Taylor's tool life equation

Frederick W. Taylor presented the results of years of cutting experiments to the American Society of Mechanical Engineers in December 1906; they were published in 1907 as “On the Art of Cutting Metals”. His central finding still sets cutting speeds today: tool life falls steeply, and predictably, as cutting speed rises.

The equation

$$V \cdot T^{\,n} = C$$

• V is the cutting speed, in m/min.
• T is the tool life, in minutes, until the chosen wear criterion is reached (for example VB = 0.3 mm; see tool wear).
• n is an exponent that depends mainly on the tool material.
• C is the cutting speed that gives a tool life of exactly one minute. It depends on the tool, the work material, the feed, the depth of cut and the wear criterion.

Taking logarithms gives log V = log C − n · log T, a straight line on log–log axes with slope −n. A small n means a flat line: a small increase in speed costs a lot of tool life. A large n means the tool tolerates speed better.

Exponent n

0.25

Constant C (m/min)

500

Cutting speed V (m/min)

Tool life at this speed

—

Speed for 15 min life

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Finding n and C from two tests

Two tests at different speeds, run to the same wear criterion, define the line.

$$n = \frac{\ln(V_1/V_2)}{\ln(T_2/T_1)} \qquad C = V_1 \cdot T_1^{\,n}$$

Example. At V₁ = 250 m/min the tool reaches VB = 0.3 mm after T₁ = 15 min. At V₂ = 200 m/min it lasts T₂ = 40 min.

• n = ln(250/200) / ln(40/15) = 0.2231 / 0.9808 ≈ 0.228
• C = 250 · 15^0.228 ≈ 463 m/min

Check: 463 / 40^0.228 ≈ 200 m/min. ✓

With these values, a 10 % increase in speed from 200 to 220 m/min shortens tool life from 40 to about 26 minutes: T = (C/V)^1/n.

V · Tⁿ = C

Typical values of n

The sources agree for HSS and ceramics. For carbides, Kalpakjian gives one wide range for all grades; the ACSCE module splits it by coating, and its coated values are the only ones we found. Groover’s C values apply to turning steel at f = 0.25 mm/rev and d = 2.5 mm.

Kalpakjian & SchmidACSCE moduleGroover

The extended equation

Feed and depth of cut also affect tool life. The extended form adds them with their own exponents:

$$V \cdot T^{\,n} \cdot f^{\,x} \cdot d^{\,y} = C$$

x and y are found by experiment, and different textbooks attach the letters to f and d the other way round. As with n and C, the exponents hold only for the tool, work material and range of conditions they were measured in.

Which tool life to aim for

Running faster removes metal sooner but spends more edges and more time changing them. Two classic optima follow from the Taylor equation.

Maximum production rate (shortest time per part), with T_c the time to change an edge:

$$T_{\max\,\text{prod}} = \left(\frac{1}{n}-1\right) T_c$$

Minimum cost per part, with C_t the cost of one cutting edge and C_m the cost per minute of machine and operator:

$$T_{\min\,\text{cost}} = \left(\frac{1}{n}-1\right)\left(T_c + \frac{C_t}{C_m}\right)$$

The matching speed in each case is V = C / Tⁿ.

Example. Carbide with n = 0.25 and C = 500 m/min. Changing an edge takes T_c = 2 min, one edge costs C_t = 3 currency units, and the machine with operator costs C_m = 1 unit per minute.

• T_{max prod} = (1/0.25 − 1) · 2 = 6 min → V = 500 / 6^0.25 ≈ 319 m/min
• T_{min cost} = 3 · (2 + 3/1) = 15 min → V = 500 / 15^0.25 ≈ 254 m/min

carbide, n = 0.25

The speed for minimum cost is always lower than the speed for maximum production. The band between them is often called the high-efficiency range: slower than the lower bound wastes machine time, faster than the upper bound wastes tools without gaining output.