Lesson 3.1: Kinematics of Machines (Links, Pairs, Mechanisms)

Kinematics of Machines forms the foundation of Theory of Machines. GATE and PSU exams often test linkages, pairs, mobility, and basic mechanism concepts.

🔹 1. Basic Concepts

• Link: A rigid body connected to other bodies to form a mechanism.

  • Examples: Rods, bars, plates

• Pair (Kinematic Pair): Two links in contact that constrain relative motion.

  • Classification by relative motion:

    1. Lower Pair: Surface contact (Revolute, Prismatic, Helical)

    2. Higher Pair: Point or line contact (Cam & Follower, Gear Pair)

• Mechanism: A combination of links and pairs to transmit motion and force.

🔹 2. Mobility (Degrees of Freedom)

• Definition: Number of independent motions a mechanism can perform.

• Gruebler’s Equation (Planar Mechanism):

F=3(n−1)−2j1−j2F = 3(n-1) – 2j_1 – j_2F=3(n−1)−2j1​−j2​

Where:

• $F$ = Degrees of freedom

• $n$ = Number of links (including frame)

• j1j_1j1​ = Lower pairs

• j2j_2j2​ = Higher pairs

Applications:

• Slider-crank, four-bar linkages, gear trains

🔹 3. Common Mechanisms

1. Four-Bar Mechanism: 4 links, one fixed.

   • Applications: Crank-rocker, Coupler mechanism

2. Slider-Crank Mechanism: Converts rotary motion to reciprocating motion.

   • Applications: Engine pistons, compressors

3. Double Crank Mechanism: Both cranks rotate completely.

   • Applications: Rotary engines, pumps

4. Cam & Follower: Converts rotary motion to oscillatory or reciprocating motion.

   • Types of followers: Roller, Flat, Knife-edge

   • Types of cams: Disc, Cylindrical, Translating

5. Gear Mechanisms: Transmit motion and torque using toothed wheels

   • Applications: Transmission, machine tools

🔹 4. Solved Examples (PYQ Style)

Example 1 (GATE ME 2017):
Four-bar mechanism, link lengths: AB = 5 cm, BC = 10 cm, CD = 7 cm, DA = 8 cm. Find mobility.
👉 Solution:

• n = 4, j1 = 4 (revolute pairs), j2 = 0

$$F=3(4-1)-2\ast 4-0=9-8=1\text{(One DOF → valid mechanism)}$$

Example 2 (PSU Exam):
Slider-crank, crank = 0.1 m, connecting rod = 0.3 m, crank rotates at 120 rpm. Find linear velocity of slider at crank angle 30°.
👉 Solution:

• Use v = r ω cosθ + √(l² – r² sin²θ) ω ? (standard kinematics formula)

🔹 5. Practice Exercises

1. Identify all links and pairs in a simple slider-crank mechanism.

2. For a four-bar linkage with given lengths, compute mobility using Gruebler’s equation.

3. Classify kinematic pairs: revolute, prismatic, cam & follower.

4. Draw a cam profile for a follower with SHM motion.

5. Calculate slider displacement for crank rotation of 45° in a slider-crank mechanism.

🔹 6. Summary

• Links: Rigid bodies in mechanism

• Pairs: Contact between links, constrain motion

• Mechanism: Combination of links & pairs to transmit motion

• DOF: Number of independent movements, computed by Gruebler’s equation

• Applications: Engines, pumps, compressors, machine tools