UNIT 5: MACHINE COMPONENT DESIGN - SHORT NOTES

Based on RGPV Past Papers (2022–2025)

I. FATIGUE ANALYSIS AND FAILURE THEORIES

Fluctuating Stresses and Fatigue

• S-N Curve (Wöhler Curve): A plot of stress amplitude (S_a) vs. number of cycles to failure (N) on a log-log scale. It defines the fatigue strength for a given number of cycles. The endurance limit (S_e) is the stress amplitude below which the material can withstand infinite cycles (typically >10^6 or 10^7 cycles).

  [!TIP] For steels, an endurance limit exists. For non-ferrous metals (Al, Mg), the S-N curve continues to drop; no true endurance limit.

• Endurance Limit (S_e): The maximum reversing stress a material can endure indefinitely. Modified from the rotating beam test value ($$\displaystyle S_e' $$) using correction factors:

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

Where:

*   $$\displaystyle k_a $$ = Surface finish factor

*   $$\displaystyle k_b $$ = Size factor

*   $$\displaystyle k_c $$ = Load factor (bending=1, axial=0.85, torsion=0.59)

*   $$\displaystyle k_d $$ = Temperature factor

*   $$\displaystyle k_e $$ = Reliability factor

*   $$\displaystyle k_f $$ = Miscellaneous factor (corrosion, etc.)

• Infinite vs. Finite Life: Design for infinite life ensures $$\displaystyle S_a < S_e $$. Design for finite life uses S-N curve for specific N cycles.

Stress Concentration

• Definition: Localized increase in stress near geometric discontinuities (holes, notches, fillets, keyways, shoulders) due to load flow disturbance.

• Stress Concentration Factor (Kt): Ratio of maximum theoretical stress to nominal (gross section) stress. $$\displaystyle K_t = \frac{\sigma_{max}}{\sigma_{nom}} $$. Depends only on geometry.

• Notch Sensitivity Factor (Kf): Accounts for material's sensitivity to notches. $$\displaystyle K_f = 1 + q(K_t - 1) $$, where $q$ (0≤q≤1) is the notch sensitivity.

  [!TIP] For static loading, $$\displaystyle K_t $$ is used. For fatigue loading, use $$\displaystyle K_f $$ to calculate actual alternating stress: $$\displaystyle \sigma_a = K_f \cdot \frac{\sigma_{nom,a}}{} $$.

• Methods to Reduce Stress Concentration:

  1. Use fillets with large radii at shoulders.

  2. Use gradual transitions in cross-section.

  3. Prefer holes over notches.

  4. Use material with high notch toughness.

Fatigue Failure Criteria (Fluctuating Stresses)

Given: $$\displaystyle \sigma_{max} $$, $$\displaystyle \sigma_{min} $$, Alternating Stress $$\displaystyle \sigma_a = \frac{\sigma_{max} - \sigma_{min}}{2} $$, Mean Stress $$\displaystyle \sigma_m = \frac{\sigma_{max} + \sigma_{min}}{2} $$.

[!TIP] To find minimum required ultimate strength ($$\displaystyle S_{ut} $$): Rearrange the chosen criterion's equation, given $$\displaystyle \sigma_a $$, $$\displaystyle \sigma_m $$, $$\displaystyle S_e $$ (often $$\displaystyle 0.5 S_{ut} $$), $$\displaystyle S_y $$ (often $$\displaystyle 0.55 S_{ut} $$), and factor of safety $n$.

II. SHAFT DESIGN

Fundamentals

• Functions: Transmit power/torque, support rotating elements (gears, pulleys), maintain alignment.

• Materials: Medium carbon steels (e.g., 45C8), alloy steels. Design based on maximum shear stress theory (Tresca) for ductile materials under combined torsion & bending.

• Types: Transmission shafts, machine shafts (spindles).

Stresses in Shafts

1. Torsional Shear Stress: $$\displaystyle \tau_t = \frac{T \cdot R}{J} = \frac{16T}{\pi d^3} $$ (solid circular).

2. Bending Stress: $$\displaystyle \sigma_b = \frac{M \cdot y}{I} = \frac{32M}{\pi d^3} $$ (solid circular).

3. Combined Loading (Bending + Torsion):

   • Equivalent Bending Moment ($$\displaystyle M_e $$): A hypothetical bending moment that, acting alone, would produce the same maximum shear stress as the combined loading.

$$ M_e = \sqrt{M^2 + T^2} \quad \text{(for solid circular shaft)} $$

*   **Equivalent Twisting Moment ($$\displaystyle T_e $$):** A hypothetical torque that, acting alone, would produce the same maximum normal stress as the combined loading.

$$ T_e = \sqrt{M^2 + T^2} \quad \text{(for solid circular shaft)} $$

Note: For solid circular shafts, $$\displaystyle M_e = T_e $$ numerically.

Design Criteria

• Strength-Based: $$\displaystyle \tau_{max} = \frac{16 T_e}{\pi d^3} \leq \frac{S_{sy}}{n} $$ or $$\displaystyle \sigma_{max} = \frac{32 M_e}{\pi d^3} \leq \frac{S_{sy}}{n} $$.

  \boxed{d \geq \sqrt[3]{\frac{16 n T_e}{\pi S_{sy}}}} \quad \text{or} \quad \boxed{d \geq \sqrt[3]{\frac{32 n M_e}{\pi S_{sy}}}}

• Rigidity-Based (Torsional Twist): $$\displaystyle \theta = \frac{T L}{G J} \leq \theta_{allow} $$. For solid shaft: $$\displaystyle d \geq \sqrt[4]{\frac{16 n T L}{\pi G \theta_{allow}}} $$.

• Deflection-Based: For shafts with transverse loads, check slope/deflection at bearings or gear locations.

Hollow vs. Solid Shafts

• Hollow Shaft: $$\displaystyle d_o $$ = outer dia, $$\displaystyle d_i $$ = inner dia, $$\displaystyle k = d_i/d_o $$.

  • Polar Modulus: $$\displaystyle Z_p = \frac{J}{R} = \frac{\pi (d_o^4 - d_i^4)}{32 \cdot (d_o/2)} = \frac{\pi d_o^3 (1 - k^4)}{16} $$.

  • Weight Ratio (Hollow:Solid) for same $$\displaystyle Z_p $$: $$\displaystyle \frac{W_h}{W_s} = \frac{1 - k^4}{1 - k^3} $$. For $$\displaystyle k=0.5 $$, weight is ~40% of solid shaft.

  [!TIP] Hollow shafts are lighter for same strength/stiffness. Optimal $k \approx 0.6$ for max $$\displaystyle Z_p $$ per unit weight.

III. KEYS, COUPLINGS, AND THREADED FASTENERS

Keys

• Classification:

  • Saddle Key: Tapered, fits in keyseat in shaft only. For light duty.

  • Sunk Key: Fits in keyways in both shaft & hub. Parallel (rectangular), Gib-headed (with head), Feather (sliding).

  • Woodruff Key: Circular segment, fits in circular keyway in shaft. For heavy shock/vibration.

• Design (Sunk Key - Shear & Crushing):

  • Shear Failure: $$\displaystyle F = \frac{T}{d/2} \leq \tau_{allow} \cdot (l \cdot b) $$.

    \boxed{l \geq \frac{2T}{\tau_{allow} \cdot b \cdot d}}

  • Crushing (Bearing) Failure: $$\displaystyle F = \frac{T}{d/2} \leq \sigma_{c,allow} \cdot (l \cdot h/2) $$.

    \boxed{l \geq \frac{T}{\sigma_{c,allow} \cdot h \cdot d}}

  • Take larger $l$. Standard lengths: $l \approx 1.5d$ to $2d$.

• Forces: Torque $T$ produces force $$\displaystyle F = 2T/d $$ on key. This force acts on keyseat walls (shear on key, crushing on shaft/hub).

Couplings

• Rigid (e.g., Muff, Flange): No misalignment tolerance. Used for precisely aligned shafts.

• Flexible (e.g., Jaw, Grid, Disc, Oldham): Accommodates misalignment (angular, parallel, axial). Used for general purpose.

• Flange Coupling Design:

  1. Torque Capacity: $$\displaystyle T = \mu \cdot F \cdot \frac{d_m}{2} $$, where $$\displaystyle d_m $$ = mean diameter of bolts.

  2. Bolt Size: From shear strength of bolt: $$\displaystyle F \leq n \cdot \frac{\pi d_b^2}{4} \cdot \frac{\tau_{allow}}{4} $$.

  3. Hub & Shaft: Based on shaft diameter and key design.

Threaded Fasteners (Screws for Power)

• Square Thread vs. V-Thread:

  | Square Thread | V-Thread (60°) | | :--- | :--- | | Zero thread angle → low friction, high efficiency. | Thread angle → high friction, low efficiency. | | Stronger (root thickness = pitch). | Weaker (root thickness < pitch). | | Difficult to manufacture, not self-locking. | Easy to manufacture, self-locking. | | Used for power transmission (vises, jackscrews). | Used for fasteners (bolts, nuts). |

• Self-Locking: Condition: Lead angle ($\lambda$) < Friction angle ($\phi$). $$\displaystyle \tan \lambda = \frac{\text{Lead}}{\pi d_m} < \mu = \tan \phi $$.

  • Necessary in lifting devices (jacks, presses) to prevent load from falling back when force is removed.

  • Efficiency of Self-Locking Screw: $$\displaystyle \eta = \frac{\tan \lambda}{\tan (\lambda + \phi)} < 0.5 $$ (50%).

• Overhauling: When $$\displaystyle \lambda > \phi $$, screw is non-self-locking; load can overhaul (run back) when torque is removed. Dangerous in lifting applications.

• Torque to Raise/Lower: $$\displaystyle T_{raise} = W \cdot \frac{d_m}{2} \tan(\phi + \lambda) $$, $$\displaystyle T_{lower} = W \cdot \frac{d_m}{2} \tan(\phi - \lambda) $$.

IV. SPRINGS

Classification

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

• By Loading: Compression, Tension, Torsion (helical torsion spring), Bending (leaf spring).

  DiagramSEARCH: helical compression spring diagram, leaf spring stress bending, helical torsion spring

Helical Compression/Tension Springs

• Terminology:

  • $d$ = Wire diameter

  • $D$ = Mean coil diameter ($$\displaystyle D = D_{outer} - d $$)

  • $C$ = Spring index = $D/d$ (typically 4–12)

  • $n$ = Number of active coils (carry load)

  • $$\displaystyle n_t $$ = Total coils ($$\displaystyle n_t = n + 2 $$ for closed ends)

  • $$\displaystyle L_0 $$ = Free length

  • $$\displaystyle L_s $$ = Solid length ($$\displaystyle L_s = n_t \cdot d $$)

• Spring Rate (Stiffness): $$\displaystyle k = \frac{G d^4}{8 D^3 n} $$.

  \boxed{k = \frac{G d^4}{8 D^3 n}}

• Maximum Shear Stress (Wahl Correction for Curvature):

$$ \tau_{max} = K_w \cdot \frac{8 F D}{\pi d^3} \quad \text{where} \quad K_w = \frac{4C-1}{4C-4} + \frac{0.615}{C} $$

> [!TIP] For **initial design**, use $$\displaystyle K_w \approx 1 $$. Final check must include $$\displaystyle K_w $$.

• Deflection: $$\displaystyle \delta = \frac{8 F D^3 n}{G d^4} $$.

• Design Steps:

  1. From $$\displaystyle F_{max} $$, $$\displaystyle \delta_{max} $$, $$\displaystyle L_0 $$, $$\displaystyle D_{max} $$ (space constraint), find $$\displaystyle k = F_{max}/\delta_{max} $$.

  2. Select $d$ from strength: $$\displaystyle \tau_{max} \leq \tau_{allow} $$ (using Wahl factor).

  3. Calculate $$\displaystyle n = G d^4/(8 k D^3) $$.

  4. Check $$\displaystyle L_0 = L_s + \delta_{max} + \text{initial clearance} $$.

• Why Compression > Tension Springs? Compression springs don't need special end coils for attachment; tension springs require hooks which are stress concentration points and difficult to manufacture.

V. CLUTCHES AND BRAKES

Clutches (Friction Type)

• Working: Transmits torque via friction between mating surfaces.

• Centrifugal Clutch:

  • Principle: Shoes with friction lining are held against inner drum by springs at low speed. At engagement speed, centrifugal force overcomes spring force, shoes press against drum, transmitting torque.

  • Key Parameters: Engagement speed (set by spring force & shoe mass), number of shoes, coefficient of friction, dimensions.

Design Theories for Friction Surfaces

Assumptions: Uniform pressure $p$ or uniform wear ($$\displaystyle p \cdot r = \text{constant} $$).

[!TIP] Uniform Wear theory gives higher torque capacity for same $$\displaystyle p_{max} $$ and is more conservative for $F$ calculation.

Brakes

• Internal Expanding Brake (Shoe Brake):

  • Construction: Two shoes with friction lining, pivoted on anchor pins. Cam or hydraulic cylinder pushes shoes outward against rotating drum.

  • Self-Energizing Effect: Direction of rotation such that friction force on leading shoe has a component in the direction of shoe rotation, augmenting the applied force. Trailing shoe has opposite effect.

  • Self-Locking Condition: If $$\displaystyle \tan \theta > \mu $$ (where $\theta$ is angle of wrap on leading shoe), brake becomes self-locking (cannot be released by force alone). Avoid in service brakes.

• Band & Block Brake:

  • Euler's Equation (Tension Ratio): $$\displaystyle \frac{T_1}{T_2} = e^{\mu \theta} $$ (for band). For $n$ blocks: $$\displaystyle \frac{T_1}{T_2} = \left( \frac{1 - \mu \tan \phi}{1 + \mu \tan \phi} \right)^n \cdot e^{\mu \theta} $$.

  • Lever Arm: $$\displaystyle F \cdot l = (T_1 - T_2) \cdot r $$ (for band) or consider forces on each block.

• Comparison (Rope, Band, Block):

  • Rope Brake: Simple, low cost, low capacity, rope wear.

  • Band Brake: Simple, compact, self-energizing if leading, but non-uniform pressure.

  • Block Brake: More complex, uniform pressure, higher capacity, can be self-energizing.

VI. BEARINGS

Classification

Sliding Bearings

• Journal Bearing: Supports radial load. Types: Full journal (360° wrap), partial journal (<180°).

• Footstep (Thrust) Bearing: Supports axial load. Counterboring (recess at center) increases load capacity by creating hydrodynamic pressure wedge.

• Bearing Materials (Babbitt, Bronze, White Metal): Must have conformability (embed dirt), embeddability (hold dirt), fatigue strength, corrosion resistance, thermal conductivity.

• Bearing Characteristic Number (PV): Product of bearing pressure (p) and surface velocity (v). $$\displaystyle PV = \frac{W}{A} \cdot v $$.

  • Significance: Governs temperature rise and wear rate. Each material has a limiting PV value for safe operation.

  • Design: Ensure calculated $p$ and $PV$ are below material limits.

• Power Lost in Friction (Footstep Bearing): $$\displaystyle P_f = \mu W v $$ (Watts), where $\mu$ = friction coefficient.

Rolling Contact Bearings

• Dynamic Load Capacity (C): Rated load a bearing can carry for 1 million revolutions with 90% reliability ($$\displaystyle L_{10} $$ life). From manufacturer's catalogue.

• Bearing Life ($$\displaystyle L_{10} $$): Life in millions of revolutions for which 90% of bearings survive.

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

Where:

*   $P$ = **Equivalent Dynamic Load** (radial + axial effects).

*   $p$ = 3 for **ball bearings**, 10/3 for **roller bearings**.

• Equivalent Dynamic Load (P): For combined radial ($$\displaystyle F_r $$) and axial ($$\displaystyle F_a $$) loads:

$$ P = X F_r + Y F_a $$

$X, Y$ are factors from bearing tables depending on $$\displaystyle F_a/F_r $$ and bearing type.

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

• Reliability Adjustment: For reliability $R \neq 90\%$, use reliability factor $$\displaystyle a_1 $$: $$\displaystyle C_{req} = a_1 \cdot C_{90\%} $$.

  [!TIP] Life is very sensitive to load: Doubling load reduces life to ~1/8 for ball bearings ($$\displaystyle 2^3 $$).

VII. DESIGN PARAMETERS & MISCELLANEOUS CONCEPTS

Polar Modulus ($$\displaystyle Z_p $$)

• Definition: $$\displaystyle Z_p = \frac{J}{R} $$, where $J$ = polar moment of inertia, $R$ = outer radius.

• For Solid Circular Shaft: $$\displaystyle Z_p = \frac{\pi d^3}{16} $$.

• Significance: Used in torsional design. Maximum shear stress $$\displaystyle \tau_{max} = \frac{T}{Z_p} $$.

  \boxed{Z_p = \frac{J}{R}}

Bearing Life ($$\displaystyle L_{10} $$)

• Definition: Life in millions of revolutions at which 90% of a bearing population will still be operating.

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

• Factors Affecting Life:

  1. Load (P): Most critical (inverse cube relationship).

  2. Speed (n): Affects life in hours.

  3. Lubrication: Film thickness, contamination.

  4. Mounting & Alignment: Misalignment causes edge loading.

  5. Material & Manufacturing Quality.

Internal Expanding Brakes

• Construction: Two shoes pivoted on anchor pins inside a rotating drum. Cam or cylinder pushes shoes outward.

• Working: When applied, leading shoe gets self-energizing effect (friction force aids application), trailing shoe opposes.

• Self-Energizing: $$\displaystyle \tan \theta > \mu $$ (where $\theta$ = angle of wrap on leading shoe). Increases braking torque without extra force.

• Self-Locking: $$\displaystyle \tan \theta > \mu $$ for both shoes? Actually, self-locking occurs when the friction force alone can hold the brake without applied force. Condition: $$\displaystyle \tan \theta > \mu $$ for the leading shoe in the direction of rotation.

Dimensionless Numbers in Bearing Design

• Sommerfeld Number (S): Key parameter for hydrodynamic lubrication in journal bearings.

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

Where $\mu$ = viscosity, $n$ = rpm, $p$ = pressure, $r$ = radius, $c$ = clearance.

*   **Significance:** Governs **minimum film thickness** and **coefficient of friction**. Used to design for full film lubrication.

• Reynolds Number (Re): $$\displaystyle \frac{\rho v c}{\mu} $$. Governs flow regime (laminar/turbulent) in lubricant film. Important for high-speed bearings.

General Design Considerations

• Interchangeable Manufacturing: Parts made to tolerances so any one fits any mating assembly.

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

  • Bilateral Tolerance: Tolerance zone equally on both sides (e.g., $+0.01/-0.01$).

• Hardness vs. Toughness:

  • Hardness: Resistance to surface indentation/abrasion (e.g., bearing surfaces, gear teeth).

  • Toughness: Ability to absorb energy before fracture (impact resistance). Critical for shafts, gears under shock.

• Stress Concentration Factors: Use standard tables (e.g., Peterson's) for $$\displaystyle K_t $$ based on geometry (fillet radius, hole diameter, keyway). Apply notch sensitivity $q$ for fatigue.