# Modeling and simulation of a vehicle with automatic transmission

### 4. Plant model: Automatic Transmission

The transmission consist of a torque converter, planetary gearbox and a differential. The torque converter is connected to the engine on the impeller side and to the gearbox input shaft on the turbine side. The role of the torque converter is to allow the engine to be disconnected from the gearbox at standstill, and assure a linear, smooth engine coupling with the gearbox at vehicle launch. The torque converter also act as an engine torque amplification device, especially at high speed slip ratio.

The transmission is made up from several planetary gear sets. A planetary gear (also known as epicyclic gear train) consists of two gears mounted so that the center of one gear revolves around the center of the other. A carrier connects the centers of the two gears and rotates to carry one gear, called the planet gear or planet pinion, around the other, called the sun gear or sun wheel. For this simulation example we are only going to consider the gear ratio and the mechanical efficiency in each gear.

The top level Xcos block diagram of the transmission contains two subsystems, the Torque Converter and the Gearbox and Differential.

Image: Transmission – Xcos block diagram

#### 4.1 Torque Converter

For a better understanding on how a torque converter works, read the article: How a torque converter works. For this study a Sachs s244 torque converter is used, with the following performance and torque coefficient characteristics:

 itc [-] ktc [-] λtc [-] 0 2.163 0.169 0.11 2.063 0.169 0.22 1.943 0.168 0.33 1.804 0.166 0.44 1.652 0.163 0.55 1.493 0.159 0.66 1.323 0.154 0.77 1.152 0.146 0.825 1.072 0.138 0.861 1.018 0.126 0.897 0.972 0.112 0.906 0.965 0.108 0.915 0.963 0.104 0.924 0.963 0.099 0.932 0.961 0.094 0.941 0.959 0.087 0.95 0.952 0.078 0.959 0.935 0.067 0.968 0.908 0.051 0.977 0.852 0.033
where:
itc [-] – torque converter speed ratio
ktc [-] – torque converter torque coefficient
λtc [-] – torque converter performance coefficient

The torque converter torque and performance characteristics function of the speed ratio are displayed in graphical format below.

Image: Torque converter characteristics – Sachs s244

The governing equations of the torque converter [1] are:

$T_{t} = k_{tc} \cdot T_{i} \tag{5}$
where:
Tt [-] – turbine torque
Ti [-] – impeller torque

The impeller torque is calculated with the equation:

$T_{i} = \rho \cdot \lambda_{tc} \cdot N_{i}^{2} \cdot D_{tc}^{5} \tag{6}$
where:
ρ [kg/m3] – torque converter fluid density
Ni [rpm] – impeller speed
Dtc [m] – torque converter (impeller) active diameter

The torque converter speed ratio is calculated as:

$i_{tc} = \frac{N_{t}}{N_{i}} = \frac{N_{t}}{N_{e}} \tag{7}$
where:
Nt [rpm] – turbine speed
Ne [rpm] – engine speed

Equations (5), (6) and (7) are used to build the Xcos block diagram for the torque converter.

Image: Torque Converter – Xcos block diagram

Inputs

 Name Value Description TrbN_rpm – Turbine speed [rpm] EngN_rpm – Engine speed [rpm]

Parameters

 Name Value Description ImplActvDiam_m_C 0.244 Torque converter (impeller) active diameter [m] TqCnvrOilRho_kgpm3_C 900 Torque converter fluid density [kg·m3] TqCnvrNRat_z_X (table) Torque converter speed ratio axis [-] TqCnvrPrfmncCoeff_z_Z (table) Torque converter performance coefficient map [-] TqCnvrTqRat_z_Z (table) Torque converter performance coefficient map [-]

Outputs

 Name Value Description ImplTq_Nm – Impeller torque [Nm] TrbTq_Nm – Turbine torque [Nm]

#### 4.2 Gearbox and Differential

The governing equations of the gearbox model are:

$T_{g} = T_{t} \cdot i_{x} \cdot \eta_{x} \tag{8}$
where:
Tg [Nm] – gearbox torque
ix [-] – gearbox gear ratio
ηx [-] – gearbox gear efficiency

The gear ratio and gearbox efficiency in each gear are displayed in the table and images below.

 Gear [-] ix [-] ηx [-] 1 4.381 0.930 2 2.860 0.948 3 1.917 0.973 4 1.363 0.987 5 1.000 1.000 6 0.822 0.987 7 0.731 0.987

Image: Gearbox ratios

The turbine speed is calculated as:

$N_{t} = N_{w} \cdot i_{x} \cdot i_{0} \tag{9}$
where:
Nw [rpm] – wheel speed
i0 [-] – final gear (differential) gear ratio

Equations (8) and (9) are used to build the Xcos block diagram for the gearbox and differential.

Image: Gearbox and Differential – Xcos block diagram

Inputs

 Name Value Description TrbTq_Nm – Turbine torque [Nm] GearNr_z – Current gear [-] WhlN_rpm – Wheel speed [rpm]

Parameters

 Name Value Description DftlGearRat_z_C 2.769 Final drive (differential) gear ratio [-] GbxGearNr_z_X (table) Gear number axis [-] GbxGearRat_z_Z (table) Gear ratio map [-] GbxGearEff_z_Z (table) Gear efficiency map [-]

Outputs

 Name Value Description GbxTq_Nm – Gearbox torque [Nm] TrbN_rpm – Turbine speed [rpm]