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Differential

Gear train for transferring power to separate shafts spinning at different speeds

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Gears

Description

This block represents a gear train for transferring power from one drive shaft to two driven shafts. A combination of simple and bevel gear constraints enable the driven shafts to spin at different speeds, when necessary, and in different directions. One example is an automobile differential, which during a turn enables the inner and outer wheels to spin at different speeds, these depending on the turning radius of each individual wheel.

Any of the shafts can provide the input that drives the remaining two shafts. The differential converts this input into rotation, torque, and power at the driven shafts. The drive gear ratio, which you specify directly in the block dialog box, helps determine the angular velocity of each driven shaft. For more information, see Differential Gear Model

This block is a composite component with three underlying blocks:

The figure shows the connections between the three blocks.

The block models the effects of heat flow and temperature change through an optional thermal 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.

Dialog Box and Parameters

Main

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 gD of the carrier gear to the longitudinal driveshaft gear. The default is 4.

Meshing Losses

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

 Without Thermal Port

 With Thermal Port

Viscous Losses

Sun-carrier and driveshaft-casing viscous friction coefficients

Vector of viscous friction coefficients [μS μD] 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)).

Thermal Port

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 Gear Model

Ideal Gear Constraints and Gear Ratios

Differential imposes one kinematic constraint on the three connected axes:

ωD = ±(1/2)gD(ωS1 + ωS2) ,

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 ωS1ωS2 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 (ωS1 = ωS2) 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 (ωD = 0) while the lateral shafts rotate with respect to each other in opposite directions (ωS1 = –ωS2).

The torques along the lateral axes, τS1 and τS2, are constrained to the longitudinal torque τD in such a way that the power flows into and out of the gear, less any power loss Ploss, sum to zero:

ωS1τS1 + ωS2τS2 + ωDτDPloss= 0 .

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

gDτD = 2(ωS1τS1 + ωS2τS2) / (ωS1 + ωS2) .

 Fundamental Sun-Planet Bevel Gear Constraints

Nonideal Gear Constraints and Losses

In the nonideal case, τloss ≠ 0. See Model Gears with Losses.

Limitations

  • Gear inertia is assumed negligible.

  • Gears are treated as rigid components.

  • Coulomb friction slows down simulation. See Adjust Model Fidelity.

Examples

These SimDriveline™ example models contain working examples of differential gears:

Ports

PortDescription
DRotational conserving port representing the longitudinal driveshaft
S1Rotational conserving port representing one of the sun gears
S2Rotational conserving port representing one of the sun gears
HThermal conserving port for thermal modeling

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