# Differential

Gear mechanism that allows driven shafts to spin at different speeds

• Library:
• Simscape / Driveline / Gears

• ## Description

The Differential block represents a gear mechanism that allows the driven shafts to spin at different speeds. Differentials are common in automobiles, where they enable the various wheels to spin at different speeds while cornering. Ports D, S1, and S2 represent the longitudinal driving and the sun driven gear shafts of the differential. Any one of the shafts can drive the other two.

The block models the differential mechanism as a structural component based on Simple Gear and Sun-Planet Bevel Simscape™ Driveline™ blocks. The figure shows the equivalent circuit for the block. To increase the fidelity of the gear model, specify properties such as gear inertia, meshing losses, and viscous losses. By default, gear inertia and viscous losses are assumed negligible. The block enables you to specify the inertias of the gear carrier and internal planet gears only. To model the inertias of the outer gears, connect Simscape Inertia blocks to ports D, S1, and S2.

### Thermal Modeling

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```.

### Equations

Ideal Gear Constraints and Gear Ratios

The differential imposes one kinematic constraint on the three connected axes such that

`${\omega }_{S1}-{\omega }_{S2},$`

where:

• ωS1 is the velocity of driven sun gear shaft 1.

• ωS2 is the velocity of driven sun gear shaft 2.

with the upper (+) or lower (–) sign valid for the differential crown to the right or left, respectively, of the centerline. The three degrees of freedom reduce to two independent degrees of freedom. The gear pairs are (1,2) = (S, S) and (C, D). C is the carrier.

The sum of the lateral motions is the transformed longitudinal motion. The difference of side motions, ${\omega }_{S1}-{\omega }_{S2}$, is independent of the longitudinal motion. The general motion of the lateral shafts is a superposition of these two independent degrees of freedom, which have this physical significance:

• One degree of freedom (longitudinal) is equivalent to the two lateral shafts rotating at the same angular velocity, ${\omega }_{S1}={\omega }_{S2}$, and at a fixed ratio with respect to the longitudinal shaft.

• The other degree of freedom (differential) is equivalent to keeping the longitudinal driving shaft locked, ${\omega }_{D}=0$, where ωD is the velocity of the driving shaft, while the lateral shafts rotate with respect to each other in opposite directions, ${\omega }_{S1}=-{\omega }_{S2}$.

The torques along the lateral axes are constrained to the longitudinal torque such that the power flows into and out of the gear, less any power loss, sum to zero:

`${\omega }_{S1}{\tau }_{S1}+{\omega }_{S2}{\tau }_{S2}+{\omega }_{D}{\tau }_{D}-{P}_{loss}=0,$`

where:

• τS1 and τS2 are the torques along the lateral axes.

• τD is the longitudinal torque.

• Ploss is the power loss.

When the kinematic and power constraints are combined, the ideal case yields

`${g}_{D}{\tau }_{D}=2\frac{\left({\omega }_{S1}{\tau }_{S1}+{\omega }_{S2}{\tau }_{S2}\right)}{{\omega }_{S1}+{\omega }_{S2}}.$`

where gD is the gear ratio for the longitudinal driving shaft.

Ideal Fundamental Constraints

The effective differential constraint is composed of two sun-planet bevel Gear subconstraints.

• The first subconstraint is due to from the coupling of the two sun-planet bevel gears to the carrier:

`$\frac{{\omega }_{S1}-{\omega }_{C}}{{\omega }_{S2}-{\omega }_{C}}=-\frac{{g}_{SP2}}{{g}_{SP1}}.$`

where gSP1 and gSP2 are the gear ratios for the sun-planets.

• The second subconstraint is due to the coupling of the carrier to the longitudinal driveshaft:

`${\omega }_{D}=-{g}_{D}{\omega }_{C}.$`

The sun-planet gear ratios of the underlying sun-planet bevel gears, in terms of the radii, r, of the sun and planet gears are:

`${g}_{SP1}=\frac{{r}_{S1}}{{r}_{P1}}$`

and

`${g}_{SP2}=\frac{{r}_{S2}}{{r}_{P2}}.$`

The Differential block is implemented with ${g}_{SP1}={g}_{SP2}=1$, leaving gD free to adjust.

Nonideal Gear Constraints and Losses

In the nonideal case, τloss ≠ 0. For more information, see Model Gears with Losses.

## Assumptions and Limitations

• Gears are assumed rigid.

## Ports

### Conserving

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Rotational conserving port representing the longitudinal driveshaft.

Rotational conserving port representing sun gear 1.

Rotational conserving port representing sun gear 2.

Thermal conserving port associated with heat flow. Heat flow affects gear temperature, and therefore, power transmission efficiency.

#### Dependencies

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

Exposing this port also exposes related parameters.

## Parameters

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### Main

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`.

### Meshing Losses

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 ≤ 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:

• `Constant efficiency` — Related parameters are exposed.

• `Temperature-dependent meshing losses` — 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 Friction model is set to `Constant efficiency` or ```Temperature-dependent meshing losses```.

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 sun gear to the planet gear, 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_S, pth_D], above which full efficiency loss is applied, for the sun-carrier and longitudinal driveshaft-casing. 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```.

### Viscous Losses

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

### Inertia

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`.

### Thermal Port

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

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