Limited-Slip Differential

Reduce velocity difference between two connected shafts

Libraries:
Simscape / Driveline / Gears

Description

The Limited-Slip Differential block represents a limited-slip differential (LSD), which is a gear assembly that limits the velocity difference between two connected shafts. The block models the LSD mechanism as a structural component that combines a differential and a clutch.

The differential component in the LSD block is an open differential. An open differential is a gear mechanism that allows two driven shafts to spin at different speeds. In an automobile, a differential allows the inner wheels to spin more slowly than the outer wheels when the vehicle is cornering. A vehicle that has wheel shafts connected by an open differential can get stuck when one of the wheels slips and spins freely due to traction loss. This vehicle stops moving because the driveshaft supplies less power to the wheel with traction than it supplies to the spinning wheel.

In the same scenario, a vehicle that has an LSD is less likely to get stuck because it contains a clutch assembly that can transmit power to the wheel that retains traction. The clutch component in the LSD block is a friction clutch that has two sets of flat friction plates. The clutch engages when applied pressure exceeds the engagement threshold pressure. In an LSD, a spring preload that separates the sun gears presses the plates in both sets together. When the shafts experience a traction differential, the planet pinion gears exert an additional force in the direction of the high-traction shaft. If the additional pressure exceeds the engagement threshold, the clutch assembly engages. The engagement allows the driveshaft to transmit more power to the slower-spinning high-traction wheel. The additional power reduces the difference in velocity of the two shafts. Because the high-traction wheel continues to rotate, the vehicle continues to move.

The figure shows the orientation of the major components in an LSD mechanism. The driveshaft pinion gear is not visible is this representation.

The Limited-Slip Differential block models the LSD mechanism as a structural component based on the Simscape™ Driveline™ Differential and Disc Friction Clutch blocks. The differential mechanism modeled by the Differential block is a structural component based on two other Simscape Driveline blocks, the Simple Gear and the Sun-Planet Bevel. The block diagram shows the structural components of the LSD.

The ports of the Limited-Slip Differential block are associated with the driveshaft (port D) and the two driven shafts (ports S1 and S2), which connect the sun gears to the wheels.

The Limited-Slip Differential block enables you to specify inertias only for the gear carrier and internal planet gears. By default, the inertias of the outer gears are assumed negligible. To model the inertias of the outer gears, connect Simscape Inertia blocks to the D, S1, and S2 ports.

The table shows the rotation direction of the driven shaft ports for different block parameterizations and input conditions.

Rotation Direction of the Driven Shaft Ports (S1 and S2)	Crown Gear Location Relative to the Centerline	Rotation Direction of Driveshaft Port D	Relative Slippage Across the Differential
Positive	Right	Positive	0
Positive for the nonslipping port Negative the slipping port	Right	Positive	> 0
Negative	Right	Negative	0
Negative for the nonslipping port Positive the slipping port	Right	Negative	> 0
Negative	Left	Positive	0
Negative for the nonslipping port Positive the slipping port	Left	Positive	> 0
Positive	Left	Negative	0
Positive for the nonslipping port Negative the slipping port	Left	Negative	> 0

Model

To examine the mathematical models for the structural components of the Limited-Slip Differential block, see:

Thermal Model

You can model the effects of heat flow and temperature change by enabling the optional thermal port. To enable the port, set Friction model to Temperature-dependent efficiency.

Examples

Limited Slip Differential with Clutches

A comparison between the behavior of an open differential and a limited slip differential with clutch packs. The limited slip differential is modeled using components from the Gears library and Clutches library in Simscape™ Driveline™. Wheel slip is limited by clutches that engage when the torque applied to the input of the differential exceeds a threshold. The clutches lock the differential so that the output shafts of the differential spin at the same speed.

Open Model

Electro-Hydraulic Limited Slip Differential

A simple way of modeling an electro-hydraulic limited slip differential system. The model shows the velocity and the torque profile responses achieved for the left and the right wheels.

Open Model

Ports

Conserving

expand all

D — Driving shaft
rotational mechanical

Rotational mechanical conserving port associated with the driveshaft.

S1 — Sun gear 1
rotational mechanical

Rotational mechanical conserving port associated with the sun gear 1 shaft.

S2 — Sun gear 2
rotational mechanical

Rotational mechanical conserving port associated with the sun gear 2 shaft.

H — Heat flow
thermal

Thermal conserving port associated with heat flow.

Dependencies

To enable this port, in the Differential tab, set Friction model to Temperature-dependent efficiency.

Parameters

expand all

Differential

Crown gear located — Crown gear location
`To the right of centerline` (default) | `To the left of centerline`

Location of the bevel crown gear relative to the centerline of the gear assembly.

Carrier (C) to driveshaft (D) teeth ratio (NC/ND) — Crown gear to driveshaft pinion gear teeth ratio
`4` (default) | positive scalar

Fixed ratio, g_D, of the number of crown gear teeth NC to the number of driveshaft pinion gear teeth ND. This gear ratio must be strictly greater than 0. The crown gear is rigidly mounted to the carrier.

Friction model — Friction model
`No meshing losses - Suitable for HIL simulation` (default) | `Constant efficiency` | `Temperature-dependent efficiency`

Friction model for the block:

No meshing losses - Suitable for HIL simulation— Gear meshing is ideal.
Constant efficiency— Transfer of torque between the gear wheel pairs is reduced by a constant efficiency, η, such that $0 < η \leq 1$ .
Temperature-dependent efficiency— Transfer of torque between the gear wheel pairs is defined by the temperature lookup table

Sun-sun and carrier-driveshaft ordinary efficiencies — Differential torque constant transfer efficiencies
`[.85, .92]` (default) | vector

Vector of torque transfer efficiencies, [η_SS, η_CD]. Here,

η_SS is the output-to-input power ratio that describes the power flow from the driving sun gear to the driven sun gear.
η_CD is the output-to-input power ratio that describes the power flow from the crown gear to the driveshaft pinion gear.

The carrier is rigidly mounted to the crown gear. The vector elements must be in the range (0,1].

Dependencies

To enable this parameter, set Friction model to Constant efficiency.

Temperature — Temperature
`[280, 300, 320]` `K` (default) | vector

Vector of temperatures used to construct a 1-D temperature-efficiency lookup table. The vector elements must increase from left to right.

Dependencies

To enable this parameter, set Friction model to Temperature-dependent efficiency.

Sun-sun efficiency — Torque transfer efficiency from the driving sun gear to the driven sun gear
`[.95, .9, .85]` (default) | vector

Vector of output-to-input power ratios that describe the power flow from the driving sun gear to the driven sun gear, η_SS. The block uses the values to construct a 1-D temperature-efficiency lookup table.

Each element is an efficiency that relates to a temperature in the Temperature vector. The length of the vector must be equal to the length of the Temperature vector. Each element in the vector must be in the range (0,1].

Dependencies

To enable this parameter, set Friction model to Temperature-dependent efficiency.

Carrier-driveshaft efficiency — Torque transfer efficiency from the crown gear to the driveshaft pinon gear
`[.95, .9, .85]` (default) | vector

Vector of output-to-input power ratios that describe the power flow from the crown gear to the driveshaft pinion gear, η_CD. The block uses the values to construct a 1-D temperature-efficiency lookup table. The carrier gear is rigidly mounted to the crown gear.

Dependencies

To enable this parameter, set Friction model to Temperature-dependent efficiency.

Sun-carrier and driveshaft-casing power thresholds — Minimum efficiency power thresholds for the sun gear to the crown gear and driveshaft-casing gear couplings
`[.001, .001]` `W` (default) | vector

Vector of power thresholds, p_th, for sun-carrier and longitudinal driveshaft-casing [p_S, p_D], respectively. The full efficiency loss applies above these values. Below these values, a hyperbolic tangent function smooths the efficiency factor.

When you set Friction model to Constant efficiency, the block lowers the efficiency losses to zero when no power is transmitted. When you set Friction model to Temperature-dependent efficiency, the block smooths the efficiency factors between zero when at rest and the values provided by the temperature-efficiency lookup tables at the power thresholds.

Dependencies

To enable this parameter, set Friction model to Constant efficiency or Temperature-dependent efficiency.

Sun-carrier and driveshaft-casing viscous friction coefficients — Viscous friction coefficients
`[0, 0]` `N*m/(rad/s)` (default) | vector

Vector of viscous friction coefficients [μ_S, μ_D] for the sun-carrier and longitudinal driveshaft-casing gear motions, respectively.

Inertia — Inertia model
`off` (default) | `on`

Inertia model for the block:

Off — Model gear inertia.
On — Neglect gear inertia.

Sun gear inertia — Sun gear inertia
`0.001` `kg*m^2` (default) | positive scalar

Sun gear inertia.

Dependencies

To enable this parameter, set Inertia to On.

Sun gear radius — Sun gear radius
`0.01` `m` (default) | positive scalar

Sun gear radius.

Dependencies

To enable this parameter, set Inertia to On.

Carrier inertia — Carrier inertia
`0.001` `kg*m^2` (default) | positive scalar

Moment of inertia of the planet gear carrier.

Dependencies

To enable this parameter, set Inertia to On.

Planet gear spin inertia — Planet gear spin inertia
`0.001` `kg*m^2` (default) | positive scalar

Planetary gear inertia about its own axis.

Dependencies

To enable this parameter, set Inertia to On.

Planet gear mass — Planet gear mass
`0.001` `kg*m^2` (default) | positive scalar

Planet gear mass.

Dependencies

To enable this parameter, set Inertia to On.

Clutch

Number of friction surfaces — Number of contact surfaces
`4` (default) | positive scalar

Number, N, of friction-generating contact surfaces inside the clutch.

Effective torque radius — Friction force application radius
`130` `mm` (default) | positive scalar

Effective moment arm radius, r_eff, that determines the kinetic friction torque inside the clutch.

Preload force — Clutch plate engagement pressure
`300` `N` (default) | nonegative scalar

Force that the spring preload exerts on the clutch plate assemblies. Must be greater than or equal to zero.

Static friction coefficient — Static friction coefficient
`0.35` (default) | positive scalar

Static or peak value of the friction coefficient. The static friction coefficient must be greater than the kinetic friction coefficient.

Dependencies

To enable this parameter, set Friction model to No meshing losses - Suitable for HIL simulation or Fixed kinetic friction coefficient.

Static friction coefficient vector — Static friction coefficient
`[.4, .38, .36]` (default) | vector

Vector of static or peak values of the friction coefficient for a given temperature. The vector must be the same length as the Temperature. Each element must be greater than the maximum value of the corresponding row in the Kinetic friction coefficient matrix.

Dependencies

To enable this parameter, set Friction model to Temperature-dependent efficiency.

Kinetic friction coefficient relative velocity vector — Absolute surface velocity
`[0, 100, 1000]` `rad/s` (default) | vector

Vector of input values for the relative velocity. The values in the vector must increase from left to right. The minimum number of values depends on the interpolation method that you select. For linear interpolation, provide at least two values per dimension. For smooth interpolation, provide at least three values per dimension.

Kinetic friction coefficient vector — Coulomb coefficient of friction, k_K
`[.3, .22, .19]` (default) | vector

Vector of output values for the kinetic friction coefficient. All values must be greater than zero.

Dependencies

To enable this parameter, in the Differential tab, set Friction model to No meshing losses - Suitable for HIL simulation or Constant efficiency.

Kinetic friction coefficient matrix — Coulomb coefficient of friction, k_K
`[.34, .32, .3; .3, .28, .25; .25, .2, .15]` (default) | matrix

Matrix of output values for the kinetic friction coefficient. All values must be greater than zero.

Dependencies

To enable this parameter, in the Differential tab, set Friction model to Temperature-dependent efficiency.

Friction coefficient interpolation method — Output value filling method
`Linear` (default) | `Smooth`

Interpolation methods for approximating the output value when the input value is between two consecutive grid points. To optimize performance, select Linear. To produce a continuous curve with continuous first-order derivatives, select Smooth.

For more information on interpolation algorithms, see PS Lookup Table (1D).

Friction coefficient extrapolation method — Output value extending method
`Linear` (default) | `Nearest`

Extrapolation methods for approximating the output value when the input value is outside the range specified in the argument list. To produce a curve with continuous first-order derivatives in the extrapolation region and at the boundary with the interpolation region, select Linear. To produce an extrapolation that does not go above the highest point in the data or below the lowest point in the data, select Nearest.

For more information on interpolation algorithms, see PS Lookup Table (1D).

Velocity tolerance — Slip velocity locking boundary
`0.001` `rad/s` (default)

Maximum slip velocity at which the clutch can lock. The slip velocity is the signed difference between the base and follower shaft angular velocities, $w = w_{F} - w_{B}$ . When the kinetic friction torque is nonzero and the transferred torque is within the static friction torque limits, the clutch locks if the actual slip velocity falls below the velocity tolerance.

Initial state — Initial clutch state
`Unlocked` (default) | `Locked`

Clutch state at the start of simulation. The clutch can be in one of two states, locked and unlocked. A locked clutch constrains the base and follower shafts to spin at the same velocity, that is, as a single unit. An unlocked clutch allows the two shafts to spin at different velocities, resulting in slip between the clutch plates.

Thermal Port

To enable these settings, set Friction model to Temperature-dependent efficiency.

Thermal mass — Thermal mass
`50` `J/K` (default) | scalar

Thermal energy required to change the component temperature by a single temperature unit. The greater the thermal mass, the more resistant the component is to temperature change.

Dependencies

To enable this parameter, setFriction model to Temperature-dependent efficiency.

Initial temperature — Initial temperature
`300` `K` (default) | scalar

Block temperature at the start of simulation. The initial temperature sets the initial component efficiencies according to their respective efficiency vectors.

Dependencies

To enable this parameter, set Friction model to Temperature-dependent efficiency.

More About

expand all

Hardware-in-the-Loop Simulation

For optimal simulation performance, set Friction model to the default value, No meshing losses - Suitable for HIL simulation.

Extended Capabilities

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

Version History

Introduced in R2017a

expand all

R2024b: Inertia updates

You now can specify the inertia of all gear subcomponents including the sun, ring, and carrier gears. You also now can include the effects of planet procession in the planet inertia behavior.

Limited-Slip Differential

Description

Model

Thermal Model

Examples

Limited Slip Differential with Clutches

Electro-Hydraulic Limited Slip Differential

Ports

Conserving

D — Driving shaft rotational mechanical

S1 — Sun gear 1 rotational mechanical

S2 — Sun gear 2 rotational mechanical

H — Heat flow thermal

Dependencies

Parameters

Differential

Crown gear located — Crown gear location To the right of centerline (default) | To the left of centerline

Carrier (C) to driveshaft (D) teeth ratio (NC/ND) — Crown gear to driveshaft pinion gear teeth ratio 4 (default) | positive scalar

Friction model — Friction model No meshing losses - Suitable for HIL simulation (default) | Constant efficiency | Temperature-dependent efficiency

Sun-sun and carrier-driveshaft ordinary efficiencies — Differential torque constant transfer efficiencies [.85, .92] (default) | vector

Dependencies

Temperature — Temperature [280, 300, 320] K (default) | vector

Dependencies

Sun-sun efficiency — Torque transfer efficiency from the driving sun gear to the driven sun gear [.95, .9, .85] (default) | vector

Dependencies

Carrier-driveshaft efficiency — Torque transfer efficiency from the crown gear to the driveshaft pinon gear [.95, .9, .85] (default) | vector

Dependencies

Sun-carrier and driveshaft-casing power thresholds — Minimum efficiency power thresholds for the sun gear to the crown gear and driveshaft-casing gear couplings [.001, .001] W (default) | vector

Dependencies

Sun-carrier and driveshaft-casing viscous friction coefficients — Viscous friction coefficients [0, 0] N*m/(rad/s) (default) | vector

Inertia — Inertia model off (default) | on

Sun gear inertia — Sun gear inertia 0.001 kg*m^2 (default) | positive scalar

Dependencies

Sun gear radius — Sun gear radius 0.01 m (default) | positive scalar

Dependencies

Carrier inertia — Carrier inertia 0.001 kg*m^2 (default) | positive scalar

Dependencies

Planet gear spin inertia — Planet gear spin inertia 0.001 kg*m^2 (default) | positive scalar

Dependencies

Planet gear mass — Planet gear mass 0.001 kg*m^2 (default) | positive scalar

Dependencies

Clutch

Number of friction surfaces — Number of contact surfaces 4 (default) | positive scalar

Effective torque radius — Friction force application radius 130 mm (default) | positive scalar

Preload force — Clutch plate engagement pressure 300 N (default) | nonegative scalar

Static friction coefficient — Static friction coefficient 0.35 (default) | positive scalar

Dependencies

Static friction coefficient vector — Static friction coefficient [.4, .38, .36] (default) | vector

Dependencies

Kinetic friction coefficient relative velocity vector — Absolute surface velocity [0, 100, 1000] rad/s (default) | vector

Kinetic friction coefficient vector — Coulomb coefficient of friction, kK [.3, .22, .19] (default) | vector

Dependencies

Kinetic friction coefficient matrix — Coulomb coefficient of friction, kK [.34, .32, .3; .3, .28, .25; .25, .2, .15] (default) | matrix

Dependencies

Friction coefficient interpolation method — Output value filling method Linear (default) | Smooth

Friction coefficient extrapolation method — Output value extending method Linear (default) | Nearest

Velocity tolerance — Slip velocity locking boundary 0.001 rad/s (default)

Initial state — Initial clutch state Unlocked (default) | Locked

Thermal Port

Thermal mass — Thermal mass 50 J/K (default) | scalar

Dependencies

Initial temperature — Initial temperature 300 K (default) | scalar

Dependencies

More About

Hardware-in-the-Loop Simulation

Extended Capabilities

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

Version History

R2024b: Inertia updates

See Also

Simscape Blocks

Topics

D — Driving shaft
rotational mechanical

S1 — Sun gear 1
rotational mechanical

S2 — Sun gear 2
rotational mechanical

H — Heat flow
thermal

Crown gear located — Crown gear location
`To the right of centerline` (default) | `To the left of centerline`

Carrier (C) to driveshaft (D) teeth ratio (NC/ND) — Crown gear to driveshaft pinion gear teeth ratio
`4` (default) | positive scalar

Friction model — Friction model
`No meshing losses - Suitable for HIL simulation` (default) | `Constant efficiency` | `Temperature-dependent efficiency`

Sun-sun and carrier-driveshaft ordinary efficiencies — Differential torque constant transfer efficiencies
`[.85, .92]` (default) | vector

Temperature — Temperature
`[280, 300, 320]` `K` (default) | vector

Sun-sun efficiency — Torque transfer efficiency from the driving sun gear to the driven sun gear
`[.95, .9, .85]` (default) | vector

Carrier-driveshaft efficiency — Torque transfer efficiency from the crown gear to the driveshaft pinon gear
`[.95, .9, .85]` (default) | vector

Sun-carrier and driveshaft-casing power thresholds — Minimum efficiency power thresholds for the sun gear to the crown gear and driveshaft-casing gear couplings
`[.001, .001]` `W` (default) | vector

Sun-carrier and driveshaft-casing viscous friction coefficients — Viscous friction coefficients
`[0, 0]` `N*m/(rad/s)` (default) | vector

Inertia — Inertia model
`off` (default) | `on`

Sun gear inertia — Sun gear inertia
`0.001` `kg*m^2` (default) | positive scalar

Sun gear radius — Sun gear radius
`0.01` `m` (default) | positive scalar

Carrier inertia — Carrier inertia
`0.001` `kg*m^2` (default) | positive scalar

Planet gear spin inertia — Planet gear spin inertia
`0.001` `kg*m^2` (default) | positive scalar

Planet gear mass — Planet gear mass
`0.001` `kg*m^2` (default) | positive scalar

Number of friction surfaces — Number of contact surfaces
`4` (default) | positive scalar

Effective torque radius — Friction force application radius
`130` `mm` (default) | positive scalar

Preload force — Clutch plate engagement pressure
`300` `N` (default) | nonegative scalar

Static friction coefficient — Static friction coefficient
`0.35` (default) | positive scalar

Static friction coefficient vector — Static friction coefficient
`[.4, .38, .36]` (default) | vector

Kinetic friction coefficient relative velocity vector — Absolute surface velocity
`[0, 100, 1000]` `rad/s` (default) | vector

Kinetic friction coefficient vector — Coulomb coefficient of friction, k_K
`[.3, .22, .19]` (default) | vector

Kinetic friction coefficient matrix — Coulomb coefficient of friction, k_K
`[.34, .32, .3; .3, .28, .25; .25, .2, .15]` (default) | matrix

Friction coefficient interpolation method — Output value filling method
`Linear` (default) | `Smooth`

Friction coefficient extrapolation method — Output value extending method
`Linear` (default) | `Nearest`

Velocity tolerance — Slip velocity locking boundary
`0.001` `rad/s` (default)

Initial state — Initial clutch state
`Unlocked` (default) | `Locked`

Thermal mass — Thermal mass
`50` `J/K` (default) | scalar

Initial temperature — Initial temperature
`300` `K` (default) | scalar

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