This is a straightforward problem on tool life that can be solved simply by using Taylor’s Tool Life equation. Here, two cutting velocities and the corresponding tool life values are given. So, let us first formulate the basic equations to apply Taylor’s Tool Life formula.
Step-1: Apply Taylor’s Tool Life equation
Let us first denote the two set of given values as 1 and 2. Therefore, from the given values, the following can be written:
T1 = 16 min when V1 = 100 m/min
T2 = 4 min when V2 = 200 m/min
Among these two given cases, the workpiece material, tool material, cutting environment, feed and depth of cut can be assumed as unchanged. Accordingly, the Taylor’s Tool Life equation can be applied for both the cases. That means:
\[{V_1}{\left( {{T_1}} \right)^n} = C\]
\[{V_2}{\left( {{T_2}} \right)^n} = C\]
Step-2: Solve for Taylor’s exponent (n)
Equating the above two equations, the value of n can be easily calculated as solved below.
\[\begin{array}{l}
{V_1}{\left( {{T_1}} \right)^n} = {V_2}{\left( {{T_2}} \right)^n}\\
\frac{{{V_1}}}{{{V_2}}} = {\left( {\frac{{{T_2}}}{{{T_1}}}} \right)^n}\\
\left( {\frac{{100}}{{200}}} \right) = {\left( {\frac{4}{{16}}} \right)^n}\\
\left( {\frac{1}{2}} \right) = {\left( {\frac{1}{4}} \right)^n}\\
\left( {\frac{1}{2}} \right) = {\left( {\frac{1}{2}} \right)^{2n}}\\
1 = 2n\\
n = 0.5
\end{array}\]
Therefore, the calculated value of Taylor’s exponent for the given case is n = 0.5. It is one unitless constant.
Step-3: Solve for Taylor’s constant (C)
When the n value is known, the value of C can be easily calculated using any one of the above two basic equations. The solution is shown below.
\[\begin{array}{l}
{V_1}{\left( {{T_1}} \right)^n} = C\\
100 \times {\left( {16} \right)^{0.5}} = C\\
C = 400
\end{array}\]
Therefore, the intended value of C = 400. It is also one unitless constant.
Now the required values are calculated. As an extension, you can also get the Taylor’s Tool Life equation for the given case, as expressed below. The variation of tool life with velocity is also shown below in the V-T curve.
\[V.{T^{0.5}} = 400\]
