UNIT 1: MACHINE COMPONENT AND DESIGN - EXAM-FOCUS NOTES

Based on rigorous analysis of RGPV past papers (2022-2025), these notes prioritize high-yield concepts. Focus on definitions, formulas, comparisons, and design procedures.

I. FATIGUE ANALYSIS & STRESS CONCENTRATION

A. Stress Concentration

• Definition: Localized increase in stress near geometric discontinuities (holes, notches, fillets, keyways) under load.

• Primary Causes: Abrupt changes in cross-section, material defects, surface scratches.

• Stress Concentration Factor (Kt):

  • Theoretical Kt: From elasticity theory (depends only on geometry). Obtained from charts.

  • Experimental Kt: From strain gauge measurements.

  • Formula: Kt = σ_max / σ_nominal

• Notch Sensitivity (q): Material's responsiveness to stress concentration.

  • q = (Kf - 1) / (Kt - 1)

  • 0 ≤ q ≤ 1. Brittle materials → q ≈ 1; Ductile materials → q < 1.

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

  Key Formula: Kf = 1 + q(Kt - 1)

• Methods to Reduce Stress Concentration:

  • Use generous fillets.

  • Add relief notches.

  • Avoid sharp corners; use gradual transitions.

  • Prefer drilled holes over notches.

[!TIP] Exam Focus: Distinguish between Kt (static/theoretical) and Kf (dynamic/fatigue). Always use Kf in fatigue design calculations.

B. Fatigue Failure Theories & Diagrams

• S-N Curve (Wöhler Curve):

  • Plot of Stress Amplitude (S) vs. Number of Cycles to Failure (N) on log-log scale.

  • Endurance Limit (σe): Stress below which failure does not occur for infinite life (N > 10⁶ cycles). For steel, σe ≈ 0.5 S_ut.

  • Importance: Predicts fatigue life; basis for safe design against fluctuating loads.

• Factors Affecting Endurance Limit (σe):

  • Surface Finish Factor (Ka): Rough surface → lower σe.

  • Size Factor (Kb): Larger size → higher probability of flaw → lower σe.

  • Loading Factor (Kc): For bending/axial/torsion.

  • Temperature Factor (Kd): High temp → reduced σe.

  • Reliability Factor (Ke): Higher reliability → lower σe.

  • Modified Endurance Limit: σe' = Ka * Kb * Kc * Kd * Ke * σe

• Failure Criteria for Fluctuating Stresses:

  • Define: σa = (σ_max - σ_min)/2 (Alternating stress), σm = (σ_max + σ_min)/2 (Mean stress).

  • Goodman Diagram: Straight line from (σe, 0) to (S_ut, 0). Equation: σa / σe + σm / S_ut = 1 / n

  • Modified Goodman Diagram: More conservative. Uses yield strength (S_yt) instead of S_ut for mean stress term: σa / σe + σm / S_yt = 1 / n

  • Soderberg Relation: Most conservative. Line from (σe, 0) to (S_yt, S_yt). Equation: σa / σe + σm / S_yt = 1 / n

[!TIP] Common Pitfall: Soderberg and Modified Goodman both use S_yt for mean stress term, but Soderberg line is more restrictive. Always check which criterion is specified.

C. Design for Fluctuating Stresses

1. Calculate σa and σm from given load cycle.

2. Select appropriate failure criterion (Goodman/Modified/Soderberg).

3. Apply factor of safety n.

4. Solve for required material strength (S_ut, S_yt) or component dimensions.

5. If using stress concentration: σa,actual = Kf * σa,nominal, σm,actual = Kf * σm,nominal (often Kf used for both).

II. SHAFT DESIGN

A. Stresses in Shafts

• Torsional Shear Stress: τ = Tρ / J → Max at surface: τ_max = T / Zp

  • Zp = J / c = πd³/16 (Solid circular), Zp = π(D⁴ - d⁴)/(16D) (Hollow)

• Bending Stress: σ = My / I → Max at surface: σ_max = M / Z

  • Z = I / c = πd³/32 (Solid), Z = π(D⁴ - d⁴)/(32D) (Hollow)

• Combined Bending & Torsion:

  • Equivalent Bending Moment (Me): Moment that alone would produce max principal stress.

    Me = √(M² + T²) (for ductile materials, using max shear theory)

  • Equivalent Twisting Moment (Te): Torque that alone would produce max shear stress.

    Te = √(M² + T²) (same expression for solid circular shaft).

B. Design Equations & Parameters

• Design for Strength (Solid Shaft):

  σ_max = Me / Z ≤ [σ] or τ_max = Te / Zp ≤ [τ]

• Design for Stiffness (Twist):

  θ = TL / (GJ) ≤ θ_allow

  • J = πd⁴/32 (solid), J = π(D⁴ - d⁴)/32 (hollow)

• Hollow Shaft Advantages:

  • Higher strength/stiffness per unit weight.

  • Material distributed farther from center (higher Z, J for same weight).

  • Used in aerospace, high-speed machinery.

[!TIP] Exam Problem Pattern: 1) Find M from transverse loads (e.g., central load on simply supported shaft). 2) Find T from power/speed: T = (P * 60)/(2πN). 3) Compute Me/Te. 4) Design for both strength and twist. Check both criteria.

III. KEYS & COUPLINGS

A. Keys

• Classification:

  • Saddle Key: Sits on shaft keyway, no keyway in hub. Only transmits torque by pressure on key seat.

  • Sunk Key: Fits into keyways in BOTH shaft and hub. Transmits torque by shear & crushing.

    • Parallel, Gib-head, Feather, Woodruff.

  • Tangent Key: For heavy torque, transmits by shear on 2 keys at 90°.

• Saddle Key vs. Sunk Key:

  | Feature | Saddle Key | Sunk Key | | :--- | :--- | :--- | | Fitting | Keyway only in hub | Keyways in shaft & hub | | Torque Capacity | Low (only friction/pressure) | High (shear + crushing) | | Axial Movement | Allows (e.g., pulleys) | Prevents (fixed connection) | | Application | Light duty, sliding hubs | Heavy duty, fixed gears |

• Forces on Sunk Key (Design):

  1. Shear Failure: τ = T / (L * b * d) ≤ [τ]

  2. Crushing Failure: σ_cr = (2T) / (L * (d/2) * h) ≤ [σ_cr]

     • h = key height (usually = 0.5d for square key).
  • Design: Calculate L from both equations, adopt larger value.

B. Couplings

• Rigid Couplings: No misalignment accommodation.

  • Sleeve/Full-length: Simple, for aligned shafts.

  • Flange: Bolted flanges, precise alignment needed.

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

  • Muff/Clamp: Split sleeve, minor misalignment.

  • Bush Pin: Uses rubber/bush pins for flexibility.

  • Gear: High torque, some misalignment.

  • Oldham (Disc): Parallel misalignment only.

  • Universal Joint: Angular misalignment (Hooke's joint).

• Comparison: Rigid = simple, cheap, precise alignment required. Flexible = compensates misalignment, absorbs shock, more complex/costly.

IV. THREADED FASTENERS & POWER SCREWS

A. Thread Forms

• Square Thread: Parallel flanks, 0° thread angle.

  • Advantages: Lowest friction, highest efficiency, no radial thrust. Preferable for power transmission (screw jacks, lead screws).

• V-Thread (Metric/Unified): 60° (metric) or 55° (BSW) thread angle.

  • Disadvantages: High friction, lower efficiency, radial thrust. Used for fasteners (bolts, nuts).

• Self-locking: Condition: Lead angle (λ) < Friction angle (φ).

  • tan λ = Lead / (π * d_m), tan φ = μ

  • Necessity: Prevents back-driving (e.g., screw jack holds load without brake).

B. Screw Design & Analysis

• Overhauling: Screw is not self-locking (λ > φ). It can be back-driven by load (e.g., some power screws).

• Efficiency of Screw Jack:

  η = (tan λ) / (tan(λ + φ)) for raising load.

  Proof for Self-locking: For self-locking, λ < φ → tan λ < tan φ. Max efficiency when λ = φ → η_max = tan φ / (tan 2φ) = (1 - tan²φ)/2 < 0.5 for φ > 14°. Hence η < 50% for self-locking screws.

• Stresses in Power Screws:

  • Torsional Stress: τ = T / Zp (from torque applied to turn screw).

  • Compressive/Buckling Stress: For long columns (e.g., vertical screw), check Euler buckling.

  • Thread Shear Stress: Check shear across thread root.

V. SPRING DESIGN

A. Classification

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

• By Loading: Compression, Extension, Torsion.

• Sketch Stress Type:

  • Helical (comp/ext): Bending stress in wire.

  • Torsion spring: Torsional shear stress in wire.

B. Close-coiled vs. Open-coiled

• Close-coiled: Small helix angle (< 5°), wires almost parallel. Used for compression/extension springs. Stiff.

• Open-coiled: Large helix angle. Used for torsion springs. Flexible, can extend/contract axially.

C. Helical Compression/Extension Spring Design

• Spring Index (C): C = D / d (Mean coil diameter / Wire diameter). Typical: 4 ≤ C ≤ 12.

• Active Coils (Na): Coils contributing to deflection. For plain ends: Na = Nt - 2.

• Solid Length (Ls): Length when fully compressed. Ls = Nt * d.

• Free Length (Lf): Lf = Ls + δ_max + (0.1 to 0.2)Lf (gap between coils).

• Spring Rate (k): k = F / δ (N/mm).

• Deflection (δ): δ = (8FD³Na) / (Gd⁴) (Close-coiled).

• Wahl's Correction Factor (Kw): Accounts for curvature and direct shear.

  Kw = (4C - 1)/(4C - 4) + (0.615/C)

• Shear Stress (τ):

  Critical Formula: τ = (8FD) / (πd³) * Kw = (8F C) / (πd²) * Kw

  • Design: Ensure τ ≤ [τ] (usually 0.45 S_ut for static, lower for fatigue).

• Design Steps:

  1. From F_max, δ_max, G, [τ]: Use τ formula to find d.

  2. Check C = D/d (4-12). Adjust D if needed.

  3. Find Na from deflection formula.

  4. Calculate Ls, Lf, check buckling (if Lf > 5.3D, use guided rod).

D. Special Topics

• Why Extension Springs < Compression Springs?

  • Need special end hooks (stress concentration, manufacturing complex).

  • Often require initial tension (preload) to keep coils closed.

  • More prone to buckling if slender.

• Impact Loading (Drop Height):

  h = ( (δ_static * δ_impact) / δ_static ) - δ_static

  • δ_static = F/k (static deflection under load F).

  • δ_impact = maximum deflection under impact.

  • From energy: F(h + δ_impact) = (1/2)k δ_impact².

VI. CLUTCHES & BRAKES

A. Clutches

• Centrifugal Clutch:

  • Working: Shoes fly outward by centrifugal force at high speed, engage drum.

  • Key Parameters: Engagement speed (set by spring force), number of shoes, friction material, mass of shoes.

• Friction Clutches (Single/Multi-plate):

  • Uniform Pressure Theory: Assumes p = constant across friction surface. Valid for new, rigid clutches.

    T = μ p π (D² - d²) n / 4

    Mean Radius (Rm) = (2/3) * (D³ - d³)/(D² - d²)

  • Uniform Wear Theory: Assumes p * r = constant. Valid for worn clutches (pressure highest at inner radius).

    T = (μ π n p_max (D² - d²)) / 2

    Mean Radius (Rm) = (D + d)/2 (Arithmetic mean)

  • Design: Given T, μ, p_max, D/d ratio → find D, d, axial force F = p * π(D² - d²)n/4 (UP) or F = (π n p_max (D² - d²))/4 (UW).

B. Brakes

• Types & Comparison:

  | Type | Principle | Applications | Notes | | :--- | :--- | :--- | :--- | | Rope | Friction on drum via rope | Cranes, hoists | High wear, low μ | | Band | Friction band tightens on drum | Bicycles, band brakes | Can be self-energizing | | Block | Block pressed against drum | Vehicles, machinery | Simple, common | | Internal Shoe | Expanding shoes inside drum | Automobile drum brakes | Self-energizing |

• Band & Block Brake (n blocks):

  T_brake = (T1 - T2) * R

  T2 = T1 * e^(-nθμ) (θ in radians, n = number of blocks)

  Lever force F related to T1 and T2 via moment equilibrium.

• Self-energizing Brake: Friction force on one side aids the applied force. Condition: tan α > μ (α = angle of contact at pivot).

• Self-locking Brake: tan α ≤ μ. Applied force alone can hold load (no back-driving). Dangerous for stopping (cannot release).

[!TIP] Key Difference: Self-energizing (aids force, good for braking) vs. Self-locking (cannot release, bad for clutches/brakes that must disengage).

VII. BEARINGS

A. Classification

• Sliding Contact (Plain): Journal (radial), Footstep/Thrust (axial). High load, low speed.

• Rolling Contact: Ball, Roller (cylindrical, tapered, spherical). Low friction, high speed.

B. Rolling Contact Bearings

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

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

  Core Formula: L10 = (C / P)^p million revolutions

  • p = 3 for ball bearings, 10/3 for roller bearings.

  • P = Equivalent dynamic bearing load (kN).

• Life in Hours: L10h = (10⁶ / (60 * N)) * L10

  • N = rpm.

• Factors Affecting Life:

  • Load (P): Life ∝ 1/P³ (ball) → Doubling load reduces life to 1/8.

  • Speed: Affects life in hours.

  • Reliability: Higher reliability (e.g., 99% vs 90%) requires higher C (use correction factors).

  • Mounting: Inner ring rotation vs outer ring rotation affects P calculation.

C. Sliding Contact Bearings

• Footstep (Thrust) Bearing:

  • Pressure Limit: p = W / (π/4 * (D² - d²)) ≤ [p]

  • Power Loss in Friction: Pf = (μ * W * N) / 1000 kW

    • μ = friction coefficient, W = load (N), N = rpm.

• Bearing Characteristic Number (Z N / P):

  • Z = Absolute viscosity (Pa.s or N.s/m²), N = rpm, P = bearing pressure (N/m²).

  • Significance (Petroff's Equation): f = (Z N) / P (for unloaded bearing, f = coefficient of friction).

  • Optimum Value: ~0.001 - 0.002 for minimum f. Guides selection of N and P.

• Properties of Bearing Linings (Babbitt, Bronze, etc.):

  • Conformability & Embeddability (accommodate misalignment/dirt).

  • Anti-seizure (low friction with shaft).

  • Corrosion resistance.

  • High thermal conductivity.

  • Adequate fatigue strength (compressive).

D. Short Notes (Direct from Past Papers)

• Bearing Life (L10): Life that 90% of bearings achieve before fatigue. L10 = (C/P)^p million rev. Based on statistical distribution (Weibull).

• Internal Expanding Shoe Brake:

  • Construction: Two shoes pivoted inside brake drum. Leading shoe (self-energizing) and trailing shoe.

  • Application: Automobile drum brakes. Self-energizing effect reduces required pedal force.

VIII. DESIGN METHODOLOGY & MATERIAL PROPERTIES

A. Design Considerations

• Factors: Functional requirements, cost, manufacturing feasibility, safety, reliability, maintenance, environment.

• Design Types:

  • Adaptive Design: Minor modifications to existing design.

  • Optimum Design: Mathematical optimization for best solution (min weight, cost).

• Interchangeable Manufacturing: Parts made to tolerances so any part fits any assembly. Requires tolerances.

  • Unilateral Tolerance: Tolerance zone on one side of nominal size (e.g., +0.02/-0).

  • Bilateral Tolerance: Tolerance zone on both sides (e.g., ±0.01).

B. Material Properties

• Hardness vs. Toughness:

  • Hardness: Resistance to indentation/scratching (Brinell, Rockwell). High hardness → low ductility.

  • Toughness: Ability to absorb energy before fracture (area under stress-strain curve). Requires both strength and ductility.

• For Friction Surfaces (Clutches/Brakes):

  • High & stable coefficient of friction (μ).

  • High wear resistance.

  • Good heat resistance (high thermal capacity, conductivity).

  • Examples: Asbestos (old), Ceramic, Sintered metal, Organic compounds.

• Strength Relationships:

  • For steels: S_yt ≈ 0.6 S_ut, σe ≈ 0.5 S_ut (for reversed bending).

IX. SPECIAL TOPICS (SHORT NOTES)

A. Polar Modulus of Section (Zp)

• Definition: Zp = J / c (Polar moment of inertia / outer radius). Measures torsional resistance.

• For Solid Circular Shaft: Zp = πd³/16

• Significance: Used in torsional shear stress formula τ_max = T / Zp. Analogous to section modulus Z for bending.

B. Bearing Life (L10)

• Definition: Rating life where 90% of bearings survive. L10 = (C/P)^p million revolutions.

• C = Dynamic load capacity (catalog value).

• P = Equivalent dynamic load (depends on radial/axial loads).

• p = 3 (ball), 10/3 (roller).

• Life Adjustment: For reliability ≠ 90%, use life adjustment factor a1. For speed/material/operating conditions, use a2, a3.

C. Internal Expanding Brakes

• Construction: Two brake shoes (leading & trailing) pivoted inside a brake drum. Actuated by a cam or hydraulic cylinder.

• Working: When applied, shoes expand outward against drum. Leading shoe gets self-energizing effect (friction force aids actuation).

• Application: Automobile drum brakes, some industrial brakes.

• Advantage: Self-energizing reduces required input force.

D. Dimensionless Numbers (Bearings)

• Bearing Characteristic Number (Z N / P):

  • Z = Absolute viscosity (Pa.s), N = rpm, P = bearing pressure (Pa).

  • Petroff's Equation: f ≈ (Z N) / P for unloaded bearing.

  • Significance: Guides operating zone. Minimum f at Z N / P ≈ 0.001 - 0.002. Too high → high friction; too low → boundary lubrication failure.

E. Punching Force Calculation

• Principle: Shear failure along perimeter of hole.

• Formula: F_punch = τ_allow * Perimeter * Thickness

  • τ_allow = Allowable shear stress (often 0.6 S_ut or 0.8 S_ty).

  • Perimeter = π * d (for circular hole).

  • Thickness = t.

• Example: F = π * d * t * τ_allow

[!TIP] Final Exam Checklist:

1. Fatigue: Know Kf, S-N curve factors, Goodman/Soderberg equations.

2. Shafts: Compute M & T from loads/power, design for strength & twist, Zp formula.

3. Keys: Shear & crushing design for sunk key. Saddle vs Sunk comparison.

4. Springs: τ = (8FD)/(πd³)*Kw, design steps, close vs open.

5. Bearings: L10 = (C/P)^p, ZN/P concept, footstep bearing pressure & power loss.

6. Short Notes: Be ready for Polar Modulus, Bearing Life, Internal Expanding Brake, Dimensionless Numbers, Punching Force.