4. Mechanics of Grinding
Uncut Chip thickness per grit
f
t1 = mm
ZN
Where
Z = Number of active grains
N = rpm of the wheel
5. Z = π DCb ' Where
D = Diameter of the wheel
C = Surface density of active grains (mm-2)
b’ = Average grain width of cut (mm)
rg = b ' / t1
f
t1 =
π DNCrg
Power
AfU c
W= Where A cross sectional area of the job
60 Uc = Specific energy
Force per single grit
60, 000W 1000 fU c
Fc='
N= N
π DACN π DCN
6. Chip Formation during surface grinding
D
l≈ β
2
D D 2d
Cos β = ( − d ) / = 1 −
2 2 D
β2
Cos β ≈ 1 −
2
l ≈ Dd
1 '
(π NDBC ) × bmax t1max l = fdB
6
7. 6f d
t1max =
π NDrg C D
BfdU c
W= W
60
60, 000W 1000 BfdU c
Fc = =
π ND π ND Components of Grinding Force
Average force per grit
60, 000W
F =
c
'
N
π NDCB Dd
369U o f 0.8 d 0.4 rg0.2 N
Fc' =
N 0.8 D1.2C 0.8
8. Thermal aspects
Energy spent per unit surface area ground
Fcπ ND
θ sα
Bf
Since
−0.4
1
θ sα dU c and U c = U o (t1av ) and t1av = t1max
2
d 0.9 D 0.3C 0.2 N 0.2
θ sα
f 0.2
Grain chip interface temperature
vt1max
θ g = ΘU c
k ρC
19. Ideal roughness in turning
Maximum height of unevenness
where
f
H max = ψ side cutting edge angle
tanψ + cot γ '
γ end cutting edge angle
Maximum height of unevenness, when nose radius (r) is used
f2
H max =
8r
23. Optimizing cutting parameters for Minimum cost
R = R1 + R2 + R3 + R4 + R5
R = Total Cost/ piece
R1 = Material Cost/ piece
R2 = Set up and idle time Cost/ piece
R3 = Machining Cost/ piece
R4 = Tool changing Cost/ piece
R5 = Tool regrinding Cost/ piece
λ 1= Cost/ min of labour and overheads
λ 2= Cost of setting a tool for regrinding
λ3 = Cost/mm of tool ground
ts = Set-up tme and idel time/ piece, min,
tm = Machining time/piece, min,
tct = Tool changing time, min
24. Set- up and idle time cost
R2 = λ1ts
Machining cost
π LD L = Length
R3 = λ1t3 = λ1 D =Diameter
1000 fv f = feed
Tool Changing cost
V = speed
tm
R4 = λ1 tct
T
k
T = 1/ n 1/ m T = Tool life
v f
π LD
R4 = λ1tct v1/ n −1 f 1/ m −1
1000 fv
25. Tool regrinding cost
δ = h f tan vs , hf = flank wear
δ = Minimum length of tool to be reground
λ2 + λ3 = λ2 + λ3h f tanν s
tm
R5 = (λ2 + λ3 h f tan vs )
T
Vs = Clearance angle
π LD
= (λ2 + λ3 h f tan vs ) v1/ n −1 f 1/ m −1
1000k
If tool cost of new tool is A and the total length that can be reground is B mm ,
then cost per mm of the tool
A
λ3 =
⎛ B ⎞
1 + ⎜ h f ⎟
⎝ ta n v s ⎠
26. Total cost per piece
π LD π LD π LD
R = R1 + λ1ts + λ1 + λ1tct v1/ n −1 f 1/ m −1 + (λ2 + λ3 h f tan vs ) v1/ n −1 f 1/ m −1
1000 fv 1000 fv 1000 fv
Optimum speed for a given feed
∂R π LD −2 ⎛ 1 ⎞ π LD 1/ n − 2 1/ m −1
= −λ1 v + (λ1tct + λ2 + λ3h f tan vs ) × ⎜ − 1⎟ v f =0
∂v vopt 1000 f ⎝ n ⎠ 1000k v = vopt
or
n
⎡ nk λ1 ⎤
vopt =⎢ ⎥
⎢ (1 − n) f (λ1tct + λ2 + λ3 h f tanν s ) ⎥
1/ m
⎣ ⎦
27. Optimum speed for minimum cost
n
⎡ nk λ1 ⎤
vopt =⎢ ⎥
⎣ (1 − n) f (λ1tct + λ4 ) ⎦
1/ m
Optimum feed for minimum cost
m
⎡ mk λ1 ⎤
f opt =⎢ ⎥
⎣ (1 − m)v (λ1tct + λ4 ) ⎦
1/ n
f max = 8rH max lim
H maxlim= Limiting value of unevenness
28. Machining force
Fc = 1000U 0 wt10.6
Fc = k1 f 0.6
Power consumption
Variation of machining cost with v and f
W = k1vf 0.6
Maximum available power in the machine then limiting cutting speed-feed
Wlim
vf 0.6
=
k1
Selection of optimum feed
30. Optimum cutting parameters for maximum production
tm
tt = ts + tm + tct min
T
π LD π LD
= ts + + v1/ n −1 f 1/ m −1tct min
1000 fv 1000k
For optimum speed to minimize t1
∂tt π LD −2⎛ 1 ⎞ π LD 1/ n − 2 1/ m −1
= v + ⎜ − 1⎟ v f tct =0
∂v v = vopt 1000 f ⎝ n ⎠ 1000k v = vopt
n
⎡ nk ⎤
vopt =⎢
⎣ (1 − n) f 1/ mtct ⎥
⎦
31. Optimum cutting seed for maximum efficiency
Profit rate
S−R S = Amount received per piece
pr =
tt
R and tt can be expressed in terms of v as before, then
∂pr
=0
∂v v = vopt
