Spatial mechanisms · Parallel offset

Oldham Coupling

Connect parallel shafts that do not share a center. Inspect the perpendicular tongue-and-groove faces, follow the floating disc around its orbit, and compare constant output speed with changing sliding speed.

Output angle $0$

Output / input speed $0$

Disc-center speed $0$

Disc-center acceleration $0$

Required input torque $0$

Signed load power $0$

Slide closure residual $0$

Quasistatic power residual $0$

Turn & compare

Follow what changes between the shafts.

Input phase

\theta

35.00\,{}^\circ

Prescribed input speed

\omega

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

Parallel axis offset

e

0.22\,\mathrm m

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 is parallel to the input and shifted upward. Both shafts and the disc have the same angular phase; their facing tongues and grooves are perpendicular. The disc center follows the exact intersection of two sliding constraints. Its small circular orbit is traversed twice per input turn.

Separate the mating faces is an exploded inspection pose. The hubs move apart only for visibility; those displayed faces are then disengaged. The model, plots and readouts still describe the assembled ideal coupling. Neither the offset nor the display clearances imply a safe working limit.

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.

Ideal parallel shafts and frictionless perpendicular slides. The displayed offset and part dimensions are schematic, not manufacturer ratings. No angular misalignment, wear, contact force, inertia or backlash model.

Start with the essentials

Focus question: How can offset shafts turn evenly while the disc center moves?
One-sentence intuition: Two perpendicular slides absorb the lateral offset without changing the angular phase.

Core mathematical model

Two perpendicular sliding constraints

\mathbf u=(\cos\theta,\sin\theta),\quad\mathbf v=(-\sin\theta,\cos\theta),\quad\mathbf C=e\cos\theta\,\mathbf u=(e,0)+e\sin\theta\,\mathbf v

Coordinates are in the transverse upward–depth plane, with the input center at the origin.

Twice-around center orbit

\mathbf C=\left(\frac e2(1+\cos2\theta),\frac e2\sin2\theta\right),\quad |\dot{\mathbf C}|=e\omega,\quad|\ddot{\mathbf C}|=2e\omega^2

The center orbit has radius half the axis offset; it is not either shaft orbiting.

Constant transmission with changing sliding

\phi=\theta,\quad\frac{\dot\phi}{\omega}=1,\quad\dot s_1=-e\omega\sin\theta,\quad\dot s_2=e\omega\cos\theta

Travel is signed from the corresponding shaft center along its own groove direction.

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
Follow the floating center
Increase the offset and compare the two slide-speed curves.
What to observe: The center goes around twice per input turn. Sliding speeds are in quadrature while shaft speeds stay equal.
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