Machining Kinematics & MRR Formula (N = 1000Vc/πD, Vf = N·fz·z)
Calculate precision CNC spindle speeds (RPM), linear table feed rates (mm/min or ipm), and volumetric material removal rates (MRR) for milling, turning, and drilling.
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Primary Mathematical Expression
Design Schematic
Nomenclature & Variables
| Symbol | Variable Name | Metric Unit | Imperial Unit | Description |
|---|---|---|---|---|
| N | Spindle rotational speed | RPM | RPM | Rotational speed of the machine tool spindle. |
| V_c | Surface cutting speed | m/min | SFM | Peripheral linear velocity of the cutting tool relative to the workpiece. |
| D | Tool or workpiece diameter | mm | in | Effective cutting tool diameter (milling/drilling) or workpiece diameter (turning). |
| V_f | Table feed rate | mm/min | ipm | Linear traverse speed of the machine tool axis along the cut path. |
| f_z | Feed per tooth (chip load) | mm/tooth | in/tooth | Uncut linear advance per cutting tooth / flute. |
| f_n | Feed per revolution | mm/rev | in/rev | Total linear advance per complete spindle revolution (f_n = f_z * z). |
| z | Number of cutting teeth | flutes | flutes | Number of active cutting edges or flutes on the tool. |
| a_p | Axial depth of cut | mm | in | Depth of tool engagement along the spindle axis (milling) or depth of cut (turning). |
| a_e | Radial width of cut | mm | in | Stepover or radial engagement width perpendicular to the feed vector. |
| Q | Material removal rate (MRR) | cm³/min | in³/min | Volumetric rate of metal removal during active cutting. |
Step-by-Step Derivation
- 1
The circumferential distance traveled by the outer cutting edge in one revolution is $\pi \cdot D$ (mm).
- 2
Given surface cutting speed $V_c$ in meters per minute (where $1\text{ m} = 1,000\text{ mm}$), the linear distance traveled per minute is $1,000 \cdot V_c\text{ mm/min}$.
- 3
Spindle speed $N$ in RPM is derived by dividing total linear surface distance by circumference: $N = \frac{1,000 \cdot V_c}{\pi \cdot D}$. In US Customary units where $V_c$ is in Surface Feet per Minute (SFM) and $D$ in inches: $N = \frac{12 \cdot V_c}{\pi \cdot D} \approx \frac{3.82 \cdot V_c}{D}$.
- 4
Linear table feed rate $V_f$ is the product of spindle RPM, feed per tooth $f_z$, and number of flutes $z$: $V_f = N \cdot f_z \cdot z = N \cdot f_n$.
- 5
In milling, volumetric material removal rate $Q$ is the product of axial depth $a_p$, radial width $a_e$, and table feed $V_f$: $Q = \frac{a_p \cdot a_e \cdot V_f}{1,000}\text{ cm}^3/\text{min}$.
- 6
In turning, $Q = a_p \cdot f_n \cdot V_c\text{ cm}^3/\text{min}$. In drilling, $Q = \frac{\pi \cdot D^2 \cdot V_f}{4,000}\text{ cm}^3/\text{min}$.
Worked Example Calculation
A CNC machining center is shoulder milling AISI 1020 mild steel using a $\varnothing 12\text{ mm}$ 4-flute TiAlN-coated carbide end mill. The recommended cutting speed is $V_c = 220\text{ m/min}$, with a feed per tooth of $f_z = 0.06\text{ mm/tooth}$, axial depth of cut $a_p = 5.0\text{ mm}$, and radial stepover $a_e = 6.0\text{ mm}$. Calculate the required spindle RPM, table feed rate, and material removal rate.
- •Calculate Spindle Speed: N = (1,000 * 220) / (π * 12) = 220,000 / 37.6991 = 5,835.7 RPM ≈ 5,836 RPM.
- •Calculate Table Feed Rate: Vf = 5,835.7 * 0.06 * 4 = 1,400.6 mm/min (Feed per rev fn = 0.24 mm/rev).
- •Calculate Volumetric Material Removal Rate: Q = (5.0 * 6.0 * 1,400.6) / 1,000 = 42.02 cm³/min (2.56 in³/min).
- •Estimate Spindle Power with Kienzle kc ≈ 1,850 N/mm²: Pc = (42.02 * 1,850) / 60,000 ≈ 1.30 kW (1.74 HP).
Engineering Assumptions
- •Rigid machine setup, rigid workholding, and minimal tool overhang ($L/D \le 3$).
- •Constant spindle speed and uniform chip load per tooth with concentric tool runout ($< 0.005\text{ mm}$).
- •Standard radial engagement ($a_e \ge 0.5D$). When $a_e < 0.5D$, radial chip thinning compensation (RCTF) should be applied to prevent rubbing.
Design Limitations
- •High-RPM operations above 20,000 RPM require balanced toolholders (ISO 1940 G2.5).
- •Thin-walled parts or long tool overhangs ($L/D > 4$) require reduced cutting parameters to avoid chatter vibration.
Academic References & Standards
Industrial Press
Speeds and feeds data, cutting speed equations, and milling formulas
Tool Life and Cutting Parameters for Metal Cutting
American Society of Mechanical Engineers standard