An equal-side bevel differential lets two coaxial output shafts turn at different speeds while their average remains equal to carrier speed. The cutaway shows two separate side gears, two orbiting spider pinions, a cross-pin and the rotating carrier ring. Change the speed difference to see the spider pinions spin on their cross-pin.
Differential Motions Interactive Calculator
Follow the two side gears and the orbiting spider pinions. Set carrier speed and output-speed difference, then compare equal side torques and their different powers.
Equation Used
- Equal side gears; illustrated spider gears also have equal tooth counts.
- Rigid, steady, ideal open differential.
- Entered speed difference imposed rather than predicted from load.
- External drive gearing and bearing housings omitted.
- Input is transmitted carrier torque, not engine torque.
Four equal 24-tooth bevel gears for explanatory kinematics. Simplified tooth ribs; imposed speed difference; no traction, strength or loss prediction.
The carrier turns; the spiders can also spin
The gray ring and cross-pin form the input carrier. The blue left side gear and gold right side gear connect to separate half-shafts. Purple spider pinions run on the carrier cross-pin and mesh with both side gears.
When both outputs have equal speed, the spiders have no spin relative to the cross-pin. They still orbit with the carrier. When one output slows by an amount d, the other speeds up by the same amount. The spiders then spin relative to their moving support.
This explanatory model uses four equal 24-tooth, 45-degree pitch-cone gears, so spider spin magnitude equals half the output-speed difference. A real differential may use different spider tooth counts and cone angles; its side-speed averaging relation can remain the same while spider spin changes.
Explore equal speeds, a stopped side and back-driving
Set the right-minus-left speed difference to zero for equal outputs. Set it to twice the carrier speed to stop the left output while the right runs at twice carrier speed. A larger positive difference reverses the left output. Holding the carrier stationary while imposing a difference makes the outputs turn oppositely.
The selected speed difference is an imposed operating condition. The calculator does not predict it from tire grip, road curvature or load. Carrier drive gearing and bearing housings are omitted so the four internal gears remain visible.
The average-speed identity
For equal side gears, nL+nR=2nC. Defining d as half the right-minus-left difference gives nL=nC−d and nR=nC+d. For the equal spider gears illustrated here, spider spin relative to the carrier has magnitude |d|.
In an ideal, steady, frictionless open differential, transmitted carrier torque TC is shared equally: TL=TR=TC/2. Equal torque does not require equal power. Each output power is T×2πn/60; their sum equals carrier power. A negative signed output power indicates back-driving through that side.
The torque input is the torque actually transmitted at the carrier, after any external final-drive reduction. Traction or other loading can limit how much torque can be transmitted. The kinematic controls alone do not establish those limits.
Example: different output speeds, equal torque
At 500 carrier rpm and a right-minus-left difference of 200 rpm, the outputs run at 400 and 600 rpm. The spiders spin at 100 rpm relative to the carrier in this equal-gear model.
With 4000 N·m of transmitted carrier torque, each output receives 2000 N·m. The powers are approximately 83.78 kW and 125.66 kW, totaling the carrier’s 209.44 kW. The faster side carries more power even though the torques are equal.
Scope of this open-differential model
The model assumes rigid, equal side gears, ideal contact and steady motion. It omits losses, acceleration, bearing loads, traction limits, locking clutches and limited-slip friction. The entered torque does not establish a component load rating.
The cutaway preserves pitch-cone contact and carrier-relative motion. Tooth ribs are simplified visual markers, not manufactured tooth profiles. Gear strength, backlash and bearing setup require the dimensions and instructions for the actual assembly.
Differential questions
Why do the purple gears move even with no speed difference?
They orbit with the rotating carrier. Their spin relative to the cross-pin is zero when both side speeds are equal.
Can one output turn backward?
Yes. If the magnitude of the entered right-minus-left difference exceeds twice the carrier speed, one output has a negative signed speed.
Does an open differential send all torque to the faster side?
The ideal steady open differential transmits equal side torques. Different speeds give different powers, and available traction can limit both transmitted torques.
Is the input torque engine torque?
No. It is transmitted torque at the differential carrier, after any upstream gear reduction.
Construction and kinematics references
J. W. Kamman, Intermediate Dynamics: Differential Gear Set, derives the side-speed and carrier-relative relations. Hiscox, Mechanical Movements, Powers and Devices, printed page 238, figure 936 (PDF page 248), illustrates differential motions on concentric shafts using bevel gearing.
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