Spatial mechanisms · Uneven rotation

Universal Joint

Follow two intersecting shafts joined by a gold right-angle cross. Increase the bend, compare input and output phase markers, and measure the speed ripple hidden inside a complete revolution.

Output angle $0$

Output / input speed $0$

Output phase error $0$

Output acceleration $0$

Required input torque $0$

Signed load power $0$

Cross orthogonality residual $0$

Quasistatic power residual $0$

Turn & compare

Follow what changes between the shafts.

Input phase

\theta

40.00\,{}^\circ

Prescribed input speed

\omega

60.00\,{}^\circ/\mathrm s

Shaft intersection angle

\beta

35.00\,{}^\circ

Signed external output torque

\tau_o

-1.00\,\mathrm{N\,m}

Blue is input, coral output and gold the connecting member. World axes point right, up and toward the viewer. The input shaft points right; positive input rotation follows the right-hand rule. Use the gold phase marker in the separate end view or the keyboard phase slider.

The output axis tilts upward in the world horizontal–vertical plane. Each fork rotates about its own shaft. The cross axes remain perpendicular. The end-view phase comparison uses a separate shaft-normal coordinate basis for each marker; it is not a common spatial projection.

A single bent joint is not constant-velocity. Its output completes one revolution per input revolution but speeds up and slows down twice. Straightening the shafts removes the ripple. The right-angle trunnions and the zero-phase convention determine the plotted phase relation.

Input speed is imposed. Required torque balances a constant signed external output torque in a rigid, massless, frictionless model. It is a virtual-power calculation, not a motor-sizing result. The load can supply or absorb power; no energy is lost to the modeled slides.

Playback makes one input turn and stops; replay explicitly resets phase. Lengths use metres and the display sliders use degrees. CSV contains 721 half-degree kinematic samples, with radians and SI rates. Part thicknesses, bearings and supports are schematic; there is no collision, force, fatigue or wear solver.

Single rigid Hooke–Cardan joint with ideal intersecting axes and prescribed input speed. Exact kinematics and massless quasistatic load transfer; no inertia, friction, bearing forces, flexibility or interference rating.

Start with the essentials

Focus question: Why does a bent universal joint change output speed twice each turn?
One-sentence intuition: The right-angle cross constrains the two fork orientations, so equal angle increments cannot pass through unchanged.

Core mathematical model

Right-angle cross constraint

\tan\phi=\cos\beta\tan\theta,\qquad\phi=\operatorname{unwrap}\!\left[\operatorname{atan2}(\cos\beta\sin\theta,\cos\theta)\right]

Quadrant-safe unwrapping preserves a full output turn; the tangent relation alone loses quadrant information.

Instantaneous speed ratio

R=\frac{\dot\phi}{\omega}=\frac{\cos\beta}{1-\sin^2\beta\sin^2\theta},\quad R_{\min}=\cos\beta,\quad R_{\max}=\frac1{\cos\beta}

The extrema occur a quarter input turn apart and the pattern repeats each half-turn.

Output acceleration

\ddot\phi=\omega^2\frac{\cos\beta\sin^2\beta\sin2\theta}{(1-\sin^2\beta\sin^2\theta)^2}

Input angular speed is fixed; output angular acceleration can still be nonzero.

Massless quasistatic virtual power

\tau_{\rm in}=-\tau_o\frac{\mathrm d\phi}{\mathrm d\theta},\qquad\tau_{\rm in}\omega+\tau_o\dot\phi=0

No frictional loss or inertial power is included.

Common difficulties

Equal turns versus equal speed

Typical misconceptionOne output turn per input turn guarantees constant speed.

Better mental modelA universal joint has a varying instantaneous ratio; an ideal parallel Oldham coupling has a constant ratio.

Kinematics versus dynamics

Typical misconceptionA smooth animation establishes acceptable bearing loads and wear.

Better mental modelInput motion is prescribed, and the connecting parts are massless and frictionless. Dynamic inertia, real clearances and contact forces require a different model.

Run the experiment

01
Establish the aligned limit
Select aligned shafts and play one turn.
What to observe: Both shaft phase markers agree and the speed ratio stays unity.
02
Increase the bend
Compare the slowest and fastest output presets, then inspect the cross from different views.
What to observe: The output speed oscillates twice while the orthogonality residual remains near numerical precision.
03
Inspect the transfer of power
Reverse the signed load and export the cycle.
What to observe: Torque and power change sign; the prescribed motion does not.

Theory & apparatus: Carnegie Mellon · Universal Joint · Ruland · Oldham Coupling