Machine Design
Comprehensive guide to the principles and practices of designing mechanical components and systems for optimal performance, reliability, and safety.
Design Fundamentals
Machine design involves the application of engineering principles to create mechanical systems that meet specific performance requirements while ensuring safety, reliability, and cost-effectiveness.
Design Process
- Problem identification and specification
- Conceptual design and synthesis
- Analysis and optimization
- Detailed design and documentation
- Prototyping and testing
- Manufacturing and assembly
Design Criteria
- Strength and stiffness requirements
- Fatigue and wear resistance
- Reliability and safety factors
- Manufacturing considerations
- Cost and economic factors
- Environmental and sustainability aspects
Safety Factors
SF = S_ultimate/S_working
Where:
SF = safety factor
S_ultimate = ultimate strength of material
S_working = working stress in application
Typical safety factors: Static loading (2-4), Dynamic loading (4-8), Uncertain conditions (8-12)
Shaft Design and Analysis
Shafts are rotating machine elements used to transmit power and rotational motion. Proper shaft design ensures adequate strength, stiffness, and critical speed characteristics.
Combined Loading Analysis
d³ = (16/πτ_allow) × √[(K_f M)² + (K_fs T)²]
Where:
d = shaft diameter (m)
τ_allow = allowable shear stress (Pa)
K_f = fatigue stress concentration factor for bending
K_fs = fatigue stress concentration factor for torsion
M = bending moment (N·m)
T = torque (N·m)
Critical Speed
ω_c = √(g/δ_st)
Where:
ω_c = critical angular velocity (rad/s)
g = gravitational acceleration (9.81 m/s²)
δ_st = static deflection at the center of mass (m)
Deflection Limits
General machinery: δ/L ≤ 1/1000
Precision machinery: δ/L ≤ 1/2000
High-speed machinery: δ/L ≤ 1/5000
Keyway Stress
τ = 2T/(d²l)
Where:
τ = shear stress in key (Pa)
T = transmitted torque (N·m)
d = shaft diameter (m)
l = key length (m)
Gear Design and Analysis
Gears are used to transmit motion and power between rotating shafts with specific speed ratios. Proper gear design ensures adequate strength, durability, and smooth operation.
Fundamental Relationships
Gear Ratio: i = N₂/N₁ = D₂/D₁ = ω₁/ω₂
Module: m = D/N = p/π
Circular Pitch: p = πm
Where:
N = number of teeth
D = pitch diameter (m)
ω = angular velocity (rad/s)
m = module (m)
p = circular pitch (m)
Lewis Bending Equation
σ_b = W_t/(b × m × Y)
Where:
σ_b = bending stress at tooth root (Pa)
W_t = transmitted load (N)
b = face width (m)
m = module (m)
Y = Lewis form factor
Contact Stress (Hertz)
σ_c = √[W_t × E_eff/(b × R_eff)]
Where:
σ_c = contact stress (Pa)
E_eff = effective elastic modulus
R_eff = effective radius of curvature
Dynamic Factor
K_v = (A + √V)/(A)
Where:
K_v = dynamic factor
A = quality factor (50-200)
V = pitch line velocity (m/s)
Gear Ratios and Efficiency
| Gear Type | Typical Ratio | Efficiency (%) |
|---|---|---|
| Spur gears | 1:1 to 10:1 | 98-99 |
| Helical gears | 1:1 to 10:1 | 96-98 |
| Bevel gears | 1:1 to 6:1 | 95-97 |
| Worm gears | 5:1 to 100:1 | 50-90 |
Bearing Selection and Design
Bearings support rotating shafts and reduce friction between moving parts. Proper bearing selection ensures adequate load capacity, life, and reliability.
Basic Rating Life
L₁₀ = (C/P)ᵏ × 10⁶/(60n)
Where:
L₁₀ = basic rating life (hours)
C = dynamic load capacity (N)
P = equivalent dynamic bearing load (N)
k = 3 for ball bearings, 10/3 for roller bearings
n = rotational speed (rpm)
Equivalent Dynamic Load
P = X × F_r + Y × F_a
Where:
P = equivalent dynamic load (N)
F_r = radial load (N)
F_a = axial load (N)
X = radial load factor
Y = axial load factor
Modified Rating Life
L_na = a₁ × a₂ × a₃ × L₁₀
Where:
a₁ = reliability factor
a₂ = material factor
a₃ = operating condition factor
Bearing Types and Applications
| Bearing Type | Load Capacity | Applications |
|---|---|---|
| Deep groove ball | Moderate radial, light axial | General purpose, high speed |
| Angular contact ball | Combined radial and axial | Spindles, pumps |
| Cylindrical roller | High radial, no axial | Heavy radial loads |
| Tapered roller | High radial and axial | Automotive, heavy machinery |
| Thrust ball | Pure axial loads | Vertical shafts |
Fasteners and Joints
Fasteners and joints connect machine components together. Proper design ensures adequate strength, preload, and fatigue resistance under various loading conditions.
Bolt Preload and Tension
F_i = 0.75 × F_proof = 0.75 × S_p × A_t
Where:
F_i = initial preload (N)
F_proof = proof load (N)
S_p = proof strength (Pa)
A_t = tensile stress area (m²)
Joint Stiffness
k_b = A_t × E_b / l_e (bolt stiffness)
k_m = A_m × E_m / l_m (member stiffness)
Where:
k = stiffness (N/m)
A = effective area (m²)
E = elastic modulus (Pa)
l = effective length (m)
Load Distribution
C = k_b/(k_b + k_m)
F_b = F_i + C × P (bolt load)
F_m = F_i - (1-C) × P (member load)
Where:
C = joint constant
P = external applied load (N)
Fatigue Analysis
σ_a = C × P/(2 × A_t)
σ_m = (F_i + C × P)/A_t
Where:
σ_a = alternating stress (Pa)
σ_m = mean stress (Pa)
Thread Specifications
| Metric Size | Pitch (mm) | Stress Area (mm²) | Proof Load (kN) |
|---|---|---|---|
| M6 | 1.0 | 20.1 | 14.1 |
| M8 | 1.25 | 36.6 | 25.6 |
| M10 | 1.5 | 58.0 | 40.6 |
| M12 | 1.75 | 84.3 | 59.0 |
Springs and Elastic Elements
Springs store and release mechanical energy, providing force, motion control, and vibration isolation. Proper spring design ensures adequate force, deflection, and fatigue life.
Helical Compression Springs
k = Gd⁴/(8D³n)
τ = K_s × 8FD/(πd³)
Where:
k = spring rate (N/m)
G = shear modulus (Pa)
d = wire diameter (m)
D = mean coil diameter (m)
n = number of active coils
K_s = shear stress correction factor
F = applied force (N)
τ = shear stress (Pa)
Spring Index and Correction Factor
C = D/d (spring index)
K_s = (2C + 1)/(2C)
Typical spring index: C = 4 to 12
Preferred range: C = 6 to 9
Buckling Analysis
L_cr/D = 2.6√(E/G)
For L/D > L_cr/D, buckling may occur
Where:
L_cr = critical free length
L = actual free length
E = Young's modulus
Fatigue Life
τ_a = K_f × 8F_a D/(πd³)
τ_m = K_f × 8F_m D/(πd³)
Where:
τ_a = alternating shear stress
τ_m = mean shear stress
F_a = alternating force
F_m = mean force
K_f = fatigue stress concentration factor
Spring Materials
| Material | G (GPa) | S_ut (MPa) | Applications |
|---|---|---|---|
| Music wire | 81.7 | 2000-3000 | High-stress applications |
| Oil-tempered wire | 79.3 | 1400-2000 | General purpose |
| Stainless steel 302 | 69.0 | 1600-2000 | Corrosive environments |
| Phosphor bronze | 41.4 | 900-1200 | Electrical contacts |