Machine Component and Design (ME-602) - Unit 4 Short Notes

1. Fatigue and Failure Analysis

Stress Concentration

• Definition: Localized increase in stress due to geometric discontinuities (holes, notches, fillets, sudden cross-section changes).

• Stress Concentration Factor (Kt):

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

Where $$\displaystyle \sigma_{max} $$ = max local stress, $$\displaystyle \sigma_{nom} $$ = nominal stress from simplified cross-section.

• Notch Sensitivity Factor (q):

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

$$\displaystyle K_f $$ = fatigue stress concentration factor; $q$ depends on material and notch radius.

• Dynamic Loading Consideration: Use $$\displaystyle K_f $$ (not $$\displaystyle K_t $$) in fatigue analysis. $$\displaystyle K_f $$ reduces as material ductility increases.

• Reduction Methods:

  • Fillets (large radius)

  • Gradual transitions (tapered steps)

  • Relief notches/counterbores

  • Surface treatments (shot peening, carburizing)

  • Avoid sharp corners in high-stress regions.

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Exam Alert: $$\displaystyle K_t $$ is theoretical (elastic), $$\displaystyle K_f $$ is actual (includes material notch sensitivity). For ductile materials, $q \approx 0$, so $$\displaystyle K_f \approx 1 $$.

S-N Curve and Endurance Limit

• S-N Curve: Plot of stress amplitude ($$\displaystyle S_a $$) vs. number of cycles to failure ($N$). Shows endurance limit $$\displaystyle \sigma_e $$ for ferrous metals (asymptote).

• Endurance Limit ($$\displaystyle \sigma_e $$): Maximum completely reversed stress for infinite life ($$\displaystyle >10^6 $$ cycles).

  For steels: $$\displaystyle \sigma_e \approx 0.5 \sigma_u $$ (for rotating beam test).

• Modifying Factors (for design $$\displaystyle \sigma_e' $$):

$$\sigma_e' = \sigma_e \cdot K_a \cdot K_b \cdot K_c \cdot K_d$$

• $$\displaystyle K_a $$: Size factor (larger diameter → lower $$\displaystyle \sigma_e $$)

• $$\displaystyle K_b $$: Surface finish factor (machined < ground)

• $$\displaystyle K_c $$: Reliability factor (99% reliability < 50%)

• $$\displaystyle K_d $$: Temperature factor (reduced at high T)

• Fatigue Life Determination: From S-N curve, for given stress amplitude $$\displaystyle S_a $$, find $N$.

[!TIP]

Common Pitfall: Non-ferrous metals (Al, Cu) have no true endurance limit; use S-N curve directly.

Fatigue Failure Criteria

• Goodman Diagram (Complete & Modified):

  • Complete Goodman:

$$\frac{\sigma_a}{\sigma_e} + \frac{\sigma_m}{\sigma_u} = \frac{1}{n}$$

$$\displaystyle \sigma_a $$ = alternating stress, $$\displaystyle \sigma_m $$ = mean stress, $n$ = safety factor.

• Modified Goodman (more accurate for ductile materials):

$$\frac{\sigma_a}{\sigma_e} + \frac{\sigma_m}{\sigma_y} = \frac{1}{n}$$

$$\displaystyle \sigma_y $$ = yield strength.

• Soderberg Line (most conservative):

$$\frac{\sigma_a}{\sigma_e} + \frac{\sigma_m}{\sigma_y} = \frac{1}{n}$$

Uses $$\displaystyle \sigma_e $$ (not $$\displaystyle \sigma_u $$) for mean stress term.

• Design for Infinite Life: Ensure combined stress point lies below chosen failure line with factor of safety.

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Exam Pattern: Soderberg is safest; Goodman is most used. Modified Goodman replaces $$\displaystyle \sigma_u $$ with $$\displaystyle \sigma_y $$ for better accuracy with ductile materials.

2. Shaft Design

Stresses in Shafts

• Torsional shear stress: $$\displaystyle \tau = \frac{T \cdot r}{J} = \frac{16T}{\pi d^3} $$ (solid circular)

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

• Axial stress (if applicable): $$\displaystyle \sigma_a = \frac{P}{A} $$

Combined Loading & Design Equations

• Equivalent Bending Moment (for von Mises):

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

• Equivalent Twisting Moment (for Tresca):

$$T_e = \sqrt{M^2 + T^2} \quad \text{(same form, different theory)}$$

• Design by Maximum Shear Stress (Tresca):

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

$$\displaystyle Z_p $$ = polar modulus.

• Design by Distortion Energy (von Mises):

$$\sigma_{eq} = \sqrt{\sigma_b^2 + 3\tau^2} \leq \frac{\sigma_y}{n}$$

Polar Modulus ($$\displaystyle Z_p $$)

• Definition: $$\displaystyle Z_p = \frac{J}{c} $$ (resistance to torsion).

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

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

Hollow Shafts

• Advantages: Greater strength/stiffness per unit weight; material used efficiently.

• Design: Use $$\displaystyle J = \frac{\pi}{32}(D^4 - d_i^4) $$ for torsion; $$\displaystyle Z_p $$ as above.

Torsional Rigidity Check

• Angular twist $\phi$ (radians):

$$\phi = \frac{T L}{G J} \leq \phi_{allow}$$

• Solve for $d$ or $D$ if twist limit given.

[!TIP]

Key Insight: For combined bending & torsion, von Mises (distortion energy) is more accurate for ductile materials; Tresca is simpler but conservative.

3. Keys and Couplings

Keys

• Classification:

  • Saddle Key: Key sits in shaft keyway, hub rests on top. No keyway in hub. For light loads.

  • Sunk Key: Key fits into keyways in both shaft and hub (parallel, tapered, Woodruff).

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

  • Round Key: Circular cross-section, for low torque.

• Saddle vs. Sunk Key:

• Key Design (sunk key, rectangular):

  • Shear failure (key fails in transverse shear):

$$\tau = \frac{T}{b L d} \leq \frac{\tau_{allow}}{n}$$

• Crushing failure (key crushed between shaft & hub):

$$\sigma_c = \frac{2T}{b L D} \leq \frac{\sigma_{allow}}{n}$$

• $b$ = key width, $L$ = key length, $d$ = shaft diameter, $D$ = hub diameter.

Couplings

• Rigid vs. Flexible:

• Selection Criteria: Torque, speed, misalignment type, damping needed, maintenance, cost.

[!TIP]

Design Tip: For key design, check both shear and crushing. Usually crushing governs for steel keys on steel shafts.

4. Springs

Classification by Shape & Stress

Helical Spring Design

• Parameters:

  • $d$ = wire diameter

  • $D$ = mean coil diameter

  • $n$ = number of active coils

  • $$\displaystyle C = D/d $$ = spring index (typically 4–12)

• Spring Rate (Stiffness):

$$k = \frac{G d^4}{8 D^3 n} \quad \left[\frac{N}{m}\right]$$

• Deflection under load $F$:

$$\delta = \frac{8 F D^3 n}{G d^4} = \frac{F}{k}$$

• Maximum Shear Stress (Wahl correction factor $$\displaystyle K_W $$ for curvature):

$$\tau_{max} = \frac{8 F D}{\pi d^3} \cdot K_W$$

$$K_W = \frac{4C - 1}{4C - 4} + \frac{0.615}{C}$$

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

• Free Length: $$\displaystyle L_0 = L_s + \delta_{max} + \text{initial tension (if any)} $$

• Initial Tension: In close-coiled extension springs, coils are pre-stressed during manufacturing.

Close vs. Open Coiled

Buckling of Compression Springs

• Critical Load (for long slender springs):

$$F_{cr} = \frac{\pi^2 E I}{(K L)^2}$$

$K$ = end condition factor (1 for fixed-fixed, 0.5 for fixed-free), $L$ = free length.

• Design Rule: Keep $$\displaystyle L_0 < 4D $$ to avoid buckling, or use guide rods.

Impact Loading (Drop Weight)

• Energy balance: $$\displaystyle mgh = \frac{1}{2} k \delta^2 $$

  $h$ = drop height, $\delta$ = instantaneous compression.

• Maximum force during impact: $$\displaystyle F_{max} = k \delta $$

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Common Error: Forgetting Wahl factor $$\displaystyle K_W $$ in stress calculation for $$\displaystyle C < 5 $$ leads to underestimation of stress. Always use $$\displaystyle K_W $$.

5. Screws and Threads

Power Screws

• Overhauling: Screw lowers under load without friction (efficiency > 50%). Non-self-locking.

• Self-locking: Screw holds position under load without braking. Condition:

$$\text{Lead angle } \lambda < \text{friction angle } \phi \quad \text{or} \quad \eta < 50\%$$

• Efficiency Derivation (for square thread, raising load):

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

$$\displaystyle \lambda = \tan^{-1}(L / \pi D_m) $$, $$\displaystyle \phi = \tan^{-1}(\mu) $$, $$\displaystyle D_m $$ = mean diameter.

• Square vs. V-threads:

  • Square: Highest efficiency, no radial thrust, used in power transmission (screws, lead screws).

  • V-thread (metric, ACME, buttress): Lower efficiency due to friction, but stronger, easier to manufacture. Used for fastening.

Thread Forms

[!TIP]

Self-locking Check: If $$\displaystyle \eta > 50\% $$, screw is overhauling (not self-locking). Most machine screws are self-locking ($\lambda$ small).

6. Clutches and Brakes

Clutches

• Friction Materials: High $\mu$, wear-resistant, good heat dissipation, low cost. Examples: asbestos-ceramic, organic (resin-based), metallic (copper-steel).

• Design Theories:

  • Uniform Pressure ($$\displaystyle p = \text{constant} $$): Assumes new clutch, pressure uniform.

    Mean radius:

$$R_m = \frac{2}{3} \cdot \frac{r_2^3 - r_1^3}{r_2^2 - r_1^2}$$

• Uniform Wear ($$\displaystyle p \cdot r = \text{constant} $$): Assumes wear uniform, pressure $\propto 1/r$. More realistic for worn clutches.

  Mean radius:

$$R_m = \frac{r_2 - r_1}{\ln(r_2/r_1)}$$

• Torque Capacity:

  • Uniform pressure: $$\displaystyle T = \mu p \pi (r_2^2 - r_1^2) R_m \cdot n $$ ($n$ = number of surfaces)

  • Uniform wear: $$\displaystyle T = \mu p \pi (r_2^2 - r_1^2) R_m \cdot n $$ (same form, different $$\displaystyle R_m $$)

• Axial Force: $$\displaystyle W = p \cdot \pi (r_2^2 - r_1^2) $$ (uniform pressure)

Centrifugal Clutch

• Working: Friction shoes engage drum via centrifugal force as speed increases. Disengages at low speed.

• Design Parameters: Engagement speed ($\omega$), shoe mass ($m$), radius ($r$), friction coefficient ($\mu$), spring force (to disengage).

Brakes

• Block Brake: Simple, but high wear, poor heat dissipation.

• Band Brake: Flexible, self-energizing possible, but adjustment needed.

• Internal Expanding (Drum) Brake: Leading/trailing shoes. Self-energizing when leading shoe friction aids rotation.

• Self-energizing Condition: For leading shoe, $$\displaystyle \tan \phi > \mu $$ (where $\phi$ is angle of force application). Can cause self-locking (brake grabs).

• Braking Torque (general): $$\displaystyle T = \mu W R_e $$

  $$\displaystyle R_e $$ = effective radius (from pressure distribution).

• Band & Block Brake Lever Force:

$$F_2 = F_1 \cdot e^{n \mu \theta} \quad \text{(band)}$$

For block brake with lever: $$\displaystyle T = (F_1 - F_2) R $$ (if self-energizing, $$\displaystyle F_2 $$ may be negative).

[!TIP]

Critical: In leading/trailing shoe brakes, leading shoe is self-energizing; trailing shoe is not. Both used together for balance.

7. Bearings

Sliding Contact Bearings

• Journal Bearing: Supports radial load. Types: full journal (360°), partial journal (<180°), fitted bearing (clearance fit).

• Footstep (Thrust) Bearing:

  • Load Capacity: $$\displaystyle W = p \cdot A $$

    $A$ = projected area (for counterbored shaft: $$\displaystyle A = \frac{\pi}{4}(D^2 - d_i^2) $$)

  • Power Loss (friction):

$$P = \frac{\mu W N r}{1000} \quad [kW]$$

$N$ = rpm, $r$ = mean radius, $\mu$ = friction coefficient.

• Bearing Materials: Babbitt (white metal), bronze, plastics. Properties: conformability, embeddability, fatigue strength, corrosion resistance, thermal conductivity.

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

  • $Z$ = absolute viscosity (Pa·s or N·s/m²)

  • $N$ = speed (rpm)

  • $P$ = bearing pressure (MPa)

  • Significance: Determines lubrication regime:

    • $$\displaystyle Z.N/P < 1000 $$: Boundary lubrication

    • $$\displaystyle 1000 < Z.N/P < 10000 $$: Mixed lubrication

    • $$\displaystyle Z.N/P > 10000 $$: Hydrodynamic lubrication

Rolling Contact Bearings

• Types:

  • Ball bearings: Point contact, lower load capacity, higher speed.

  • Roller bearings: Line contact (cylindrical, tapered, spherical), higher load capacity.

• Advantages over Sliding:

  • Low friction (no sliding)

  • Less maintenance (pre-lubricated)

  • Interchangeable (standardized)

  • Higher speed capability

  • Lower starting torque

• Dynamic Load Capacity (C): Load for which 90% of bearings achieve $$\displaystyle L_{10} $$ life of $$\displaystyle 10^6 $$ revolutions. Manufacturer-rated.

• Life Calculation:

  • Basic rating life (revolutions):

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

$$\displaystyle p = 3 $$ (ball bearings), $$\displaystyle p = 10/3 $$ (roller bearings)

• Adjusted life for reliability $R$:

$$L = a_1 \cdot \left(\frac{C}{P}\right)^p$$

$$\displaystyle a_1 $$ = reliability factor (e.g., 0.21 for 99%, 1.0 for 90%)

• Life in hours:

$$L_h = \frac{10^6}{60 \cdot n} \cdot L$$

• Equivalent Dynamic Load (P) for combined loads:

$$P = X F_r + Y F_a$$

$X, Y$ from bearing tables (depend on $$\displaystyle F_a/F_r $$ ratio).

• Factors Affecting Life: Load, speed, lubrication quality, contamination, misalignment, mounting errors, material.

[!TIP]

Life Calculation: Always use equivalent dynamic load P (not just $$\displaystyle F_r $$). For pure radial load, $$\displaystyle P = F_r $$; for combined, use $X, Y$ factors. $$\displaystyle L_{10} $$ means 90% survive that life.

8. Design Fundamentals (General Principles)

Design Approaches

• Adaptive Design: Modify existing design for new requirements (e.g., change size, material).

• Optimum Design: Mathematical optimization (minimize cost, weight) subject to constraints.

Manufacturing Considerations

• Interchangeable Manufacturing: Parts made to tolerances so any one fits any assembly. Reduces cost, allows mass production.

Tolerances

• Unilateral Tolerance: Tolerance zone on one side of nominal size (e.g., $$\displaystyle 20^{+0.1}_{0} $$ mm).

• Bilateral Tolerance: Tolerance zone on both sides (e.g., $$\displaystyle 20^{\pm0.05} $$ mm).

Material Properties

• Hardness: Resistance to localized plastic deformation (indentation, scratching). Measured by Brinell, Rockwell.

• Toughness: Energy absorbed before fracture (impact resistance). Measured by Izod, Charpy.

Punching/Shearing Force

• Shearing Force:

$$F = \tau \cdot A = \tau \cdot (\text{perimeter} \times \text{thickness})$$

$\tau$ = ultimate shear strength, $A$ = shear area.

9. Special Topics (Short Notes)

Polar Modulus of Shaft

• Definition: $$\displaystyle Z_p = J/c $$, geometric property for torsional strength.

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

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

• Significance: Torsional shear stress $$\displaystyle \tau = \frac{T}{Z_p} $$.

Bearing Life

• $$\displaystyle L_{10} $$ Life: 90% of bearings survive at least $$\displaystyle L_{10} $$ revolutions under given load.

• Reliability Adjustment: $$\displaystyle L = a_1 \cdot (C/P)^p $$, $$\displaystyle a_1 $$ from table (e.g., 0.21 for 99% reliability).

• Life in Hours: $$\displaystyle L_h = \frac{10^6}{60 \cdot n} \cdot L $$

Internal Expanding Brakes

• Construction: Two shoes (leading & trailing) inside rotating drum. Self-adjusting mechanism.

• Working: Hydraulic/pneumatic force pushes shoes outward against drum. Leading shoe self-energizes.

• Self-energizing: Friction force on leading shoe acts to increase normal force, amplifying braking torque.

Dimensionless Numbers in Bearing Design

• Bearing Characteristic Number: $Z.N/P$ – indicates lubrication regime.

• Reynolds Number (for lubrication film): $$\displaystyle Re = \frac{\rho N D}{\mu} $$ – governs hydrodynamic lubrication development.

[!TIP]

Exam Focus: $Z.N/P$ is critical for sliding bearings. $Re$ is more for fluid film analysis. Know both definitions and significance.

Final Note: This compilation strictly follows the approved blueprint and past paper analysis. All formulas are boxed for quick reference. Diagrams are indicated for self-sketching practice (e.g., S-N curve, Goodman diagram, clutch/brake sketches, bearing types).