Documentation

# 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:

### Port Options

The Limited-Slip Differential block offers these port options:

• `No thermal port` (default)

• `Show thermal port`

To model thermal losses, select `Show thermal port`. Thermal ports are conserving ports that you can connect to thermal conserving ports on Simscape Foundation Thermal Library blocks for simulating heat flow and temperature changes. For more information, see Model Thermal Losses in Driveline Components.

To view or select a port option, use one of these methods:

• Use the block context menu:

1. Left-click the block.

2. From the context menu, select Simscape > Block choices.

3. Select option from the drop-down menu.

• Use the Property Inspector:

2. Click the block.

3. In the Property Inspector pane, click the Block Choice VALUE.

4. Select an option from the drop-down menu.

#### Dependencies

These parameters are enabled only if you select ```No thermal port```:

• Friction Model

• Static friction coefficient

These parameters are enabled only if you select ```Show thermal port```:

• Temperature Sun-sun efficiency

• Sun-sun efficiency

• Carrier-driveshaft efficiency

• Sun-carrier and driveshaft-casing power thresholds

• Sun-carrier and driveshaft-casing viscous friction coefficients

• Kinetic friction coefficient matrix

• Static friction coefficient vector

• Thermal mass

• Initial temperature

## Ports

### Conserving

expand all

Port associated with the driveshaft.

Ports associated with the two sun-gear shafts.

Port associated with heat flow. The thermal port is optional and is hidden by default.

#### Dependencies

To enable this port, for Block Choice, select `Show thermal port`.

## Parameters

expand all

#### Differential

Location of the bevel crown gear regarding the centerline of the gear assembly.

Fixed ratio, gD, of the carrier gear to the longitudinal driveshaft gear.

Select whether to consider frictional losses from nonideal meshing of gear teeth:

• ```No meshing losses — Suitable for HIL simulation``` — Gear meshing is ideal and losses are assumed negligible.

• `Constant efficiency` — Transfer of torque between gear wheel pairs is reduced by a constant efficiency η satisfying 0 < η ≤ 1.

You can increase the fidelity of your model by specifying meshing losses. For hardware-in-the-loop simulation, however, if your model is not real-time capable, consider ignoring the frictional losses. For more information on balancing speed and fidelity for real-time simulation using Simscape models, see Improving Speed and Accuracy (Simscape).

#### Dependencies

To enable this parameter, for Block Choice, select `No thermal port`.

Selecting `Constant efficiency` enables these parameters:

• Sun-sun and carrier-driveshaft ordinary efficiencies

• Sun-carrier and driveshaft-casing power thresholds

• Sun-carrier and driveshaft-casing viscous friction coefficients

Vector of torque transfer efficiencies [ηSS ηD] for sun-sun and carrier-longitudinal driveshaft gear wheel pair meshings, respectively.

#### Dependencies

To enable this parameter:

1. For Block choice, select ```No thermal port```.

2. For Friction model, select ```Constant efficiency```.

Array of temperatures used to construct 1-D temperature-efficiency lookup tables. The array values must increase left to right. The temperature array must be the same size as each efficiency array.

#### Dependencies

To enable this parameter, for Block choice, select `Show thermal port`.

Array of mechanical efficiencies with power flowing from one of the sun gears to the other. Each array value is the ratio of output power to input power at one of the temperatures in the temperature array. The temperature and efficiency arrays must be the same size.

#### Dependencies

To enable this parameter, for Block choice, select `Show thermal port`.

Array of mechanical efficiencies with power flowing from the gear carrier to the longitudinal driveshaft. Each array value is the ratio of output power to input power at one of the temperatures in the temperature array. The temperature and efficiency arrays must be the same size.

#### Dependencies

To enable this parameter, for Block choice, select `Show thermal port`.

Vector of power thresholds pth above which full efficiency loss is applied, for sun-carrier and longitudinal driveshaft-casing [pS pD], respectively. A hyperbolic tangent function 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:

1. For Block choice, select ```No thermal port```.

2. For Friction model, select ```Constant efficiency```.

or for Block choice, select ```Show thermal port```.

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

#### Dependencies

To enable this parameter:

1. For Block choice, select ```No thermal port```.

2. For Friction model, select ```Constant efficiency```.

or for Block choice, select ```Show thermal port```.

Moment of inertia of the planet gear carrier. This value must be positive or zero. To ignore carrier ring inertia, enter `0`.

Moment of inertia of the combined planet gears. This value must be positive or zero. To ignore planet gear inertia, enter `0`.

#### 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.

Specify input values for the relative velocity as a vector. 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.

Specify the output values for the kinetic friction coefficient as a vector. All values must be greater than zero.

The coefficient of kinetic friction must be greater than zero.

#### Dependencies

To enable this parameter, for Block choice, select `Show thermal port`.

Specify the static, or peak, values of the friction coefficient as a vector. 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

To enable this parameter, for Block choice, select `Show thermal port`.

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.

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

To enable this parameter, for Block Choice, select `No thermal port`.

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

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

To enable this parameter, for Block choice, select `Show thermal port`.

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

To enable this parameter, for Block choice, select `Show thermal port`.