# Limited-Slip Differential

Reduce velocity difference between two connected shafts

• Library:
• Simscape / Driveline / Gears

## Description

The Limited-Slip Differential block represents a limited-slip differential (LSD), which is a gear assembly that can reduce the velocity difference between two connected shafts. The block models the LSD mechanism as a structural component that combines a differential gear 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 that are connected by an open differential can get stuck when only one of the wheels slips and then 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 wheels experience a traction differential, the planet pinion gears exert an additional force in the direction of the high-traction wheel. 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 Limited-Slip Differential block models the LSD mechanism as a structural component based on the Simscape™ Driveline™ Differential and Disk 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 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 CenterlineRotation Direction of Driveshaft Port (D)Relative Slippage Across the Differential
PositiveRightPositive0
• Positive for the nonslipping port

• Negative the slipping port

RightPositive> 0
NegativeRightNegative0
• Negative for the nonslipping port

• Positive the slipping port

RightNegative> 0
NegativeLeftPositive0
• Negative for the nonslipping port

• Positive the slipping port

LeftPositive> 0
PositiveLeftNegative0
• Positive for the nonslipping port

• Negative the slipping port

LeftNegative> 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 exposing an optional thermal port. To expose the port, in the Meshing Losses settings, set the Friction parameter to ```Temperature-dependent efficiency```.

## Ports

### Conserving

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Port associated with the driveshaft.

Rotational conserving port representing sun gear 1 shaft.

Rotational conserving port representing sun gear 2 shaft.

Thermal conserving port associated with heat flow.

#### Dependencies

This port is exposed when, in the Differential settings, the Friction model parameter is set to ```Temperature-dependent efficiency```.

Exposing this port makes related parameters visible.

## Parameters

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### Differential

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

Fixed ratio, gD, of the carrier gear to the longitudinal driveshaft gear. This gear ratio must be strictly greater than `0`.

Friction model for the block:

• ```No meshing losses - Suitable for HIL simulation``` — Gear meshing is ideal.

• `Constant efficiency` — Transfer of torque between gear wheel pairs is reduced by a constant efficiency, η, such that $0<\eta \le 1$.

• `Temperature-dependent efficiency` — Transfer of torque between gear wheel pairs is defined by table lookup based on the temperature.

#### Dependencies

If this parameter is set to:

• ```No meshing losses - Suitable for HIL simulation``` — Related parameters are exposed.

• `Constant efficiency` — Related parameters are exposed.

• ```Temperature-dependent efficiency``` — A thermal port and related parameters are exposed.

Array of torque transfer efficiencies [ηSS, ηD] for sun-sun and carrier-longitudinal driveshaft gear wheel pair meshings, respectively. The array element values must be greater than `0` and less than or equal to `1`.

#### Dependencies

This parameter is exposed when the Friction model parameter is set to ```Constant efficiency```.

Array of temperatures used to construct a 1-D temperature-efficiency lookup table. The array values must increase from left to right.

#### Dependencies

This parameter is exposed when Friction model is set to `Temperature-dependent efficiency`.

Array of mechanical efficiencies, ratios of output power to input power, for the power flow from the sun gear to the planet gear, ηSS. The block uses the values to construct a 1-D temperature-efficiency lookup table.

Each array element values is the efficiency at the temperature of the corresponding element in the Temperature array. The number of elements in the Efficiency array must be the same as the number of elements in the Temperature array. The value of each Efficiency array element must be greater than `0` and less than or equal to `1`.

#### Dependencies

This parameter is exposed when the Friction model parameter is set to ```Temperature-dependent efficiency```.

Array of mechanical efficiencies, ratios of output power to input power, for the power flow from the carrier to the driveshaft, ηCD. The block uses the values to construct a 1-D temperature-efficiency lookup table.

Each array element values is the efficiency at the temperature of the corresponding element in the Temperature array. The number of elements in the Efficiency array must be the same as the number of elements in the Temperature array. The value of each Efficiency array element must be greater than `0` and less than or equal to `1`.

#### Dependencies

This parameter is exposed when the Friction model parameter is set to ```Temperature-dependent efficiency```.

Array of power thresholds pth above which full efficiency loss is applied, for sun-carrier and longitudinal driveshaft-casing [pS pD], respectively. Below these values, a hyperbolic tangent function smooths the efficiency factor. For a model without thermal losses, the function lowers the efficiency losses to zero when no power is transmitted. For a model that considers thermal losses, the function smooths the efficiency factors between zero at rest and the values provided by the temperature-efficiency lookup tables at the power thresholds.

#### Dependencies

This parameter is exposed when the Friction model parameter is set to ```Constant efficiency``` or ```Temperature-dependent efficiency```.

Array of viscous friction coefficients [μS, μD ] for the sun-carrier and longitudinal driveshaft-casing gear motions, respectively.

Inertia model for the block:

• `Off` — Model gear inertia.

• `On` — Neglect gear inertia.

#### Dependencies

When this parameter is set to `On` exposes related parameters.

Moment of inertia of the planet gear carrier. This value must be positive.

#### Dependencies

This parameter is exposed when the Inertia parameter is set to `On`.

Moment of inertia of the combined planet gears. This value must be positive.

#### Dependencies

This parameter is exposed when the Inertia parameter is set to `On`.

### Clutch

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

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

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

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

#### Dependencies

This parameter is exposed when, in the Differential settings, the Friction model parameter is set to ```No meshing losses - Suitable for HIL simulation``` or ```Fixed kinetic friction coefficient```.

Dimensionless Coulomb static friction coefficient, kS, applied to the normal force across the clutch when the clutch is locked. The static friction coefficient, kS, must be larger than the kinetic friction coefficient, kK.

#### Dependencies

This parameter is exposed when, in the Differential settings, the Friction model parameter is set to ```No meshing losses - Suitable for HIL simulation``` or ```Constant efficiency```.

Vector of static, or peak, values of the friction coefficient. The vector must have the same number of elements as the temperature vector. Each value must be greater than the value of the corresponding element in the kinetic friction coefficient vector.

#### Dependencies

This parameter is exposed when, in the Differential settings, the Friction model parameter is set to ```Temperature-dependent efficiency```.

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.

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

#### Dependencies

This parameter is exposed when, in the Differential settings, the Friction model parameter is set to ```No meshing losses - Suitable for HIL simulation``` or ```Constant efficiency```.

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

#### Dependencies

This parameter is exposed when, in the Differential settings, the Friction model parameter is set to ```Temperature-dependent efficiency```.

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 the PS Lookup Table (1D) block reference page.

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 extrapolation algorithms, see the PS Lookup Table (1D) block reference page.

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

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

These settings are exposed only when, in the Differential settings, the Friction model parameter is set to `Temperature-dependent efficiency`.

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

#### Dependencies

This parameter is exposed when, in the Differential settings, the Friction model parameter is set to ```Temperature-dependent efficiency```.

Component temperature at the start of simulation. The initial temperature alters the component efficiency according to an efficiency vector that you specify, affecting the starting meshing or friction losses.

#### Dependencies

This parameter is exposed when, in the Differential settings, the Friction model parameter is set to ```Temperature-dependent efficiency```.

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