Overview
This lecture provides a focused revision of the "Simple Mechanisms" chapter from Theory of Machines, emphasizing key concepts, formulas, and practical examples relevant for engineering exams such as GATE.
Introduction to Mechanisms
- Mechanisms are assemblies of rigid, flexible, or fluid links that transfer motion or force between components.
- Common examples include fans, folding tables, internal combustion (IC) engines, belt drives, and fluid systems.
Kinematic Links and Their Types
- A kinematic link is a part of a mechanism with relative motion compared to other parts.
- Links can be rigid (solid, minimal deformation), flexible (e.g., belts), or fluid (e.g., hydraulic systems).
- Mechanisms are schematically represented as line diagrams for analysis.
Degree of Freedom (DOF) and Kutzbach/Gruebler Criteria
- Degree of Freedom (DOF): Number of independent input parameters needed to define a mechanism's position.
- For planar mechanisms, the formula is:
F = 3(L - 1) - 2J - H
where L = number of links, J = lower pairs (one DOF), H = higher pairs (two DOF).
- Kutzbach’s equation is used for general planar mechanisms.
- Gruebler’s criterion: For a simple planar mechanism with only lower pairs and one DOF,
1 = 3L - 1 - 2J leads to 3L - 2J - 4 = 0 (minimum 4 links, 4 joints).
Simple Mechanisms and Their Types
- Simple mechanisms have 4 links and 4 lower pairs.
- Three main types:
- Four-bar mechanism (all turning pairs)
- Single slider-crank mechanism (3 turning, 1 sliding pair)
- Double slider-crank mechanism (2 turning, 2 sliding pairs)
Four-Bar Mechanism and Grashof’s Law
- Four-bar mechanism: Four links joined with turning pairs; motion depends on which link is fixed.
- Grashof’s Law: If the sum of the shortest and longest links ≤ sum of the other two, at least one link can rotate fully (crank).
- Inversions: By fixing different links, different motion types arise (crank-crank, crank-rocker, rocker-rocker).
Inversions and Examples
- Inversion: Different mechanisms obtained by fixing different links in a kinematic chain.
- Four-bar mechanism inversions: double crank, crank-rocker, and double rocker.
- Special link proportions yield deltoid or parallelogram linkages.
Single Slider-Crank Mechanism & Inversions
- Four links, with one sliding pair.
- Depending on the fixed link, examples include IC engines, reciprocating compressors, rotary engines, crank and slotted lever (quick return mechanism), and hand pumps.
- Quick return ratio = β/α, where β and α are angles described by crank travel during forward and return strokes.
Double Slider-Crank Mechanism & Inversions
- Four links, two turning pairs and two sliding pairs.
- Inversions include elliptical trammel (drawing ellipses), Scotch yoke (convert rotation to reciprocating motion), and Oldham’s coupling (connect rotating shafts).
Mechanical Advantage and Transmission Angle
- Mechanical advantage = output torque/input torque = (input angular velocity)/(output angular velocity)⁻¹.
- At toggle positions (extremes), mechanical advantage can approach infinity or zero, depending on input/output designations.
- Transmission angle is the angle between the coupler and output link in a four-bar mechanism, affecting force transmission efficiency.
Key Terms & Definitions
- Kinematic Link — A component of a mechanism with relative motion compared to others.
- Degree of Freedom (DOF) — Independent inputs needed to define mechanism motion.
- Lower Pair — Joint with one DOF (e.g., turning or sliding).
- Higher Pair — Joint with two DOF (point or line contact).
- Inversion — A new mechanism formed by fixing a different link in a chain.
- Crank — Link that completes full rotation.
- Rocker — Link that oscillates (partial rotation).
- Mechanical Advantage — Ratio of output to input effort in a mechanism.
- Transmission Angle — Angle between the coupler and output link in a four-bar mechanism.
Action Items / Next Steps
- Practice identifying links, pairs, and DOFs in sample mechanisms.
- Memorize Kutzbach and Gruebler equations.
- Review quick return mechanism geometry and calculations.
- Prepare for the next session on velocity and acceleration analysis.