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Gear train for transferring power to separate shafts spinning at different speeds
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.
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 g_{D} 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 [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)).
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.
Gear inertia is negligible. It does not impact gear dynamics.
Gears are rigid. They do not deform.
Coulomb friction slows down simulation. See Adjust Model Fidelity.