Machining Processes
Turning, Milling, Drilling — Tool Life, Cutting Forces, MRR & Solved Problems
Last Updated: March 2026
Key Takeaways 📌
- Machining removes material from a workpiece using a cutting tool to achieve desired shape, size, and surface finish.
- Taylor’s tool life: VTn = C — relates cutting speed (V) to tool life (T). Higher speed → shorter tool life.
- Merchant’s circle: Relates cutting force, thrust force, friction force, shear force, and chip geometry in orthogonal cutting.
- MRR (turning): MRR = πDNfd = Vfd — material removal rate determines productivity.
- Three primary operations: Turning (cylindrical parts on lathe), Milling (flat/complex surfaces), Drilling (holes).
- Machining is the highest-weightage manufacturing topic in GATE ME (4–6 marks).
1. Machining Fundamentals
Machining is a subtractive manufacturing process — material is removed from a workpiece by a cutting tool to produce the desired shape. The three basic parameters that define any machining operation are:
Cutting Parameters
Cutting speed (V): Speed of the tool relative to the workpiece at the cutting point (m/min). V = πDN/1000 for turning.
Feed (f): Distance the tool advances per revolution (mm/rev for turning) or per tooth (mm/tooth for milling).
Depth of cut (d): Thickness of the material layer removed in one pass (mm).
| Parameter | Effect of Increasing |
|---|---|
| Cutting speed ↑ | Higher MRR, shorter tool life, better surface finish |
| Feed ↑ | Higher MRR, shorter tool life, rougher surface finish |
| Depth of cut ↑ | Higher MRR, higher cutting forces, similar tool life impact |
2. Turning (Lathe Operations)
Turning is the most fundamental machining operation — a single-point cutting tool removes material from a rotating cylindrical workpiece to produce cylindrical, conical, or contoured shapes. Performed on a lathe.
Cutting Speed (Turning)
V = πDN / 1000 (m/min)
Where D = workpiece diameter (mm), N = spindle speed (rpm)
MRR — Turning
MRR = V × f × d = πDNfd / 1000 (mm³/min or cm³/min)
Where f = feed (mm/rev), d = depth of cut (mm)
Machining Time (Turning)
Tm = L / (fN) (minutes)
Where L = length of cut (mm), f = feed (mm/rev), N = rpm
Common lathe operations: straight turning, facing, taper turning, threading, boring, knurling, parting/grooving.
3. Milling
Milling uses a multi-point rotary cutter to remove material from a workpiece that is fed against the cutter. It can produce flat surfaces, slots, pockets, contours, and complex 3D shapes.
| Type | Description | Cutting Action |
|---|---|---|
| Up milling (conventional) | Cutter rotates against the feed direction | Chip thickness increases from zero to maximum. Higher forces lift workpiece. |
| Down milling (climb) | Cutter rotates in the same direction as feed | Chip thickness decreases from maximum to zero. Better surface finish, less tool wear. |
MRR — Milling
MRR = w × d × vf
Where: w = width of cut (mm), d = depth of cut (mm), vf = feed rate (mm/min) = fz × z × N
fz = feed per tooth, z = number of teeth, N = cutter rpm
4. Drilling
Drilling produces cylindrical holes using a rotating drill bit. The standard twist drill has two cutting edges (lips), two flutes for chip removal, and a chisel edge at the centre.
MRR — Drilling
MRR = (πD²/4) × f × N
Where D = drill diameter, f = feed (mm/rev), N = rpm
Drilling Time
Tm = (L + A) / (fN)
Where L = hole depth, A = drill point advance = D/(2 tan(θ/2)), θ = drill point angle (typically 118°)
5. Chip Formation & Types
| Chip Type | Characteristics | Occurs With |
|---|---|---|
| Continuous chip | Long, ribbon-like, smooth. Best surface finish. | Ductile materials, high speed, sharp tool, small feed |
| Continuous with built-up edge (BUE) | Material welds to tool tip, breaks off periodically. Poor surface finish. | Low speed, ductile material, high friction |
| Discontinuous (segmented) | Small, separate segments. Acceptable surface finish. | Brittle materials, low speed, large feed, large depth of cut |
Chip Thickness Ratio (Cutting Ratio)
r = t₁/t₂ = sinφ / cos(φ − α)
Where: t₁ = uncut chip thickness, t₂ = chip thickness after cutting, φ = shear angle, α = rake angle
r < 1 always (chip is thicker than the uncut layer).
6. Merchant’s Circle — Cutting Force Analysis
Merchant’s circle diagram is a graphical method for analysing forces in orthogonal (2D) cutting. All forces pass through a single point and can be represented on a force circle.
Key Force Relationships
Resultant force R connects all force components on the circle.
Cutting force: Fc = R cos(β − α) / cos(φ + β − α) × … (from the circle geometry)
Shear angle (Merchant’s equation):
2φ + β − α = 90° → φ = 45° + α/2 − β/2
Where: φ = shear angle, α = rake angle, β = friction angle (tan β = μ = F/N)
Shear Force and Cutting Force
Fs = τs × As, where As = bt₁/sinφ (shear plane area)
Fc = Fs × cos(β − α) / cos(φ + β − α)
Ft = Fs × sin(β − α) / cos(φ + β − α)
Where Fc = cutting (tangential) force, Ft = thrust (feed) force
Power
Cutting power = Fc × V (watts, if F in N and V in m/s)
7. Taylor’s Tool Life Equation
Taylor’s Equation
VTn = C
Where: V = cutting speed (m/min), T = tool life (min), n = tool life exponent, C = constant (speed for T = 1 min)
| Tool Material | n (typical) |
|---|---|
| HSS (High Speed Steel) | 0.08–0.2 |
| Carbide | 0.2–0.4 |
| Ceramic | 0.5–0.7 |
| Diamond/CBN | 0.6–0.9 |
Extended Taylor’s equation: VTnfn₁dn₂ = C — includes the effects of feed and depth of cut. Typically n₁ < n and n₂ < n₁, meaning cutting speed has the strongest effect on tool life.
Optimum Cutting Speed (Minimum Cost)
Vopt = C / [(1/n − 1) × Tc]n
Where Tc = tool change time. This gives the speed that minimises total cost per part.
8. Material Removal Rate — Summary
| Operation | MRR Formula |
|---|---|
| Turning | MRR = πDNfd / 1000 = Vfd |
| Milling | MRR = w × d × vf = w × d × fz × z × N |
| Drilling | MRR = (πD²/4) × f × N |
9. Worked Numerical Examples
Example 1: Tool Life — Taylor’s Equation
Problem: A tool has Taylor constants n = 0.25 and C = 300. Find the tool life at V = 150 m/min.
Solution
VTn = C → 150 × T0.25 = 300 → T0.25 = 2 → T = 24 = 16 minutes
Example 2: MRR and Machining Time — Turning
Problem: A workpiece of diameter 60 mm and length 200 mm is turned at N = 500 rpm, f = 0.2 mm/rev, d = 2 mm. Find MRR and machining time.
Solution
V = πDN/1000 = π × 60 × 500/1000 = 94.25 m/min
MRR = Vfd = 94.25 × 1000 × 0.2 × 2 = 37,700 mm³/min ≈ 37.7 cm³/min
Tm = L/(fN) = 200/(0.2 × 500) = 200/100 = 2 minutes
Example 3: Merchant’s Shear Angle
Problem: In orthogonal cutting, rake angle α = 10° and friction coefficient μ = 0.5. Find the shear angle using Merchant’s equation.
Solution
β = tan⁻¹(μ) = tan⁻¹(0.5) = 26.57°
φ = 45° + α/2 − β/2 = 45° + 5° − 13.28° = 36.72°
10. Common Mistakes Students Make
- Confusing n and C in Taylor’s equation: n is the exponent (typically 0.1–0.7); C is the constant (cutting speed for 1-minute tool life). A higher n means tool life is less sensitive to speed changes.
- Forgetting to convert V to consistent units: Taylor’s equation uses V in m/min. MRR formulas need consistent units (mm/min or m/min). Mixing units gives wrong results.
- Using the wrong MRR formula for different operations: Turning uses Vfd, milling uses w×d×vf, drilling uses (πD²/4)×f×N. They are not interchangeable.
- Confusing up milling and down milling: Up milling = cutter rotates against feed direction (chip starts thin). Down milling = cutter rotates with feed direction (chip starts thick, better finish). GATE often asks which produces better surface finish (answer: down milling).
11. Frequently Asked Questions
What is Taylor’s tool life equation?
VTn = C relates cutting speed (V) to tool life (T). Higher cutting speed gives shorter tool life. The exponent n depends on tool material (0.08–0.2 for HSS, 0.2–0.4 for carbide, 0.5+ for ceramics). The constant C is the cutting speed that gives exactly 1 minute of tool life. It is the most important formula in machining economics.
What is Merchant’s circle?
Merchant’s circle is a graphical force analysis method for orthogonal cutting. It relates the cutting force, thrust force, friction force, normal force, and shear force on a single circle. Merchant’s equation (2φ + β − α = 90°) predicts the shear angle from the rake angle and friction angle. The shear angle determines chip thickness, cutting forces, and power consumption.