Links, Pairs and Degrees of Freedom
Start here. Take three bars in a plane and add joints one at a time, watching nine freedoms fall to one.
Skip to the animationEvery rigid body in a plane starts with three degrees of freedom and every joint takes some away, so a mechanism's mobility is simply what is left over — and one degree of freedom, meaning a single input determines every position, is what most machines are designed to have.
The vocabulary
- Link
- A rigid body in the mechanism. Rigid is an idealisation, and a good one at normal loads.
- Kinematic pair
- Two links in contact such that their relative motion is constrained. This is what 'joint' means precisely.
- Kinematic chain
- Links joined so that the motion of one determines the motion of the others.
- Mechanism
- A kinematic chain with one link fixed to the ground.
- Machine
- A mechanism arranged to transmit useful force or work, not just motion.
The counting argument
- 1A free rigid body in a plane needs three numbers to locate it:
x,yandθ. That is 3 degrees of freedom, or 6 in three dimensions. - 2So
nlinks have3nfreedoms — but one link is fixed to the ground and contributes none, leaving3(n − 1). - 3Each lower pair removes 2 freedoms and leaves 1: a pin forces two links to share a point but allows rotation; a slider allows one translation.
- 4Each higher pair — point or line contact, such as a cam on a follower or gear teeth — removes only 1 and leaves 2.
- 5Mobility is what is left:
F = 3(n − 1) − 2j − h.
Applied to a four-bar linkage: n = 4, so 3(4−1) = 9; four pins remove 4 × 2 = 8; mobility is 1. One input — one motor on one crank — determines the position of every other link, which is exactly what makes a mechanism predictable.
Reading the answer
| F | Means | Example |
|---|---|---|
| < 0 | Over-constrained; redundant members | A truss with an extra bar — fine if perfectly sized, self-stressing if not |
| 0 | A structure — nothing moves | A bridge truss, and correctly so |
| 1 | A constrained mechanism | Four-bar linkage, slider-crank, most machines |
| 2 | Needs two independent inputs | Robot arm in a plane, excavator boom |
| > 2 | Under-constrained without more inputs | A chain that flops unless every joint is driven |
The same equation is used in structural analysis with the sign of success reversed: there F = 0 is the goal and F = 1 means the structure is a mechanism and will collapse.
Pairs, classified
| Pair | Contact | DOF | Example |
|---|---|---|---|
| Revolute (pin) | Surface | 1 | Hinge, crank pin |
| Prismatic (slider) | Surface | 1 | Piston in a cylinder |
| Screw (helical) | Surface | 1 | Lead screw — rotation and translation are locked together |
| Cylindrical | Surface | 2 | Shaft free to turn and slide in a bearing |
| Spherical (ball) | Surface | 3 | Ball joint |
| Cam / gear | Line or point | 2 | Higher pair — rolling with sliding |
Lower pairs have surface contact, which spreads the load and wears slowly. Higher pairs have line or point contact, so the contact stress is high — which is why gear and cam design is dominated by surface durability rather than by bending.
What the count cannot see
- How far anything moves. A linkage may have F = 1 and swing through three degrees. Mobility says motion is possible, not that it is useful.
- Whether full rotation is possible. That is Grashof's law: the shortest plus longest link must not exceed the sum of the other two. It is a statement about *lengths*, and no counting argument can produce it.
- Special geometry. A parallelogram linkage counts as over-constrained and works perfectly, because equal link lengths make one constraint redundant rather than conflicting.
- What the motion is for. The same chain, grounded at a different link, becomes a different machine entirely.
Grubler counts constraints and never measures anything. Holding both facts — that it is powerful and that it is blind — is the actual skill.
Inversions
Fixing a different link of the same chain gives an inversion. The relative motion between links is unchanged — only the observer's frame moves — but the machine can be completely different.
| Slider-crank, grounded at | Becomes |
|---|---|
| The frame | Every reciprocating engine and compressor |
| The connecting rod | Oscillating cylinder engine |
| The crank | Rotary engine; Whitworth quick-return mechanism |
| The slider | Hand pump |
All four have identical mobility, which is a compact demonstration of both what the count gives you and what it leaves entirely open.
The numbers you will be asked for
- Grubler / Kutzbach, planar
F = 3(n − 1) − 2j − h
n links, j lower pairs, h higher pairs. The n−1 is the fixed link.
- Spatial mobility
F = 6(n − 1) − Σ(6 − f_i)
Six freedoms per body in three dimensions, less what each joint removes.
- Grashof's condition
s + l ≤ p + q
Shortest plus longest against the other two. About lengths, which mobility never sees.
- Links in a simple chain
j = (3n/2) − 2 for F = 1
Which is why single-DOF chains have an even number of links.
Advantages and disadvantages
Advantages
- One count tells you whether you have built a structure, a mechanism or a jammed assembly.
- It needs no dimensions at all — only the topology of what is connected to what.
- It applies unchanged to structures, where F = 0 is the target instead.
- It catches over-constraint early, before tolerances turn it into self-stress.
Disadvantages
- Blind to link lengths, so it cannot predict range of motion or whether a crank can rotate fully.
- Mis-classifies special geometries such as parallelogram linkages.
- Says nothing about force, torque, or whether the mechanism is any good at its job.
- Assumes rigid links and ideal joints, so clearance and flexibility are invisible to it.
Watch it work
Check yourself
question 1 / 5
One question at a time. Pick an answer to see why it is right or wrong, then move on — there is no score to keep and nothing is saved.