Gear mechanism that allows driven shafts to spin at different speeds

Gears

This 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 S, D1, and D2 represent the driving and driven shafts of the differential. Any of the shafts can drive the remaining 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 block diagram of this structural component.

To increase the fidelity of the gear model, you can 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
through an optional thermal conserving port. To expose the thermal
port, right-click the block and select **Simscape** > **Block choices** > **Show thermal
port**. Exposing the thermal port causes
new parameters specific to thermal modeling to appear in the block
dialog box.

**Crown wheel located**Select the placement of the bevel crown gear with respect to the center-line of the gear assembly. The default is

`To the right of the center-line`

.**Carrier (C) to driveshaft (D) teeth ratio (NC/ND)**Fixed ratio

*g*_{D}of the carrier gear to the longitudinal driveshaft gear. The default is`4`

.

Parameters for meshing losses vary with the block variant chosen—one with a thermal port for thermal modeling and one without it.

**Sun-carrier and driveshaft-casing viscous friction coefficients**Vector of viscous friction coefficients [

*μ*_{S}*μ*_{D}] for the sun-carrier and longitudinal driveshaft-casing gear motions, respectively. The default is`[0 0]`

.From the drop-down list, choose units. The default is newton-meters/(radians/second) (

`N*m/(rad/s)`

).

**Carrier inertia**Moment of inertia of the planet gear carrier. This value must be positive or zero. Enter

`0`

to ignore carrier inertia. The default value is`0`

kg*m^2.**Planet gear inertia**Moment of inertia of the combined planet gears. This value must be positive or zero. Enter

`0`

to ignore gear inertia. The default value is`0`

kg*m^2.

**Thermal mass**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. The default value is

`50`

J/K.**Initial temperature**Component temperature at the start of simulation. The initial temperature influences the starting meshing or friction losses by altering the component efficiency according to an efficiency vector that you specify. The default value is

`300`

K.

Differential imposes one kinematic constraint on the three connected axes:

*ω*_{D} =
±(1/2)*g*_{D}(*ω*_{S1} + *ω*_{S2})
,

with the upper (+) or lower (–) sign valid for the differential crown to the right or left, respectively, of the center line. 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 *ω*_{S1} – *ω*_{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 (

*ω*_{S1}=*ω*_{S2}) and at a fixed ratio with respect to the longitudinal shaft.The other degree of freedom (differential) is equivalent to keeping the longitudinal shaft locked (ω

_{D}= 0) while the lateral shafts rotate with respect to each other in opposite directions (*ω*_{S1}= –*ω*_{S2}).

The torques along the lateral axes, *τ*_{S1} and *τ*_{S2},
are constrained to the longitudinal torque *τ*_{D} in
such a way that the power flows into and out of the gear, less any
power loss *P*_{loss}, sum to
zero:

*ω*_{S1}*τ*_{S1} + *ω*_{S2}*τ*_{S2} + *ω*_{D}*τ*_{D} – *P*_{loss}=
0 .

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

*g*_{D}*τ*_{D} =
2(*ω*_{S1}*τ*_{S1} + *ω*_{S2}*τ*_{S2})
/ (*ω*_{S1} + *ω*_{S2})
.

Fundamental Sun-Planet Bevel Gear Constraints

In the nonideal case, *τ*_{loss} ≠
0. See Model Gears with Losses.

Gears are assumed rigid.

Coulomb friction slows down simulation. See Adjust Model Fidelity.

These Simscape Driveline example models contain working examples of differential gears:

Port | Description |
---|---|

D | Rotational conserving port representing the longitudinal driveshaft |

S1 | Rotational conserving port representing one of the sun gears |

S2 | Rotational conserving port representing one of the sun gears |

H | Thermal conserving port for thermal modeling |

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