Goubet’s universal shaft coupling joins shafts through two spherical shells. Each shell contains a ball attached to a shaft and a trunnion ring recessed in the ball. The two pivot directions in each assembly let the shaft tilt while transmitting rotation. The cutaway animation exposes both rings and follows the motion of an ideal, coplanar double-joint arrangement.
Goubet's Universal Shaft Coupling Interactive Calculator
Set the angle of each shaft to the central shell axis and the input speed. Watch the ball-and-ring joints turn, then compare the maximum and minimum output speeds. The calculator assumes perpendicular pivot axes within each ring and aligned shell trunnions, as illustrated. Equal joint angles cancel speed variation in this ideal arrangement.
Equation Used
βA and βB are the non-negative angles from the central shell axis to the two coplanar shaft axes. The shafts bend in opposite directions from that axis, so their total bend is βA+βB. The two shell trunnion axes are parallel. The ball-to-ring pivot axis is perpendicular to the ring-to-shell pivot axis. These ideal Cardan constraints determine the speed relation; the historical drawing alone does not specify bearing clearances or operating ratings. Ripple is 100(nmax−nmin)/nin when nin is positive; at zero rpm both shafts stop and the displayed ripple is the geometric ratio range. Dimensions in the cutaway are illustrative. The animation displays rotation at the selected rpm.
Two ball-and-ring universal joints
Hiscox labels the shafts A, the spherical shells B, the recessed trunnion rings C and the balls D. The two shells form the central connecting body. A ring pivots in its shell, while the ball pivots in the ring about a second axis. The shaft is attached to the ball. The drawing shows a complete joint at each end of the central body.
The silver pins in the animation belong to the shell-to-ring pivots. The gold pins show the perpendicular ball-to-ring pivots. Both axes meet at the centre of the ball. The front shell surfaces are cut away so the rings and pins remain visible. A stripe on each shaft shows its rotation.
Hiscox describes 45 degrees of articulation for each shell and 90 degrees for the pair. Those are the historical geometric angles; they are not a speed, torque or service-life rating. The animation illustrates the same double-shell topology with simplified dimensions.
Speed through the ideal double joint
For one ideal joint, if φ is the central shell angle and ψ is the shaft angle, tan ψ=tan φ/cos β, with angles followed continuously through each revolution. Differentiation gives dψ/dφ=cos β/[1−sin²β cos²φ]. The input shaft turns at the selected constant speed; the central body and output shaft follow from these constraints.
For the pair, the instantaneous output/input speed ratio is R=[cos βB/cos βA] × [1−sin²βA cos²φ]/[1−sin²βB cos²φ]. Its extreme values are cos βA/cos βB and the reciprocal. Equal joint angles make R=1 throughout the cycle for the phasing and coplanar arrangement shown.
The positive and negative speed deviations need not have equal magnitudes. The calculator therefore reports separate maximum and minimum speeds rather than a symmetric small-angle approximation. This relation does not include friction, backlash, elastic twist or shaft dynamics.
Example with unequal angles
With a 20-degree input joint, a 30-degree output joint and a 30 rpm input, q=cos(20°)/cos(30°)=1.0851. Output speed varies from about 27.65 to 32.55 rpm. The peak-to-peak ratio variation is about 16.35 percent, and the total shaft bend is 50 degrees.
Changing the output angle to 20 degrees makes both joint angles equal. In the illustrated ideal phasing, output speed stays at 30 rpm although the intermediate shell still turns non-uniformly. Setting both angles to zero gives a straight drive. At zero input speed the entire assembly stops.
What the arrangement demonstrates
The coupling illustrates how two universal joints can share a large overall shaft deflection. It is useful for understanding the difference between articulation and constant-velocity transmission: two joints permit more angular accommodation, but their angles and relative pivot orientation determine whether their speed variations cancel.
A mechanism model can explain these geometric relationships without establishing that a particular historical machine used the coupling or that a surviving part is suitable for service. Selecting a coupling for a working installation also requires its manufacturer’s load, speed, angle and lubrication specifications.
Limits of the cutaway and calculator
The model uses intersecting, perpendicular pivot axes and rigid parts. It does not calculate stresses, clearances, lubrication, wear or interference between finite-width components. Shaft supports and detailed fastening hardware are omitted to expose the joint geometry.
The output-speed result applies to the coplanar, aligned-trunnion arrangement shown. A different orientation of the two rings or a spatial shaft layout changes the relationship. Equal numerical angles alone do not establish constant velocity for every double-joint assembly.
Frequently asked questions
What is inside each spherical shell?
A shaft-mounted ball and a recessed trunnion ring. The ring pivots relative to the shell, and the ball pivots relative to the ring about a perpendicular axis.
Does Goubet’s arrangement have one joint or two?
The Hiscox drawing shows two joined shells, each containing a complete universal joint.
Why does the output speed vary when the input speed is constant?
Each angled joint has a non-uniform angular relationship. With unequal joint angles, the two variations do not cancel in the illustrated arrangement.
When does this model give constant output speed?
When both joint angles are equal, with the coplanar shaft axes and aligned shell trunnions used here. Different phasing or spatial layouts need their own analysis.
Is 90 degrees a safe working angle?
Hiscox describes 45 degrees per joint and 90 degrees for the pair as articulation geometry. A safe operating angle depends on the actual coupling design, speed, load and clearances.
Why is a ratio variation still shown at zero rpm?
It describes the geometry’s possible speed-ratio range. At zero rpm there is no rotation and both numerical speed results are zero.
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