Differential gear set with adjustable driveshaft gear ratio and friction losses
The Differential block represents a differential gear that couples rotational motion about the longitudinal driveshaft axis to rotational motion about two lateral or side axes. The differential is composed of a simple gear and two symmetric sun-planet bevel gears. The simple gear comprises a differential crown gear attached to the carrier of one of the sun-planet bevel gears, plus a bevel gear attached to the driveshaft. For model details, see Differential Gear Model.
Any axis can be the input. In normal use, the longitudinal driveshaft is the input, and motion, torque, and power flow out through the side shafts. In general, the output axes have different angular velocities. The longitudinal motion is divided by the drive gear ratio that you specify and then split equally between the two lateral shafts.
|Differential Crown Location|
Relative to Center-line
|Side Axes Rotation|
Relative to Longitudinal Axis Rotation
Differentials in drivelines often have a controllable clutch connecting the two output shafts. You can add this clutch control by connecting a clutch block to the Differential block.
D, S1, and S2 are rotational conserving ports representing, respectively, the longitudinal driveshaft and the two side shafts.
The dialog box has one active area, Parameters, with three tabs.
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.
Fixed ratio gD of the carrier gear to the longitudinal driveshaft gear. The default is 4.
Select how to implement friction losses from nonideal meshing of gear teeth. The default is No meshing losses.
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 satisfying 0 < η ≤ 1. If you select this option, the panel changes from its default.
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)).
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 ωS1 – ωS2 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 + ωDτD – Ploss= 0 .
When the kinematic and power constraints are combined, the ideal case yields:
gDτD = 2(ωS1τS1 + ωS2τS2) / (ωS1 + ωS2) .
In the nonideal case, τloss ≠ 0. See Model Gears with Losses.
Gear ratios must be positive. Gear inertia and compliance are ignored. Coulomb friction reduces simulation performance. See Adjust Model Fidelity.
These SimDriveline™ example models contain working examples of differential gears: