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Cheatsheet: Design of Machine Elements

Table of Contents
1. Design Philosophy and Criteria
2. Fatigue Design
3. Shaft Design
4. Keys and Couplings
5. Welded Joints
6. Threaded Fasteners
7. Springs
8. Bearings
9. Brakes and Clutches
10. Gears
11. Belt Drives
12. Flywheels
13. IC Engine Components
View more

1. Design Philosophy and Criteria

1.1 Factor of Safety (FOS)

Definition	Formula
Factor of Safety (Tensile)	FOS = S_ut / σ or S_yt / σ
Factor of Safety (Shear)	FOS = S_sy / τ or S_us / τ

S_ut = Ultimate tensile strength, S_yt = Yield tensile strength
S_sy = Yield shear strength, S_us = Ultimate shear strength
S_sy ≈ 0.577 S_yt (Distortion Energy Theory), S_sy ≈ 0.5 S_yt (Maximum Shear Stress Theory)

1.2 Theories of Failure

Theory	Failure Criterion
Maximum Principal Stress (Rankine)	σ₁ ≤ S_yt/FOS (Brittle materials)
Maximum Shear Stress (Tresca)	(σ₁ - σ₃)/2 ≤ S_yt/(2×FOS)
Distortion Energy (von Mises)	σ_e = √[σ₁² + σ₂² + σ₃² - σ₁σ₂ - σ₂σ₃ - σ₃σ₁] ≤ S_yt/FOS
Maximum Strain (St. Venant)	ε₁ ≤ S_yt/(E×FOS)

For 2D stress: σ_e = √[σ_x² + σ_y² - σ_xσ_y + 3τ_xy²]
Ductile materials: von Mises or Tresca; Brittle materials: Maximum Principal Stress

1.3 Stress Concentration Factor

Parameter	Definition
K_t	Theoretical stress concentration factor = σ_max / σ_nom
K_f	Fatigue stress concentration factor = 1 + q(K_t - 1)
q	Notch sensitivity (0 ≤ q ≤ 1); q = 0 (no sensitivity), q = 1 (full sensitivity)

2. Fatigue Design

2.1 Endurance Limit

Material	Endurance Limit (S_e)
Steel	S_e ≈ 0.5 S_ut (S_ut ≤ 1400 MPa)
Iron	S_e ≈ 0.4 S_ut
Aluminum	No true endurance limit; use finite life approach

2.2 Modified Endurance Limit

Se = Ka × Kb × Kc × Kd × Ke × Kf × Se'

Factor	Description
K_a	Surface finish factor = a S_ut^b; Ground: a=1.58, b=-0.085; Machined: a=4.51, b=-0.265; Hot rolled: a=57.7, b=-0.718
K_b	Size factor = (d/7.62)^-0.107 for 2.79 ≤ d ≤ 51 mm (bending/torsion); K_b = 1 for axial loading
K_c	Load factor: 1 (bending), 0.85 (axial), 0.59 (torsion)
K_d	Kd = 1 for T ≤ 450°C (for steels); for T > 450°C use temperature reduction charts.
K_e	Reliability factor: 50%=1.0, 90%=0.897, 95%=0.868, 99%=0.814, 99.9%=0.753
K_f	Miscellaneous effects factor

2.3 Fluctuating Stress Analysis

Parameter	Formula
Mean Stress	σ_m = (σ_max + σ_min) / 2
Alternating Stress	σ_a = (σ_max - σ_min) / 2
Stress Ratio	R = σ_min / σ_max
Amplitude Ratio	A = σ_a / σ_m

2.4 Fatigue Failure Criteria

Criterion	Equation
Soderberg Line	σ_a/S_e + σ_m/S_yt = 1/FOS
Goodman Line	σ_a/S_e + σ_m/S_ut = 1/FOS
Gerber Parabola	σ_a/S_e + (σ_m/S_ut)² = 1/FOS

Goodman: Most commonly used, conservative
Soderberg: Most conservative
Gerber: Fits experimental data better but less conservative

3. Shaft Design

3.1 Basic Shaft Design Equations

Loading	Design Equation
Torsion Only	τ = 16T/(πd³); d = ∛[16T/(πτ)]
Bending Only	σ = 32M/(πd³); d = ∛[32M/(πσ)]
Combined Bending & Torsion	Te = √(M² + T²) d = ∛[(16Te)/(πτ)]

3.2 ASME Code for Shaft Design

Theory	Equivalent Stress
Maximum Shear Stress	d = ∛{(16/π)[(K_bM)² + (K_tT)²]^0.5/S_sy}
Distortion Energy	d = ∛{(16/π)[(K_bM)² + 0.75(K_tT)²]^0.5/S_y}

K_b = combined shock and fatigue factor (bending); K_t = combined shock and fatigue factor (torsion)
Gradual load: K_b = 1.5, K_t = 1.0; Shock load: K_b = 2.0, K_t = 1.5

3.3 Critical Speed of Shaft

Type	Formula
Single Load	N_c = (60/2π)√(g/δ) rpm; δ = static deflection
Rayleigh-Ritz Method	N_c = (60/2π)√[g Σ(W_iδ_i)/Σ(W_iδ_i²)]
Dunkerley's Method	1/N_c² = 1/N₁² + 1/N₂² + ... + 1/N_n²

Operating speed should be 20-30% away from critical speed

4. Keys and Couplings

4.1 Keys

Type	Design Equations
Sunk Key (Shear)	τ = 4T/(d×w×l); l = 4T/(d×w×τ)
Sunk Key (Crushing)	σ_c = 8T/(d×t×l); l = 8T/(d×t×σ_c)

w = width of key, t = thickness of key (in shaft), l = length of key, d = shaft diameter
Standard proportions: w = d/4, t = 2w/3 for square keys; w = d/4, t = w/2 for flat keys

4.2 Splines

Parameter	Description
Torque Capacity	T = (π/8) × n × p_c × h × l × D_m

n = number of splines, p_c = crushing stress, h = depth of spline, l = length, D_m = mean diameter

4.3 Rigid Couplings

Type	Key Design Feature
Sleeve/Muff Coupling	Outer diameter D = 2d + 13 mm; Length L = 3.5d
Clamp/Compression Coupling	Uses friction to transmit torque; T = μ × W × d/2
Flange Coupling	Bolt circle diameter: D_b = 3d; Number of bolts: n ≥ 3

4.4 Flexible Couplings

Bushed pin coupling: Accommodates misalignment; pins are rubber bushed
Oldham coupling: Accommodates parallel misalignment up to 5 mm
Universal coupling: Accommodates angular misalignment; ω₂/ω₁ = cos α/(1 - sin²α sin²θ)

5. Welded Joints

5.1 Throat Thickness

Parameter	Value
Throat thickness (t)	t = s/√2 = 0.707s (for 45° fillet weld)
Throat area	A = t × l (l = length of weld)

5.2 Weld Joint Strength

Loading	Design Equation
Parallel Fillet (Shear)	τ = P/(√2 × s × l); s = P/(√2 × l × τ)
Transverse Fillet	σ = P/(√2 × s × l)
Torsion (Circular)	τ = 16T/(π × √2 × s × d²)

s = weld size, l = weld length, P = load, T = torque
Allowable stress: τ_weld = 0.707 × τ_parent

5.3 Eccentric Welded Joints

Component	Formula
Primary Shear	τ' = P/A
Secondary Shear	τ'' = (P × e × r)/(I_G)
Resultant Shear	τ = √[(τ')² + (τ'')² + 2τ'τ''cos θ]

I_G = polar moment of inertia of weld group about centroid

6. Threaded Fasteners

6.1 Thread Terminology

Parameter	Definition
Pitch (p)	Axial distance between adjacent threads; p = 1/n (n = threads per inch)
Lead (L)	Axial advance per revolution; L = n_s × p (n_s = number of starts)
Core/Root Diameter (d_c)	d_c = d - 1.2268p (ISO Metric)
Pitch Diameter (d_p)	d_p = d - 0.6495p (ISO Metric)

6.2 Power Screw Mechanics

Parameter	Formula
Torque to Raise Load	T = (W × d_m/2) × [(L + πμd_m)/(πd_m - μL)]
Torque to Lower Load	T = (W × d_m/2) × [(πμd_m - L)/(πd_m + μL)]
Efficiency	η = tan λ / tan(λ + φ); λ = lead angle, φ = friction angle
Self-Locking Condition	λ < φ

d_m = mean diameter = (d + d_c)/2, tan λ = L/(πd_m), tan φ = μ

6.3 Bolt Loading

Loading Type	Stress Equation
Pure Tension	σ_t = 4F/(πd_c²)
Initial Tightening	F_i = σ_p × π d_c²/4 (σ_p = proof stress)
External Load (Static)	σ_t = [F_i + F × K_b/(K_b + K_m)] / A_c

K_b = bolt stiffness = A_bE_b/L_b; K_m = member stiffness = A_mE_m/L_m
For steel bolts: proof stress σ_p ≈ 0.85 S_yt

6.4 Bolted Joint Under Eccentric Load

Component	Formula
Direct Shear per Bolt	F' = P/n
Torsional Shear per Bolt	F'' = (P × e × r_i)/(Σr_i²)
Resultant Force	F_R = √[(F')² + (F'')² + 2F'F''cos θ]

7. Springs

7.1 Helical Compression Spring

Parameter	Formula
Shear Stress (Direct)	τ = 8WD/(πd³) × K_w
Wahl Factor	K_w = (4C-1)/(4C-4) + 0.615/C
Spring Index	C = D/d (6 ≤ C ≤ 12)
Deflection	δ = 8WD³n/(Gd⁴)
Stiffness	k = W/δ = Gd⁴/(8D³n)
Solid Length	L_s = (n_t + 1)d or n_td
Free Length	L_f = L_s + δ_max + 0.15 δ_max

W = axial load, D = mean coil diameter, d = wire diameter, n = active coils, G = shear modulus
n_t = total coils = n + 2 (plain ends), n + 1 (squared ends)

7.2 Helical Tension Spring

Parameter	Value
Shear Stress	τ = 8WD/(πd³) × K_w + initial tension stress
Deflection	δ = 8WD³n/(Gd⁴) + hook deflection

7.3 Helical Torsion Spring

Parameter	Formula
Bending Stress	σ_b = 32M/(πd³) × K_i
Stress Factor	K_i = (4C² - C - 1)/(4C(C - 1))
Angular Deflection	θ = 64MDn/(Ed⁴) radians

7.4 Leaf Spring

Type	Design Equation
Semi-Elliptic (Full Width)	σ_b = 6WL/(nbt²); δ = (3WL³)/(8Enbt³)
Semi-Elliptic (Graduated)	σ_b = 12WL/(nbt²); δ = 12WL³/(Enbt³)

n = number of leaves, b = width, t = thickness, L = half span length
Clip stress: τ = W/A_clip

7.5 Spring Materials and Properties

Material	Shear Modulus G (GPa)
Steel	79-84
Stainless Steel	69-79
Phosphor Bronze	41-44

8. Bearings

8.1 Sliding Contact Bearings

8.1.1 Hydrodynamic Lubrication Parameters

Parameter	Formula/Value
Bearing Pressure	p = W/(L × D)
Bearing Characteristic Number	S = (μN/p)(r/c)²
Sommerfeld Number	S = (μN/p)(r/c)² × (L/D)
Coefficient of Friction	f = k(μN/p)(r/c) where k = constant

μ = dynamic viscosity, N = speed (rps), r = shaft radius, c = radial clearance, L = bearing length, D = bearing diameter
Clearance ratio: c/r ≈ 0.001 to 0.002

8.1.2 Heat Generation and Dissipation

Parameter	Formula
Heat Generated	H_g = μ × W × V (V = rubbing velocity)
Heat Dissipated	H_d = C_h × A × (T_b - T_a)

C_h = heat dissipation coefficient (470-1200 W/m²°C), A = projected bearing area
For equilibrium: H_g = H_d

8.2 Rolling Contact Bearings

8.2.1 Bearing Load Ratings

Parameter	Definition
Basic Static Load (C₀)	Load for max contact stress = 4000 MPa (ball), 4200 MPa (roller)
Basic Dynamic Load (C)	Load for L₁₀ life = 1 million revolutions
Equivalent Static Load	P₀ = X₀F_r + Y₀F_a
Equivalent Dynamic Load	P = XF_r + YF_a

F_r = radial load, F_a = axial load; X, Y = radial and axial factors (from bearing catalog)

8.2.2 Bearing Life Calculation

Formula	Description
L₁₀ = (C/P)^k	Life in million revolutions; k = 3 (ball bearings), k = 10/3 (roller bearings)
L_10h = (10⁶/60N) × L₁₀	Life in hours; N = speed in rpm
L₅₀ = 5 × L₁₀	Median life (50% reliability)

8.2.3 Equivalent Load for Combined Loading

Bearing Type	Load Factors
Deep Groove Ball	If F_a/F_r ≤ e: X=1, Y=0; If F_a/F_r > e: X=0.56, Y varies
Angular Contact Ball	X and Y depend on contact angle (15°, 25°, 40°)
Cylindrical Roller	P = F_r (radial load only)
Tapered Roller	X and Y depend on F_a/(VF_r) ratio

9. Brakes and Clutches

9.1 Block Brake

Parameter	Formula
Braking Torque	T = μ × N × r
Normal Force (Short Shoe)	N = F × a/b (lever mechanism)
Heat Generated	Q = μ × N × V × t (V = rubbing velocity)

9.2 Band Brake

Type	Tension Ratio
Simple Band	T₁/T₂ = e^μβ
Differential Band	T = (T₁ - T₂) × r

T₁ = tension on tight side, T₂ = tension on slack side, β = angle of wrap (radians)
Braking torque: T = (T₁ - T₂) × r

9.3 Disc Brake/Clutch

Assumption	Torque Formula
Uniform Pressure	T = (2/3) × μ × W × [(r_o³ - r_i³)/(r_o² - r_i²)]
Uniform Wear	T = (1/2) × μ × W × (r_o + r_i)

r_o = outer radius, r_i = inner radius, W = axial force
For n pairs of friction surfaces: multiply torque by n

9.4 Cone Clutch

Parameter	Formula
Torque (Uniform Pressure)	T = (2/3) × μ × W × [(r_o³ - r_i³)/(r_o² - r_i²)] × (1/sin α)
Torque (Uniform Wear)	T = (1/2) × μ × W × (r_o + r_i) × (1/sin α)

α = semi-cone angle (12.5° to 15° commonly used)

9.5 Centrifugal Clutch

Parameter	Formula
Centrifugal Force	F_c = m × ω² × r
Torque	T = n × μ × F_c × R

n = number of shoes, m = mass of each shoe, R = inside radius of drum

10. Gears

10.1 Gear Terminology

Term	Formula/Definition
Module (m)	m = d/T (d = pitch circle diameter, T = number of teeth)
Circular Pitch (p_c)	p_c = πd/T = πm
Diametral Pitch (P_d)	P_d = T/d = 1/m (in inch units)
Addendum (a)	a = m (for 20° full depth)
Dedendum (d_d)	d_d = 1.25m (for 20° full depth)
Clearance (c)	c = 0.25m
Center Distance	C = (d₁ + d₂)/2 = m(T₁ + T₂)/2
Velocity Ratio	i = N₂/N₁ = T₂/T₁ = d₂/d₁

10.2 Lewis Bending Equation

Parameter	Formula
Beam Strength	F_b = σ_b × b × m × Y
Dynamic Load (Buckingham)	F_d = F_t + F_i = F_t + (21V(bC + F_t))/(21V + √(bC + F_t))

Y = Lewis form factor (depends on tooth number), b = face width, m = module, σ_b = bending stress
F_t = tangential/transmitted load, V = pitch line velocity (m/s), C = deformation factor
For design: F_b ≥ F_d

10.3 Wear Strength (Buckingham)

Parameter	Formula
Wear Load	F_w = d₁ × b × Q × K
Ratio Factor	Q = 2i/(i + 1)
Load Stress Factor	K = (σ_c² s)/(1.4E) × [(1/E₁) + (1/E₂)]

σ_c = surface endurance limit, φ = pressure angle (20° standard)
For steel-steel: K ≈ 0.16 N/mm² (approximate value)

10.4 Helical Gear

Parameter	Formula
Normal Module	m_n = m_t cos ψ
Axial Pitch	p_a = πm/tan ψ
Virtual Teeth Number	T_v = T/cos³ψ
Tangential Force	F_t = F × cos φ_n × cos ψ
Axial Force	F_a = F_t tan ψ
Radial Force	F_r = F_t tan φ_n / cos ψ

ψ = helix angle (15° to 30°), φ_n = normal pressure angle

10.5 Bevel Gear

Parameter	Formula
Cone Distance	A₀ = d/(2 sin δ)
Pitch Angle	tan δ₁ = T₁/T₂ = sin γ/(i + cos γ)
Virtual Teeth	T_v = T/cos δ
Lewis Equation	F_b = σ_b × b × m × Y × (A₀ - b/2)/A₀

δ = pitch angle, γ = shaft angle (90° for right angle bevel gears)

10.6 Worm Gear

Parameter	Formula
Velocity Ratio	i = T_wheel/T_worm
Lead	L = πm × T_worm
Lead Angle	tan λ = L/(πd_worm)
Efficiency	η = tan λ / tan(λ + φ)
Self-Locking	λ < φ (or tanλ ≤ μ)

Center distance: C = (d_worm + d_wheel)/2
Tangential force on worm = Axial force on wheel

11. Belt Drives

11.1 Flat Belt

Parameter	Formula
Velocity Ratio	i = N₂/N₁ = d₁/d₂ (no slip)
Length (Open Belt)	L = π(d₁ + d₂)/2 + 2C + (d₂ - d₁)²/(4C)
Length (Cross Belt)	L = π(d₁ + d₂)/2 + 2C + (d₁ + d₂)²/(4C)
Angle of Contact (Open)	θ = 180° - 2α; sin α = (d₂ - d₁)/(2C)
Angle of Contact (Cross)	θ = 180° + 2α; sin α = (d₁ + d₂)/(2C)

11.2 Tension Relationship

Condition	Formula
Without Centrifugal Tension	T₁/T₂ = e^μθ
With Centrifugal Tension	(T₁ - T_c)/(T₂ - T_c) = e^μθ
Centrifugal Tension	T_c = m × V² (m = mass per unit length)
Power Transmitted	P = (T₁ - T₂) × V

T₁ = tension on tight side, T₂ = tension on slack side, μ = coefficient of friction, θ = angle of wrap (radians)
Maximum power occurs when T₁ = σ_max × A and T_c = T₁/3

11.3 V-Belt

Parameter	Formula
Tension Ratio	T₁/T₂ = e^{μθ cosec β}
Power per Belt	P = (T₁ - T₂) × V × K_a

β = groove angle (40° for V-belts), K_a = arc of contact factor
Number of belts required: n = Total Power / (Power per belt)

11.4 Belt Material Properties

Material	Allowable Stress (MPa)
Leather	2.5 - 3.5
Fabric	2.0 - 2.5
Rubber	2.0 - 3.0

12. Flywheels

12.1 Energy Storage

Parameter	Formula
Kinetic Energy	E = (1/2) × I × ω²
Energy Fluctuation	ΔE = (1/2) × I × ω_m² × (2k_s)
Coefficient of Fluctuation	k_s = (ω_max - ω_min)/ω_m = (N_max - N_min)/N_m
Mass Moment of Inertia	I = m × k² (k = radius of gyration)

12.2 Rim Design

Parameter	Formula
Rim Mass (Approx)	m ≈ ΔE/(k_s × V²)
Hoop Stress	σ_h = ρ × V²
Rim Dimensions	m = π × D × A × ρ (A = cross-sectional area)

ρ = density of flywheel material, V = mean rim velocity
For cast iron: σ_h ≤ 8 MPa; For steel: σ_h ≤ 50 MPa

12.3 Energy Calculation from Turning Moment Diagram

Maximum fluctuation: ΔE = Maximum area above mean line + Maximum area below mean line
Mean torque: T_mean = Total work done per cycle / (2π)
For engines: Use crank-effort diagram to find maximum and minimum energies

13. IC Engine Components

13.1 Piston Design

Parameter	Typical Ratio
Thickness (t)	t = 0.032D + 1.5 mm (D = cylinder bore)
Piston Height (H)	H = 1.0D to 1.5D
Piston Pin Diameter	d_p = 0.3D to 0.4D
Compression Ring Width	h = 0.7√D to D/10

13.2 Connecting Rod

Component	Design Aspect
Length	L = 4 to 5 times crank radius
Small End	Designed for bearing pressure; p = 15-20 MPa
Big End	Bearing pressure: p = 8-15 MPa
Shank	I-section for weight reduction; check for buckling

13.3 Crankshaft Design

Loading	Critical Consideration
Bending	Maximum at crank pin due to connecting rod force
Torsion	Transmitted torque through crankshaft
Combined Stress	Use equivalent bending moment method

Equivalent twisting moment: T_e = √(M² + T²)
Crank pin diameter: d = ∛[(16T_e)/(πτ)]

The document Cheatsheet: Design of Machine Elements is a part of the Mechanical Engineering Course Mechanical Engineering SSC JE (Technical).

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FAQs on Cheatsheet: Design of Machine Elements

1. What is the importance of design philosophy in machine elements?

Ans. Design philosophy in machine elements is crucial as it establishes the fundamental principles and criteria guiding the creation and evaluation of mechanical components. It ensures that designs are functional, safe, efficient, and meet specific requirements while considering factors such as material selection, manufacturing processes, and intended usage.

2. How is fatigue design significant in engineering applications?

Ans. Fatigue design is significant as it addresses the failure of materials due to cyclic loading over time. Engineers must consider the material's endurance limit and design components to withstand repeated stress without failure, ensuring reliability and safety in applications such as automotive, aerospace, and structural engineering.

3. What are the key considerations in shaft design?

Ans. Key considerations in shaft design include determining the shaft's diameter, length, and material, as well as analysing the torque and bending moments it will experience. Engineers must also account for factors like critical speed, alignment, and the type of loads (static or dynamic) to ensure optimal performance and durability.

4. Why are keys and couplings essential in machine design?

Ans. Keys and couplings are essential as they facilitate the transfer of torque and rotational motion between components, such as shafts and gears. They prevent slippage and maintain the alignment of rotating parts, thereby enhancing the efficiency and reliability of mechanical systems.

5. What role do bearings play in machinery?

Ans. Bearings play a vital role in machinery by reducing friction between moving parts, supporting loads, and enabling smooth rotational or linear motion. They are crucial for enhancing the lifespan of components, improving efficiency, and ensuring stability in various applications, from automotive to industrial machinery.

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