# 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 driveshaft and the sun gear
shafts of the differential, respectively. Any one of the shafts can drive the other
two.

The block models the differential mechanism as a structural component based on the Simple Gear and Sun-Planet Bevel Simscape™ Driveline™ blocks. The figure demonstrates the equivalent block diagram for the Differential 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 to be negligible. The block enables you to specify the inertias of the gear
carrier and internal planet gears. 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 enabling the optional thermal port. To enable
the port, set **Friction model** 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:

*ω*is the velocity of sun gear shaft 1._{S1}*ω*is the velocity of sun gear shaft 2._{S2}

Negative values imply that the differential is left of 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:

The longitudinal degree of freedom 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 differential degree of freedom is equivalent to keeping the longitudinal driving shaft locked, $${\omega}_{D}=0$$, where

*ω*is the velocity of the driving shaft, while the lateral shafts rotate with respect to each other in opposite directions, $${\omega}_{S1}=-{\omega}_{S2}$$._{D}

The lateral axis torques are constrained by the longitudinal axis torque such that the net power flow sums to zero:

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

where:

*τ*and_{S1}*τ*are the torques along the lateral axes._{S2}*τ*is the longitudinal torque._{D}*P*is the power loss._{loss}

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

$${g}_{D}{\tau}_{D}=2\frac{({\omega}_{S1}{\tau}_{S1}+{\omega}_{S2}{\tau}_{S2})}{{\omega}_{S1}+{\omega}_{S2}},$$

where *g _{D}* is the
gear ratio for the longitudinal driveshaft.

**Ideal Fundamental Constraints**

The effective Differential block constraint is composed of two sun-planet bevel gear subconstraints.

The first subconstraint is due to 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

*g*and_{SP1}*g*are the gear ratios for the sun-planet gears._{SP2}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-planet gears are:

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

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

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

**Nonideal Gear Constraints and Losses**

In the nonideal case, *τ _{loss}* ≠ 0. For more information, see Model Gears with Losses.

### Assumptions and Limitations

The gears are assumed to be rigid.

Coulomb friction slows down simulation. For more information, see Adjust Model Fidelity.

## Ports

### Conserving

## Parameters

## More About

## Extended Capabilities

## Version History

**Introduced in R2011a**