---
title: "Grashof's law and four-bar classification"
subject: "Theory of Mechanism and Machine I"
chapter: 2
key: "semester_6/theory_of_mechanism_and_machine_i:grashof-law-classify-four-bar"
source: https://www.bctengineer.com/q/semester_6/theory_of_mechanism_and_machine_i/grashof-law-classify-four-bar
site: BCT Engineering
---

# Grashof's law and four-bar classification

Theory of Mechanism and Machine I · Chapter 2: Linkages and Mechanisms

- Asked in: year not given, 6 marks
- Video: [Four Bar Chain Mechanism | Grashof's Law | Inversion Of Four Bar Chain](https://www.youtube.com/watch?v=UNEvKKXyIdI) — Sarang Vispute
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/theory_of_mechanism_and_machine_i/grashof-law-classify-four-bar
- Whole chapter: https://www.bctengineer.com/md/semester_6/theory_of_mechanism_and_machine_i/chapter-2

## Question

State Grashof's law for a four-bar linkage. A four-bar linkage ABCD has link lengths AB = 30 mm, BC = 70 mm, CD = 80 mm and DA = 60 mm. Check whether it satisfies Grashof's law and name the type of mechanism obtained when each of the four links in turn is fixed.

## Short answer

Grashof's law: s + l ≤ p + q for a four-bar linkage, where s=shortest, l=longest. Given AB=30, BC=70, CD=80, DA=60 mm: s+l=110 < p+q=130, so Grashof condition satisfied. Inversions: AB fixed (shortest) - double crank; BC fixed (adjacent) - crank rocker; DA fixed (adjacent) - crank rocker; CD fixed (opposite) - double rocker.

## Answer

#### Grashof's law
In a planar four-bar linkage, if the sum of the shortest and longest links is less than or equal to the sum of the other two links, then the shortest link can make a full rotation relative to the other links. If the sum is greater, no link can rotate fully (all inversions are double-rockers).

$$
s + l \le p + q
$$

where $s$ = shortest, $l$ = longest, $p, q$ = the other two links.

#### Check
$s = 30$ mm (AB), $l = 80$ mm (CD), $p = 60$ mm, $q = 70$ mm.

$$
s + l = 30 + 80 = 110\ \text{mm} \quad < \quad p + q = 60 + 70 = 130\ \text{mm}
$$

The linkage satisfies Grashof's law, so it is a Grashof chain and the shortest link AB can rotate fully.

#### Inversions

| Fixed link | Relation to shortest link | Mechanism |
|---|---|---|
| AB (30 mm) | Shortest link fixed | Double-crank (drag link) |
| BC (70 mm) | Adjacent to shortest | Crank-rocker (AB is crank, CD rocker) |
| DA (60 mm) | Adjacent to shortest | Crank-rocker (AB is crank, CD rocker) |
| CD (80 mm) | Opposite to shortest | Double-rocker (coupler AB rotates fully) |

- Shortest link fixed: both links adjacent to it rotate completely.
- A link adjacent to the shortest link fixed: the shortest link is the crank and the link opposite the shortest link is the rocker.
- Link opposite the shortest link fixed (shortest link is the coupler): both side links only oscillate, while the coupler makes full turns.

**Answer: $s + l = 110$ mm $< p + q = 130$ mm, so Grashof; double-crank (AB fixed), crank-rocker (BC or DA fixed), double-rocker (CD fixed).**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Grashof's law says that a link can rotate fully when:
   A. Sum of shortest and longest links <= sum of the other two links
   B. Sum of shortest and longest links >= sum of the other two links
   C. Shortest link is longer than half the frame
   D. Sum of the two shortest links equals the longest link
   Answer: A. If s + l <= p + q, the shortest link can make a full rotation relative to the others. (Key point: Grashof's law statement)

2. For links 30, 70, 80 and 60 mm, which result is correct?
   A. s + l = 110 > p + q = 130, not Grashof
   B. s + l = 110 < p + q = 130, Grashof
   C. s + l = 100 < p + q = 140, Grashof
   D. s + l = 90 < p + q = 150, Grashof
   Answer: B. Shortest is 30 and longest is 80, so s + l = 110, and p + q = 60 + 70 = 130. Since 110 < 130 the linkage is a Grashof chain. (Key point: Grashof check for given linkage)

3. In a Grashof four-bar chain, fixing the shortest link (AB) gives a:
   A. Crank-rocker
   B. Double-rocker
   C. Double-crank (drag link)
   D. Slider-crank
   Answer: C. With the shortest link as frame, both links adjacent to it rotate completely, giving a double-crank. (Key point: Shortest link fixed)

4. Fixing link CD (80 mm), opposite to the shortest link AB, gives a:
   A. Double-crank
   B. Crank-rocker
   C. Drag link
   D. Double-rocker
   Answer: D. When the shortest link is the coupler, both side links only oscillate while the coupler turns fully: a double-rocker. (Key point: Link opposite the shortest fixed)

## Related in Theory of Mechanism and Machine I

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- [Four-bar angular velocities by IC method](https://www.bctengineer.com/question.md?key=semester_6%2Ftheory_of_mechanism_and_machine_i%3Afour-bar-velocity-by-ic-method) — Theory of Mechanism and Machine I, chapter 7

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