Mechanisms, Mobility & Kinematic Inversions

Theory of Machines contributes ~6–9 marks to GATE ME, and mechanisms is the entry point. Mobility (degrees of freedom), Grashof's law and inversions are quick, high-confidence marks.

Links, pairs & mobility

A kinematic link is a rigid body; a kinematic pair is a joint. Lower pairs have surface contact (revolute, prismatic, screw — 1 DOF each in planar work); higher pairs have line/point contact (cam, gear).

Kutzbach (Grubler) mobility, planar: DOF = 3(n − 1) − 2j₁ − j₂, where n = links, j₁ = lower (single-DOF) pairs, j₂ = higher pairs. DOF = 1 → a constrained mechanism (single input); DOF = 0 → a structure; DOF < 0 → a preloaded (redundant) structure.

Grashof's law (four-bar): if s + l ≤ p + q (s = shortest, l = longest, p, q = others), at least one link fully rotates. Fixing different links gives inversions:

• Four-bar inversions: crank-rocker, double-crank (drag-link), double-rocker.
• Slider-crank inversions: reciprocating engine, oscillating-cylinder engine, Whitworth quick-return, crank-and-slotted-lever.

Exam Tricks & Tips

• 🎯 DOF = 3(n−1) − 2j₁ − j₂: count links and joints carefully — a single joint connecting k links counts as (k−1) pairs.
• 🎯 DOF = 1 means one input fully controls the mechanism (most machines); DOF = 0 is a structure (a truss), not a mechanism.
• 🎯 Grashof: shortest + longest ≤ other two → a link can fully rotate; if it fails, you get a double-rocker.
• 🎯 The four-bar has 4 links, 4 revolute pairs → DOF = 1 (a benchmark to check your counting).
• 🎯 Inversion = fixing a different link of the SAME chain — the relative motion is unchanged, only the frame changes.
• ❌ Common mistake: counting a compound (multiple) joint as one lower pair — a pin joining 3 links is 2 pairs, and miscounting throws off the whole DOF.

Expected exam pattern

A 1-mark DOF calculation, a Grashof-classification or "identify the inversion" MCQ, and occasionally a quick-return-ratio question. Distinguishing lower vs higher pairs is a common conceptual point.

Quick recap

DOF = 3(n−1) − 2j₁ − j₂; DOF 1 = mechanism, 0 = structure. Grashof s+l ≤ p+q → a link rotates fully. Four-bar (4 links, 4 pins) has DOF 1. Inversions fix different links of one chain (engine, Whitworth, drag-link). Count compound joints as (k−1) pairs.

Cover the answer, recall, then check. 11 cards on mechanisms for GATE ME.

Q1. State the Kutzbach criterion for planar mobility.
A1. DOF = 3(n − 1) − 2j₁ − j₂, where n = links, j₁ = lower pairs (1-DOF), j₂ = higher pairs.

Q2. What do DOF = 1, 0, and negative signify?
A2. 1 → constrained mechanism (one input); 0 → structure; negative → statically indeterminate/preloaded structure.

Q3. Difference between lower and higher pairs?
A3. Lower pairs have surface (area) contact (revolute, prismatic, screw); higher pairs have line/point contact (cam-follower, gear teeth).

Q4. State Grashof's law for a four-bar chain.
A4. If s + l ≤ p + q (shortest + longest ≤ other two), at least one link can make a complete revolution.

Q5. Name the four-bar inversions.
A5. Crank-rocker, double-crank (drag-link), and double-rocker — obtained by fixing different links of the same chain.

Q6. Name slider-crank inversions.
A6. Reciprocating engine, oscillating-cylinder engine, Whitworth quick-return, and crank-and-slotted-lever mechanisms.

Q7. What is a kinematic inversion?
A7. Fixing (grounding) a different link of the same kinematic chain; relative motion between links is unchanged.

Q8. DOF of a standard four-bar linkage?
A8. 1 (n = 4 links, j₁ = 4 revolute pairs: 3(4−1) − 2·4 = 9 − 8 = 1).

Q9. How do you count a joint where 3 links meet?
A9. As 2 lower pairs (in general, k links at one joint = k − 1 pairs).

Q10. What is a structure in mobility terms?
A10. A rigid assembly with DOF = 0 — no relative motion possible (e.g. a simple triangular truss).

Q11. Which quick-return mechanism gives a larger stroke ratio, Whitworth or slotted-lever?
A11. The Whitworth mechanism generally gives a higher quick-return (time) ratio; both convert rotation into a fast-return linear stroke.

Theory of Machines starts by counting: how many independent motions does a linkage have? Grubler's/Kutzbach's mobility criterion answers that, and inversions explain how one four-bar loop becomes an engine, a pump, or a quick-return shaper. GATE tests this counting and classification directly.

Core concept: a mechanism is a kinematic chain with one link fixed; its mobility (degrees of freedom) is the number of independent inputs needed to define the motion of every link.

Deep explanation

Beginner — links, pairs and chains

• A kinematic pair is a joint constraining two links. Lower pairs (revolute, prismatic, screw) contact over a surface; higher pairs (cam-follower, gear teeth) contact along a line/point.
• A kinematic chain becomes a mechanism when one link is grounded.
• Grashof's law (for four-bar): if s + l ≤ p + q (shortest + longest ≤ other two), at least one link can fully rotate (crank).

Intermediate — Kutzbach/Grubler mobility

For planar mechanisms:
M = 3(n − 1) − 2j₁ − j₂,
where n = number of links, j₁ = number of lower (1-DOF) pairs, j₂ = number of higher (2-DOF) pairs.

• M = 1: single input drives the mechanism (most machines).
• M = 0: a structure (no motion).
• M < 0: a redundant/preloaded structure.
  For a four-bar: n = 4, j₁ = 4, j₂ = 0 → M = 3(3) − 8 = 1 ✓.

Advanced — inversions

Fixing a different link of the same chain yields a different mechanism — an inversion. From the slider-crank chain:

• Fix the frame → reciprocating engine/compressor.
• Fix the connecting rod → oscillating cylinder / Whitworth quick-return.
• Fix the crank → rotary engine / Whitworth mechanism.
• Fix the slider → hand pump.
  The four-bar chain inverts into the beam engine, coupling-rod (locomotive) mechanism, and Watt's mechanism.

Worked example

A planar mechanism has 6 links, 7 lower pairs and 1 higher pair. Find its degrees of freedom.
M = 3(n − 1) − 2j₁ − j₂ = 3(6 − 1) − 2(7) − 1 = 15 − 14 − 1 = 0.
So it is a structure (statically determinate frame), not a moving mechanism — a single input would not produce motion.

GATE relevance

Mobility (Kutzbach) calculation and inversion identification are frequent 1-mark TOM questions. Grashof classification (crank-rocker, double-crank, double-rocker) and recognising the quick-return mechanism recur in both concept and numerical form.

Exam tricks & shortcuts

• Count carefully: a joint where three links meet counts as two pairs.
• For a simple four-bar or slider-crank, M = 1 — use it as a sanity check.
• Mnemonic for Grashof: "Shortest plus longest ≤ other two → a crank exists."

Key formulas

• Grübler/Kutzbach (planar): DOF = 3(n − 1) − 2j₁ − j₂ (n = links, j₁ = lower pairs, j₂ = higher pairs).
• Mechanism if DOF = 1; structure if DOF = 0.
• Grashof's law (4-bar): s + l ≤ p + q for continuous rotation (s = shortest, l = longest).
• Inversions: obtained by fixing different links of the same kinematic chain.

Kutzbach's equation gives mobility; Grashof's law tells whether a four-bar link can fully rotate.

Worked Example

Problem: A planar mechanism has 6 links connected by 7 lower (single-degree-of-freedom) pairs and no higher pairs. Use Kutzbach's criterion to find its degrees of freedom (mobility). Also verify the result for a simple four-bar mechanism.

Solution:
Kutzbach's criterion for a planar mechanism:
DOF = 3(n − 1) − 2j − h,
where n = number of links, j = number of lower pairs (1-DOF joints), h = number of higher pairs.

For the given mechanism (n = 6, j = 7, h = 0):
DOF = 3(6 − 1) − 2(7) − 0 = 15 − 14 = 1.

A DOF of 1 means a single input (one driven link) fully determines the motion — a properly constrained mechanism.

Verify for a four-bar mechanism (n = 4 links, j = 4 turning pairs, h = 0):
DOF = 3(4 − 1) − 2(4) − 0 = 9 − 8 = 1 ✓.
The four-bar also has one DOF, consistent with a single crank input.

Answer: DOF = 1 (a single-input constrained mechanism); the four-bar check gives the same result.