Machining Parameters: Method

Machining Parameters: Speeds, Feeds, Power, Tool Life and Machining Economics

Scope

The tool works one cut of turning, milling or drilling from its geometry, feed and speed: the cutting speed and spindle speed, the manuals' recommended speed range for the work and tool material, the removal rate and cutting time, the power, force and torque by the unit-power method, the tool life by Taylor's law, the cost and time per piece with the two economic speeds, the ideal roughness of a turned surface and the chip geometry of orthogonal cutting. It is a process-planning calculator, not a cutting-force model: the forces come from an entered specific cutting energy, and the tool-life constants are entered for the tool and work pair.

Cutting speed and spindle speed

The cutting speed v is the surface speed at the diameter D of the work (turning), the cutter (milling) or the drill: v = π D N with N the spindle speed. The shop manuals write the inverse as RPM = CS × 4 / D for CS in ft/min and D in inches (TC 9-524, chapters 4, 7 and 8) and RPM = CS × 320 / D for CS in m/min and D in millimetres, both with 12/π = 3.82 rounded to 4 and 1000/π = 318 rounded to 320; NAVEDTRA 12204 writes rpm = CS × 12 / (D × 3.1416) and the shorthand rpm = fpm / (0.2618 D). The tool prints the exact value and the manuals' shorthand beside it (the shorthand runs 4.7 % high). Both manuals are U.S. government works; their speed tables are reproduced for the work materials of the material select: TC 9-524 Table 7-2 (turning and threading with high-speed-steel tool bits), Table 8-1 (plain and end milling cutters, roughing and finishing) and the header speeds of Table 4-4 (twist drills), NAVEDTRA 12204 Table 8-1 (turning) and p. 5-22 (drilling). The ranges are for high-speed steel; the manuals' factors for other tool materials are applied as printed: carbon-steel tools one half (TC 9-524 Table 8-1 note 1; NAVEDTRA 12204 p. 8-15), cast alloy 1.6 times (NAVEDTRA), carbide 2 to 3 times (TC 9-524 Table 7-2 note; Table 8-1 says double, Table 4-4 says 200 to 300 percent higher, NAVEDTRA 2 to 4 times), ceramic 4 to 16 times (NAVEDTRA). The lower end of a range is for roughing, worn machines and dry cutting, the upper end for finishing with coolant. The tool says whether the entered speed lies below, within or above the range; the feeds of TC 9-524 Table 7-3 (turning), Table 4-4 (drilling) and the chip loads of Table 8-3 and NAVEDTRA Table 11-7 (milling) are quoted in the input tooltips for the same purpose.

Feed, removal rate and time

Turning. The tool advances f per revolution, so the feed rate is f N and the cutting time t_m = n_p (L + A) / (f N) for n_p passes over the length L with an approach and over-travel allowance A. The volume removed per revolution is the annulus π/4 [D² − (D − 2d)²] f = π (D − d) d f, so the exact removal rate is MRR = π (D − d) d f N with the mean diameter D − d; the handbook form π D d f N with the outside diameter overstates it by d/(D − d) and is printed beside it. Milling. With z teeth and a chip load f_t per tooth the table feed is f_r = f_t z N (TC 9-524: IPM = CPT × N × RPM), the removal rate MRR = w d f_r for a width w and depth d, and the time t_m = n_p (L + A + O) / f_r. The allowance A is the distance the cutter must travel before the full cut is established and after it: for peripheral (slab or side) milling the chord of the cutter at the depth of cut, √(d (D − d)); for face or end milling at full width the cutter axis travels D/2 before the full width is engaged and D/2 after the far edge, so A = D; any extra over-travel O is entered. The geometric mean chip thickness of a peripheral cut, f_t √(d/D), is printed for comparison with the chip load. Drilling. The feed rate is f N, the removal rate (π D²/4) f N, and the point of a drill of included angle θ has the height A = (D/2) / tan(θ/2) (0.300 D for the standard 118° point), which must clear the full-diameter depth L: t_m = n_h (L + A + O) / (f N) for n_h holes.

Power, force and torque

The unit power or specific cutting energy u is the energy needed to remove a unit volume of the work material; the power at the cut is P_c = u × MRR, the motor power P_m = P_c / η for a drive efficiency η, the tangential cutting force F_c = P_c / v and the spindle torque T = P_c / ω = F_c D / 2. Unit power is tabulated in hp·min/in³ (1 hp·min/in³ = 2.73 W·s/mm³ = 2.73 GJ/m³) for sharp tools at moderate feeds; it rises at small feeds (the size effect) and with tool wear, and the handbooks give a correction for both. The values depend on the material's hardness and condition, so the tool takes u as an entered input and only falls back on an illustrative value for the material of the select when the field is blank, saying so in the output; the handbook tables (Machinery's Handbook, the Machining Data Handbook) are the place to read the value for a given alloy, hardness, feed and tool. The force F_c is the tangential component only; the feed and radial components are not estimated.

Tool life: Taylor's law

Taylor's experiments at Midvale and Bethlehem, reported in his 1907 presidential address to the ASME, established that the cutting speed a tool stands for a given time before it must be reground follows v Tⁿ = C: for his high-speed tool on an annealed steel forging he found V = 90 / T^1/8 with V in ft/min and T in minutes (paragraph 700), equivalently a 40-minute tool runs at 0.92 and an 80-minute tool at 0.84 of the 20-minute speed (paragraphs 684, 685 and 701). The exponent n and the constant C (the speed for a one-minute life) are entered: n = 1/8 is Taylor's high-speed steel, modern high-speed steels fall near 0.1 to 0.2, carbides near 0.2 to 0.5 and ceramics higher (Kalpakjian and Schmid, Chapter 21), while C depends strongly on the work material and the feed and must come from the tool maker's data or a tool-life test. The tool prints the life at the entered speed, T = (C / v)^1/n, the pieces per cutting edge T / t_m and the edges consumed per piece, and draws the Taylor line on logarithmic axes with the operating point and the economic speeds.

Cost and time per piece; the economic speeds

With a machine-and-operator rate C_o per minute, a tool-change time t_c, a cost C_t per cutting edge and a handling time t_h per piece, the cost per piece is C_p = C_o t_h + C_o t_m + (t_m / T)(C_o t_c + C_t) and the time per piece t_p = t_h + t_m + (t_m / T) t_c, the last terms being the tool changes charged to the piece. At fixed feed and depth the cutting time varies as 1/v and the life as (C/v)^1/n, so t_m/T grows as v^{1/n − 1}: a higher speed cuts the machining cost and raises the tool cost. Setting dC_p/dv = 0 gives the minimum-cost speed v = C [n/(1 − n) · C_o/(C_o t_c + C_t)]ⁿ, whose tool life is T = (1/n − 1)(t_c + C_t/C_o); setting dt_p/dv = 0 gives the maximum-production speed v = C [n/((1 − n) t_c)]ⁿ with T = (1/n − 1) t_c (Gilbert 1950). The maximum-production speed is always the higher of the two when the tool costs anything, and the interval between them is the high-efficiency range in which a shop chooses its speed. The closed forms are checked in the tool's test script against a numerical minimisation of the cost and time functions. The charts of cost and time against speed show how flat the minimum is: a speed 20 percent off the optimum costs only a few percent.

Surface finish

A round-nosed turning tool leaves circular-arc feed marks; for a feed f and nose radius r the peak-to-valley height is R_t = f² / (8 r) and the arithmetic-average roughness R_a = f² / (32 r) (the parabolic approximation of the arc, within a percent for f below about r/2). This ideal roughness is a lower bound: built-up edge, tool wear, vibration and side flow add to it. Halving the feed quarters the ideal roughness, which is why finishing cuts use the fine feeds of TC 9-524 Table 7-3.

Chip geometry

In orthogonal cutting the chip of thickness t_c is thicker than the uncut layer t₀; the chip-thickness ratio r = t₀ / t_c (less than one, measured from a chip) and the rake angle α fix the shear angle through tan φ = r cos α / (1 − r sin α), the shear strain on the shear plane γ = cot φ + tan(φ − α), and the velocities: the chip moves at r v and the material crosses the shear plane at v cos α / cos(φ − α). Merchant (1945) predicted the shear angle from the minimum-energy condition as φ = 45° + α/2 − β/2 with the friction angle β = arctan μ on the rake face; measured angles are usually smaller, which is why the tool shows both when a chip ratio and a friction coefficient are entered.

Units

Diameters, lengths, depths, feeds and the nose radius convert with the unit switch (inches or millimetres); the spindle speed is in rev/min in both systems; the cutting speed, the Taylor constant and the recommended range are in ft/min (USCS) or m/min (SI), the unit power in hp·min/in³ or W·s/mm³, the removal rate in in³/min or cm³/min, the force in lb or N, the torque in lb-in or N-m and the roughness in µin or µm. The plain-number fields (cutting speed, unit power, Taylor constant, costs, times, angles) are read in the unit of the selected system and are not converted by the switch; re-enter them after switching.

Limits

Single-point turning, plain milling at full engagement and twist drilling only: no boring bars, no thread cutting, no grinding; no feed or thrust force (the drilling thrust in particular is not estimated); no tool deflection, chatter or power limit of the machine beyond the comparison the analyst makes with the motor rating; the tool-life law holds for the feed and depth at which its constants were fitted; the economics assume one tool and one cut per piece and a cost rate independent of the speed.