Question

The relation between number of links $ (l) $ and number of joints $ (j) $ in a kinematic chain is generally

• A. \( l = \frac{(j+2)}{2} \)]
• B. \( l = \frac{2(j+2)}{3} \)
• C. \( l = \frac{3(j+3)}{4} \)
• D. \( l = j+4 \)

Hint:

For a planar kinematic chain with one degree of freedom and only single-degree-of-freedom joints (revolute or prismatic), the relationship between the number of links (\( l \)) and the number of joints (\( j \)) is given by Grübler's criterion: \( l = \frac{2(j+2)}{3} \).

Answer 1

The Correct Option is B

Step 1: Understand the concept of a kinematic chain and its degrees of freedom.
A kinematic chain is an assembly of rigid bodies (links) connected by joints to provide constrained motion. The relationship between the number of links (\( l \)) and the number of joints (\( j \)) in a kinematic chain is crucial for determining its mobility or degrees of freedom.
Step 2: Recall Kutzbach's Criterion for a planar mechanism.
For a planar mechanism, Kutzbach's Criterion for the degrees of freedom (\( F \)) is given by: \[ F = 3(l-1) - 2j_1 - j_2 \] where:
\( l \) = number of links \( j_1 \) = number of joints with one degree of freedom (e.g., revolute or prismatic joints) \( j_2 \) = number of joints with two degrees of freedom (e.g., higher pairs like cam-follower)
Step 3: Apply the condition for a constrained kinematic chain.
For a constrained kinematic chain (which is the usual context for 'relation between links and joints' in a general sense), the degrees of freedom \( F \) must be equal to 1. This means the mechanism can be driven by a single input. In many basic cases, we consider only single degree of freedom joints (\( j_1 \)) and no higher pairs (\( j_2 = 0 \)). Substituting \( F=1 \) and \( j_2=0 \) (so \( j_1 = j \)) into Kutzbach's criterion: \[ 1 = 3(l-1) - 2j \] \[ 1 = 3l - 3 - 2j \]
Step 4: Rearrange the equation to express \( l \) in terms of \( j \).
\[ 1 + 3 + 2j = 3l \] \[ 4 + 2j = 3l \] \[ 3l = 2j + 4 \] \[ 3l = 2(j+2) \] \[ l = \frac{2(j+2)}{3} \] This relation is known as Grübler's criterion for planar mechanisms, which is a specific case of Kutzbach's criterion for mechanisms with one degree of freedom and only revolute/prismatic joints. This formula is generally used to define the relationship between links and joints for a constrained kinematic chain. The final answer is $\boxed{\text{2}}$.