1 degree of freedom — 1 rotation, 0 translations. One rotation about the axis (A, x).
Examples: Shaft in bearings, door hinge.
Kinematic joint symbols (ISO 3952)
The symbols of the kinematic pairs of ISO 3952-1, with their degrees of freedom and the twist and wrench each joint allows, then the gears, pulleys, actuators and cams. Each symbol is drawn by the editor itself, in two colors: one per part. The French names are given too: the same symbols are taught in France as “liaisons normalisées”.
Notation: x is the axis of the joint (or the contact line), y the contact normal, z completes the basis. V gives the velocity of part 2 relative to part 1 (angular velocity ω, velocity v), T the action of 1 on 2 (force F, moment M); a 0 is a blocked motion, or an action the joint can’t transmit. Ideal joints: no clearance, no friction.
| Joint | French name | Rotations | Translations | Degrees of freedom |
|---|---|---|---|---|
| Revolute pair (R) | Pivot | 1 | 0 | 1 |
| Prismatic pair (P) | Glissière | 0 | 1 | 1 |
| Screw pair (H) | Hélicoïdale | 1 | 1, tied to the rotation | 1 |
| Cylindrical pair (C) | Pivot glissant | 1 | 1 | 2 |
| Spherical pair with pin | Rotule à doigt | 2 | 0 | 2 |
| Spherical pair (S) | Rotule | 3 | 0 | 3 |
| Planar contact pair (E) | Appui plan | 1 | 2 | 3 |
| Ball-and-cylinder pair | Linéaire annulaire | 3 | 1 | 4 |
| Cylinder-and-plane pair | Linéaire rectiligne | 2 | 2 | 4 |
| Ball-and-plane pair | Ponctuelle | 3 | 2 | 5 |
| Permanent connection | Encastrement | 0 | 0 | 0 |
Revolute pair (R) · Pivot
{V2/1} =
A
| ωx | 0 |
| 0 | 0 |
| 0 | 0 |
{T1→2} =
A
| Fx | 0 |
| Fy | My |
| Fz | Mz |
Same form at any point of the axis, in the basis (x, y, z).
Prismatic pair (P) · Glissière
1 degree of freedom — 0 rotations, 1 translation. One translation along x.
Examples: Drawer, carriage on a rail.
{V2/1} =
A
| 0 | vx |
| 0 | 0 |
| 0 | 0 |
{T1→2} =
A
| 0 | Mx |
| Fy | My |
| Fz | Mz |
Same form at any point, in the basis (x, y, z).
Screw pair (H) · Hélicoïdale
1 degree of freedom — a rotation and the translation it drives. A rotation about the axis (A, x) and a translation along x, tied by the lead p (mm per turn, right-hand thread).
Examples: Screw and nut, lead screw, ball screw.
{V2/1} =
A
| ωx | p/2π·ωx |
| 0 | 0 |
| 0 | 0 |
{T1→2} =
A
| Fx | −p/2π·Fx |
| Fy | My |
| Fz | Mz |
Same form at any point of the axis, in the basis (x, y, z).
Cylindrical pair (C) · Pivot glissant
2 degrees of freedom — 1 rotation, 1 translation. A rotation about the axis (A, x) and a translation along x, independent.
Examples: Shaft in a long bore, piston rod in its cylinder.
{V2/1} =
A
| ωx | vx |
| 0 | 0 |
| 0 | 0 |
{T1→2} =
A
| 0 | 0 |
| Fy | My |
| Fz | Mz |
Same form at any point of the axis, in the basis (x, y, z).
Spherical pair with pin · Rotule à doigt
2 degrees of freedom — 2 rotations, 0 translations. Two rotations about the center O: the rotation about the pin x is blocked.
Examples: Gear lever, joystick.
{V2/1} =
O
| 0 | 0 |
| ωy | 0 |
| ωz | 0 |
{T1→2} =
O
| Fx | Mx |
| Fy | 0 |
| Fz | 0 |
Same form at the center O, in the basis (x, y, z).
Spherical pair (S) · Rotule
3 degrees of freedom — 3 rotations, 0 translations. Three rotations about the center O.
Examples: Steering ball joint, tow hitch.
{V2/1} =
O
| ωx | 0 |
| ωy | 0 |
| ωz | 0 |
{T1→2} =
O
| Fx | 0 |
| Fy | 0 |
| Fz | 0 |
Same form at the center O, in the basis (x, y, z).
Planar contact pair (E) · Appui plan
3 degrees of freedom — 1 rotation, 2 translations. One rotation about the normal y and two translations in the plane (x, z).
Examples: Block lying flat on a table, base plate on a bench.
{V2/1} =
A
| 0 | vx |
| ωy | 0 |
| 0 | vz |
{T1→2} =
A
| 0 | Mx |
| Fy | 0 |
| 0 | Mz |
Same form at any point, in the basis (x, y, z).
Ball-and-cylinder pair · Linéaire annulaire
4 degrees of freedom — 3 rotations, 1 translation. Three rotations about the center O and one translation along the cylinder’s axis x.
Examples: Ball in a cylinder; a single ball bearing is often modeled this way.
{V2/1} =
O
| ωx | vx |
| ωy | 0 |
| ωz | 0 |
{T1→2} =
O
| 0 | 0 |
| Fy | 0 |
| Fz | 0 |
Same form at the center O, in the basis (x, y, z).
Cylinder-and-plane pair · Linéaire rectiligne
4 degrees of freedom — 2 rotations, 2 translations. Two rotations (about the normal y and about the contact line x) and two translations in the tangent plane (x, z).
Examples: Cylinder or roller on a plane, contact between gear teeth.
{V2/1} =
A
| ωx | vx |
| ωy | 0 |
| 0 | vz |
{T1→2} =
A
| 0 | 0 |
| Fy | 0 |
| 0 | Mz |
Same form at any point of the plane (A, x, y), in the basis (x, y, z).
Ball-and-plane pair · Ponctuelle
5 degrees of freedom — 3 rotations, 2 translations. Three rotations and two translations in the tangent plane (x, z): only the contact along the normal y is kept.
Examples: Ball on a plane, knife-edge follower on a cam.
{V2/1} =
A
| ωx | vx |
| ωy | 0 |
| ωz | vz |
{T1→2} =
A
| 0 | 0 |
| Fy | 0 |
| 0 | 0 |
Same form at any point of the normal (A, y), in the basis (x, y, z).
Permanent connection · Encastrement
0 degrees of freedom. No relative motion: the two parts make one body.
Examples: Parts bolted, welded or glued together.
{V2/1} =
A
| 0 | 0 |
| 0 | 0 |
| 0 | 0 |
{T1→2} =
A
| Fx | Mx |
| Fy | My |
| Fz | Mz |
Same form at any point, in the basis (x, y, z).
Frame, gears and pulleys
Frame · bâti
The part deemed fixed: body 0 of the mechanism (ground).
External gear pair · Engrenage cylindrique extérieur
Two toothed wheels on parallel axes, turning opposite ways.
Internal gear pair · Engrenage cylindrique intérieur
A pinion inside a ring gear: both turn the same way.
Bevel gear pair · Engrenage conique
Intersecting axes, usually at a right angle.
Worm and wheel · Système roue et vis sans fin
Perpendicular, non-intersecting axes, large ratio, often self-locking.
Rack and pinion · Système roue-crémaillère
Pinion rotation ↔ rack translation.
Belt and pulleys · Poulies et courroie
Flexible drive between parallel shafts, both turn the same way.
Friction wheels · Roues de friction cylindriques
Smooth wheels rolling on each other without slipping.
Bevel friction wheels · Roues de friction coniques
Friction wheels on intersecting axes.
Single wheels and pulleys
A single wheel is a part, not a joint: two gears that touch mesh, a pulley joins others through a belt, a chain or a cable.
Gear · Roue dentée
Ring gear (internal teeth) · Couronne (denture intérieure)
Rack · Crémaillère
Pulley · Poulie
Sprocket · Pignon de chaîne
Wheel on floor · Roue sur sol
Floor · Sol
Actuators and cams
Linear actuator · Vérin linéaire
Hydraulic or pneumatic cylinder: body and rod in a cylindrical pair.
Screw jack · Vérin à vis
Input shaft (crank or motor), screw and nut: the rod moves out.
Cam and follower · Came et poussoir
Profiled disc and follower (flat, knife-edge or roller), translating or oscillating.
Examples to open in the editor
- Four-bar linkage — Crank-rocker (Grashof): the crank turns all the way round, the rocker swings. h = 3 in space, 0 as a planar model.
- Slider-crank — The mechanism of a piston engine: the crank’s rotation becomes the piston’s translation.
- Two-stage gear reducer — Three shafts, two pairs of 20- and 30-tooth gears: ratio (20/30)² ≈ 0.444.
- Eccentric cam — A flat follower on a disc 8 mm off center: it moves by e·sin θ, simple harmonic motion.
- Tipping body — A screw jack, turned with a hand crank, lifts the arm until the rod reaches its stop.
- Shaft on two revolute pairs — The classic overconstrained case, h = 5, and the joints to use instead.