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Casting, machining, welding, forming processes and production quality fundamentals.
| Casting Type | Process | Advantages | Limitations |
|---|---|---|---|
| Sand Casting | Sand mold, single-use | Low cost, any size/shape, all metals | Rough surface, low accuracy, slow |
| Die Casting | Permanent metal mold (steel) | High production rate, smooth surface, thin walls | High tooling cost, limited to low-melting alloys |
| Investment Casting | Wax pattern, ceramic shell | Excellent surface finish, intricate shapes | Expensive, slow cycle, size limited |
| Shell Molding | Resin-bonded sand shell | Better surface than sand, semi-permanent | Higher cost than sand, size limits |
| Centrifugal Casting | Rotating mold, centrifugal force | Denser metal, cylindrical parts | Limited to symmetric shapes |
| Lost-Foam | EPS foam pattern vaporized by metal | No parting lines, complex shapes | Surface finish depends on foam quality |
| Permanent Mold | Reusable metal mold (gravity/poured) | Better finish, faster than sand | Limited to simple shapes, high startup cost |
| Pattern Type | Description | Use Case |
|---|---|---|
| Single piece | One solid pattern | Simple parts, low production |
| Split pattern | Two halves matching at parting line | Most common, moderate complexity |
| Match plate | Patterns mounted on plate | High production, machine molding |
| Cope & Drag | Separate cope/drag patterns | Large castings, mechanized foundries |
| Loose piece | Removable inserts for undercuts | Complex geometry |
| Skeleton | Ribbed frame (saves material) | Large, simple castings |
Pattern material: Wood (prototypes), Aluminum (production), Steel (high-volume), Plastic (complex). Pattern always larger than final part due to allowances.
| Allowance | Formula / Value | Purpose |
|---|---|---|
| Shrinkage allowance | 1–2% of linear dims (varies by metal) | Compensate for solidification shrinkage |
| Draft allowance | 1°–3° taper | Easy pattern withdrawal from mold |
| Machining allowance | 2–5 mm (depends on machining) | Extra material for finish machining |
| Shake/rapping allowance | ~0.5 mm | Pattern rapping enlarges cavity |
| Distortion allowance | Case-specific | Compensate for warping during cooling |
| Defect | Cause | Remedy |
|---|---|---|
| Porosity / Blow holes | Trapped gases in metal | Proper venting, dry molds, lower pouring temp |
| Shrinkage cavity | Insufficient feeding during solidification | Risers, proper gating, chills |
| Sand inclusions | Sand eroded into cavity | Proper ramming, mold wash, coatings |
| Misrun / Cold shut | Metal solidifies before filling | Increase pouring temp, improve fluidity, redesign gating |
| Hot tears / cracks | Tensile stress during solidification | Reduce shrinkage stress, flexible cores, collapsible molds |
| Wash | Erosion of mold surface by metal | Reinforce mold, reduce velocity, change gating |
| Shift / Mismatch | Cope-drag misalignment | Align pins, proper clamping |
| Fin / Flash | Metal in mold joint gap | Tighter mold joints, better parting surface |
import math
# Solidification time (Chvorinov's Rule)
# t = C × (V/A)²
# V = volume of casting, A = cooling surface area
# C = mold constant (depends on metal, mold material)
# Example: Calculate solidification time for a sphere
d = 0.10 # m (100 mm diameter sphere)
V = math.pi * d**3 / 6 # m³
A = math.pi * d**2 # m² (surface area)
C = 2.0 # min/cm² (typical for steel in sand mold)
# Convert V/A to cm
modulus = (V/A) * 100 # cm
t = C * modulus**2
print(f"V/A ratio = {modulus:.2f} cm")
print(f"Solidification time = {t:.2f} min")
# Riser design: Riser solidification time > Casting time
# (V/A)_riser > (V/A)_casting
# Typically: (V/A)_riser ≥ 1.2 × (V/A)_castingt = C(V/A)². The casting with the largest V/A ratio solidifies last. Risers must have a larger V/A ratio than the casting to ensure the riser feeds molten metal until the casting is fully solidified. Spherical risers are most efficient; cylindrical with hemispherical bottom is practical.| Process | Full Name | Heat Source | Shielding | Application |
|---|---|---|---|---|
| SMAW | Shielded Metal Arc Welding | Electric arc | Flux coating on electrode | General fabrication, construction, field work |
| GMAW/MIG | Gas Metal Arc Welding | Electric arc | External gas (Ar, CO₂) | Sheet metal, automotive, production welding |
| GTAW/TIG | Gas Tungsten Arc Welding | Electric arc (non-consumable W) | External gas (Ar, He) | Precision, thin materials, non-ferrous metals |
| FCAW | Flux-Cored Arc Welding | Electric arc | Flux in tubular wire + gas | Outdoor, heavy fabrication |
| SAW | Submerged Arc Welding | Electric arc | Granular flux blanket | Heavy plate, shipbuilding, pipe |
| PAW | Plasma Arc Welding | Plasma arc | Plasma gas + shielding gas | Precision, deep penetration, aerospace |
| OFW | Oxy-Fuel Welding | Flame (acetylene + O₂) | Flame itself | Repair, thin sheet, brazing |
| LBW | Laser Beam Welding | Focused laser | Shielding gas | Precision, electronics, automotive |
| EBW | Electron Beam Welding | Electron beam | Vacuum chamber | Aerospace, nuclear, reactive metals |
| Resistance | Spot/Seam/Projection | Resistance heating (I²R) | Electrode pressure | Automotive, sheet metal joining |
| Parameter | Details |
|---|---|
| Current type | DCEN, DCEP, or AC (depends on electrode) |
| Electrode designation | E6010, E7018, E308L-16, etc. |
| E = Electrode | |
| 60/70 = Tensile strength (ksi) | |
| 1 = All positions, 2 = Flat/Horiz., 3 = Flat | |
| 0/1 = Cellulosic, 8 = Low hydrogen | |
| -16 = Rutile coating, -15 = Lime | |
| Arc length | ≈ electrode diameter |
| Travel speed | 150–300 mm/min |
| Typical current | 60–250 A (depends on electrode size) |
| Polarity for E7018 | DCEP (electrode positive) for deep penetration |
| Aspect | GMAW (MIG) | GTAW (TIG) |
|---|---|---|
| Electrode | Consumable wire (continuous feed) | Non-consumable tungsten |
| Filler metal | Wire electrode itself | Separate filler rod (added manually) |
| Shielding | Ar, CO₂, or Ar/CO₂ mix | Ar or He (inert only) |
| Skill level | Easier to learn | High skill required |
| Speed | Fast (high deposition rate) | Slow (low deposition) |
| Surface finish | Good | Excellent |
| Thin materials | Can be tricky | Excellent for thin sheets |
| Cost | Moderate equipment | Higher equipment cost |
| Best for | Production, thick plate | Aerospace, precision, non-ferrous |
| Joint Type | Description | Application |
|---|---|---|
| Butt joint | Two plates edge-to-edge | Pipes, plates, pressure vessels |
| Lap joint | Overlapping plates | Sheet metal, spot welding |
| T-joint | Perpendicular plates | Structural, stiffeners |
| Corner joint | Plates at 90° (inside or outside) | Boxes, frames |
| Edge joint | Parallel edges joined | Sheet metal, flanges |
| Defect | Cause | Detection |
|---|---|---|
| Porosity | Gas trapped (moisture, rust) | RT, UT |
| Slag inclusion | Slag not removed between passes | RT, UT |
| Undercut | Excessive current, wrong angle | Visual, MT |
| Lack of fusion | Low heat, dirty surfaces | UT, RT |
| Cracks (hot) | Solidification stresses, S content | RT, MT, UT |
| Cracks (cold) | Hydrogen embrittlement | UT (delayed) |
| Spatter | Wrong parameters, gas flow | Visual |
| Burn-through | Excessive heat, gap too large | Visual |
─── Lathe (Turning) Operations ───
Cutting speed: V = πDN / 1000 [m/min]
D = workpiece diameter (mm)
N = spindle speed (RPM)
Feed rate: f = feed per revolution [mm/rev]
Material Removal Rate: MRR = V × f × d [cc/min]
d = depth of cut [mm]
Machining time: T = L / (f × N) [min]
L = total length of cut (mm)
Operations:
• Facing — Axial feed on flat end
• Turning — Longitudinal feed (parallel to axis)
• Taper — Angle between tool path and axis
• Drilling — Axial hole making (on lathe center)
• Boring — Enlarging existing hole
• Knurling — Creating diamond/cross pattern
• Threading — Cutting screw threads (V/N = pitch)
• Parting — Cutting off finished workpiece─── Milling Operations ───
Cutting speed: V = πDN / 1000 [m/min]
D = cutter diameter (mm)
Table feed: V_f = f_z × Z × N [mm/min]
f_z = feed per tooth [mm/tooth]
Z = number of teeth (flutes)
N = spindle RPM
Material Removal Rate (slab milling):
MRR = w × d × V_f [cc/min]
w = width of cut, d = depth of cut
Types:
• Slab/Plain milling — Horizontal cutter, parallel surface
• Face milling — Axial cutter, perpendicular surface
• End milling — Vertical cutter, slots/profiles
• Side milling — Peripheral cutting, deep slots
• Form milling — Special shaped cutter (gears, etc.)
• Gang milling — Multiple cutters simultaneously
Up milling vs Down milling:
Up: Cutter rotates against feed → smoother
Down: Cutter rotates with feed → better finish, need backlash control| Operation | Tool | Description | Tolerance (IT) |
|---|---|---|---|
| Drilling | Twist drill | Creating holes from solid | IT10–IT12 |
| Reaming | Reamer | Precision sizing + finishing | IT7–IT9 |
| Boring | Boring bar/single-point | Enlarging existing holes | IT7–IT9 |
| Honing | Hone (abrasive stones) | Super-finishing holes | IT5–IT7 |
| Lapping | Lap (abrasive paste) | Ultra-fine surface finish | IT3–IT5 |
| Counterboring | Counterbore | Flat-bottomed enlarged end | — |
| Countersinking | Countersink | Conical enlarged end (screw heads) | — |
| Tapping | Tap | Cutting internal threads | — |
─── Taylor's Tool Life Equation ───
VTⁿ = C
V = Cutting speed (m/min)
T = Tool life (minutes)
n = Tool life exponent (depends on tool material)
C = Constant (speed for 1-min life)
n and C values:
┌──────────────────────┬──────┬────────────┐
│ Tool Material │ n │ C │
├──────────────────────┼──────┼────────────┤
│ HSS (High Speed Steel)│ 0.08│ ~70-100 │
│ Carbide │ 0.20│ ~200-500 │
│ Ceramic │ 0.40│ ~300-800 │
│ CBN (Cubic Boron Nitride)│0.50│ ~500+ │
│ Diamond │ 0.70│ ~1000+ │
└──────────────────────┴──────┴────────────┘
Higher n → speed has LESS effect on tool life
Higher C → tool can run at higher speed
Machinability Rating:
M = V_target / V_standard × 100%
(B1112 steel = 100% baseline)import math
# Lathe turning example
D = 80 # mm (workpiece diameter)
L = 200 # mm (length to turn)
d = 2 # mm (depth of cut)
f = 0.3 # mm/rev (feed)
V = 60 # m/min (cutting speed for steel with carbide)
# Spindle speed
N = 1000 * V / (math.pi * D)
# N = 1000 × 60 / (π × 80) = 238.7 RPM → use 240 RPM
# Machining time
T = L / (f * N)
# T = 200 / (0.3 × 240) = 2.78 min
# MRR
MRR = V * f * d * 1000 / 1e6 # convert mm/min to cc/min
print(f"N = {N:.0f} RPM, Time = {T:.2f} min, MRR = {V*f*d:.1f} cc/min")
# Taylor's equation — find speed for desired tool life
n = 0.2 # carbide
C = 400 # m/min
T_desired = 60 # minutes
V_opt = C / T_desired**(1/n)
# V = 400 / 60^5 = 400 / 77760000 = ... (very sensitive to n!)
# For carbide: V ≈ C/T^5 → tool life VERY sensitive to speedRa ≈ f²/(8r) where f is feed (mm/rev) and r is tool nose radius (mm). To improve surface finish: reduce feed, increase nose radius, use smaller depth of cut, or switch to a finishing tool with a wiper insert.| Process | Operation | Temperature | Application |
|---|---|---|---|
| Rolling | Reduce thickness by compressive forces between rolls | Hot/Cold | Plates, sheets, structural sections |
| Forging | Plastic deformation by hammer/press | Hot/Cold | Cranks, connecting rods, turbine blades |
| Extrusion | Push billet through die to create cross-section | Hot/Cold | Tubes, rods, complex profiles |
| Drawing | Pull wire/rod through die (reduces cross-section) | Cold | Wires, rods, tubes |
| Deep Drawing | Flat blank into cup-shaped part | Cold | Cans, pots, automobile panels |
| Stamping | Sheet metal formed in press with die | Cold | Car body panels, appliances |
| Bending | Plastic deformation to create angle | Cold | Frames, brackets, channels |
| Shearing | Cutting sheet metal between blades | Cold | Blanking, piercing, notching |
| Operation | Description | Key Formula |
|---|---|---|
| Blanking | Cutting out a piece from sheet (punching a blank) | F = πDt × τ_s (τ_s = shear strength) |
| Piercing/Punching | Creating a hole in sheet (slug is waste) | F = πDt × τ_s (same formula) |
| Clearance | Gap between punch and die | C = 0.0032 × t × √τ_s (per side) |
| Drawing ratio | D_blank / d_punch | First draw: ≤ 1.8, Redraw: ≤ 1.4 |
| Bending allowance | Length of neutral axis in bend | BA = α(r + Kt) (K = 0.33–0.5) |
| Springback | Elastic recovery after bending | Δθ = 3σ_yL/(E×t) × θ (approx) |
─── Rolling Formulas ───
Draft: Δh = h₁ − h₂
Reduction: r = (h₁ − h₂) / h₁ × 100%
Contact length: L = √(R × Δh)
R = roll radius, Δh = draft
Rolling force: F = σ_avg × w × L
σ_avg = average flow stress
w = strip width
Power: P = 2π × N × T × η
T = Torque = F × L/2
Friction in rolling:
tan(α) = μ (α = bite angle)
μ ≈ 0.1–0.4 (hot rolling, higher)
μ ≈ 0.05–0.1 (cold rolling, lower)
For rolling to succeed: bite angle ≤ friction angle
Δh_max = μ²R| Property | Hot Working | Cold Working |
|---|---|---|
| Temperature | Above recrystallization temp | Below recrystallization temp |
| Surface finish | Poor (oxidation, scaling) | Excellent (smooth, bright) |
| Dimensional accuracy | Low (thermal contraction) | High (minimal distortion) |
| Grain structure | Recrystallized, isotropic | Elongated, direction-dependent |
| Strength | Lower (due to heat) | Higher (strain hardening) |
| Ductility | High (easy to form) | Reduced (work hardened) |
| Residual stresses | Low | High |
| Tooling cost | Higher (heat-resistant tools) | Lower |
| Energy required | Lower (softer material) | Higher (stronger material) |
| Applications | Large parts, ingots, blooms | Sheet metal, wires, precision parts |
Recrystallization temperature: approximately 0.4 × T_melting (Kelvin). For steel: ~550 °C. For aluminum: ~150 °C. For copper: ~200 °C.
─── Limits, Fits & Tolerances ───
Basic size = Nominal dimension (design size)
Upper deviation = Max limit − Basic size (es or ES)
Lower deviation = Min limit − Basic size (ei or EI)
Tolerance = Upper limit − Lower limit
= |Upper dev − Lower dev|
Fundamental deviation:
H = hole basis (lower deviation = 0)
h = shaft basis (upper deviation = 0)
IT Grade Tolerances (ISO):
IT01, IT0, IT1 ... IT18
IT01 = most precise (tiny tolerance)
IT18 = coarsest (large tolerance)
General: T = i × K
where i (in μm) = 0.45 × D^(1/3) + 0.001 × D
D = geometric mean of range limits (mm)
Types of Fits:
Clearance fit: Hole > Shaft (always gap)
Transition fit: May be clearance or interference
Interference fit: Hole < Shaft (always tight)
Examples:
H7/g6 → Running fit (sliding), clearance
H7/k6 → Transition fit (light press)
H7/p6 → Press fit, interference| Fit | Hole | Shaft | Clearance | Application |
|---|---|---|---|---|
| Loose running | H11 | c11 | Large | Pivot pins, loose linkages |
| Free running | H9 | d9 | Moderate | Sliding fits, guide parts |
| Running | H8 | f7 | Small | Shaft in bearing, gears |
| Push fit | H7 | g6 | Very small | Hand-removable assemblies |
| Transition | H7 | k6 | ± | Couplings, gears on shafts |
| Light press | H7 | n6 | Slight interference | Pulleys, armatures |
| Press fit | H7 | p6 | Interference | Permanent assemblies, gears |
| Heavy press | H7 | s6 | Large interference | Rings, bearings in housing |
| Instrument | Least Count | Range | Best For |
|---|---|---|---|
| Steel rule | 0.5 mm | 0–300 mm | Rough measurement |
| Vernier caliper | 0.02 mm (or 0.001") | 0–150 mm | External/internal/depth measurement |
| Micrometer | 0.01 mm (0.001 mm with vernier) | 0–25 mm (varies) | Precision external measurement |
| Bore gauge | 0.001 mm | 6–150 mm | Internal diameter measurement |
| Dial gauge | 0.01 mm | ±0.5 to ±10 mm | Runout, flatness comparison |
| Height gauge | 0.02 mm | 0–600 mm | Marking, vertical measurement |
| Surface plate | Grade A/B | Various | Reference plane, inspection |
| CMM | 0.001 mm | Varies | 3D coordinate measurement, GD&T |
─── Vernier Caliper Reading ───
Main scale division = 1 mm
Vernier scale divisions = 50
Least count = 1/50 = 0.02 mm
Reading = Main scale reading + (Vernier coinciding line × LC)
Example: MSR = 25 mm, coinciding line = 7
Reading = 25 + 7 × 0.02 = 25.14 mm
─── Micrometer Reading ───
Main scale division = 0.5 mm
Thimble divisions = 50
Least count = 0.5/50 = 0.01 mm
Reading = MSR + (Thimble reading × 0.01)
+ Vernier if available (× 0.001)
Example: MSR = 12.5, TR = 23
Reading = 12.5 + 23 × 0.01 = 12.73 mm| G-Code | Function | Description |
|---|---|---|
| G00 | Rapid positioning | Fastest traverse to position (no cutting) |
| G01 | Linear interpolation | Straight line cutting at feed rate |
| G02 | Circular interpolation (CW) | Clockwise arc cutting |
| G03 | Circular interpolation (CCW) | Counter-clockwise arc cutting |
| G04 | Dwell | Pause for specified time |
| G17/G18/G19 | Plane selection | XY / XZ / YZ plane |
| G20/G21 | Unit selection | Inches / Millimeters |
| G28 | Reference return | Move to machine home position |
| G40/G41/G42 | Cutter comp | Cancel / Left / Right compensation |
| G43 | Tool length comp | Positive tool length offset |
| G54–G59 | Work offset | Select work coordinate system |
| G73/G83 | Peck drilling | Deep hole drilling cycle |
| G81 | Standard drilling | Simple drill cycle |
| G90/G91 | Abs/Incremental | Absolute / Incremental positioning |
| G96 | CSS | Constant surface speed (V = constant) |
| M-Code | Function | Description |
|---|---|---|
| M00 | Program stop | Stops spindle, coolant; operator restart |
| M02 | Program end | End of program (rewind) |
| M03 | Spindle CW | Start spindle clockwise |
| M04 | Spindle CCW | Start spindle counter-clockwise |
| M05 | Spindle stop | Stop spindle rotation |
| M06 | Tool change | Automatic tool changer |
| M07/M08 | Coolant | Mist / Flood coolant ON |
| M09 | Coolant off | Turn off all coolant |
| M30 | Program end | End + reset to start |
| M98 | Subprogram call | Call subprogram (P=number, L=repeats) |
| M99 | Subprogram return | Return to main program |
| M97 | Local sub call | Branch to local subprogram |
# CNC G-Code: Milling a rectangular pocket
# Units: mm, Absolute positioning
G21 G90 # Metric, Absolute
G17 # XY plane selection
G54 # Work offset 1
T01 M06 # Select tool 1 (10mm end mill)
S3000 M03 # Spindle speed 3000 RPM, CW
G43 H01 Z10 # Tool length comp, rapid to Z=10
M08 # Coolant ON
# Rapid to start position
G00 X10 Y10 Z2
# Plunge into material
G01 Z-5 F100 # 5mm depth of cut at 100mm/min
# Pocket profile (4 passes for roughing)
G01 X90 F500 # Bottom edge
G01 Y60 # Right edge
G01 X10 # Top edge
G01 Y10 # Left edge (back to start)
# Finish pass (slower, cleaner)
G01 X90 F200
G01 Y60
G01 X10
G01 Y10
# Retract and end
G00 Z10 # Rapid retract
M05 # Spindle stop
M09 # Coolant off
M30 # Program end| Machine | Axes | Motion | Best For |
|---|---|---|---|
| 3-axis vertical mill | X, Y, Z | Table moves XY, spindle moves Z | Flat parts, pockets, contours |
| 4-axis (indexing) | X, Y, Z + A/B | Additional rotary axis | Sides of parts, engravings |
| 5-axis mill | X, Y, Z + A, B/C | Simultaneous 5-axis motion | Complex surfaces, turbine blades, impellers |
| CNC Lathe (2-axis) | X, Z | Cross-slide (X) + carriage (Z) | Turned parts, shafts, threads |
| CNC Turning Center | X, Z + C (sub-spindle) | Multi-axis turning with live tools | Complete parts in one setup |
| Wire EDM | X, Y, U, V | Wire-guided erosion cutting | Precision dies, complex profiles |
| Process | Abbreviation | Energy Type | Material Removal Mechanism | Best For |
|---|---|---|---|---|
| Electric Discharge Machining | EDM (Wire/Sinker) | Electrical spark | Melting + vaporization by spark erosion | Hard metals, complex cavities, dies |
| Laser Beam Machining | LBM | Coherent light (laser) | Focused laser melts/vaporizes material | Cutting thin sheets, drilling, marking |
| Electron Beam Machining | EBM | High-velocity electrons | Electrons melt/vaporize in vacuum | Micro-holes, precision cuts, reactive metals |
| Water Jet Machining | WJM | High-pressure water (with abrasives) | Mechanical erosion by abrasive particles | Cutting any material (soft/hard), thick plates |
| Wire EDM | WEDM | Electrical spark (wire electrode) | Spark erosion using thin wire | Precision profiles, dies, thick parts |
| Ultrasonic Machining | USM | Ultrasonic vibration + abrasive | Abrasive particles hammering | Brittle materials: glass, ceramics, diamond |
| Chemical Machining | CHM | Chemical etchant | Selective chemical dissolution | Thin metal parts, aerospace skin panels |
| Electrochemical Machining | ECM | Electrolysis (anodic dissolution) | Faraday's law — metal removed as ions | Complex cavities, turbine blades, no tool wear |
| Plasma Arc Machining | PAM | Ionized gas (plasma) | High-temp plasma melts material | Cutting thick metal plates, shipbuilding |
─── EDM Parameters ───
Material removal rate (MRR) in EDM:
MRR = k × I × T_on [mm³/min]
k = material constant
I = discharge current
T_on = pulse on-time (μs)
Surface finish:
Ra ∝ 1/T_on (shorter pulses = better finish)
Overcut (gap):
gap ≈ 0.02–0.1 mm (depends on voltage, dielectric)
Key parameters:
• Voltage: 50–400 V
• Current: 1–500 A
• Pulse on-time: 1–2000 μs
• Pulse off-time: 1–500 μs
• Dielectric fluid: Kerosene, deionized water
• Electrode: Copper, graphite, brass, tungsten
Applications:
• Die-sinking EDM: Complex mold cavities
• Wire EDM: Thin wire (0.05–0.3mm) cuts profiles
• Micro-EDM: Holes < 1mm diameter
Limitations:
• Works only on ELECTRICALLY CONDUCTIVE materials
• Slow MRR compared to conventional machining
• Surface has recast layer (heat-affected zone)─── ECM (Electrochemical Machining) ───
Faraday's Law: M = Z × I × t
M = mass removed (g), Z = electrochemical equivalent
I = current (A), t = time (s)
MRR = (I × A_w) / (ρ × F × v)
A_w = atomic weight, F = Faraday's constant (96500 C/mol)
ρ = density, v = valency
Advantages: No tool wear, high MRR, complex shapes,
no thermal damage, mirror-like finish
─── LBM (Laser Beam Machining) ───
Power density: 10⁴ – 10¹¹ W/cm²
Lasers: CO₂ (gas), Nd:YAG (solid), Fiber
─── WJM / AWJM (Abrasive Water Jet) ───
Pressure: 2000–4000 MPa (30,000–60,000 psi)
Water + abrasive (garnet, Al₂O₃) at 300–900 m/s
Cutting: Any material, no heat-affected zone
MRR ≈ 100–500 mm³/min─── Statistical Quality Control ───
Population parameters:
μ = population mean
σ = population standard deviation
Sample statistics:
x̄ = sample mean = Σxᵢ / n
s = sample std dev = √[Σ(xᵢ − x̄)² / (n−1)]
R = range = x_max − x_min
Normal Distribution (68-95-99.7 rule):
μ ± 1σ → 68.27% of data
μ ± 2σ → 95.45% of data
μ ± 3σ → 99.73% of data
Process capability:
Cp = (USL − LSL) / (6σ) [potential capability]
Cpk = min[(USL−μ)/(3σ), (μ−LSL)/(3σ)] [actual capability]
Cp ≥ 1.33 → Capable process
Cp ≥ 1.67 → Excellent
Cp < 1.0 → Not capable (parts outside spec)| Chart Type | Monitors | Formula (CL) | UCL / LCL |
|---|---|---|---|
| X̄ chart (mean) | Process mean (average) | CL = x̄̄ (grand mean) | x̄̄ ± A₂R̄ |
| R chart (range) | Process variability | CL = R̄ (avg range) | D₄R̄ / D₃R̄ |
| S chart (std dev) | Process variability | CL = s̄ | B₄s̄ / B₃s̄ |
| p chart (fraction) | Proportion defective | CL = p̄ | p̄ ± 3√(p̄(1−p̄)/n) |
| c chart (count) | Number of defects | CL = c̄ | c̄ ± 3√c̄ |
| np chart | Number of defectives | CL = np̄ | np̄ ± 3√(np̄(1−p̄)) |
Control chart constants (for n=5): A₂ = 0.577, D₃ = 0, D₄ = 2.114, d₂ = 2.326. A point outside UCL/LCL → process out of control. 7 consecutive points on one side → shift detected. Trend of 7 rising/falling → trend detected.
─── Acceptance Sampling ───
Single sampling plan:
N = lot size
n = sample size
c = acceptance number
Procedure: Draw n items from lot N.
If defectives ≤ c → Accept lot
If defectives > c → Reject lot
OC (Operating Characteristic) Curve:
Shows probability of acceptance vs fraction defective.
Pa = probability lot is accepted.
Key points:
AQL (Acceptable Quality Level): acceptable defect rate
LTPD (Lot Tolerance Percent Defective): rejectable rate
α (Producer's risk): P(reject | good lot) ≈ 5%
β (Consumer's risk): P(accept | bad lot) ≈ 10%
AOQ (Average Outgoing Quality):
AOQ = Pa × p × (N−n)/N
AOQL (Average Outgoing Quality Limit):
Maximum AOQ over all p values| Tool | Purpose | Description |
|---|---|---|
| Pareto chart | Identify vital few | Bar chart + cumulative line (80/20 rule) |
| Fishbone (Ishikawa) | Root cause analysis | 6M: Man, Machine, Method, Material, Measurement, Mother Nature |
| Check sheet | Data collection | Structured tally form for defect types |
| Histogram | Distribution shape | Frequency distribution of quality data |
| Scatter plot | Correlation | Relationship between two variables |
| Control chart | Process monitoring | UCL/LCL with time-series data |
| Stratification | Data segmentation | Separate data into meaningful groups |
─── Six Sigma ───
3σ process: 2700 defects per million (DPMO = 2700)
4σ process: 63 DPMO
6σ process: 3.4 DPMO (near perfection)
DMAIC Cycle:
D — Define: Project goals, customer requirements
M — Measure: Current process performance
A — Analyze: Root causes of defects
I — Improve: Implement solutions
C — Control: Sustain improvements