Answer:
Among all three tools, the ceramic tool is taking the least time for the production of a batch, however, machining from the HSS tool is taking the highest time.
Explanation:
The optimum cutting speed for the minimum cost
![V_{opt}= \frac{C}{\left[\left(T_c+\frac{C_e}{C_m}\right)\left(\frac{1}{n}-1\right)\right]^n}\;\cdots(i)](https://tex.z-dn.net/?f=V_%7Bopt%7D%3D%20%5Cfrac%7BC%7D%7B%5Cleft%5B%5Cleft%28T_c%2B%5Cfrac%7BC_e%7D%7BC_m%7D%5Cright%29%5Cleft%28%5Cfrac%7B1%7D%7Bn%7D-1%5Cright%29%5Cright%5D%5En%7D%5C%3B%5Ccdots%28i%29)
Where,
C,n = Taylor equation parameters
=Tool changing time in minutes
=Cost per grinding per edge
= Machine and operator cost per minute
On comparing with the Taylor equation
,
Tool life,
![T= \left[ \left(T_t+\frac{C_e}{C_m}\right)\left(\frac{1}{n}-1\right)\right]}\;\cdots(ii)](https://tex.z-dn.net/?f=T%3D%20%5Cleft%5B%20%5Cleft%28T_t%2B%5Cfrac%7BC_e%7D%7BC_m%7D%5Cright%29%5Cleft%28%5Cfrac%7B1%7D%7Bn%7D-1%5Cright%29%5Cright%5D%7D%5C%3B%5Ccdots%28ii%29)
Given that,
Cost of operator and machine time![=\$40/hr=\$0.667/min](https://tex.z-dn.net/?f=%3D%5C%2440%2Fhr%3D%5C%240.667%2Fmin)
Batch setting time = 2 hr
Part handling time:
min
Part diameter:
![=73\times 10^{-3} m](https://tex.z-dn.net/?f=%3D73%5Ctimes%2010%5E%7B-3%7D%20m)
Part length: ![l=250 mm=250\times 10^{-3} m](https://tex.z-dn.net/?f=l%3D250%20mm%3D250%5Ctimes%2010%5E%7B-3%7D%20m)
Feed: ![f=0.30 mm/rev= 0.3\times 10^{-3} m/rev](https://tex.z-dn.net/?f=f%3D0.30%20mm%2Frev%3D%200.3%5Ctimes%2010%5E%7B-3%7D%20m%2Frev)
Depth of cut: ![d=3.5 mm](https://tex.z-dn.net/?f=d%3D3.5%20mm)
For the HSS tool:
Tool cost is $20 and it can be ground and reground 15 times and the grinding= $2/grind.
So,
![\$20/15+2=\$3.33/edge](https://tex.z-dn.net/?f=%5C%2420%2F15%2B2%3D%5C%243.33%2Fedge)
Tool changing time,
min.
![C= 80 m/min](https://tex.z-dn.net/?f=C%3D%2080%20m%2Fmin)
![n=0.130](https://tex.z-dn.net/?f=n%3D0.130)
(a) From equation (i), cutting speed for the minimum cost:
![V_{opt}= \frac {80}{\left[ \left(3+\frac{3.33}{0.667}\right)\left(\frac{1}{0.13}-1\right)\right]^{0.13}}](https://tex.z-dn.net/?f=V_%7Bopt%7D%3D%20%5Cfrac%20%7B80%7D%7B%5Cleft%5B%20%5Cleft%283%2B%5Cfrac%7B3.33%7D%7B0.667%7D%5Cright%29%5Cleft%28%5Cfrac%7B1%7D%7B0.13%7D-1%5Cright%29%5Cright%5D%5E%7B0.13%7D%7D)
m/min
(b) From equation (ii), the tool life,
![T=\left(3+\frac{3.33}{0.667}\right)\left(\frac{1}{0.13}-1\right)\right]}](https://tex.z-dn.net/?f=T%3D%5Cleft%283%2B%5Cfrac%7B3.33%7D%7B0.667%7D%5Cright%29%5Cleft%28%5Cfrac%7B1%7D%7B0.13%7D-1%5Cright%29%5Cright%5D%7D)
min
(c) Cycle time: ![T_c=T_h+T_m+\frac{T_t}{n_p}](https://tex.z-dn.net/?f=T_c%3DT_h%2BT_m%2B%5Cfrac%7BT_t%7D%7Bn_p%7D)
where,
Machining time for one part
Number of pieces cut in one tool life
min, where
is the rpm of the spindle.
![\Rightarrow T_m= \frac{\pi D l}{fV_{opt}}](https://tex.z-dn.net/?f=%5CRightarrow%20T_m%3D%20%5Cfrac%7B%5Cpi%20D%20l%7D%7BfV_%7Bopt%7D%7D)
![\Rightarrow T_m=\frac{\pi \times 73 \times 250\times 10^{-6}}{0.3\times 10^{-3}\times 47.7}=4.01 min/pc](https://tex.z-dn.net/?f=%5CRightarrow%20T_m%3D%5Cfrac%7B%5Cpi%20%5Ctimes%2073%20%5Ctimes%20250%5Ctimes%2010%5E%7B-6%7D%7D%7B0.3%5Ctimes%2010%5E%7B-3%7D%5Ctimes%2047.7%7D%3D4.01%20min%2Fpc)
So, the number of parts produced in one tool life
![n_p=\frac {T}{T_m}](https://tex.z-dn.net/?f=n_p%3D%5Cfrac%20%7BT%7D%7BT_m%7D)
![\Rightarrow n_p=\frac {53.4}{4.01}=13.3](https://tex.z-dn.net/?f=%5CRightarrow%20n_p%3D%5Cfrac%20%7B53.4%7D%7B4.01%7D%3D13.3)
Round it to the lower integer
![\Rightarrow n_p=13](https://tex.z-dn.net/?f=%5CRightarrow%20n_p%3D13)
So, the cycle time
min/pc
(d) Cost per production unit:
![C_c= C_mT_c+\frac{C_e}{n_p}](https://tex.z-dn.net/?f=C_c%3D%20C_mT_c%2B%5Cfrac%7BC_e%7D%7Bn_p%7D)
![\Rightarrow C_c=0.667\times6.74+\frac{3.33}{13}=\$4.75/pc](https://tex.z-dn.net/?f=%5CRightarrow%20C_c%3D0.667%5Ctimes6.74%2B%5Cfrac%7B3.33%7D%7B13%7D%3D%5C%244.75%2Fpc)
(e) Total time to complete the batch= Sum of setup time and production time for one batch
.
(f) The proportion of time spent actually cutting metal
![=\frac{50\times4.01}{457}=0.4387=43.87\%](https://tex.z-dn.net/?f=%3D%5Cfrac%7B50%5Ctimes4.01%7D%7B457%7D%3D0.4387%3D43.87%5C%25)
Now, for the cemented carbide tool:
Cost per edge,
![\$8/6=\$1.33/edge](https://tex.z-dn.net/?f=%5C%248%2F6%3D%5C%241.33%2Fedge)
Tool changing time, ![T_t=1min](https://tex.z-dn.net/?f=T_t%3D1min)
![C= 650 m/min](https://tex.z-dn.net/?f=C%3D%20650%20m%2Fmin)
![n=0.30](https://tex.z-dn.net/?f=n%3D0.30)
(a) Cutting speed for the minimum cost:
[from(i)]
(b) Tool life,
[from(ii)]
(c) Cycle time:
![T_c=T_h+T_m+\frac{T_t}{n_p}](https://tex.z-dn.net/?f=T_c%3DT_h%2BT_m%2B%5Cfrac%7BT_t%7D%7Bn_p%7D)
![\Rightarrow T_m=\frac{\pi \times 73 \times 250\times 10^{-6}}{0.3\times 10^{-3}\times 363}=0.53min/pc](https://tex.z-dn.net/?f=%5CRightarrow%20T_m%3D%5Cfrac%7B%5Cpi%20%5Ctimes%2073%20%5Ctimes%20250%5Ctimes%2010%5E%7B-6%7D%7D%7B0.3%5Ctimes%2010%5E%7B-3%7D%5Ctimes%20363%7D%3D0.53min%2Fpc)
![n_p=\frac {7}{0.53}=13.2](https://tex.z-dn.net/?f=n_p%3D%5Cfrac%20%7B7%7D%7B0.53%7D%3D13.2)
[ nearest lower integer]
So, the cycle time
![T_c=2.5+0.53+\frac{1}{13}=3.11 min/pc](https://tex.z-dn.net/?f=T_c%3D2.5%2B0.53%2B%5Cfrac%7B1%7D%7B13%7D%3D3.11%20min%2Fpc)
(d) Cost per production unit:
![C_c= C_mT_c+\frac{C_e}{n_p}](https://tex.z-dn.net/?f=C_c%3D%20C_mT_c%2B%5Cfrac%7BC_e%7D%7Bn_p%7D)
![\Rightarrow C_c=0.667\times3.11+\frac{1.33}{13}=\$2.18/pc](https://tex.z-dn.net/?f=%5CRightarrow%20C_c%3D0.667%5Ctimes3.11%2B%5Cfrac%7B1.33%7D%7B13%7D%3D%5C%242.18%2Fpc)
(e) Total time to complete the batch
.
(f) The proportion of time spent actually cutting metal
![=\frac{50\times0.53}{275.5}=0.0962=9.62\%](https://tex.z-dn.net/?f=%3D%5Cfrac%7B50%5Ctimes0.53%7D%7B275.5%7D%3D0.0962%3D9.62%5C%25)
Similarly, for the ceramic tool:
![T_t-1min](https://tex.z-dn.net/?f=T_t-1min)
![C= 3500 m/min](https://tex.z-dn.net/?f=C%3D%203500%20m%2Fmin)
![n=0.6](https://tex.z-dn.net/?f=n%3D0.6)
(a) Cutting speed:
![V_{opt}= \frac {3500}{\left[ \left(1+\frac{1.67}{0.667}\right)\left(\frac{1}{0.6}-1\right)\right]^{0.6}}](https://tex.z-dn.net/?f=V_%7Bopt%7D%3D%20%5Cfrac%20%7B3500%7D%7B%5Cleft%5B%20%5Cleft%281%2B%5Cfrac%7B1.67%7D%7B0.667%7D%5Cright%29%5Cleft%28%5Cfrac%7B1%7D%7B0.6%7D-1%5Cright%29%5Cright%5D%5E%7B0.6%7D%7D)
![\Rightarrow V_{opt}=2105 m/min](https://tex.z-dn.net/?f=%5CRightarrow%20V_%7Bopt%7D%3D2105%20m%2Fmin)
(b) Tool life,
![T=\left[ \left(1+\frac{1.67}{0.667}\right)\left(\frac{1}{0.6}-1\right)\right]=2.33 min](https://tex.z-dn.net/?f=T%3D%5Cleft%5B%20%5Cleft%281%2B%5Cfrac%7B1.67%7D%7B0.667%7D%5Cright%29%5Cleft%28%5Cfrac%7B1%7D%7B0.6%7D-1%5Cright%29%5Cright%5D%3D2.33%20min)
(c) Cycle time:
![T_c=T_h+T_m+\frac{T_t}{n_p}](https://tex.z-dn.net/?f=T_c%3DT_h%2BT_m%2B%5Cfrac%7BT_t%7D%7Bn_p%7D)
![\Rightarrow T_m=\frac{\pi \times 73 \times 250\times 10^{-6}}{0.3\times 10^{-3}\times 2105}=0.091 min/pc](https://tex.z-dn.net/?f=%5CRightarrow%20T_m%3D%5Cfrac%7B%5Cpi%20%5Ctimes%2073%20%5Ctimes%20250%5Ctimes%2010%5E%7B-6%7D%7D%7B0.3%5Ctimes%2010%5E%7B-3%7D%5Ctimes%202105%7D%3D0.091%20min%2Fpc)
![n_p=\frac {2.33}{0.091}=25.6](https://tex.z-dn.net/?f=n_p%3D%5Cfrac%20%7B2.33%7D%7B0.091%7D%3D25.6)
![\Rightarrow n_p=25 pc/tool\; life](https://tex.z-dn.net/?f=%5CRightarrow%20n_p%3D25%20pc%2Ftool%5C%3B%20life)
So,
![T_c=2.5+0.091+\frac{1}{25}=2.63 min/pc](https://tex.z-dn.net/?f=T_c%3D2.5%2B0.091%2B%5Cfrac%7B1%7D%7B25%7D%3D2.63%20min%2Fpc)
(d) Cost per production unit:
![C_c= C_mT_c+\frac{C_e}{n_p}](https://tex.z-dn.net/?f=C_c%3D%20C_mT_c%2B%5Cfrac%7BC_e%7D%7Bn_p%7D)
![\Rightarrow C_c=0.667\times2.63+\frac{1.67}{25}=$1.82/pc](https://tex.z-dn.net/?f=%5CRightarrow%20C_c%3D0.667%5Ctimes2.63%2B%5Cfrac%7B1.67%7D%7B25%7D%3D%241.82%2Fpc)
(e) Total time to complete the batch
.
(f) The proportion of time spent actually cutting metal
![=\frac{50\times0.091}{251.5}=0.0181=1.81\%](https://tex.z-dn.net/?f=%3D%5Cfrac%7B50%5Ctimes0.091%7D%7B251.5%7D%3D0.0181%3D1.81%5C%25)