Gear mechanism that allows driven shafts to spin at different speeds

**Library:**Simscape / Driveline / Gears

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**.

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
```

.

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

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

where:

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

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

*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 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:

*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 driving shaft.

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

*g*and_{SP1}*g*are the gear ratios for the sun-planets._{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 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 *g _{D}* free to
adjust.

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

Gears are assumed rigid.

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