An open bevel differential has two coaxial output gears and spider pinions carried between them. This calculator compares signed shaft speeds, ideal torque sharing and an illustrative backlash band.
Bevel Gear Differential Pitch System Interactive Calculator
Adjust carrier speed, left output speed, input torque and backlash to compare the output-speed balance, spider rotation, ideal torque split and example backlash band. The cutaway shows the bevel gears inside the carrier.
Equation Used
- Both output speeds use the same positive direction along the common shaft axis.
- Equal output gears; the illustrated side-to-spider tooth ratio is 1:1.
- Input torque means torque applied to the carrier, after any final-drive reduction.
- A negative right-output speed indicates reverse rotation.
- The backlash comparison is an illustrative band, not a damage or service prediction.
Compare signed shaft speeds and ideal equal output torques. Spider spin assumes equal side-gear and spider tooth counts; the backlash band is an illustrative comparison, not a gear-set specification.
Bevel gears inside a rotating carrier
The two output shafts are coaxial. Their side gears face inward, and spider pinions mesh with both side gears while riding on a cross pin in the carrier. The illustration uses four equal-size 45° pitch cones; the teeth are simplified for viewing.
When the output speeds are equal, the spider pinions orbit with the carrier without turning relative to the cross pin. A speed difference makes them turn on that pin. In the carrier frame, the two equal side gears have equal and opposite relative speeds.
KHK’s differential example shows the opposing bevel gears and supporting carrier. The cutaway exposes the side gears, spider pinions and carrier so their relationships are visible. The external final-drive ring and pinion are omitted; carrier speed and torque are entered directly.
Where the arrangement is useful
An open differential permits two outputs to have different speeds while retaining a fixed average-speed relationship. Vehicle axles are a familiar use. This ideal calculator does not model tire traction, a limited-slip clutch, gear losses or the strength of a specific axle.
Speed balance and ideal torque sharing
For equal side gears, nL + nR = 2 nC. Therefore nR = 2 nC − nL. Speeds are signed in one common axial direction. The right output can reverse when the entered left speed is greater than twice carrier speed.
The magnitude of spider spin relative to its cross pin is |nL − nC| × Zside / Zspider. The displayed result assumes Zside = Zspider, as in the illustration. For other tooth counts, apply the ratio rather than treating this number as universal.
For a lossless, quasi-static open differential, each output torque is Tin / 2, where Tin is carrier torque. Power is conserved because (Tin / 2)(nL + nR) = Tin nC. Transient inertial torque and friction are outside this model.
The final output is the distance of the entered backlash from an example band of 0.10–0.20 mm. It is zero inside that band. Backlash requirements depend on the actual gear set and assembly; KHK describes mounting-distance adjustment and contact checks rather than a single universal limit.
Worked speed-split example
With carrier speed 600 rpm and left output 750 rpm, the right output is 450 rpm. Both outputs average to 600 rpm. Equal side and spider tooth counts give 150 rpm spider spin relative to the carrier. A 400 N·m carrier torque gives an ideal 200 N·m on each output.
At 0.15 mm backlash, the example-band deviation is zero. This does not certify a particular gear assembly. With carrier speed zero and a nonzero left speed, the right output reverses at the same speed; the animation preserves that case.
What this calculator does not establish
The speed equation does not establish tooth strength, contact pattern, bearing loads, traction or lubrication requirements. Equal output torque does not mean equal output power when speeds differ. A limited-slip or locking differential requires another model.
Differential questions
Why can the right speed be negative?
The equation uses signed speeds. With the carrier held, rotating one output makes the other turn in the opposite direction.
Does spider spin always equal half the output-speed difference?
Only when side and spider tooth counts are equal. Otherwise multiply by the side-to-spider tooth-count ratio.
Does zero backlash-band deviation mean the mesh is correct?
No. It only means the entered value lies in this page’s illustrative band. Use the actual gear-set specification and verify tooth contact and mounting dimensions.
Why does torque not change the speed animation?
The two entered speeds determine the ideal kinematics. Torque changes the calculated load split, not the independently specified speed pair.
Manufacturer reference
KHK Bevel Gears technical information, differential example on printed page 336 and assembly guidance on page 341. The photograph supports the opposing side gears and spider arrangement; the assembly guidance explains why backlash and mounting conditions need gear-specific checks.
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