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 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 exposing an optional thermal port. To expose the port, in the Meshing Losses tab, set the Friction model parameter to Temperature-dependent efficiency.

Ports

Conserving

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`D` — Driving shaft interface
mechanical rotational

Port associated with the driveshaft.

`S1` — Sun gear 1
rotational mechanical

Rotational conserving port representing sun gear 1 shaft.

`S2` — Sun gear 2
rotational mechanical

Rotational conserving port representing sun gear 2 shaft.

`H` — Heat flow
thermal

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

`Crown gear location` — Driveshaft 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)` — Carrier-to-driveshaft gear ratio
`4` (default) | positive scalar

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

`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 gear wheel pairs is reduced by a constant efficiency, η, such that $0 < η \leq 1$ .
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.

`Sun-sun and carrier-driveshaft ordinary efficiencies` — Torque transfer efficiency
`[.85, .92]` (default) | array | `0` < η ≤ `1`

Array of torque transfer efficiencies [η_SS, η_D] 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.

`Temperature` — Temperature
`[280, 300, 320]` `K` (default) | array of increasing values

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.

`Sun-sun efficiency` — Sun-sun gear efficiency
`[.95, .9, .85]` (default) | array | `0` < η_SS ≤ `1`

Array of mechanical efficiencies, ratios of output power to input power, for the power flow from the sun gear to the planet gear, η_SS. 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.

`Carrier-driveshaft efficiency` — Carrier gear-longnitudinal driveshaft efficiency
`[.95, .9, .85]` (default) | array | `0` < η_CD ≤ `1`

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

Dependencies

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

`Sun-carrier and driveshaft-casing power thresholds` — Full efficiency power thresholds
`[.001, .001]` `W` (default) | array

Array of power thresholds p_th above which full efficiency loss is applied, for sun-carrier and longitudinal driveshaft-casing [p_S p_D], 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.

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

Array 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.

Dependencies

When this parameter is set to On exposes related parameters.

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

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.

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

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 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

Preload force that the spring 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

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.

`Static friction coefficient` — Coulomb coefficient of friction, k_S
`0.35` (default) | positive scalar

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

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.

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

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.

`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

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.

`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

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

`Friction coefficient interpolation method` — Output value approximation 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 the PS Lookup Table (1D) block reference page.

`Friction coefficient extrapolation method` — Output value approximation 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 extrapolation algorithms, see the PS Lookup Table (1D) block reference page.

`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, 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.

`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

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

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

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.

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

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.

Model Examples