Also show that the optimum cutting speed for maximum production rate is always more than optimum cutting speed for minimum cost.

Subject: Production Process 2

Topic: Metal Cutting

Difficulty: Medium

Calculation of different Cost elements:

Let ; D= Diameter of job in mm

L= length of the job in mm

f=feed in mm/rev

N= rotational speed.= $\frac{1000 V}{(π ×D}$

$t_m$ =machining speed of work piece. =$\frac{L}{f ×N}$

$t_h$= handling or loading time.

$K_1$= Operating cost.

$t_{c}$= tool changing time for one edge

We know that;

Cost of manufacturing one piece= handling cost per piece+ Machining cost per pc.+ Tool change cost per pc.+ Tool cost per pc ……..1

Handling Cost per pc.

Handling Cost per pc (Rs/pc)=$ t_h \times K_{1}$ = $t_h . K_{1}$………2

Machining cost per pc.

Machining cost per pc. = $t_{m}\times K_{1}$

=$K_1 . t_m$…………………3

=$K1. \frac{ L}{f ×N}$

= $K_1. \frac{L π D}{f ×1000 V}$

= $K_1 .\frac{L π D}{f \times 1000}\frac{1}{ V}$

= $K_1. \frac{K}{V}$ ………………………4

Tool change cost per pc.

$t_m$=machining time (min)

T=tool life (min)

Number of edges required to manufacture one piece = $\frac{t_m}{T}$

tool changing cost per pc. = $tc \times \frac{t_{m}}{T} \times K_{1}$

= $K_1 \times t_c \times \frac{t_{m}}{T}$ ……………..5

= $K_1 \times t_c \times \frac{K}{V}\times \frac{1}{T}$ …………6 (from eqn. 3 & 4)

Taylors tool life eqn.

$4VT^{n}$=C

∴ $T=\frac{C^{1⁄n}}{V^{1⁄n}}$ ……………7

put in eqn. 6

tool changing cost per pc. = $K_1\times t_c \times \frac{K}{V }×\frac{V^{1⁄n}}{C^{1⁄n}}$

=$ K_1 \times t_c \times \frac{K}{(C^{1⁄n}} \times V^{1⁄n} x V-1$

= $K_1 \times t_c \times \frac{K}{C^{1⁄n}} \times V^{\frac{1-n}{n}}$……….8

Tool cost per pc.

Let; $K_2$=tool cost per edge.

tool cost per piece = (no. of tool edges required to produce one pc.) x (tool cost per edge.)

= $\frac{t_m}{T} \times K_2$

= $ K_2 \times \frac{t_{m}}{T} $

=$K_2 x \frac{K}{C^{1⁄n}} × V^\frac{1-n}{n}$ ……….9

Expression of optimum cutting speed for minimum cost of production: from eqn. 1, 2, 4, 8, 9

Cost of manufacturing one piece= $t_h . K_1 + K_1. \frac{K}{V} + K_1 \times t_c \times \frac{K}{C^{1⁄n}} ×V^\frac{1-n}{n}+ K_2 x \frac{K}{C^{1⁄n}} ×V^\frac{1-n}{n}$= $ t_h . K_1 + K_1. \frac{K}{V} + \frac{K}{C^{1⁄n}} (K_1. t_c+ K_2) .V^\frac{1-n}{n}$

for min cost of production

$\frac{.d}{dV}$ (RHS)=0

$0+(-K_1. \frac{K}{V^2}) + \frac{K}{C^(1⁄n)}(K_1. t_c+ K_2) .V^\frac{1-2n}{n}$ = 0

⇛$V^\frac{1-2n}{n} .V^2= K_1 . C^{1⁄n} \frac{1}{(K_1 t_c+K_2)} . \frac{n}{1-n}$

⇛$V_{min}=C.[\frac{n}{1-n}×\frac{K_1}{K_1 t_c+K_2}]^n$

this is an eqn. of cutting speed for min cost of production.

To find corresponding tool life

$VT^n$=C

$T_{min}=[\frac{C}{V_min} ]^{\frac{1}{n}}$

$\left[\frac{c}{c[\frac{n}{1-n}\times \frac{K_{1}}{k_{1}t_{c}+k_{2}}]}\right]^{\frac{1}{n}}$

$T_{min}=\frac{n}{1-n} × \frac{K_1 t_c+K_2}{K_1}$

this is the eqn. for tool life for min cost of production

$ VT^{n}=C$

$T_{max}=[\frac{C}{V_{max}}]^{1/n}$

$T_{max}=\left[\frac{c}{c[\frac{1}{t_C}\times \frac{n}{1-n}]^{n}}\right]^{\frac{1}{n}}$

$T_{max}= \frac{t_c (1-n)}{n}$

this is the eqn. for tool life for max. Rate of production.