A single Cardan joint connects intersecting shafts through two yokes and a cross. At nonzero shaft angle, constant input speed produces a cyclic output-speed ratio; this calculator evaluates that ideal kinematic relation.
Universal Angle Coupling Interactive Calculator
Enter the angle between intersecting shaft axes and an input-yoke phase measured from the calculator's reference plane. Results are ideal rigid single-Cardan kinematics—not a torque, vibration, life or suitability prediction.
Equation Used
- The equation represents one ideal rigid single Cardan joint with intersecting shaft axes.
- The drawing projects intersecting shaft axes and a rigid orthogonal cross onto the screen. Bearing details and clearances are not represented.
- Input angular speed is treated as constant and nonzero.
- No clearance, elasticity, friction, bearing loss, torque load, inertia or damping is included.
- Percent outputs are relative to input angular speed, not measured vibration or allowable limits.
- Only ideal phase-cancelled equal-angle paired joints can cancel the kinematic ripple; installation suitability is not evaluated.
Single Cardan Joint Kinematics
A cross-shaft universal joint has an input yoke, output yoke and orthogonal cross trunnions. Patent literature describes the same cross-and-yoke topology and notes that a single Cardan joint produces variable output angular velocity from constant input velocity when the shafts operate at an angle.
The calculation is purely kinematic. It does not model a named joint series, bearings, strength, vibration or service life.
What This Calculator Represents
The tool evaluates the output-to-input angular-speed ratio at one entered input phase and reports the theoretical maximum, minimum and span over a complete revolution. The mean angular displacement ratio over a full revolution remains one-to-one.
Ideal Single-Cardan Velocity Ratio
rmax = sec(theta)
rmin = cos(theta)
ratio span = rmax - rmin
With this page's phase origin, maxima occur at 0° and 180° and minima at 90° and 270°. The two extrema therefore repeat twice per input revolution.
Worked Kinematic Example
At an entered shaft angle of 18°, cos(theta)=0.9511. The ideal ratio therefore ranges from 0.9511 to 1.0515. Relative to input speed, that is a 4.9% lower departure, 5.1% upper departure and 10.0 percentage-point span.
Model Boundary
- Calculated: ideal instantaneous speed ratio and its extrema for one rigid single Cardan joint.
- Illustrated: a spatial projection of intersecting shaft axes and a rigid orthogonal cross, driven by the ideal Cardan phase relation.
- Not calculated: actual RPM, torque, forces, bearing motion, friction, backlash, elasticity, vibration response, resonance, lubrication, wear, fatigue, temperature, allowable angle, rating, life, installation tolerance or safety.
Patents note that an equal-angle, correctly phased second Cardan joint can cancel the first joint's ideal velocity error, but this calculator does not solve a two-joint installation.
Frequently Asked Questions
Is a single Cardan joint constant velocity?
Only at zero shaft angle in this ideal model.
Why are there two ratio cycles per revolution?
The equation contains cos²(phi), which repeats every 180°.
Does ratio span predict vibration amplitude?
No.
Does the calculator give torque capacity or bearing life?
No.
Will any two U-joints cancel the ripple?
No. Cancellation requires the appropriate equal-angle geometry and phasing.
Does the entered phase describe output-yoke angle?
No; it is the input-yoke phase under this calculator's stated reference convention.
References
- Zhang et al., Study of the influence of universal gearing of steering system on vehicle handling stability: Cardan-joint angular-velocity-ratio relation.
- U.S. Patent 5,498,208: single Cardan output velocity varies with operating angle; an equal-angle in-phase second joint can cancel the oscillation.
- Japanese Patent JP2015011009A: cross-shaft universal-joint yokes, cross and deliberate angular-velocity fluctuation.
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