FP4} |

F_{mt4} = 15.391 N | |||

σ_{F4} = 224.03 MPa | |||

Stress on tooth face flank | σ_{HP4} = 1441.1 MPa | σ_{H4} < σ_{HP4} | |

σ_{H4} = 954.44 MPa |

Parameters | Name | Lower Value, mm | Upper Value, mm |
---|---|---|---|

a | Vertical chamfer of the ring gear | 1 | 7 |

b | Horizontal chamfer of the ring gear | 1 | 13 |

h | Overhang height of the ring gear | 1 | 3 |

r | Inner radius of the drive shaft | 2.5 | 13.5 |

Response | Name | Description |
---|---|---|

m_{all} | Overall mass of optimized design elements | Objective function—mass reduction of the optimized design elements |

m_{1} | Mass of the ring gear | Objective function—mass reduction of the ring gear |

m_{2} | Mass of the drive shaft | Objective function—mass reduction of the drive shaft |

No. | a, mm | b, mm | h, mm | r, mm | No. | a, mm | b, mm | h, mm | r, mm |
---|---|---|---|---|---|---|---|---|---|

1. | 4 | 7 | 2 | 8 | 14. | 6.11 | 2.77 | 1.30 | 4.13 |

2. | 4 | 7 | 2 | 2.5 | 15. | 6.11 | 2.77 | 1.30 | 11.87 |

3. | 4 | 7 | 2 | 13.5 | 16. | 6.11 | 2.77 | 2.70 | 4.13 |

4. | 4 | 7 | 1 | 8 | 17. | 6.11 | 2.77 | 2.70 | 11.87 |

5. | 4 | 7 | 3 | 8 | 18. | 1.89 | 11.23 | 1.30 | 4.13 |

6. | 1 | 7 | 2 | 8 | 19. | 1.89 | 11.23 | 1.30 | 11.87 |

7. | 7 | 7 | 2 | 8 | 20. | 1.89 | 11.23 | 2.70 | 4.13 |

8. | 4 | 1 | 2 | 8 | 21. | 1.89 | 11.23 | 2.70 | 11.87 |

9. | 4 | 13 | 2 | 8 | 22. | 6.11 | 11.23 | 1.30 | 4.13 |

10. | 1.89 | 2.77 | 1.30 | 4.13 | 23. | 6.11 | 11.23 | 1.30 | 11.87 |

11. | 1.89 | 2.77 | 1.30 | 11.87 | 24. | 6.11 | 11.23 | 2.70 | 4.13 |

12. | 1.89 | 2.77 | 2.70 | 4.13 | 25. | 6.11 | 11.23 | 2.70 | 11.87 |

13. | 1.89 | 2.77 | 2.70 | 11.87 |

Responses | R^{2} | RMSE |
---|---|---|

Overall mass | 0.99532 | 0.03193 |

Mass of the ring gear | 0.99589 | 0.00149 |

Mass of the drive shaft | 0.99988 | 0.00700 |

Parameter | Value, mm |
---|---|

a | 7 |

b | 13 |

h | 3 |

r | 13.5 |

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## Share and Cite

**MDPI and ACS Style**

Karakašić, M.; Konjatić, P.; Glavaš, H.; Grgić, I.
Influence of Open Differential Design on the Mass Reduction Function. *Appl. Sci.* **2023**, *13*, 13300.
https://doi.org/10.3390/app132413300

**AMA Style**

Karakašić M, Konjatić P, Glavaš H, Grgić I.
Influence of Open Differential Design on the Mass Reduction Function. *Applied Sciences*. 2023; 13(24):13300.
https://doi.org/10.3390/app132413300

**Chicago/Turabian Style**

Karakašić, Mirko, Pejo Konjatić, Hrvoje Glavaš, and Ivan Grgić.
2023. "Influence of Open Differential Design on the Mass Reduction Function" *Applied Sciences* 13, no. 24: 13300.
https://doi.org/10.3390/app132413300