Machining or metal cutting is one important aspect of the production system. Ultimate objecting of machining is to give intended shape, size and finish by gradually removing material from workpiece. Relevant steps such as removal of material, setting the job and cutting tool, and dispatching the machined job consume substantial amount of time, which are at least not negligible. For effective planning of the entire production, overall machining or cutting time must be incorporated.

Now-a-days the primary goal of industries is to manufacture the product at a faster rate but at minimal cost and that too without sacrificing product quality. As long as conventional machining is utilized, in order to fulfill first requirement (faster production rate), the cutting speed and feed rate should have to be increased. However, this may lead to reduced cutting tool life due to faster wear rate and higher heat generation. Hence, cutting tool is required to change frequently, which will ultimately impose a loss for the industry as a result of idle time for changing tools. Cost of tool is also not negligible. Therefore abrupt increase of cutting speed and feed rate is not a feasible solution; rather, an optimization is necessary.

**Various time elements associated with machining operation**

Basically overall or total machining time (T_{m}) is the summation of three different time elements closely associated with the machining or metal cutting process. These three elements include—actual cutting time (T_{c}), total tool changing time (T_{ct}) and other handling or idle time (T_{i}). Mathematically, total time for machining (T_{m}) can be expressed as follows. You can learn more about these time elements: Economics of machining – cutting time, tool changing time & idle time.

T_{m }= T_{c} + T_{ct} + T_{i}

**Formulating problem – expressing machining time as a function of cutting velocity**

If, L_{c} is the total length of cut (mm), N is the spindle speed (rpm) and s is the feed rate (mm/rev), then estimated cutting time can be expressed as:

Actual Cutting Time (T_{c}) = \({{{L_c}} \over {N.s}}\)

In most of the cases, where either workpiece or cutting tool is rotating, the spindle speed (N) and cutting velocity (V_{c}) are interchangeable. However, cutting velocity also depends on the diameter of the job/cutter (D). Cutting velocity can be expressed, in terms of speed and diameter of job or cutter (whichever is rotating), as follows. For better understanding of this conversion, you may read: Cutting speed and cutting velocity in machining.

Cutting velocity (V_{c}) = \({{\pi DN} \over {1000}}\)

Now, \({T_c} = {L \over {N.s}}\)

or, \({T_c} = {L \over {\left( {{{1000{V_C}} \over {\pi D}}} \right)S}}\)

or, \({T_c} = {{\pi DL} \over {1000{V_C}}}\)

Again from **Taylor’s Tool Life Formula**, we know:

\({V_c}{(TL)^n} = C\)

So, \(TL = {\left( {{C \over {{V_c}}}} \right)^n}\)

Since total tool changing time (T_{ct}) depends on the number of changing requirement, so tool life will influence this time element. So, if TCT be the time required for single tool changing, then total tool changing time (T_{ct}) for the entire machining operation of T_{c} duration can be expressed as:

$${T_{ct}} = {{{T_c}} \over {TL}} \times TCT$$

Therefore, total machining time can be expressed as:

\({T_m} = {T_c} + {T_{ct}} + {T_i}\)

or, \({T_m} = {T_c} + {{{T_c}} \over {TL}} \times TCT + {T_i}\)

or, \({T_m} = {{\pi DL} \over {1000{V_c}s}} + \left\{ {{{{{\pi DL} \over {1000{V_c}s}}} \over {{{\left( {{C \over {{V_c}}}} \right)}^{1/n}}}} \times TCT} \right\} + {T_i}\)

or, \({T_m} = {{\pi DL} \over {1000{V_c}s}} + TCT{{\pi DL} \over {1000s{C^{1/n}}}}{V_c}^{{{1 – n} \over n}} + {T_i}\)

**Finding out optimum cutting velocity for maximum production rate by differentiation (Gilbert’s Model)**

It is to be noted that this model (known as Gilbert’s Model of Machining Economics) is applicable only for such cutting tools that closely follow Taylor’s Tool Life Equation. So, tool life and other times are dependent on cutting velocity only. There exists another model, called Economic Model for Minimum Production Cost, where cost elements are considered instead of considering time. Now in order to maximize production rate, we need to differentiate total machining time (T_{m}) with respect to cutting velocity. So,

\({{d{T_m}} \over {d{V_c}}} = {{\pi DL} \over {1000s}}\left( { – {V_c}^{ – 2}} \right) + TCT{{\pi DL} \over {1000s{C^{1/n}}}}\left( {{{1 – n} \over n}} \right){\left( {{V_c}} \right)^{{{1 – 2n} \over n}}} + 0\)

or, \(\left\{ {{{\pi DL} \over {1000s}}} \right\}{1 \over {{V_c}^2}} = TCT\left( {{{\pi DL} \over {1000s{C^{1/n}}}}} \right)\left( {{{1 – n} \over n}} \right){\left( {{V_c}} \right)^{{{1 – 2n} \over n}}}\)

or, \(\left( {{n \over {1 – n}}} \right){{{C^{1/n}}} \over {TCT}} = {\left( {{V_c}} \right)^{{{1 – 2n} \over n} + 2}}\)

or, \({{{C^{1/n}}} \over {\left( {{{1 – n} \over n}} \right)TCT}} = {\left( {{V_c}} \right)^{1/n}}\)

or, \({V_c} = {C \over {{{\left\{ {\left( {{1 \over n} – 1} \right)TCT} \right\}}^n}}}\)

**Finding out optimum tool life for maximum production rate**

Expression for optimum tool life can be obtained by putting optimum cutting velocity (as derived above) in the Taylor’s Tool Life Formula. So,

\(TL = {\left( {{C \over {{V_c}}}} \right)^{1/n}}\)

\(TL = \left( {{1 \over n} – 1} \right)TCT\)

**Final formulas of optimum velocity and tool life**

$${V_c,opt} = {C \over {{{\left\{ {\left( {{1 \over n} – 1} \right)TCT} \right\}}^n}}}$$

$$T{L_{opt}} = \left( {{1 \over n} – 1} \right)TCT$$

**Reference**

- Book: Metal Cutting: Theory And Practice by A. Bhattacharya (New Central Book Agency).
- Book: Machining and Machine Tools by A. B. Chattopadhyay (Wiley).