Two correctly arranged Hooke joints can cancel their individual speed variations. This calculator models a coplanar two-joint shaft: equal operating angles and aligned intermediate yokes give constant ideal output speed. The animation exposes both crosses and yokes so the intermediate rotation is visible.
Double Hooke Joint Constant Velocity Interactive Calculator
Inspect two connected Cardan joints in a projected 3D view. Adjust each bend angle, intermediate-yoke phasing and input speed. Both joint crosses and the intermediate rotation determine the actual output motion.
Equation Used
- Rigid shafts and perpendicular cross trunnions without clearance.
- Shaft axes are coplanar and fixed.
- Zero phasing error means parallel intermediate fork axes.
- Shaft length and yoke thickness are schematic.
- 720 positions sample output-speed extrema.
- Zero input rpm stops motion while preserving manual position inspection.
Ideal spatial joint kinematics with coplanar fixed shaft axes. Output extrema are sampled; torque capacity and vibration are not calculated.
One shaft drives the next through each cross
Each joint contains a cross with two perpendicular trunnion axes. One pair runs in the driving fork and the other pair in the driven fork. The yokes rotate about their own shaft axes while the cross articulates between them.
The first joint determines the intermediate shaft orientation. The second joint is solved from that orientation, including the entered twist of its inner yoke. The intermediate shaft therefore supplies the position and speed used by the second joint.
In the illustrated coplanar arrangement, matching the two bend angles makes the outer shafts parallel. Correctly aligned inner yokes cancel the ideal output-speed variation. Unequal angles or phasing error generally leave a ripple.
Explore angle and phasing
Set the two operating angles and the intermediate-yoke phasing error. The model updates the actual fork orientations, live intermediate and output speeds, and the full-revolution speed-ratio curves.
The input-speed slider scales the rpm results. Setting it to zero stops automatic motion while leaving the cycle-position control available. The geometric ratio curve remains meaningful as a positional relationship.
The drawing uses schematic shaft lengths and yoke dimensions. It explains joint kinematics rather than selecting a commercial joint or specifying a driveline.
Sequential spatial constraints
Let n1 be the intermediate shaft axis and u0 the first input-fork trunnion direction. The first driven trunnion direction is proportional to n1×u0. Normalizing it gives a unit direction perpendicular to both the intermediate shaft and the input trunnion.
Rotate that direction about n1 by the entered phasing error to obtain the second input-fork direction. Cross it with the final shaft axis and normalize again to obtain the final driven trunnion direction. Output angle is read in a plane perpendicular to the output shaft. Analytic derivatives of these normalized vectors give the instantaneous speed ratios.
For zero phasing error, the solution simplifies to φ=atan2(cosβ2 sinθ, cosβ1 cosθ), with angles unwrapped continuously. The output-to-input speed ratio is cosβ1 cosβ2 divided by (cos²β1 cos²θ + cos²β2 sin²θ). Equal operating angles reduce this ratio to exactly one.
Minimum and maximum speeds are sampled at 720 positions in one input revolution. Peak-to-peak ratio variation is 100(rmax−rmin) percent. It is a geometric measure relative to input speed, not a measured vibration level.
Equal bends with a fluctuating middle shaft
The starting configuration uses two 20° joints, zero phasing error and 60 rpm input. The ideal output remains at 60 rpm. The intermediate shaft varies from about 56.4 to 63.9 rpm; the second joint cancels that variation at the output.
Changing the second angle or twisting the intermediate yoke breaks that cancellation. The green output-ratio curve then varies over a revolution. The grey curve shows the intermediate speed ratio.
At two zero bend angles, both joints transmit a constant speed even if the yoke setting introduces an angular offset. At zero input rpm, all actual rpm readouts are zero.
Scope of the model
The axes lie in one plane and stay fixed in space. The calculation treats the cross bearings and shafts as ideal rigid connections without clearance or elastic deflection.
The entered angle range is an exploration range, not a supplier operating limit. The model does not calculate rated torque, bearing stress, wear, balance, resonance or shaft support loads.
Playback is slowed by a factor of ten. A perfectly flat ideal output-speed curve is not a claim that a manufactured assembly has zero vibration or error.
Double-Hooke-joint questions
Why does the middle shaft speed vary when the output is constant?
The first joint introduces a positional speed variation; with the correct geometry and phasing, the second joint cancels it.
What does zero phasing error mean here?
The two inner fork axes are parallel on the intermediate shaft. The input is a deviation from that aligned condition.
Why can’t both joints use the original input angle?
The second joint is driven by the intermediate shaft, whose orientation is generally nonuniform. It must be evaluated sequentially.
Does zero ripple mean the assembly is safe at any torque?
No. The result concerns ideal speed transmission only; torque and operating limits require the actual joint specifications.
What happens when input speed is zero?
Automatic motion stops and actual rpm results are zero. Manual position inspection and geometric ratios remain available.
References
Belden Universal, Connecting Multiple U-Joints, explains the coplanar arrangement, matching operating angles and aligned inward yokes required for compensated output.
Belden Universal, Universal Joint Phasing, describes alignment of the yokes on the intermediate shaft. The vector constraints and their analytic derivatives implement the illustrated ideal geometry.
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