4.4 Theory of Machines

4.4 Theory of Machines

1. Degree of Freedom

1. Definition (DOF or Mobility):

   • Number of independent coordinates needed to define configuration.

   • Number of independent motions possible.

2. Kutzbach Criterion (Grübler's Equation):

   • For planar mechanisms: $DOF = 3(L-1) - 2J - H$

   • $L$ = number of links (including fixed link).

   • $J$ = number of lower pairs (revolute/prismatic joints).

   • $H$ = number of higher pairs.

3. Types of Mechanisms:

   • $DOF = 1$: Constrained mechanism (most common).

   • $DOF = 0$: Structure (no motion).

   • $DOF > 1$: Requires multiple inputs.

4. Grashof's Law:

   • For four-bar linkage to have at least one revolving link:

     • $s + l \leq p + q$

     • $s$ = shortest link, $l$ = longest link.

     • $p, q$ = other two links.

2. Linkage Mechanisms

1. Basic Components:

   • Link: Rigid body with at least two nodes.

   • Joint: Connection between links.

2. Joint Types:

   • Lower Pair: Surface contact (revolute, prismatic).

   • Higher Pair: Point/line contact (cam, gear).

3. Four-Bar Linkage:

   • Most fundamental mechanism.

   • Components: Fixed link, Crank, Coupler, Follower.

   • Types based on which link is fixed.

4. Slider-Crank Mechanism:

   • Converts rotary to reciprocating motion (engine).

   • Variations: Rotary engine, quick-return mechanisms.

5. Other Common Mechanisms:

   • Scotch Yoke: Produces simple harmonic motion.

   • Geneva Mechanism: Intermittent motion.

   • Pantograph: Scaling mechanism.

   • Straight-line Mechanisms: Peaucellier, Watt.

3. Kinematics and Kinetics of Motion

1. Kinematics:

   • Study of motion without considering forces.

   • Describes position, velocity, acceleration.

   • Absolute Motion: Relative to fixed reference.

   • Relative Motion: Between moving points.

2. Kinetics:

   • Study of forces causing motion.

   • Uses Newton's laws, work-energy, impulse-momentum.

3. Types of Motion:

   • Translation: All points have same velocity.

   • Rotation: All points move in circles about axis.

   • General Plane Motion: Combination of translation + rotation.

4. Velocity and Acceleration Analysis

1. Methods:

   • Relative Velocity Method:

     • $V_B = V_A + V_{B/A}$

     • $V_{B/A}$ is velocity of B relative to A.

   • Instantaneous Center (IC) Method:

     • Point with zero instantaneous velocity.

     • Velocity proportional to distance from IC.

2. Velocity Diagrams:

   • Graphical construction using vector polygons.

   • Scale: actual velocity = drawing length × scale factor.

3. Acceleration Analysis:

   • Relative Acceleration:

     • $a_B = a_A + a_{B/A}$

     • $a_{B/A} = a_{B/A}^t + a_{B/A}^n$

   • Components:

     • Tangential: $a^t = \alpha r$ (due to angular acceleration).

     • Normal/Centripetal: $a^n = \omega^2 r$ (towards center).

4. Coriolis Acceleration:

   • Appears when point moves along rotating link.

   • $a_c = 2\omega v$

   • Direction: perpendicular to both $\omega$ and $v$.

5. Forces in Mechanisms

1. Static Force Analysis:

   • For slow-moving mechanisms (neglect inertia).

   • Uses equilibrium equations ($\sum F=0, \sum M=0$).

2. Dynamic Force Analysis:

   • Considers inertia forces at higher speeds.

   • D'Alembert's Principle: Convert dynamics to statics by adding inertia forces.

3. Inertia Forces:

   • For translating mass: $F = -ma$ (opposes acceleration).

   • For rotating mass: Torque = $-I\alpha$.

4. Force Transmission:

   • Transmission Angle: Angle between coupler and follower.

   • Optimal: near 90° for smooth force transfer.

   • Poor transmission: near 0° or 180°.

5. Mechanical Advantage:

   • Ratio of output to input force.

   • Related to velocity ratio: $MA = \frac{1}{VR}$ (ideal, no friction).

6. Shaking Forces:

   • Unbalanced forces transmitted to foundation.

   • Cause vibration, noise, wear.

   • Reduced by balancing techniques.

7. Flywheel Analysis:

   • Smooths speed fluctuations in cyclic machines.

   • Coefficient of Fluctuation:

     • $C_s = \frac{\omega_{max} - \omega_{min}}{\omega_{avg}}$

   • Energy Equation:

     • $\Delta E = \frac{1}{2} I (\omega_{max}^2 - \omega_{min}^2)$