THEORY OF MACHINES (ME-403) - Unit 4 Short Notes

Unit 4: Theory of Machines – Comprehensive Short Notes

(Aligned with RGPV Past Papers: Jun 2025, Dec 2024, Jun 2024, Jun 2023, Nov 2023, Jun 2022)

1. Fundamental Concepts

Mechanisms vs. Machines

• Mechanism: A set of links connected by joints to transmit motion/force. Focuses on kinematics (motion only).

  Example: Slider-crank mechanism in an engine.

• Machine: A mechanism + additional elements (like frame, fasteners) to perform work with energy conversion.

  Example: Complete engine assembly.

[!TIP]

Exam Key: Mechanism = motion transformer; Machine = work producer.

Degree of Freedom (DOF)

• Definition: Number of independent coordinates required to define the configuration of a mechanism.

• Kutzbach’s Criterion (Planar):

$$ DOF = 3(n - 1) - 2j - h $$

Where:

$n$ = total links (including frame),

$j$ = number of lower pairs (1 DOF joints like revolute/prismatic),

$h$ = number of higher pairs (2 DOF joints like cam-follower).

• Gruebler’s Criterion:

$$ DOF = 3(n - 1) - 2j $$

Assumes no redundant constraints and all joints are 1 DOF.

[!CAUTION]

Gruebler’s fails for mechanisms with redundant constraints (e.g., parallel linkages). Use Kutzbach for general cases.

Grashof’s Law & Four-Bar Classification

• Law: For a 4-bar linkage with link lengths $s$ (shortest), $l$ (longest), $p, q$ (intermediate):

$$ s + l \leq p + q $$

• Classification:

  | Condition | Type | Crank Possibility | |-----------------------------|---------------------------|-----------------------------------------------| | $$\displaystyle s + l < p + q $$ | Grashof | At least one crank can rotate continuously. | | $$\displaystyle s + l = p + q $$ | Parallelogram/Delta | Special cases; change point. | | $$\displaystyle s + l > p + q $$ | Non-Grashof | No continuous crank rotation; double rocker. |

• Inversions: Fixing different links yields distinct mechanisms:

  1. Crank-rocker (fixed longest/shortest).

  2. Double crank (fixed intermediate).

  3. Double rocker (non-Grashof).

Inversions of Mechanisms

• Definition: Different mechanisms obtained by fixing (grounding) different links in a kinematic chain.

• Four-Bar Inversions:

  • Input crank fixed → Oscillating output.

  • Coupler fixed → Drag-link (e.g., automobile wiper).

  • Output rocker fixed → Crank-rocker input.

• Double Slider Crank Inversions:

  1. Scotch yoke (slider fixed → crank oscillates).

  2. Oldham’s coupling (one slider fixed → parallel misalignment compensation).

  3. Elliptical trammel (both sliders fixed → point on coupler traces ellipse).

[!TIP]

Past Paper Focus: Whitworth quick return (crank and slotted lever) is an inversion of crank and slotted lever mechanism.

2. Kinematic & Dynamic Analysis

Velocity & Acceleration Analysis

• Instantaneous Center (IC) Method:

  • Definition: Point in a body with zero instantaneous velocity.

  • Kennedy’s Theorem: Three bodies in plane have 3 ICs collinear.

  • Velocity at any point: $$\displaystyle v = \omega \times r_{IC} $$.

  • Steps:

    1. Locate all fixed ICs (IC between link & frame).

    2. Use Kennedy’s theorem for moving ICs.

    3. Compute $\omega$ from known velocity: $$\displaystyle \omega = v/r $$.

  [!DIAGRAM: CANVAS]

  Slider-crank with ICs: $$\displaystyle I_{12} $$ (crank-ground), $$\displaystyle I_{23} $$ (coupler-ground), $$\displaystyle I_{13} $$ (crank-coupler).

• Relative Motion Method:

  • Velocity: $$\displaystyle \vec{v}_B = \vec{v}_A + \vec{\omega}_{AB} \times \vec{r}_{B/A} $$.

  • Acceleration:

$$ \vec{a}_B = \vec{a}_A + \vec{\alpha}_{AB} \times \vec{r}_{B/A} + \vec{\omega}_{AB} \times (\vec{\omega}_{AB} \times \vec{r}_{B/A}) $$

+ **Coriolis component** if $$\displaystyle \vec{v}_{B/A} $$ exists: $$\displaystyle 2\vec{\omega}_{AB} \times \vec{v}_{B/A} $$.  

Coriolis Acceleration

• Expression: $$\displaystyle \vec{a}_c = 2\vec{\omega} \times \vec{v}_{rel} $$.

• Magnitude: $$\displaystyle a_c = 2\omega v_{rel} $$.

• Direction: Perpendicular to $$\displaystyle \vec{v}_{rel} $$, rotated by $$\displaystyle 90^\circ $$ in direction of $\vec{\omega}$.

[!TIP]

Past Paper Alert: Derivation of Coriolis component is frequent (Jun 2025, Jun 2022).

D’Alembert’s Principle

• Statement: Inertia forces and external forces on a body form a system in dynamic equilibrium.

$$ \sum (\vec{F} - m\vec{a}) = 0 \quad \text{and} \quad \sum (\vec{T} - I\vec{\alpha}) = 0 $$

• Significance: Converts dynamics problem into static equivalent; used for force analysis in mechanisms.

Force Analysis in Mechanisms

• Steps:

  1. Draw free-body diagrams of each link.

  2. Apply D’Alembert’s principle (add inertia forces $-m\vec{a}$, $-I\vec{\alpha}$).

  3. Resolve forces at joints (two-component unknowns).

  4. Use loop equations for torque/power balance.

• Friction: Include in joint reactions; e.g., friction force $$\displaystyle F_f = \mu N $$.

3. Mechanism Synthesis

• Number Synthesis: Determines possible kinematic chains for given DOF and joints (using graph theory).

• Motion Synthesis: Designs linkages for specified motion (position, velocity, acceleration).

• Analytical Method (Freudenstein’s Equation):

  For four-bar with input $$\displaystyle \theta_2 $$, output $$\displaystyle \theta_4 $$:

$$ K_1 \cos \theta_4 - K_2 \cos \theta_2 + K_3 \cos(\theta_2 - \theta_4) = K_4 $$

Where $$\displaystyle K_1 = \frac{d^2 + c^2 - a^2 - b^2}{2ab} $$, etc. (standard form).

Satisfy at specified $$\displaystyle \theta_2, \theta_4 $$ to find link ratios.

• Graphical Method: Use relative pole technique for precision points.

[!TIP]

Past Paper: Freudenstein’s equation for position/velocity/acceleration synthesis (Jun 2025).

4. Specific Mechanisms

Four-Bar Linkage

• Type Determination: Apply Grashof’s criterion.

• Kinematic Analysis:

  • Position: Loop-closure equations (complex numbers or trigonometry).

  • Velocity/Acceleration: Use IC method or relative motion.

Slider-Crank Mechanism

• Displacement: $$\displaystyle x = r\cos\theta + \sqrt{l^2 - r^2\sin^2\theta} $$ (for crank length $r$, connecting rod $l$).

• Velocity: $$\displaystyle v = -r\omega\sin\theta + \frac{r^2\omega\sin\theta\cos\theta}{\sqrt{l^2 - r^2\sin^2\theta}} $$.

• Acceleration: Differentiate velocity.

Quick Return Mechanisms

• Whitworth Mechanism:

  • Time Ratio: $$\displaystyle Q = \frac{\text{Cut time}}{\text{Return time}} = \frac{360^\circ - 2\alpha}{360^\circ + 2\alpha} $$, where $$\displaystyle \alpha = \cos^{-1}\left(\frac{R}{d}\right) $$, $R$ = crank radius, $d$ = distance from crank center to slotted lever pivot.

  • Design: Given $Q$, $R$, find $d$: $$\displaystyle d = R / \cos\alpha $$, $$\displaystyle \alpha = \frac{180^\circ(Q-1)}{2(Q+1)} $$.

• Crank and Slotted Lever: Similar time ratio formula.

Double Slider Crank

• Applications: Elliptical trammel, Oldham’s coupling.

• Inversions: See Section 1.

5. Cam Systems

Classification & Terminology

• Cams: Disk, plate, cylindrical, end face, conical.

• Followers:

  | Type | Motion Axis | Pressure Angle | |-------------------|----------------------|-------------------| | Knife-edge | Through cam center | High wear | | Roller | Offset/through | Low wear | | Flat-faced | Through/offset | Low wear |

• Key Terms:

  • Base circle: Smallest cam radius.

  • Pitch circle: Circle through follower pitch point.

  • Pressure angle $\phi$: Angle between follower motion and normal to cam profile.

$$\tan\phi = \frac{\text{transverse component}}{\text{radial component}}$$

• Stroke: Maximum follower displacement $h$.

Follower Motions & Displacement Diagrams

1. Uniform Velocity: $$\displaystyle s = \frac{h}{\beta}\theta $$ (linear diagram).

   • $$\displaystyle v_{max} = \frac{h\omega}{\beta} $$, $$\displaystyle a = 0 $$ (except at transitions → infinite).

2. Simple Harmonic Motion (SHM):

$$ s = \frac{h}{2}\left(1 - \cos\frac{\pi\theta}{\beta}\right) $$

• $$\displaystyle v = \frac{\pi h \omega}{2\beta}\sin\frac{\pi\theta}{\beta} $$,

• $$\displaystyle a = \frac{\pi^2 h \omega^2}{2\beta^2}\cos\frac{\pi\theta}{\beta} $$.

• $$\displaystyle v_{max} = \frac{\pi h \omega}{2\beta} $$, $$\displaystyle a_{max} = \frac{\pi^2 h \omega^2}{2\beta^2} $$.

3. Uniform Acceleration/Deceleration:

   • Rise/fall in $\beta/2$ each: $$\displaystyle s = \frac{2h}{\beta^2}\theta^2 $$ (first half), $$\displaystyle s = h - \frac{2h}{\beta^2}(\beta-\theta)^2 $$ (second half).

   • $$\displaystyle v_{max} = \frac{2h\omega}{\beta} $$, $$\displaystyle a_{max} = \frac{4h\omega^2}{\beta^2} $$.

Cam Profile Design

• Knife-edge: Inverse of displacement diagram (offset if follower axis offset).

• Roller: Add roller radius to base circle → pitch curve; offset by roller radius normal to pitch curve.

• Flat-faced: Tangential to pitch curve; ensure no undercut.

[!CAUTION]

Undercutting: Occurs if $$\displaystyle \phi > \phi_{max} $$ (typically $$\displaystyle 30^\circ $$ for slow, $$\displaystyle 15^\circ $$ for high speed). Reduces contact area.

Design Considerations

• Pressure Angle: Keep $$\displaystyle \phi < \phi_{allow} $$ to avoid high side thrust.

• Undercutting: Avoid by increasing base circle or modifying motion.

• Critical Path Motion: Path where follower velocity/acceleration peaks; design for minimum $\phi$.

• Torque on Cam Shaft: $$\displaystyle T = F_t \cdot r_{pitch} $$, where $$\displaystyle F_t $$ = tangential force.

6. Gear Systems

Fundamentals of Gearing

• Law of Gearing: Common normal to tooth profiles at point of contact passes through pitch point (instantaneous center).

  Proof: For constant velocity ratio, $$\displaystyle \frac{\omega_1}{\omega_2} = \frac{r_2}{r_1} = \text{constant} $$.

• Conjugate Profiles: Profiles satisfying law of gearing (e.g., involute, cycloidal).

Involute Gear Geometry

• Parameters (for module $m$, pressure angle $\phi$):

  • Addendum $$\displaystyle a = m $$, Dedendum $$\displaystyle d = 1.25m $$ (standard).

  • Circular pitch $$\displaystyle p_c = \pi m $$.

  • Base circle radius $$\displaystyle r_b = r \cos\phi $$.

• Path of Contact: Line along which contact occurs; length $$\displaystyle = \sqrt{r_{a1}^2 - r_{b1}^2} + \sqrt{r_{a2}^2 - r_{b2}^2} - (r_{b2} - r_{b1})\tan\phi $$.

• Arc of Contact: Path of contact / $\cos\phi$.

• Contact Ratio $\epsilon$:

$$ \epsilon = \frac{\text{Path of contact}}{p_c} $$

$$\displaystyle \epsilon > 1 $$ ensures smooth operation (typical $$\displaystyle \epsilon = 1.2–2 $$).

Interference & Undercutting

• Interference: Addendum tip of pinion contacts gear root before engagement ends.

  • Avoidance: Minimum pinion teeth $$\displaystyle T_{min} = \frac{2r_a}{m}\sin^2\phi $$ (for full-depth teeth).

  • For $$\displaystyle 20^\circ $$, $$\displaystyle T_{min} \approx 18 $$.

• Undercutting: Gear tooth trimmed near root to avoid interference; weakens tooth.

Gear Types

• Spur: Parallel axes, straight teeth.

• Helical: Helical teeth, gradual engagement ($$\displaystyle \phi_b = \tan^{-1}(\tan\phi/\cos\psi) $$).

• Bevel: Intersecting axes (conical).

• Worm: High reduction, non-intersecting perpendicular axes.

Gear Trains

• Simple: Each gear on separate shaft; $$\displaystyle \omega_{out}/\omega_{in} = \prod (N_{in}/N_{out}) $$.

• Compound: Multiple gears on same shaft; same as simple but intermediate shafts compound.

• Epicyclic (Sun-Planet):

  • Velocity Ratio: Use Tabular Method or Aronson’s Method.

    Tabular: Fix ring → count teeth for arm speed; then add arm speed.

  • Torque: $$\displaystyle T_{sun} \cdot \omega_{sun} + T_{ring} \cdot \omega_{ring} + T_{arm} \cdot \omega_{arm} = 0 $$ (power balance).

7. Power Transmission Elements

Belt Drives

• Open Belt: $$\displaystyle T_1/T_2 = e^{\mu\theta} $$ (where $\theta$ in radians, angle of lap on smaller pulley).

• Cross Belt: $$\displaystyle T_1/T_2 = e^{\mu\pi} $$ (if $$\displaystyle \theta = \pi $$).

• Belt Stress: $$\displaystyle \sigma = \frac{T}{b \cdot t} $$ (max tension $$\displaystyle T_1 $$).

• Power Transmitted: $$\displaystyle P = (T_1 - T_2)v $$.

• Maximum Power Condition: $$\displaystyle T_1 = T_{max} $$, $$\displaystyle T_2 = T_{max}/e^{\mu\theta} $$, $$\displaystyle v = \sqrt{\frac{T_{max}}{3\rho}} $$ (where $\rho$ = density).

• Slip & Creep:

  • Slip: $$\displaystyle s = \frac{v_1 - v_2}{v_1} \times 100\% $$.

  • Creep: Local deformation; $$\displaystyle s = \frac{\Delta l}{l} $$.

Friction Devices

• Clutches:

  • Plate Clutch (uniform pressure): $$\displaystyle T = \frac{2}{3}\mu W \frac{r_o^3 - r_i^3}{r_o^2 - r_i^2} $$.

  • Plate Clutch (uniform wear): $$\displaystyle T = \mu W \frac{r_o^2 + r_i^2}{2(r_o - r_i)} $$.

    [!BOX]

    Uniform Wear Torque: $$\displaystyle T = \frac{\mu W}{2}(r_o + r_i) $$.

  • Cone Clutch: $$\displaystyle T = \frac{\mu W \sin\alpha}{\sin\alpha + \mu\cos\alpha} \cdot \frac{r_o^2 + r_i^2}{2(r_o - r_i)} $$.

• Brakes:

  • Band & Block:

$$ \frac{T_0}{T_n} = \left(\frac{1 + \mu\tan\theta}{1 - \mu\tan\theta}\right)^n $$

Where $n$ = number of blocks, $2\theta$ = angle subtended by each block.  

> [!TIP]  

> **Past Paper**: Derivation of band & block tension ratio (Jun 2025).  

• Block Brake: $$\displaystyle T = \mu W R \frac{1 + \mu f}{1 - \mu f} $$ (with lever arm $f$).

• Disk Brake: $$\displaystyle T = 2\mu F R_{eff} $$ (two pads).

• Power Screws:

  • Torque (raising): $$\displaystyle T = W \frac{d_m}{2} \tan(\phi + \lambda) $$, where $\lambda$ = lead angle, $\phi$ = friction angle.

  • Efficiency: $$\displaystyle \eta = \frac{\tan\lambda}{\tan(\phi + \lambda)} $$ (square thread).

• Dynamometers:

  • Prony Brake: $$\displaystyle P = W L \omega $$ (where $L$ = lever length, $W$ = force, $\omega$ = angular speed).

8. Vibrations & Balancing

Free Vibrations

• Longitudinal: Axial vibrations in rods/shafts; $$\displaystyle \omega_n = \sqrt{\frac{EA}{\rho A L^2}} = \sqrt{\frac{E}{\rho}} \cdot \frac{\pi}{L} $$.

• Transverse: Beam vibrations; $$\displaystyle \omega_n = \sqrt{\frac{EI}{\rho A L^4}} \cdot \beta^2 $$ (depends on end conditions).

• Torsional: Shaft twist; $$\displaystyle \omega_n = \sqrt{\frac{GJ}{\rho J_p L^2}} = \sqrt{\frac{G}{\rho}} \cdot \frac{\pi}{L} $$.

Balancing of Rotating Masses

• Static Balancing: Single plane; $$\displaystyle \sum m_i r_i = 0 $$ (vector sum).

• Dynamic Balancing: Two planes;

$$ \sum m_i r_i = 0, \quad \sum m_i r_i x_i = 0 $$

Where $$\displaystyle x_i $$ = axial distance from reference plane.

• Procedure for Multi-plane:

  1. Choose two balancing planes (X, Y).

  2. Write force and moment equations about X, Y.

  3. Solve for $$\displaystyle M_X, \theta_X $$ and $$\displaystyle M_Y, \theta_Y $$.

  [!BOX]

  Balance Mass: $$\displaystyle M = \sqrt{M_X^2 + M_Y^2} $$, $$\displaystyle \theta = \tan^{-1}(M_Y/M_X) $$.

9. Dynamic Analysis Applications

• Force Analysis in Mechanisms: Combine inertia forces (from acceleration analysis) with external loads; use D’Alembert’s to find joint reactions and input torque.

• Torque in Clutches/Brakes: As above.

• Power in Power Screws/Belts: $$\displaystyle P = T\omega $$ or $$\displaystyle P = (T_1 - T_2)v $$.

• Critical Speed: For shafts, $$\displaystyle N_c = \frac{1}{2\pi}\sqrt{\frac{g}{\delta}} $$ (where $\delta$ = deflection).

Key Formulas Summary

Past Paper Hotspots

1. Jun 2025: Whitworth design, Freudenstein synthesis, cam max velocity/acceleration, band brake derivation, plate clutch pressure, balancing mass.

2. Dec 2024: Grashof type determination, epicyclic gear teeth numbers, belt stress (open/cross), Coriolis derivation.

3. Jun 2024: Double slider crank inversions, tangent cam design, cam classification, belt power transmission.

4. Jun 2023: Inversions, D’Alembert, cam displacement diagrams, gear contact ratio, Prony brake.

5. Nov 2023: Constrained motion, Whitworth, slider-crank torque, cam profile SHM, belt selection, gear types.

6. Jun 2022: Grashof, slip/creep, quick return distance, 4-bar type/velocity, epicyclic speed, cam acceleration, balancing.

[!FINAL TIP]

Exam Strategy: For numericals, always sketch the mechanism first. For derivations, state assumptions (e.g., uniform wear in clutches). For synthesis, clearly list given specs and unknowns.

These notes consolidate all high-frequency topics from RGPV past papers. Focus on derivations (Coriolis, band brake, law of gearing), numericals (belt stress, cam motions, balancing), and diagram-based questions (IC method, cam profiles, gear geometry).