Kinematics of mechanisms – total number of instantaneous centers For a mechanism (kinematic chain) having n links, what is the total number of instantaneous centers of rotation that can be identified (Kennedy’s theorem framework)?

Introduction / Context:
Instantaneous centers (ICs) are pivotal in graphical kinematics to determine velocities without calculus. In a mechanism with multiple links, every pair of links has one IC, enabling systematic velocity mapping via Kennedy’s theorem of three centers in relative motion.

Given Data / Assumptions:

• Planar mechanism with n rigid links connected by pairs.
• All pairs of links possess a unique instantaneous center.
• Links are numbered and considered in pairs.

Concept / Approach:

The count of ICs equals the number of unique unordered pairs of links, which is a combination count C(n, 2) = n(n − 1)/2. This is independent of the particular mechanism layout; location of ICs depends on geometry, but the count is purely combinatorial.

Step-by-Step Solution:

Verification / Alternative check:

Check small cases: n = 2 → 1 IC; n = 3 → 3 ICs (a triangle); n = 4 → 6 ICs, matching known results for four-bar linkages.

Why Other Options Are Wrong:

(a) and (b) undercount; (d) n/2 is not combinatorial. (e) 2n − 3 is a formula seen for joints in trees, not for ICs.

Common Pitfalls:

Confusing number of joints with number of ICs; counting ordered pairs rather than unordered; forgetting that even non-adjacent links have an IC (possibly at infinity).

Final Answer:

n(n − 1)/2