Question

21. What will be the minimum number of links required to form a kinematic chain considering all the pairs as lower pairs

• A. 3
• B. 4
• C. 5
• D. 6

Hint:

Remember that a three-link system forms a rigid structure (triangle), while a four-link system with lower pairs provides one degree of freedom.
This makes the four-bar chain the simplest kinematic chain used in mechanical linkages.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A kinematic chain is defined as an assembly of links connected by kinematic pairs designed to provide constrained relative motion.
A lower pair is a joint where the contact between two mating elements is a surface or area contact, such as turning, sliding, or screw joints.
To allow relative motion between links, the system must have at least one degree of freedom, preventing it from behaving as a locked, rigid structure.
Key Formula or Approach:
For a kinematic chain consisting entirely of lower pairs, the relation between the number of links (\(l\)) and the number of joints or lower pairs (\(j\)) is governed by Grubler's mobility criterion for plane mechanisms:
\[ F = 3(l - 1) - 2j \]
Alternatively, the structural relationship between the number of links (\(l\)) and the number of lower pairs (\(p\)) is given by:
\[ l = 2p - 4 \]

Step 2: Detailed Explanation:
Let us analyze the system using different numbers of links to find the minimum requirements for a kinematic chain.
If we consider \(l = 3\) links connected by \(p = 3\) lower pairs, we can test the structural relation:
\[ 3 = 2(3) - 4 \implies 3 \neq 2 \]
Applying Grubler's criterion for a three-link assembly with three turning joints (\(j = 3\)):
\[ F = 3(3 - 1) - 2(3) = 6 - 6 = 0 \]
A system with zero degrees of freedom represents a rigid frame or structure, which means no relative motion can occur between the links.
Next, let us evaluate an assembly with \(l = 4\) links connected by \(p = 4\) lower pairs:
\[ 4 = 2(4) - 4 \implies 4 = 4 \]
Applying Grubler's criterion for a four-link assembly with four turning joints (\(j = 4\)):
\[ F = 3(4 - 1) - 2(4) = 9 - 8 = 1 \]
Since the degree of freedom is exactly 1, this assembly forms a valid kinematic chain with constrained relative motion.
The four-bar kinematic chain is the most fundamental kinematic chain consisting entirely of lower pairs.

Step 3: Final Answer:
The minimum number of links required to form a kinematic chain having only lower pairs is 4.