Mechanisms & Inversions
Degrees of Freedom, Grashof’s Law, Four-Bar Linkage & Slider-Crank Inversions
Last Updated: March 2026
Key Takeaways 📌
- A mechanism is a system of rigid bodies (links) connected by joints that transforms input motion into desired output motion.
- Degrees of Freedom (DOF): F = 3(n−1) − 2j₁ − j₂ (Grubler/Kutzbach criterion for planar mechanisms).
- Grashof’s law: For a four-bar linkage, at least one link can make a full revolution if s + l ≤ p + q (shortest + longest ≤ sum of others).
- Four-bar linkage is the most fundamental mechanism — it has 4 inversions depending on which link is fixed.
- Slider-crank mechanism converts rotary to linear motion (or vice versa) — used in IC engines, pumps, compressors.
- A kinematic chain becomes a mechanism when one link is fixed (grounded).
1. Kinematic Links, Pairs & Chains
A link (or element) is a rigid body that has two or more pairing elements (connection points). Links are classified by the number of joints they have: binary (2 joints), ternary (3), quaternary (4).
A kinematic pair is a connection between two links that allows relative motion. The two most common planar pairs are:
| Pair Type | Motion | DOF Removed | Symbol |
|---|---|---|---|
| Revolute (pin/hinge) | Rotation only | 2 (removes 2 translational DOF) | j₁ |
| Prismatic (slider) | Translation only | 2 | j₁ |
| Higher pair (cam, gear) | Rolling + sliding | 1 | j₂ |
A kinematic chain is a closed group of links connected by pairs. When one link of a chain is fixed (grounded), it becomes a mechanism. Different choices of fixed link produce different mechanisms — these are called inversions.
2. Degrees of Freedom — Grubler’s Criterion
Grubler’s (Kutzbach) Criterion — Planar Mechanisms
F = 3(n − 1) − 2j₁ − j₂
Where:
- F = degrees of freedom (mobility)
- n = total number of links (including the fixed link/frame)
- j₁ = number of lower pairs (revolute, prismatic) — each removes 2 DOF
- j₂ = number of higher pairs (cam, gear contact) — each removes 1 DOF
| F Value | Meaning | Example |
|---|---|---|
| F = 1 | Constrained mechanism — one input drives the entire mechanism | Four-bar linkage, slider-crank |
| F = 0 | Structure (frame) — no motion possible | Triangular truss |
| F = 2 | Two inputs needed — differential mechanism | Automobile differential |
| F < 0 | Super-structure — over-constrained, statically indeterminate | Redundant frame |
Example: Four-bar linkage: n = 4 links, j₁ = 4 revolute pairs, j₂ = 0.
F = 3(4−1) − 2(4) − 0 = 9 − 8 = 1 ✓ (one DOF — constrained mechanism).
3. Grashof’s Law
Grashof’s law determines whether a four-bar linkage can have at least one link that makes a complete revolution.
Grashof’s Condition
Let s = shortest link, l = longest link, p and q = other two links.
s + l ≤ p + q → Grashof linkage (at least one link can rotate fully)
s + l > p + q → Non-Grashof linkage (no link can rotate fully — all links rock)
If the Grashof condition is satisfied, the type of mechanism depends on which link is fixed:
| Fixed Link | Mechanism Type | Description |
|---|---|---|
| Link adjacent to shortest | Crank-rocker | Shortest link rotates fully (crank); opposite link oscillates (rocker) |
| Link opposite to shortest | Double-crank (drag link) | Both links adjacent to the fixed link rotate fully |
| Shortest link itself | Double-rocker | Shortest link = coupler; both grounded links oscillate. The coupler rotates. |
4. Four-Bar Linkage & Its Inversions
The four-bar linkage is the simplest and most important closed-loop mechanism. It has four rigid links (including the frame) connected by four revolute joints. It has 4 inversions — one for each link being fixed.
Applications of four-bar linkages: windshield wiper mechanism, sewing machine, bicycle pedal mechanism, rock crusher, aircraft landing gear retraction.
Key terminology:
- Frame (fixed link): The ground or stationary link.
- Crank: A link that can make complete rotations relative to the frame.
- Rocker (lever): A link that oscillates (swings back and forth) without completing a full rotation.
- Coupler: The floating link connecting the crank and rocker (not attached to the frame).
5. Slider-Crank Mechanism & Its Inversions
The slider-crank mechanism converts rotary motion to linear reciprocating motion (or vice versa). It is the fundamental mechanism in IC engines, reciprocating pumps, and compressors.
The slider-crank chain has 4 links and 3 revolute pairs + 1 prismatic pair. It has 4 inversions:
| Inversion | Fixed Link | Application |
|---|---|---|
| 1st | Cylinder (frame) is fixed; crank rotates, slider reciprocates | IC engine, reciprocating pump, reciprocating compressor |
| 2nd | Crank is fixed; other links move around it | Whitworth quick-return mechanism, rotary engine |
| 3rd | Connecting rod is fixed | Oscillating cylinder engine, crank-and-slotted-lever mechanism |
| 4th | Slider (piston) is fixed; cylinder moves | Hand pump, pendulum pump |
Slider-Crank — Piston Displacement
x = r(1 − cosθ) + (r²sin²θ)/(2l) (approximate, for l >> r)
Where: x = piston displacement from TDC, r = crank radius, l = connecting rod length, θ = crank angle
6. Worked Examples
Example 1: DOF Calculation
Problem: A mechanism has 5 links and 5 revolute joints. Find the DOF.
F = 3(5−1) − 2(5) = 12 − 10 = 2 DOF
This mechanism needs 2 independent inputs to fully define its motion.
Example 2: Grashof’s Law
Problem: A four-bar linkage has link lengths 30, 40, 50, and 60 mm. Is it a Grashof linkage? If the 30 mm link is grounded, what type of mechanism is it?
s = 30, l = 60, p = 40, q = 50
s + l = 30 + 60 = 90. p + q = 40 + 50 = 90.
s + l = p + q → Grashof linkage (boundary case — change-point mechanism).
If 30 mm link is fixed (shortest link fixed) → Double-rocker with coupler rotating.
7. Common Mistakes Students Make
- Forgetting to count the fixed link: In Grubler’s criterion, n includes the frame (fixed link). A four-bar linkage has n = 4, not 3.
- Confusing inversions with different mechanisms: Inversions of the same kinematic chain are fundamentally the same chain with different links fixed. The relative motion between links is the same — only the absolute motion (relative to ground) changes.
- Applying Grashof’s law to non-four-bar mechanisms: Grashof’s law applies only to four-bar linkages with revolute joints. It does not apply to slider-crank mechanisms or mechanisms with more links.
- Miscounting joints in DOF calculation: A joint where three links meet is two joints (not one), because it represents two kinematic pairs. Count pairs, not physical pin locations.
8. Frequently Asked Questions
What is Grashof’s law?
Grashof’s law states that in a four-bar linkage, at least one link can make a complete revolution relative to another if the sum of the shortest and longest links is less than or equal to the sum of the other two links: s + l ≤ p + q. If this condition is violated, all links can only oscillate (no full rotation is possible).
What is the difference between a mechanism and a machine?
A mechanism transforms motion — it changes the direction, speed, or type of motion (rotary to linear, for example). A machine uses a mechanism to do useful work — it transforms energy and applies forces. An IC engine mechanism converts reciprocating motion to rotary motion; the engine as a whole is a machine that converts thermal energy to mechanical work.