Mumbai University > Mechanical Engineering > Sem 4 > Production Process II

Marks: 4M

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

=$K_1.\frac{L}{f ×N}$

=$K1. \frac{L π D}{f ×1000 V}$

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

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

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

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

Taylors tool life eqn.

$VT^{n}$=C

∴ T=$\frac{C^{1⁄n}}{V^{1⁄n}}$ ……………7

put in eqn. 6

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

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

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

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

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

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

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

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

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

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

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

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