This is machine translation

Translated by Microsoft
Mouseover text to see original. Click the button below to return to the English version of the page.

Note: This page has been translated by MathWorks. Click here to see
To view all translated materials including this page, select Country from the country navigator on the bottom of this page.

Limited Slip Differential

Limited differential as a planetary bevel gear

  • Library:
  • Powertrain Blockset / Drivetrain / Final Drive Unit

    Vehicle Dynamics Blockset / Powertrain / Drivetrain / Final Drive Unit

Description

The Limited Slip Differential block implements a differential as a planetary bevel gear train. The block matches the driveshaft bevel gear to the crown (ring) bevel gear. You can specify:

  • Carrier-to-driveshaft ratio

  • Crown wheel location

  • Viscous and damping coefficients for the axles and carrier

  • Type of slip coupling

Use the block in system-level driveline analysis to account for the power transfer from the transmission to the wheels. The block is suitable for use in hardware-in-the-loop (HIL) and optimization workflows. All the parameters are tunable.

In a limited slip differential, to prevent one of the wheels from slipping, the differential splits the torque applied to the left and right axles. With different torque applied to the axles, the wheels can move at different angular velocities, preventing slip. The block implements three methods for coupling the different torques applied to the axes:

  • Pre-loaded ideal clutch

  • Slip speed-dependent torque data

  • Input torque dependent torque data

The block uses a coordinate system that produces positive tire and vehicle motion for standard engine, transmission, and differential configurations. The arrows indicate positive motion.

Efficiency

To account for the block efficiency, use the Efficiency factors parameter. This table summarizes the block implementation for each setting.

SettingImplementation

Constant

Constant efficiency that you can set with the Constant efficiency factor, eta parameter.

Driveshaft torque, temperature and speed

Efficiency as a function of base gear input torque, air temperature, and driveshaft speed. Use these parameters to specify the lookup table and breakpoints:

  • Efficiency lookup table, eta_tbl

  • Efficiency torque breakpoints, Trq_bpts

  • Efficiency speed breakpoints, omega_bpts

  • Efficiency temperature breakpoints, Temp_bpts

For the air temperature, you can either:

  • Select Input temperature to create an input port.

  • Set a Ambient temperature, Tamb parameter value.

To select the interpolation method, use the Interpolation method parameter. For more information, see Interpolation Methods (Simulink).

Power Accounting

For the power accounting, the block implements these equations.

Bus Signal DescriptionEquations

PwrInfo

PwrTrnsfrd — Power transferred between blocks

  • Positive signals indicate flow into block

  • Negative signals indicate flow out of block

PwrDriveshft

Mechanical power from driveshaft

ηTdωd

PwrAxl1

Mechanical power from axle 1

ηT1ω1

PwrAxl2

Mechanical power from axle 2

ηT2ω2

PwrNotTrnsfrd — Power crossing the block boundary, but not transferred

  • Positive signals indicate an input

  • Negative signals indicate a loss

PwrMechLoss

Total power loss

W˙loss= (Pt+Pd+Pc)+PsPt=η(Tdωd+T1ω1+T2ω2)

PwrDampLoss

Power loss due to damping

Pd=(b1|ω1|+b2|ω2|+bd|ωd|)

PwrCplngLoss

Power loss due to clutch

Pc=Tc|ω¯|

PwrStored — Stored energy rate of change

  • Positive signals indicate an increase

  • Negative signals indicate a decrease

PwrStoredShft

Rate change of stored internal energy

Ps=(ω1ω˙1J1+ω2ω˙2J2+ωdω˙dJd)

Dynamics

The Limited Slip Differential block implements these differential equations to represent the mechanical dynamic response for the crown gear, left axle, and right axle.

Mechanical Dynamic ResponseDifferential Equation
Crown Gear

ω˙dJd=ηTd-ωdbd-Ti

Left Axle

ω˙1J1=ηT1-ω1b1-Ti1

Right Axle

ω˙2J2=ηT2-ω2b2-Ti2

The block assumes rigid coupling between the crown gear and axles. These constraint equations apply.

ηT1=N2Ti12TcηT2=N2Ti+12Tc

ωd=N2(ω1+ω2)

The equations use these variables.

N

Carrier-to-driveshaft gear ratio

Jd

Rotational inertia of the crown gear assembly

bd

Crown gear linear viscous damping

ωd

Driveshaft angular speed

ϖ

Slip speed

J1

Axle 1 rotational inertia

b1

Axle 1 linear viscous damping

ω1

Axle 1 speed

J2

Axle 2 rotational inertia

b2

Axle 2 linear viscous damping

ω2

Axle 2 angular speed

η

Efficiency

Td

Driveshaft torque

T1

Axle 1 torque

T2

Axle 2 torque

Ti

Axle internal resistance torque

Ti1

Axle 1 internal resistance torque

Ti2

Axle 2 internal resistance torque

μ

Coefficient of friction

Reff

Effective clutch radius

Ro

Annular disk outer radius

Ri

Annular disk inner radius

Fc

Clutch force

Tc

Clutch torque

μ

Coefficient of friction

Table blocks in the Limited Slip Differential have these parameter settings:

  • Interpolation methodLinear

  • Extrapolation methodClip

Ideal Clutch Coupling

The ideal clutch coupling model uses the axle slip speed and friction to calculate the clutch torque. The friction coefficient is a function of the slip speed.

Tc=FcNμ(|ϖ|)Refftanh(4|ϖ|)

The disc radii determine the effective clutch radius over which the clutch force acts.

Reff=2(Ro3-Ri3)3(Ro2-Ri2)

The angular velocities of the axles determine the slip speed.

ϖ=ω1ω2

Slip Speed Coupling

To calculate the clutch torque, the slip speed coupling model uses torque data that is a function of slip speed. The angular velocities of the axles determine the slip speed.

ϖ=ω1ω2

Input Torque Coupling

To calculate the clutch torque, the input torque coupling model uses torque data that is a function of input torque.

The Open Differential block assumes rigid coupling between the crown gear and axles. These constraint equations apply.

ηT1= ηT2=N2Ti

ωd==N2(ω1+ω2)

Ports

Inputs

expand all

Applied input torque, typically from the engine crankshaft, in N·m.

Axle 1 torque, T1, in N·m.

Axle 2 torque, T2, in N·m.

Temperature, in K.

Dependencies

To create this port:

  • Set Efficiency factors to Driveshaft torque, speed and temperature.

  • Select Input temperature.

Output

expand all

Bus signal containing these block calculations.

SignalDescriptionUnits

Driveshft

DriveshftTrq

Driveshaft torque

N·m

DriveshftSpd

Driveshaft speed

rad/s

Axl1

Axl1Trq

Axle 1 torque

N·m

Axl1Spd

Axle 1 speed

rad/s

Axl2

Axl2Trq

Axle 2 torque

N·m

Axl2Spd

Axle 2 speed

rad/s

Cplng

CplngTrq

Torque coupling

N·m

CplngSlipSpd

Slip speed

rad/s

PwrInfo

PwrTrnsfrd

PwrDriveshft

Mechanical power from driveshaft

W

PwrAxl1

Mechanical power from axle 1

W

PwrAxl2

Mechanical power from axle 2

W

PwrNotTrnsfrd

PwrMechLoss

Total power loss

W

PwrDampLoss

Power loss due to damping

W

PwrCplngLoss

Power loss due to clutch

W

PwrStoredShft

PwrStoredShft

Rate change of stored internal energy

W

Driveshaft angular speed, ωd, in rad/s.

Axle 1 angular speed, ω1, in rad/s.

Axle 2 angular speed, ω2, in rad/s.

Parameters

expand all

Block Options

To account for the block efficiency, use the Efficiency factors parameter. This table summarizes the block implementation for each setting.

SettingImplementation

Constant

Constant efficiency that you can set with the Constant efficiency factor, eta parameter.

Driveshaft torque, temperature and speed

Efficiency as a function of base gear input torque, air temperature, and driveshaft speed. Use these parameters to specify the lookup table and breakpoints:

  • Efficiency lookup table, eta_tbl

  • Efficiency torque breakpoints, Trq_bpts

  • Efficiency speed breakpoints, omega_bpts

  • Efficiency temperature breakpoints, Temp_bpts

For the air temperature, you can either:

  • Select Input temperature to create an input port.

  • Set a Ambient temperature, Tamb parameter value.

To select the interpolation method, use the Interpolation method parameter. For more information, see Interpolation Methods (Simulink).

For more information, see Interpolation Methods (Simulink).

Dependencies

To enable this parameter, set Efficiency factors to Driveshaft torque, speed and temperature.

Select to create input port Temp for the temperature.

Dependencies

To enable this parameter, set Efficiency factors to Driveshaft torque, speed and temperature.

Open Differential

Specify the crown wheel connection to the driveshaft.

Carrier-to-driveshaft gear ratio, N.

Rotational inertia of the crown gear assembly, Jd, in kg·m^2. You can include the driveshaft inertia.

Crown gear linear viscous damping, bd, in N·m·s/rad.

Axle 1 rotational inertia, J1, in kg·m^2.

Axle 1 linear viscous damping, b1, in N·m·s/rad.

Axle 2 rotational inertia, J2, in kg·m^2.

Axle 2 linear viscous damping, b2, in N·m·s/rad.

Axle 1 initial velocity, ωo1, in rad/s.

Axle 2 initial velocity, ωo2, in rad/s.

Constant efficiency, η.

Dependencies

To enable this parameter, set Efficiency factors to Constant.

Dimensionless array of values for efficiency as a function of:

  • M input torques

  • N input speed

  • L air temperatures

Each value specifies the efficiency for a specific combination of torque, speed, and temperature. The array size must match the dimensions defined by the torque, speed, and temperature breakpoint vectors.

Dependencies

To enable this parameter, set Efficiency factors to Driveshaft torque, speed and temperature.

Vector of input torque, breakpoints for efficiency, in N·m.

Dependencies

To enable this parameter, set Efficiency factors to Driveshaft torque, speed and temperature.

Vector of speed, breakpoints for efficiency, in rad/s.

Dependencies

To enable this parameter, set Efficiency factors to Driveshaft torque, speed and temperature.

Vector of ambient temperature breakpoints for efficiency, in K.

Dependencies

To enable this parameter, set Efficiency factors to Driveshaft torque, speed and temperature.

Ambient air temperature, Tair, in K.

Dependencies

To enable this parameter:

  • Set Efficiency factors to Driveshaft torque, speed and temperature.

  • Clear Input temperature.

Slip Coupling

Specify the type of torque coupling.

Number of disks.

Dependencies

To enable the ideal clutch parameters, select Pre-loaded ideal clutch for the Coupling type parameter.

The effective radius, Reff, used with the applied clutch friction force to determine the friction force. The effective radius is defined as:

Reff=2(Ro3-Ri3)3(Ro2-Ri2)

The equation uses these variables.

Ro

Annular disk outer radius

Ri

Annular disk inner radius

Dependencies

To enable the clutch parameters, select Pre-loaded ideal clutch for the Coupling type parameter.

Nominal preload force, in N.

Dependencies

To enable the clutch parameters, select Pre-loaded ideal clutch for the Coupling type parameter.

Friction coefficient vector.

Dependencies

To enable the clutch parameters, select Pre-loaded ideal clutch for the Coupling type parameter.

Slip speed vector, in rad/s.

Dependencies

To enable the clutch parameters, select Pre-loaded ideal clutch for the Coupling type parameter.

Torque vector, in N·m.

Dependencies

To enable the slip speed parameters, select Slip speed dependent torque data for the Coupling type parameter.

Slip speed vector, in rad/s.

Dependencies

To enable the slip speed parameters, select Slip speed dependent torque data for the Coupling type parameter.

Torque vector, in N·m.

Dependencies

To enable the input torque parameters, select Input torque dependent torque data for the Coupling type parameter.

Torque vector, in N·m.

Dependencies

To enable the input torque parameters, select Input torque dependent torque data for the Coupling type parameter.

Coupling time constant, in s.

References

[1] Deur, J., Ivanović, V., Hancock, M., and Assadian, F. "Modeling of Active Differential Dynamics." In ASME proceedings. Transportation Systems. Vol. 17, pp: 427-436.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Introduced in R2017a