Machine Component and Design (ME-602) - Unit 3 Short Notes

UNIT 3: MACHINE COMPONENT AND DESIGN - EXAM-FOCUSED SHORT NOTES

1.0 FATIGUE AND FAILURE ANALYSIS

1.1 Stress Concentration

• Definition: Localized increase in stress due to geometric discontinuities (holes, notches, fillets, keyways), material defects, or abrupt load changes.

• Stress Concentration Factor (Kt): Ratio of maximum theoretical stress to nominal stress.

$$K_t = \frac{\sigma_{max}}{\sigma_{nom}}$$

*   **Theoretical Kt:** From elastic stress analysis (depends only on geometry).

*   **Experimental Kt:** From strain gauge measurements.

• Dynamic Loading Consideration: Use fatigue stress concentration factor (Kf) instead of Kt.

$$K_f = 1 + q(K_t - 1)$$

where `q` is the notch sensitivity factor.

• Methods to Reduce Stress Concentration:

  • Use generous fillets at corners.

  • Provide grooves or relief notches.

  • Avoid sharp corners; use gradual tapers.

  • Align holes perpendicular to stress axis.

  • Select ductile materials (higher notch sensitivity q).

[!TIP] Exam Focus: Kt is always ≥ 1. For static loading, use Kt. For fatigue, use Kf. Kf ≤ Kt.

1.2 Notch Sensitivity

• Definition: Material's susceptibility to having its fatigue strength reduced by a stress concentration. Not all materials are equally sensitive.

• Notch Sensitivity Factor (q): Ranges from 0 (fully insensitive, like very ductile metals) to 1 (fully sensitive, like brittle materials).

• Key Relationship: Kf bridges theoretical stress (Kt) and actual fatigue stress.

$$\boxed{K_f = 1 + q(K_t - 1)}$$

1.3 S-N Curve (Wöhler Curve)

• Definition: Plot of stress amplitude (S) vs. number of cycles to failure (N) on log-log scale.

• Key Features:

  • Endurance Limit (σe): Stress below which failure does not occur for infinite life (typically >10^6 cycles for ferrous metals).

  • Endurance Strength: Stress at a specified finite number of cycles (e.g., 10^6 cycles).

• Factors Affecting S-N Curve (Marin Factors): Surface finish (k_a), size (k_b), loading type (k_c), temperature (k_d), reliability (k_e).

$$\sigma_e' = \sigma_e \times k_a \times k_b \times k_c \times k_d \times k_e$$

where `σe'` is the modified endurance limit.

1.4 Fatigue Failure Criteria & Diagrams

Used to design for fluctuating stresses (σ_max, σ_min) with a Factor of Safety (n).

• σ_a = Alternating stress = (σ_max - σ_min)/2

• σ_m = Mean stress = (σ_max + σ_min)/2

• Application: Rearrange to find minimum required ultimate strength (σ_ut) or yield strength (σ_y) for safe design.

[!TIP] Common Pitfall: Soderberg is safest but often overly conservative. Modified Goodman is widely used. Always check which criterion the question specifies.

1.5 Factors Affecting Fatigue Strength (Marin Factors)

• Surface Finish Factor (k_a): Accounts for surface roughness. Ground/polished > Machined > Hot-rolled > As-forged.

  • Empirical: $$\displaystyle k_a = A \cdot (S_{ut})^b $$ (A, b from tables).

• Size Factor (k_b): Larger volumes have higher probability of defects. k_b < 1 for large diameters.

• Load Factor (k_c): Accounts for load type (k_c=1.0 for bending, 0.85 for axial, 0.75 for torsion).

• Temperature Factor (k_d): k_d < 1 for elevated temperatures.

• Reliability Factor (k_e): Higher reliability (e.g., 99% vs 50%) requires lower stress. k_e < 1.

2.0 SHAFT DESIGN

2.1 Fundamentals

• Types: Transmission shafts (power), Machine shafts (support parts), Axles (support rotating parts, no torque).

• Stresses:

  • Torsional Shear: $$\displaystyle \tau_t = \frac{T \cdot r}{J_p} = \frac{16T}{\pi d^3} $$ (solid)

  • Bending (Normal): $$\displaystyle \sigma_b = \frac{M \cdot y}{I} = \frac{32M}{\pi d^3} $$ (solid)

  • Axial: $$\displaystyle \sigma_a = \frac{P}{A} $$

• Equivalent Moments (for combined bending & torsion):

  • Equivalent Twisting Moment (T_e): Used for shear stress design (Tresca).

$$T_e = \sqrt{M^2 + T^2}$$

*   **Equivalent Bending Moment (M_e):** Used for **normal stress** design (Rankine).

$$M_e = \sqrt{M^2 + \left(\frac{T}{2}\right)^2} \quad \text{or} \quad M_e = \frac{1}{2}\left[ M + \sqrt{M^2 + T^2} \right]$$

2.2 Design for Strength & Stiffness

• Maximum Shear Stress Theory (Tresca): Failure when max shear stress reaches yield shear stress.

$$\tau_{max} = \frac{T_e}{2Z_p} \leq \frac{S_y}{n}$$

• Maximum Normal Stress Theory (Rankine): Failure when max normal stress reaches yield tensile stress.

$$\sigma_{max} = \frac{M_e}{Z} \leq \frac{S_y}{n}$$

• ASME Code: Uses a stress concentration factor (Kt or Kf) and considers both steady and alternating stresses. Design equation often involves an allowable stress S_a or S_m.

• Diameter Calculation: Solve for d from the chosen theory's inequality using section modulus Z or polar modulus Z_p.

2.3 Hollow vs. Solid Shafts

• For Equal Weight (Material): Hollow shaft has higher strength and stiffness.

  • Strength ratio: $$\displaystyle \frac{Z_{p,hollow}}{Z_{p,solid}} = \frac{1 - (d_i/d_o)^4}{1 - (d_i/d_o)^3} \cdot \frac{1}{1 - (d_i/d_o)^4} $$ (simplifies to >1 for d_i/d_o > 0.58).

• Design: Subject to torque T, max shear stress τ_max, and max twist θ.

$$\tau_{max} = \frac{T \cdot (d_o/2)}{J_p} = \frac{16T}{\pi (d_o^4 - d_i^4)} \cdot d_o$$

$$\theta = \frac{T \cdot L}{G \cdot J_p}$$

Solve for `d_o` and `d_i` simultaneously.

2.4 Section Properties

• Polar Modulus (Z_p): Resistance to torsion.

  • Solid Circular: $$\displaystyle Z_p = \frac{J_p}{c} = \frac{\pi d^3}{16} $$

  • Hollow Circular: $$\displaystyle Z_p = \frac{\pi (d_o^4 - d_i^4)}{16 d_o} $$

  • Significance: $$\displaystyle \tau_{max} = \frac{T}{Z_p} $$

3.0 KEYS, COUPLINGS, AND THREADED COMPONENTS

3.1 Keys

• Classification:

  • Sunk Keys: Fit into keyways on shaft & hub (Parallel, Saddle, Gib-head).

  • Tangent Keys: Two keys at 90° for heavy torque.

  • Round Keys: For low torque, no keyway needed on shaft.

• Saddle Key vs. Sunk Key:

  | Feature | Saddle Key | Sunk Key | | :--- | :--- | :--- | | Fit | Sits in hub keyway, rests on shaft | Fits into keyways on both shaft & hub | | Torque Capacity | Low (friction only) | High (shear & crushing) | | Axial Movement | Possible | Prevented | | Application | Light duty, sliding hubs | General power transmission |

• Key Design Stresses:

  • Shear Failure: $$\displaystyle \tau = \frac{T}{l \cdot b \cdot (d/2)} \leq \frac{S_{sy}}{n} $$

  • Crushing (Bearing) Failure: $$\displaystyle \sigma_c = \frac{2T}{l \cdot b \cdot (d/2)} \leq \frac{S_{sy}}{n} $$

  • l = key length, b = width, d = shaft diameter.

• Forces on Key: Transmits torque T via shear on key area and bearing pressure on keyway walls.

3.2 Shaft Couplings

• Rigid Couplings: No misalignment accommodation.

  • Sleeve: Simple, for small shafts, precise alignment.

  • Clamp/Compression: Split, for larger shafts.

  • Flange: Most common, uses bolts.

• Flexible Couplings: Accommodate misalignment (angular, parallel, axial) & damp vibration.

  • Muff/Coupling: Elastic elements (rubber, leather).

  • Disc: Stainless steel discs, high torque, no lubrication.

  • Jaw/Spider: Elastomer spider in jaws.

  • Oldham: For parallel offset, three parts.

  • Universal: For angular misalignment, high torque.

• Distinction: Rigid = precise alignment, no flexibility. Flexible = allows misalignment, absorbs shock.

3.3 Threaded Components & Power Screws

• Thread Forms:

  • Square Thread: High efficiency, low friction, hard to manufacture. Preferable for power transmission (jacks, lead screws).

  • V-Thread (Metric/Unified): Easy to manufacture, high friction, self-locking. Used for fasteners.

• Self-Locking: Condition: $$\displaystyle \tan(\lambda) < \mu $$, where λ = lead angle, μ = friction coefficient.

  • Necessity: Prevents back-driving (load lowers screw when force removed). Essential for lifting screws (jacks, vises).

• Overhauling: When $$\displaystyle \tan(\lambda) > \mu $$, screw back-drives under load. Unacceptable for lifting.

• Efficiency Proof (Self-Locking < 50%):

$$\eta = \frac{\tan(\lambda)}{\tan(\lambda + \phi)} < 0.5 \quad \text{when} \quad \tan(\lambda) < \mu = \tan(\phi)$$

For self-locking, `λ < φ`. Max efficiency for self-locking occurs when `λ → φ`, then `η → 0.5`.

4.0 SPRING DESIGN

4.1 Classification

• By Shape: Helical (compression, extension, torsion), Leaf, Spiral, Disc (Belleville).

• Stress Indication:

  • Helical (close-coil): Bending in wire.

  • Torsion Spring: Torsion in wire.

  • Leaf Spring: Bending in leaves.

4.2 Helical Spring Fundamentals

• Close-coiled: Helix angle small (~5°), used for compression/extension. Wires under torsion.

• Open-coiled: Helix angle large, used for torsion springs.

• Spring Index (C): $$\displaystyle C = D/d $$ (mean coil dia / wire dia). Typical: 4-12.

• Spring Rate (k): $$\displaystyle k = F/\delta $$ (N/mm).

• Deflection (δ): $$\displaystyle \delta = \frac{8FD^3n}{Gd^4} $$ (for close-coil).

• Solid Length: $$\displaystyle L_s = n_s \cdot d $$ (all coils touching).

4.3 Design Calculations

• Shear Stress (with Wahl Correction Factor C_f for curvature):

$$ \tau = C_f \cdot \frac{8FD}{\pi d^3} \quad \text{where} \quad C_f = \frac{4C-1}{4C-4} + \frac{0.615}{C} $$

*   `C_f` accounts for stress gradient; increases as `C` decreases.

• Design Steps:

  1. From load F and deflection δ, find d and n using $\delta$ formula.

  2. Check shear stress τ with C_f. Iterate if needed.

  3. Check buckling for compression springs: $$\displaystyle L_o/D < \text{limit} $$ (use slenderness ratio).

  4. Check solid length, free length, etc.

4.4 Applications & Limitations

• Why Compression > Extension?

  • Compression springs don't need special end coils for attachment.

  • No risk of buckling (for compression, need guides).

  • Extension springs require initial tension and hooks/loops (stress concentrations).

• Buckling: Long, slender compression springs can buckle if unsupported. Critical free length $$\displaystyle L_o $$ depends on end condition and $D$.

5.0 CLUTCHES AND BRAKES

5.1 Clutches

• Centrifugal Clutch:

  • Working: Shoes (with friction lining) mounted on spider. At low speed, springs hold shoes in. As speed increases, centrifugal force throws shoes outward to engage drum.

  • Key Parameters: Engagement speed (set by spring force), centrifugal force, shoe geometry, coefficient of friction.

• Single Plate Clutch (Theory):

  • Uniform Pressure (p = constant): Assumes new clutch. Pressure p uniform across friction surface.

$$T = \mu p \pi (r_o^2 - r_i^2) \cdot \frac{r_o + r_i}{2}$$

$$F = p \cdot \pi (r_o^2 - r_i^2)$$

*   **Uniform Wear (p.r = constant):** Assumes worn clutch. Pressure inversely proportional to radius.

$$T = \frac{2}{3} \mu F (r_o^2 - r_i^2) \cdot \frac{r_o^3 - r_i^3}{r_o^2 - r_i^2}$$

$$F = 2\pi p_i r_i^2 \quad (\text{where } p_i \text{ is pressure at inner radius})$$

*   **Design:** Given `T`, `μ`, pressure limit `p_max`, and `r_o/r_i` ratio, solve for `r_o`, `r_i`, and axial force `F`.

5.2 Brakes

• Classification:

  • Block Brakes: Simple, shoe presses on drum (rope, band types).

  • Band Brake: Flexible band around drum. Self-energizing if leading shoe.

  • Internal Expanding (Shoe) Brake: Two shoes inside drum (automobiles). Can be self-energizing.

  • Disc Brake: Caliper with pads on disc. Excellent heat dissipation.

• Comparison:

  | Type | Advantages | Disadvantages | | :--- | :--- | :--- | | Rope/Band | Simple, low cost | Low torque, wear uneven | | Block | Robust | High wear, needs adjustment | | Internal Expanding | Self-energizing, good torque | Complex, heat in drum | | Disc | Excellent cooling, stable | Expensive, caliper design |

• Self-Energizing Brake: Friction force on leading shoe aids the applied force. Condition for self-locking (no release): $$\displaystyle \tan(\alpha) > \mu $$, where α = angle of wrap on leading shoe.

• Internal Expanding Brake (Short Note): Two semicircular shoes pivoted on anchor pin. When force applied, shoes push against drum. Leading/trailing shoe arrangement. Self-energizing effect on leading shoe increases braking torque.

5.3 Friction Materials

• Properties: High & stable μ, good wear resistance, high heat resistance (fade resistance), adequate strength, quiet operation.

• Common Materials: Asbestos-based (older), Sintered metals (metal powders), Organic composites (resin + fibers), Ceramics (high performance).

6.0 BEARINGS

6.1 Classification & Fundamentals

• Sliding Contact: Journal (radial), Thrust/Footstep (axial). Hydrodynamic (full film) or Boundary lubrication.

• Rolling Contact: Ball, Roller (cylindrical, tapered, spherical). Anti-friction bearings.

• Journal Bearing: Supports radial load on rotating shaft. Requires lubrication film.

6.2 Sliding Contact Bearings (Footstep/Thrust)

• Design Theories:

  • Uniform Pressure: Assumes pressure p uniform over entire bearing surface.

$$p = \frac{W}{\pi (D^2 - d_i^2)/4}$$

    Mean radius $$\displaystyle R_m = (D + d_i)/4 $$.

*   **Uniform Wear:** Assumes `p.r = constant` (wear proportional to pressure-sliding distance). More realistic for worn bearings.

$$p_{max} = \frac{2W}{\pi D (D - d_i)}$$

    Mean radius $$\displaystyle R_m = (D^2 + d_i^2)/(4D) $$.

• Footstep Bearing Design (with counterbore):

  • Load Capacity (Pressure Limit): Use p_max from uniform wear theory.

  • Power Lost in Friction:

$$P_f = \mu W \cdot (2\pi n) \cdot R_m \quad (\text{Watts})$$

    where `R_m` is mean radius from chosen theory, `n` = rpm.

6.3 Rolling Contact Bearings

• Dynamic Load Capacity (C): Load a bearing can withstand for 1 million revolutions with 90% reliability (L10 life).

• Bearing Life (L10): Life that 90% of bearings exceed.

$$\boxed{L_{10} = \left( \frac{C}{P} \right)^p}$$

*   `L10` = life in millions of revolutions.

*   `P` = equivalent dynamic bearing load.

*   `p` = exponent (3 for ball bearings, 10/3 for roller bearings).

*   **Life in hours:** $$\displaystyle L_{10h} = \frac{10^6}{60 \cdot n} \cdot L_{10} $$, where `n` = rpm.

• Factors Affecting Life: Load (primary), speed, lubrication quality, contamination, mounting/alignment, material/heat treatment.

• Bearing Characteristic Number: Dimensionless group related to minimum oil film thickness in hydrodynamic lubrication. Higher number = better lubrication.

• Reliability Adjustment: For reliability R ≠ 90%, use adjustment factor a_R:

$$C_{required} = a_R \cdot C_{from\ L10\ equation}$$

`a_R` > 1 for higher reliability (e.g., 99%).

• Selection Procedure:

  1. Calculate equivalent load P from radial/axial loads.

  2. Determine required life L10h from operating hours & rpm.

  3. Use $$\displaystyle C = P \cdot (L_{10})^{1/p} $$ to find required C.

  4. Apply reliability factor if needed.

  5. Select standard bearing with C ≥ C_required.

7.0 SPECIAL TOPICS (Integrated Above)

• Internal Expanding Brakes: See 5.2.

• Bearing Life: See 6.3 (L10 equation).

• Dimensionless Numbers: Bearing characteristic number, (C/P)^p in life equation.

• Polar Modulus of Shaft: See 2.4 ($$\displaystyle Z_p = J_p/c $$).