Follow a transparent car around a bend, open its rear axle, and discover how a differential shares motion, torque and a low-grip limit.
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Around a corner, the outside wheel travels farther in the same time. An open differential allows that difference while the two driven wheel speeds average to the rotating case speed. In the ideal symmetric mechanism, the output torques are equal even when their speeds differ.
A car turns left with equal-sized rear tires rolling without slip. Which rear wheel needs more turns?
The right rear wheel is outside this left turn. It covers more distance in the same time. The differential permits the required speed difference.
Imagine two friends walking side by side around a bend. The friend on the outside has farther to go. Equal-sized rear tires also cover unequal distances in a corner. With no rolling slip, the outer tire makes more turns. The differential accommodates this difference; it does not make the car steer.
The drive pinion turns the ring gear, which is fastened to the rotating differential case. Our selected pair has 20 drive teeth and 80 ring teeth. Four input turns produce one case turn. The stationary axle housing surrounds and supports the moving assembly.
Two equal side gears connect to the half-shafts through splines. Each half-shaft connects to one driven wheel. Four small bevel pinions ride on cross-shafts inside the rotating case. Their tapered pitch surfaces meet the side gears on perpendicular axes.
In straight travel with equal tire radii, the two side gears turn with the case. The spider pinions travel around the axle but do not spin relative to their own cross-shafts. In a turn they also spin on those shafts, accommodating the output difference. The carrier-frame view removes the common rotation so this second motion becomes easier to see.
The two output speeds always average to the case speed in this symmetric mechanism. At a fixed case speed, reducing one output by 20 rpm requires the other to increase by 20 rpm. The bench lets you impose those constraints directly; it is separate from the road’s no-slip constraints.
For an ideal symmetric open differential, each output receives half the case torque. Power is torque multiplied by angular speed, so a faster output can receive more power at the same torque. This is why “all torque goes to the faster wheel” is a misleading explanation.
Use rear-midpoint turn radius R, rear track t, wheel radius r and midpoint speed v. Inner and outer rear radii are R − t/2 and R + t/2. Their ground speeds are v(1 − t/(2R)) and v(1 + t/(2R)); divide by r for angular speeds. A quarter-circle path length is πR/2. Straight motion is the zero-curvature limit.
In ideal low-speed Ackermann geometry, all wheel axes meet the same instantaneous turning center. With wheelbase L, the inner and outer steering angles satisfy tan δ = L/(R ∓ t/2). Front-wheel paths are longer than the matching rear paths. This is a no-slip geometric construction, not a high-speed tire-handling model.
With forward rolling positive on both outputs, ωL + ωR = 2ωC. The same relation holds for accumulated angles with consistent initial phases. Our 40-tooth side gears and 20-tooth spider pinions require the spider’s own-axis rotation to be twice the output departure from the case, in the corresponding opposite rolling sense. Opposite camera viewpoints can reverse apparent clockwise directions without changing the scalar convention.
Ignoring friction and internal inertia, TL = TR = TC/2. The 4:1 final drive gives TC = 4Tin and ωin = 4ωC. Power conservation then gives Tinωin = TLωL + TRωR. At our default 100 N·m input torque and 3 m/s midpoint speed, the ideal case has 400 N·m, the outputs 200 N·m each, and the two output powers sum to 4000 W.
Let each supplied capacity be the maximum tire/road reaction torque that can be sustained in this simplified steady, no-slip condition. The ideal equal output torque cannot exceed the smaller capacity. The maximum sustainable case torque is twice that smaller value. This does not specify wheel spin-up, vehicle acceleration or whether available traction overcomes all resistance.
Our authored example has a 1.6 m track, 2.6 m wheelbase, 0.30 m tires and a 6 m rear-midpoint radius. In a left quarter-turn, the rear paths are about 8.168 m and 10.681 m: 4⅓ and 5⅔ wheel turns. Their mean is exactly 5 case turns. The inner front wheel steers about 26.565° and the outer about 20.925°.
The transparent car shows why tire paths differ before introducing the axle internals. Tightening the bend at the same rear-midpoint speed increases the rear-wheel speed difference. Widening it brings their ratio closer to one.
If the case is imposed at 60 rpm and one output is held at zero, the other must turn at 120 rpm. If the case is held at zero and one output turns at +30 rpm, the other turns at −30 rpm. “One wheel is off the ground” alone does not establish either complete set of constraints.
An open unit allows unequal speeds while its ideal output torques are equal. A locked unit imposes equal speeds; limited-slip designs add mechanisms that can support a torque difference. These alternatives solve different problems and need different models. The lesson does not simulate them by relabeling the open unit.
The drive-pinion/ring-gear ratio changes the common speed and torque entering the differential. The internal side-gear mechanism permits a variable difference between outputs. A fixed final-drive ratio does not force a fixed left/right wheel-speed ratio.
Mark a center near a corner of the paper. Use the ruler to mark 8 cm and 12 cm radii in several directions across a quarter circle. Join the points into two smooth arcs. Add a dotted midpoint arc at 10 cm. Predict which path is longer and whether it is twice as long.
Lay the string gently along the inner quarter arc, mark the used length, then straighten it beside the ruler. Repeat for the outer arc without stretching the string. Record both lengths, including any uncertainty from your drawing or measuring.
The calculated lengths are about 12.57 cm and 18.85 cm, a ratio of 1.5. Imagine both wheels have the same circumference, 2π cm (about 6.28 cm). Divide each path length by that circumference: two turns inside and three outside.
The mean of two and three turns is 2.5 turns. Move two paper markers along the arcs at the same swept angle. They have equal time but different distances. Connect this to the case-speed average in the digital bench, then explain what this paper activity cannot measure.
Could both wheels make 2.5 turns here and still roll without slipping?
A paper geometry activity. No tools, vehicle access, lifting, motors or powered parts are needed. It does not reproduce a working differential or prove equal output torque. The on-screen drawing is not a calibrated printable ruler.
Its path is longer for the same elapsed time The outside radius is larger. With equal tire circumferences, the greater distance requires more revolutions.
120 rpm The output average must be 100 rpm: (80 + 120)/2 = 100.
Travel with the case without spinning relative to their cross-shafts They orbit with the case. Their own-axis motion relative to the case is zero when both side gears share its speed.
200 N·m each Symmetry and the ideal constraints give half the case torque to each output. Speed differences do not change this ideal split.
The faster output Power equals torque times angular speed: about 1733 W and 2267 W. They add to 4000 W.
Raise 30 to 90 N·m The smaller capacity limits the equal output pair. Raising it from 30 to 90 raises the case limit from 60 to 180 N·m.
A side gear The side gear and half-shaft turn together. The ring gear is attached to the rotating case; the housing supports the assembly.
It approaches 1 The fixed track becomes small relative to the turning radius. The ratio (R + t/2)/(R − t/2) tends toward one; the separate final-drive ratio remains 4.
AXSM0030, September 2013, printed pp. 9 and 18. Used for component relationships; no manufacturer artwork or CAD is copied.
Dana / Spicer · S140 axle service manualCorentin Guilin, 2024–2025, §3.2 Eqs. 3.1–3.2 and §3.3 Eq. 3.21. The lesson states omitted friction and imposed constraints explicitly.
UCLouvain · Original differential dynamics studyIntroduction to Robotics preprint, §13.3.1.3, printed p. 465, Eq. 13.16. Rear-midpoint geometry supports our original path calculations.
Lynch and Park · Car-like robot kinematicsKatie DiCola, 2018. Practical linkage approximation is distinct from the exact geometry used here.
University of Waterloo · Steering linkage projectManufacturer catalogue, technical information and assembly guidance. Our tooth surfaces are original approximations, not copied engineering drawings.
KHK · Bevel gear technical informationQualitative manufacturer context. Quantitative torque statements in this lesson use the explicit ideal model and static capacity assumptions.
Eaton · Open differentialsIndependent subject review is pending.
Read the sources and model assumptionsTurn once. Count what happens. Discover gear ratios, real-world uses, and why different jobs need different gears.
Interactive explorationOpen a detailed engine, turn its crank, and take the assembly apart. Follow four different jobs hidden inside two very similar-looking turns.
Squeeze a hydraulic brake. See inside the caliper, change piston sizes, and discover how pressure becomes force.