UNIT 2: MACHINE COMPONENT DESIGN - EXAM-FOCUS SHORT NOTES

1.0 FATIGUE AND FAILURE THEORIES IN DYNAMIC LOADING

1.1 Stress Concentration

• Definition: Localized increase in stress due to geometric discontinuities (holes, notches, fillets, keyways).

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

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

*   *Theoretical:* From elasticity theory (e.g., Kirsch equations for holes).

*   *Experimental:* From strain gauge measurements.

• Notch Sensitivity (q): Material's susceptibility to stress concentration. Ranges 0 (fully sensitive) to 1 (not sensitive).

• Fatigue Stress Concentration Factor (Kf): Used in dynamic loading.

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

> [!TIP] For brittle materials, q ≈ 1, so Kf ≈ Kt. For ductile materials, q < 1, so Kf < Kt.

1.2 S-N Curve (Wöhler Curve)

• Plot: Stress amplitude (S) vs. Number of cycles to failure (N) on log-log scale.

• Key Regions:

  • Low-Cycle Fatigue (LCF): High stress, low N (<10³-10⁴). Plastic deformation occurs.

  • High-Cycle Fatigue (HCF): Low stress, high N (>10⁴-10⁶). Elastic deformation.

  • Endurance Limit (Se): Stress amplitude below which failure does not occur (for ferrous metals). Non-ferrous metals (Al, Cu) have no true endurance limit; use fatigue strength at specified N (e.g., 10⁷ cycles).

• Determining Life: For a given stress amplitude S_a, draw horizontal line to intersect S-N curve, then vertical to N-axis gives fatigue life.

• Factors Affecting S-N Curve (Modifying Se to Se'):

$$S_e' = \text{Endurance limit from test (rotating beam)}$$

$$S_e = S_e' \cdot k_a \cdot k_b \cdot k_c \cdot k_d \cdot k_e$$

*   ka: Surface finish factor (machined, ground, hot-rolled).

*   kb: Size factor (larger diameter → lower endurance).

*   kc: Load factor (bending/axial = 1.0, torsion = 0.58).

*   kd: Reliability factor (99% reliability < 90%).

*   ke: Miscellaneous (temperature, corrosion, etc.).

1.3 Fatigue Failure Criteria & Design Diagrams

Used to find safe alternating stress (σa) or safe mean stress (σm) for a given material.

[!TIP] Design Process: 1. Find σa, σm from loading. 2. Choose criterion based on material & required safety. 3. Check if point (σa, σm) lies below failure line for given n. 4. Solve for unknown (diameter, allowable load).

1.4 Factors Influencing Fatigue Strength (Summary)

$$S_e = S_e' \cdot \underbrace{k_a}_{\text{surface}} \cdot \underbrace{k_b}_{\text{size}} \cdot \underbrace{k_c}_{\text{load}} \cdot \underbrace{k_d}_{\text{reliability}} \cdot \underbrace{k_e}_{\text{misc.}}$$

• Surface Finish (ka): Ground > Machined > Hot-rolled > As-forged. Use standard tables.

• Size (kb): For axial loading, kb ≈ 1.0. For bending/torsion, kb < 1.0 for d > 8 mm.

• Load (kc): Bending/Axial = 1.0; Torsion = 0.58 (since shear fatigue limit ≈ 0.58 * bending fatigue limit).

2.0 SHAFT DESIGN

2.1 Fundamentals

• Functions: Transmit power/torque, support rotating elements, maintain alignment.

• Types: Line shaft (transmission), Machine shaft (integral with part), Axle (supports rotating element, no torque).

• Stresses:

  • Torsional shear: $$\displaystyle \tau_t = \frac{T}{J} \cdot r = \frac{16T}{\pi d^3} $$ (max at surface).

  • Bending: $$\displaystyle \sigma_b = \frac{M}{Z} = \frac{32M}{\pi d^3} $$.

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

• Failure Theory: Maximum Shear Stress Theory (Tresca) is used for ductile shafts under combined loading.

$$\tau_{max} = \sqrt{\left(\frac{\sigma_b}{2}\right)^2 + \tau_t^2} \quad \text{or} \quad \sigma_{1,2} = \frac{\sigma_b}{2} \pm \sqrt{\left(\frac{\sigma_b}{2}\right)^2 + \tau_t^2}$$

2.2 Design for Combined Loading

• Equivalent Twisting Moment (Te): For bending + torsion (no axial load).

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

Design equation: $$\displaystyle \tau_{max} = \frac{16 T_e}{\pi d^3} \leq \tau_{allow} $$.

• Equivalent Bending Moment (Me): For bending + axial load (no torsion).

$$\boxed{M_e = M + \frac{P}{4} \cdot \frac{\pi d^2}{4} \cdot \frac{d}{2} \cdot \frac{1}{Z}} \quad \text{Simplified: } M_e = M + \frac{P d}{4}}$$

Often approximated as $$\displaystyle M_e = M + 0.2 P d $$ for ductile materials.

• General Case (Bending + Torsion + Axial): Use principal stress or energy theories. Tresca is common.

2.3 Strength and Stiffness Criteria

• Strength: Based on yield strength (Sy) or ultimate strength (Sut) with FoS.

$$\tau_{max} \leq \frac{S_y}{n} \quad \text{or} \quad \sigma_{max} \leq \frac{S_y}{n}$$

• Stiffness (Angular Twist):

$$\theta = \frac{T L}{J G} \quad \text{(radians)}$$

Where $$\displaystyle J = \frac{\pi d^4}{32} $$ (polar moment), G = modulus of rigidity.

$$\boxed{\theta = \frac{32 T L}{\pi G d^4}}$$

Limit: $$\displaystyle \theta \leq \theta_{allow} $$ (typically 0.5° to 2° per meter).

• Combined Design: For hollow shafts, optimize d_i/d_o ratio to satisfy both strength and stiffness with minimum weight.

2.4 Special Considerations

• Stress Concentration: Use fatigue stress concentration factor Kf at keyways, shoulders, etc.

• Hollow vs Solid Shaft: For same weight, hollow shaft has larger outer diameter → higher strength (τ ∝ 1/d³) and stiffness (θ ∝ 1/d⁴).

• Critical Speed: Speed at which shaft vibrates violently. Should be > 1.3 × operating speed. $$\displaystyle N_c \propto \sqrt{\frac{g}{\delta}} $$ (δ = deflection).

3.0 KEYS AND COUPLINGS

3.1 Keys

• Purpose: Transmit torque between shaft & hub (gear, pulley).

• Classification:

  • Sunk Keys: Fit into keyways (parallel, gib, feather). Positive location.

  • Saddle Keys: Sit on shaft, no keyway in shaft. No positive location.

  • Tangent Keys: For heavy torque, 2 keys at 90°.

  • Round Keys: For low torque, no keyway needed in hub.

• Saddle Key vs Sunk Key:

• Forces on Sunk Key: Transmits torque T.

  • Shear Failure: $$\displaystyle T = \tau_{allow} \cdot (b \cdot L) \cdot \frac{d}{2} \quad \Rightarrow \quad L = \frac{2T}{\tau_{allow} b d} $$

  • Crushing Failure: $$\displaystyle T = \sigma_{c,allow} \cdot \left(\frac{h}{2}\right) \cdot L \cdot \frac{d}{2} \quad \Rightarrow \quad L = \frac{4T}{\sigma_{c,allow} h d} $$

  • Where b = width, h = height (usually h = 0.5d for square key).

  • Design: Use both equations, take larger L. Check bearing pressure on keyway.

3.2 Couplings

• Purpose: Connect two shafts, transmit torque, allow misalignment/assembly.

• Rigid Couplings: No misalignment accommodation.

  • Sleeve: Simple, for small shafts.

  • Clamp/Compression: For larger shafts, no keyway needed.

  • Flange: Most common, uses bolts, precise alignment.

• Flexible Couplings: Accommodate misalignment, damp vibration.

  • Elastomeric: Jaw, Oldham (with plastic/spring), Disc.

  • Metallic: Grid, Gear, Universal Joint (for angular misalignment), Chain.

  • Selection: Based on misalignment type (parallel, angular, axial), torque, speed, space.

4.0 SPRING DESIGN

4.1 Classification

• By Shape: Helical (compression, extension, torsion), Leaf, Disc, Volute, Torsion bar.

• By Stress:

  • Bending Stress: Helical compression/extension, leaf springs.

  • Torsional Stress: Helical torsion springs.

  DiagramSEARCH: helical compression spring stress diagram, leaf spring bending

4.2 Helical Compression/Extension Springs

• Geometry:

  • d = wire diameter

  • D = mean coil diameter (centerline of wire)

  • N = total coils, Na = active coils (Na = N for closed ends, N-2 for open ends)

  • Spring Index: $$\displaystyle C = D/d $$. Optimal: 4 ≤ C ≤ 12. C < 4 → hard to coil, high stress; C > 12 → prone to buckling, tangling.

• Design Equations:

  • Spring Rate (Stiffness): $$\displaystyle k = \frac{G d^4}{8 D^3 N_a} \quad \left[\frac{N}{m}\right] $$

  • Deflection: $$\displaystyle \delta = \frac{8 P D^3 N_a}{G d^4} = \frac{P}{k} $$

  • Maximum Shear Stress (Wahl Correction): $$\displaystyle \tau = \frac{8 P D}{\pi d^3} \cdot K $$

    Where Wahl Factor $$\displaystyle K = \frac{4C - 1}{4C - 4} + \frac{0.615}{C} $$ (accounts for curvature & direct shear).

• Lengths:

  • Solid length: $$\displaystyle L_s = N \cdot d $$

  • Free length: $$\displaystyle L_f = L_s + \delta_{max} + \text{clash allowance (0.1-0.2)} $$

  • Pitch: $$\displaystyle p = \frac{L_f - N d}{N-1} $$

• Buckling of Compression Springs: For slender springs (Lf/D > ~4), use buckling factor.

$$\delta_{allow} = \text{ buckling factor} \times \text{ solid length deflection}$$

4.3 Design Problems (Step-wise)

1. Given P, δ_max, G, D (from casing), find Na from spring rate equation.

2. Find τ_max from stress equation with Wahl factor.

3. Check τ_max ≤ τ_allow (usually 0.4-0.6 Sy). If not, increase d.

4. Recalculate Na with new d (since k ∝ d⁴, Na ∝ 1/d⁴).

5. Calculate Ls, Lf, pitch. Check pitch ≥ 0.3d for manufacturing.

6. Check buckling if Lf/D large.

4.4 Special Springs

• Close-coiled vs Open-coiled:

  • Close-coiled: Helix angle ≈ 0°. Used for compression/extension springs. Stiff.

  • Open-coiled: Helix angle large. Used for torsion springs. Can be extended/compressed with little force.

• Helical Torsion Springs: Loaded by moment M. Stress is torsional in wire. Deflection is angular (θ).

$$\theta = \frac{64 M D N_a}{G d^4}, \quad \tau = \frac{32 M}{\pi d^3}$$

• Why Extension Springs < Compression Springs?

  1. Initial Tension: Must be coiled with initial tension to keep coils closed → reduces usable range.

  2. Manufacturing: Harder to coil without hooks/loops.

  3. Applications: Compression springs more common in suspensions, valves, etc.

5.0 CLUTCHES AND BRAKES

5.1 Friction Clutches

• Purpose: Engage/disengage power transmission.

• Centrifugal Clutch:

  • Working: Shoes (with friction lining) on rotating member. Centrifugal force > spring force → shoes move out → engage drum.

  • Key Parameters: Shoe mass (m), engagement radius (r), coefficient (μ), spring force (Fs). Engagement speed: $$\displaystyle \omega = \sqrt{\frac{F_s}{m r}} $$.

• Theories of Friction:

  • Uniform Pressure (p = constant): New, rigid clutch.

$$T = n \cdot \mu \cdot p \cdot \pi (D^2 - d^2) \cdot \frac{D+d}{4}$$

    Mean radius: $$\displaystyle r_m = \frac{2}{3} \cdot \frac{D^3 - d^3}{D^2 - d^2} $$

*   **Uniform Wear (p.r = constant):** Worn clutch (more realistic).

$$T = n \cdot \mu \cdot p \cdot \pi D d \cdot \frac{D+d}{4}$$

    Mean radius: $$\displaystyle r_m = \frac{D+d}{2} $$

> [!TIP] For single-plate clutch, n = 1 (2 friction surfaces). For multi-plate, n = number of friction surfaces.

• Design: Given T, μ, pmax, find D, d (D/d ratio usually 1.25-1.5), then axial force F = p_max × area (uniform pressure) or F = (2T)/(μ n π D d) (uniform wear).

5.2 Friction Brakes

• Band & Block Brake:

  • Construction: Band around drum, blocks between band & drum, fulcrum.

  • Tension Relation: $$\displaystyle T_1 / T_2 = e^{\mu \theta} $$ (θ in radians). T1 = tight side, T2 = slack side.

  • Torque: $$\displaystyle T_{brake} = (T_1 - T_2) \cdot r_d $$

  • Design: Find force on lever using moments about fulcrum.

• Internal Expanding Shoe Brake (Drum Brake):

  • Sketch: Two shoes (leading & trailing) inside drum, actuated by cam or hydraulic cylinder.

  • Self-energizing: Leading shoe gets additional force from drum rotation → higher braking torque for same actuation force.

  • Self-locking: When $\mu \theta \geq 1$ (θ in radians), brake holds without external force (dangerous, usually avoided).

• Self-energizing vs Self-locking:

  • Self-energizing: Braking force assists application (μθ < 1). Desirable.

  • Self-locking: Brake cannot be released by removing force (μθ ≥ 1). Undesirable for service brakes.

5.3 Friction Materials

• Properties: High & stable μ, good wear resistance, high heat capacity, fade resistance, low maintenance.

• Materials: Asbestos (phased out), Ceramic (high temp), Metallic (sintered metal), Organic (resin-bonded), Carbon (racing).

• Effect of μ: Higher μ → higher torque for same force, but more wear & heat. Lower μ → smoother, less wear, but needs larger force.

6.0 BEARINGS

6.1 Classification

• Sliding Contact: Journal (radial), Thrust/Footstep (axial), Pivot (oscillating).

• Rolling Contact: Ball, Roller (cylindrical, tapered, spherical, needle).

• Rolling vs Sliding Advantages:

  • Lower friction, start-up torque.

  • Interchangeable, standard sizes.

  • Less lubrication required.

  • Higher speed capability.

  • Easier maintenance.

6.2 Sliding Bearing Design (Journal & Footstep)

• Essential Properties of Bearing Material:

  1. Conformability: Adjust to shaft misalignment.

  2. Embeddability: Trap dirt/dust.

  3. Low Friction: With shaft material.

  4. High Thermal Conductivity: Dissipate heat.

  5. Corrosion Resistance.

  6. High Load Capacity (PV limit).

  7. Compatibility: No galling/welding.

  8. Fatigue Strength.

  • Materials: Babbitts (Sn/Sb/Cu), Bronze (Cu/Sn), Cast iron, Sintered metals, Plastics (PTFE).

• Bearing Characteristic Number (Z.N/p):

  • Z = absolute viscosity (Pa.s), N = rpm, p = bearing pressure (Pa).

  • Significance: Minimum value required for hydrodynamic lubrication (full fluid film). Below this → boundary/mixed lubrication → high wear.

  • Lubrication Regimes: Hydrodynamic (full film), Hydrostatic (externally pressurized), Boundary (thin film, asperity contact).

• Footstep (Thrust) Bearing Design:

  • Pressure Distribution:

    • Uniform pressure: $$\displaystyle p = \frac{P}{\frac{\pi}{4}(D^2 - d^2)} $$

    • Varying (conical): $p \propto 1/r$.

  • Load Capacity (Uniform p): $$\displaystyle P_{max} = p_{allow} \cdot \frac{\pi}{4}(D^2 - d^2) $$

  • Friction Power Loss:

$$P_f = \frac{\mu \cdot P \cdot V}{1000} \quad \text{(Watts)}$$

    Where $$\displaystyle V = \frac{\pi D N}{60} $$ (surface speed m/s).

*   **Problem:** Solid shaft (d=0) vs Counterbored (d>0). Calculate P and Pf for given p_allow, N, D, d.

6.3 Rolling Contact Bearings

• Dynamic Load Capacity (C):

  • Definition: Load a bearing can carry for 1 million revolutions with 90% reliability (L10 life).

  • Importance: Primary selection criterion from manufacturer's catalog.

• Bearing Life (L10):

  • Life Equation: $$\displaystyle L_{10} = \left(\frac{C}{P}\right)^p $$ (millions of revolutions)

    • p = 3 for ball bearings.

    • p = 10/3 ≈ 3.33 for roller bearings.

  • Life in Hours: $$\displaystyle L_{10h} = \frac{10^6}{60 N} \cdot \left(\frac{C}{P}\right)^p $$

• Reliability Adjustment:

  • For reliability R ≠ 90%, use factor a1: $$\displaystyle C_{adj} = \frac{C}{a_1} $$ (a1 > 1 for R > 90%).

  • Problem: Given P, N, L10h, R → find required C.

• Loading on Bearings (Combined Radial & Axial):

  • Equivalent Dynamic Load: $$\displaystyle P_e = X \cdot F_r + Y \cdot F_a $$

  • X, Y: Load factors from manufacturer's tables (depend on F_a/F_r ratio and bearing type).

  • Design: Use Pe in life equation.

• Factors Affecting Life:

  • Load: Life ∝ (1/P)³.

  • Speed: Affects lubrication, heat.

  • Lubrication: Film thickness, contamination.

  • Misalignment, Installation, Environment.

7.0 SPECIAL TOPICS & SHORT NOTES

7.1 Screw Threads and Power Transmission

• Square vs V-threads:

  | Square Thread | V-Thread | | :--- | :--- | | Thread angle = 0° | Thread angle = 60° (usually) | | No radial thrust → efficient | Radial thrust → higher friction, needs stronger nut | | High efficiency (η ≈ 0.7-0.8) | Lower efficiency (η ≈ 0.3-0.5) | | Difficult to manufacture, needs separate nut | Easy to manufacture, self-locking | | Use: Power screws (lead screws, vises) | Use: Fasteners (bolts, screws) |

• Self-locking Property:

  • Condition: Lead angle (λ) < Friction angle (φ), where $$\displaystyle \tan \phi = \mu $$.

  • Necessity: In lifting devices (screw jacks, vises) to prevent load from back-driving when force is removed.

• Efficiency of Screw:

$$\eta = \frac{\tan \lambda}{\tan(\lambda + \phi)}$$

*   **Proof η < 50% for self-locking:** If λ < φ, then tan(λ+φ) > tan(2λ) > 2 tan λ (for small angles), so η < 0.5.

• Overhauling: When screw drives load down by itself when force is removed (λ > φ). Occurs in high-efficiency screws (square/buttress). Dangerous in lifting applications.

7.2 Dimensionless Numbers in Bearing Design

• Bearing Characteristic Number (Z.N/p): Predicts lubrication regime. Minimum value for hydrodynamic lubrication.

• Sommerfeld Number (S): For hydrodynamic journal bearings.

$$S = \frac{\mu N}{p} \left(\frac{r}{c}\right)^2$$

Where r = journal radius, c = radial clearance. Used to find minimum film thickness, friction factor.

7.3 Other Short Note Topics

• Polar Modulus (Zp or J): For circular shaft, $$\displaystyle Z_p = J = \frac{\pi d^3}{16} $$. Used in torsion: $$\displaystyle \tau_{max} = \frac{T}{Z_p} $$.

• Bearing Life (L10): Life at which 90% of bearings survive. $$\displaystyle L_{10} = (C/P)^p $$ million rev. Factors: load, speed, lubrication, contamination, alignment.

• Internal Expanding Brakes:

  DiagramSEARCH: internal expanding shoe brake drum leading trailing
  Leading shoe (rotation direction) gets self-energizing effect; trailing shoe does not. More braking torque than band brake for same size.

Final Exam Strategy:

1. Numerical Focus: Shafts (Te/Me), Springs (d, Na, Lf), Bearings (L10, Pe), Clutches (T, F, D/d) are guaranteed 7-14 marks.

2. Theory Comparisons: Always present in tables (Goodman vs Soderberg, Saddle vs Sunk, Rigid vs Flexible).

3. Diagrams: Sketches for keys, couplings, clutches, brakes, springs are essential for 5-7 marks.

4. Formulas: Memorize boxed equations. Understand when to use Kt vs Kf, Se vs Se', X/Y factors.

5. Common Pitfalls:

   • Forgetting Wahl factor (K) in spring stress.

   • Using Sut instead of Sy in Soderberg.

   • Confusing active vs total coils in springs.

   • Not converting units consistently (MPa = N/mm², GPa = kN/mm²).

   • Misidentifying leading/trailing shoe in drum brake.