Original Author: 楊志慶 Juhchin Yang Version 2.0 44 2021-10-15
Original Author: 楊志慶 Juhchin Yang Version 2.044
2021-10-15
Original Author: 楊志慶 Juhchin Yang Version 2.045
3D Cutting Force Theory
Original Author: 楊志慶 Juhchin Yang Version 2.046
Recap of Previous Learning 2D Merchant’s Model 3D Single Point Insert Orthogonal/ Normal/ Effective Rake Angles Chip Flow Angle
Original Author: 楊志慶 Juhchin Yang Version 2.047
Fl
Fz
Fc Fn
Fp
Fz
Fs
Fk
Fm
PN
Oblique Cutting Force Systems
Fc-Fl-Fz System (Tool in Use) Fc is along with tool moving direction
(parallel to velocity)
Fn-Fp-Fz System (Tool in Hand): Fp is along with the cutting edge direction;
Fz is the same as that in Fc-Fl-Fz system
Fk-Fm-Fs System (Rake Face) Fk is normal to rake face; Fm is along
cutting edge; Fs is on the rake face
N-P System (Normal-Parallel) N is normal to rake face (same as Fk ); P is
on rake face along the chip flow direction
n: Normal Rake Angle (+)s: Inclination Angle (-)c: Chip Flow Rake Angle
Original Author: 楊志慶 Juhchin Yang Version 2.048
View from Fz :
Relationship Among 3D Force Systems
𝑭𝒏𝑭𝒑
= 𝐶𝑜𝑠 𝜆 𝑆𝑖𝑛 𝜆𝑆𝑖𝑛 𝜆 𝐶𝑜𝑠 𝜆
𝑭𝒄𝑭𝒍
or𝑭𝒄𝑭𝒍
= 𝐶𝑜𝑠 𝜆 𝑆𝑖𝑛 𝜆𝑆𝑖𝑛 𝜆 𝐶𝑜𝑠 𝜆
𝑭𝒏𝑭𝒑
Fl
Fc
Fn
Fp
s𝑭𝒏𝑭𝒑𝑭𝒛
= 𝐶𝑜𝑠 𝜆 𝑆𝑖𝑛 𝜆 0𝑆𝑖𝑛 𝜆 𝐶𝑜𝑠 𝜆 0 0 0 1
𝑭𝒄𝑭𝒍𝑭𝒛
or
𝑭𝒄𝑭𝒍𝑭𝒛
=𝐶𝑜𝑠 𝜆 𝑆𝑖𝑛 𝜆 0𝑆𝑖𝑛 𝜆 𝐶𝑜𝑠 𝜆 0
0 0 1
𝑭𝒏𝑭𝒑𝑭𝒛
In 3D matrix:
Fl
Fz
Fc
Fs
FnFp
Fz
Fk
Fm
P
N
s (Negative Inclination Angle)n (Positive Rake Angle) = - n (Chip Flow Angle)
Original Author: 楊志慶 Juhchin Yang Version 2.049
Fs
Fn
Fk
Fz n
View from Fm (same as Fp)
In 3D Matrix:
Relationship Among 3D Force Systems
Fl
Fz
Fc
Fs
FnFp
Fz
Fk
Fm
P
N
s (Negative Inclination Angle)n (Positive Rake Angle) = - n (Chip Flow Angle)
𝑭𝒌𝑭𝒔
= 𝐶𝑜𝑠 𝛼 𝑆𝑖𝑛 𝛼𝑆𝑖𝑛 𝛼 𝐶𝑜𝑠 𝛼
𝑭𝒏𝑭𝒛
𝑭𝒏𝑭𝒛
= 𝐶𝑜𝑠 𝛼 𝑆𝑖𝑛 𝛼𝑆𝑖𝑛 𝛼 𝐶𝑜𝑠 𝛼
𝑭𝒌𝑭𝒔
or
𝑭𝒌𝑭𝒎𝑭𝒔
= 𝑭𝒏𝑭𝒑𝑭𝒛
𝐶𝑜𝑠 𝛼 0 𝑆𝑖𝑛 𝛼0 1 0
𝑆𝑖𝑛 𝛼 0 𝐶𝑜𝑠 𝛼
𝑭𝒏𝑭𝒑𝑭𝒛
= 𝑭𝒌𝑭𝒎𝑭𝒔
or𝐶𝑜𝑠 𝛼 0 𝑆𝑖𝑛 𝛼
0 1 0𝑆𝑖𝑛 𝛼 0 𝐶𝑜𝑠 𝛼
Original Author: 楊志慶 Juhchin Yang Version 2.050
View from Fk (same as N)
Fl
Fz
Fc
Fs
FnFp
Fz
Fk
Fm
P
N
Relationship Among 3D Force Systems
FkFmFs
𝟏 𝟎𝟎 𝑺𝒊𝒏 𝜼𝟎 𝑪𝒐𝒔 𝜼
𝑵
𝑷
s (Negative Inclination Angle)n (Positive Rake Angle) = - n (Negative Chip Flow Angle)
P
Fm
Fs
Rake Face
Original Author: 楊志慶 Juhchin Yang Version 2.051
View from Fz :
View from Fm (same as Fp)
View from Fk (same as N)Fl
Fz
Fc
Fs
FnFp
Fz
Fk
Fm
P
N
Relationship Among 3D Force Systems
FkFmFs
= 𝟏 𝟎𝟎 𝑺𝒊𝒏 𝜼𝟎 𝑪𝒐𝒔 𝜼
𝑵
𝑷
s (Negative Inclination Angle)n (Positive Rake Angle) = - n (Negative Chip Flow Angle)
PFm
Fs
Rake Face
𝑭𝒄𝑭𝒍𝑭𝒛
= 𝑪𝒐𝒔 𝝀 𝑺𝒊𝒏 𝝀 𝟎𝑺𝒊𝒏 𝝀 𝑪𝒐𝒔 𝝀 𝟎
𝟎 𝟎 𝟏
𝑭𝒏𝑭𝒑𝑭𝒛
𝑭𝒏𝑭𝒑𝑭𝒛
= 𝑪𝒐𝒔 𝜶 𝟎 𝑺𝒊𝒏 𝜶
𝟎 𝟏 𝟎𝑺𝒊𝒏 𝜶 𝟎 𝑪𝒐𝒔 𝜶
𝑭𝒌𝑭𝒎𝑭𝒔
Fl
Fc
Fn
Fp
s
𝑭𝒄𝑭𝒍𝑭𝒛
𝑪𝒐𝒔 𝝀 𝑺𝒊𝒏 𝝀 𝟎𝑺𝒊𝒏 𝝀 𝑪𝒐𝒔 𝝀 𝟎
𝟎 𝟎 𝟏
𝑪𝒐𝒔 𝜶 𝟎 𝑺𝒊𝒏 𝜶𝟎 𝟏 𝟎
𝑺𝒊𝒏 𝜶 𝟎 𝑪𝒐𝒔 𝜶
𝟏 𝟎𝟎 𝑺𝒊𝒏 𝜼𝟎 𝑪𝒐𝒔 𝜼
𝑵
𝑷
View from FzView from Fm (same as Fp)
Fs
Fn
Fk
Fz n
Original Author: 楊志慶 Juhchin Yang Version 2.052
Fl
Fz
Fc
Fs
FnFp
Fz
Fk
Fm
P
N
Relationship Among 3D Force Systems
s (Negative Inclination Angle)n (Positive Rake Angle) = - n (Negative Chip Flow Angle)
𝑭𝒄𝑭𝒍𝑭𝒛
𝑪𝒐𝒔 𝝀 𝑪𝒐𝒔 𝜶 𝑺𝒊𝒏 𝝀 𝑺𝒊𝒏 𝜼 𝑪𝒐𝒔 𝝀 𝑺𝒊𝒏 𝜶 𝑪𝒐𝒔 𝜼
𝑺𝒊𝒏 𝝀 𝑪𝒐𝒔 𝜶 𝑪𝒐𝒔 𝝀 𝒔𝒊𝒏 𝜼 𝑺𝒊𝒏 𝝀 𝑺𝒊𝒏 𝜶 𝑪𝒐𝒔 𝜼
𝑺𝒊𝒏 𝜶 𝑪𝒐𝒔 𝜶 𝑪𝒐𝒔 𝜼
𝑵
𝑷
𝑭𝒄𝑭𝒍𝑭𝒛
𝑪𝒐𝒔 𝝀 𝑺𝒊𝒏 𝝀 𝟎𝑺𝒊𝒏 𝝀 𝑪𝒐𝒔 𝝀 𝟎
𝟎 𝟎 𝟏
𝑪𝒐𝒔 𝜶 𝟎 𝑺𝒊𝒏 𝜶𝟎 𝟏 𝟎
𝑺𝒊𝒏 𝜶 𝟎 𝑪𝒐𝒔 𝜶
𝟏 𝟎𝟎 𝑺𝒊𝒏 𝜼𝟎 𝑪𝒐𝒔 𝜼
𝑵
𝑷
𝑪𝒐𝒔 𝝀 𝑪𝒐𝒔 𝜶 𝑺𝒊𝒏 𝝀 𝑪𝒐𝒔 𝝀 𝑪𝒐𝒔 𝜶𝑺𝒊𝒏 𝝀 𝑪𝒐𝒔 𝜶 𝑪𝒐𝒔 𝝀 𝑺𝒊𝒏 𝝀 𝑺𝒊𝒏 𝜶
𝑺𝒊𝒏 𝜶 𝟎 𝑪𝒐𝒔 𝜶
𝟏 𝟎𝟎 𝑺𝒊𝒏 𝜼𝟎 𝑪𝒐𝒔 𝜼
𝑵
𝑷
Original Author: 楊志慶 Juhchin Yang Version 2.053
Physical Definition of N and P
𝑭𝒄
𝑭𝒍
𝑭𝒛
𝑪𝒐𝒔 𝝀 𝑪𝒐𝒔 𝜶 𝑺𝒊𝒏 𝝀 𝑺𝒊𝒏 𝜼 𝑪𝒐𝒔 𝝀 𝑺𝒊𝒏 𝜶 𝑪𝒐𝒔 𝜼
𝑺𝒊𝒏 𝝀 𝑪𝒐𝒔 𝜶 𝑪𝒐𝒔 𝝀 𝒔𝒊𝒏 𝜼 𝑺𝒊𝒏 𝝀 𝑺𝒊𝒏 𝜶 𝑪𝒐𝒔 𝜼
𝑺𝒊𝒏 𝜶 𝑪𝒐𝒔 𝜶 𝑪𝒐𝒔 𝜼
𝑵
𝑷
Fl
Fz
Fc
Fs
FnFp
Fz
Fk
Fm
P
N
WhereKn , Kfr : Specific Normal and Frictional Cutting Pressure
Kn = N (A)chip
Kfr = P N
N: Normal Force (Force normal to the Rake Face)P: Frictional Force (Force sliding on the Rake face)
𝑵
𝑷
𝑲𝒏
𝑲𝒏𝑲𝒇𝒓
(A)chip
Original Author: 楊志慶 Juhchin Yang Version 2.054
Fl
Fz
Fc
Fs
FnFp
Fz
Fk
FmP
N
Discussion 3D Oblique Cutting
Effect of Nose Radius Application Opportunity
Slotting End Turing Broaching Drilling Internal Boring External Turing Face Milling
L
Original Author: 楊志慶 Juhchin Yang Version 2.055
Comparison of Various Cutting OperationsTurning Boring Milling
Ff = Fa (Force in Axial Direction)Fr = Fr (Force in Radial Direction)
Ff = Fr (Force in Radial Direction)Fr = Fa (Force in Axial Direction)
FrFa
Fr
FaFr
Fa
Original Author: 楊志慶 Juhchin Yang Version 2.056
Comparison of Cutting Tool Angle
Tips: Identify Main Cutting Edge Identify Feed Direction Identify Chip Flow Direction Identify Axial/Radial/Tangential Forces
Original Author: 楊志慶 Juhchin Yang Version 2.057
Definition of Positive And Negative Angle1. Check Rotation Direction2. Define the axis of rotation
for axial angle evaluation3. Define a vertical line
pathing center for radial angle evaluation
4. Along the rotation direction, If tool tip is in front of tool
back, the angle is positive If tool back is in front of tip;
it is negative angle
Inclination Angle : ‐ +
Positive (+)
Positive (+)
Positive (+)
Negative (‐)Negative (‐)
Negative (‐)
Original Author: 楊志慶 Juhchin Yang Version 2.058
Basic Turning/Boring Force Equation
Ft Cutting or Tangential ForceFr Radial or Feed ForceFa Axial or Thrust Force
Inclination Angleb Back Rake Angle (BRA)s Side Rake Angle (SRA)n Normal Rake AngleL Side Cutting Edge Angle (SCEA)𝝓 = Effective Lead Angle
𝑭𝒕
𝑭𝒂
𝑭𝒓
𝑪𝒐𝒔 𝜶𝒃 𝑪𝒐𝒔 𝜶𝒏 𝑪𝒐𝒔 𝝓 𝑺𝒊𝒏 𝜶𝒏 𝑺𝒊𝒏 𝜶𝒃 𝑺𝒊𝒏 𝝓
𝑪𝒐𝒔 𝜶𝒃 𝑺𝒊𝒏 𝜶𝒏 𝑪𝒐𝒔 𝝓 𝑪𝒐𝒔 𝜶𝒏
𝑺𝒊𝒏 𝜶𝒃 𝑺𝒊𝒏 𝝓 𝑪𝒐𝒔 𝜶𝒃
𝑲𝒏
𝑲𝒏𝑲𝒇𝒓
(A)chip
𝑻𝒂𝒏 n 𝑻𝒂𝒏(s)𝑪𝒐𝒔(L) + 𝑻𝒂𝒏 b 𝑺𝒊𝒏 L 𝑪𝒐𝒔(𝝀)
𝑻𝒂𝒏 𝝀 𝑻𝒂𝒏(b)𝑪𝒐𝒔(L) - 𝑻𝒂𝒏 s 𝑺𝒊𝒏 L
(A)chip : Uncut Chip Area (aka Chip Load)Kn , Kfr : Specific Normal and Frictional Cutting Pressure
Kn = N (A)chip
Kfr = P N
Feed
Depth of Cut
Axis of Rotation
𝝓Fa
Fr R
(A)chip
Ref: Fu, H., Devor, R. & Kapoor, S., "A Mechanistic Model for the Prediction of the Force System in Face Milling Operations", Journal of Engineering for Industry, Vol. 106, 1984, pp.81
Lead Angle
Original Author: 楊志慶 Juhchin Yang Version 2.059
Comparison of Boring/Face Milling Force Equation
b Back Rake Angle (Radial Rake Angle in Boring r )s Side Rake Angle (Axial Rake Angle in Boring a)L Side Cutting Edge Angle in Turning/Boring
𝑭𝒕
𝑭𝒂
𝑭𝒓
𝑪𝒐𝒔 𝜶𝒃 𝑪𝒐𝒔 𝜶𝒏 𝑪𝒐𝒔 𝝓 𝑺𝒊𝒏 𝜶𝒏 𝑺𝒊𝒏 𝜶𝒃 𝑺𝒊𝒏 𝝓
𝑪𝒐𝒔 𝜶𝒃 𝑺𝒊𝒏 𝜶𝒏 𝑪𝒐𝒔 𝝓 𝑪𝒐𝒔 𝜶𝒏
𝑺𝒊𝒏 𝜶𝒃 𝑺𝒊𝒏 𝝓 𝑪𝒐𝒔 𝜶𝒃
𝑲𝒏
𝑲𝒏𝑲𝒇𝒓
(A)chip
𝑻𝒂𝒏 n 𝑻𝒂𝒏(s)𝑪𝒐𝒔(L) + 𝑻𝒂𝒏 b 𝑺𝒊𝒏 L 𝑪𝒐𝒔(𝝀)𝑻𝒂𝒏 𝝀 𝑻𝒂𝒏(b)𝑪𝒐𝒔(L) - 𝑻𝒂𝒏 s 𝑺𝒊𝒏 L
𝑭𝒕
𝑭𝒓
𝑭𝒂
𝑪𝒐𝒔 𝜶𝒃 𝑪𝒐𝒔 𝜶𝒏 𝑪𝒐𝒔 𝝓 𝑺𝒊𝒏 𝜶𝒏 𝑺𝒊𝒏 𝜶𝒏 𝑺𝒊𝒏 𝝓
𝑪𝒐𝒔 𝜶𝒃 𝑺𝒊𝒏 𝜶𝒏 𝑪𝒐𝒔 𝝓 𝑪𝒐𝒔 𝜶𝒏
𝑺𝒊𝒏 𝜶𝒃 𝑺𝒊𝒏 𝝓 𝑪𝒐𝒔 𝜶𝒃
𝑲𝒏
𝑲𝒏𝑲𝒇𝒓
(A)chip
b Back Rake Angle (Axial Rake Angle in Boring a )s Side Rake Angle (Radial Rake Angle in Boring r)L Side Cutting Edge Angle in Turning/Boring
𝑻𝒂𝒏 n 𝑻𝒂𝒏(s)𝑪𝒐𝒔(L) + 𝑻𝒂𝒏 b 𝑺𝒊𝒏 L 𝑪𝒐𝒔(𝝀)𝑻𝒂𝒏 𝝀 𝑻𝒂𝒏(b)𝑪𝒐𝒔(L) - 𝑻𝒂𝒏 s 𝑺𝒊𝒏 L L
Boring/Turning
Face Milling
Original Author: 楊志慶 Juhchin Yang Version 2.060
Basic Face Milling Force Equation𝑭𝒕
𝑭𝒓
𝑭𝒂
𝑪𝒐𝒔 𝜶𝒂 𝑪𝒐𝒔 𝜶𝒏 𝑪𝒐𝒔 𝝓 𝑺𝒊𝒏 𝜶𝒏 𝑺𝒊𝒏 𝜶𝒂 𝑺𝒊𝒏 𝝓
𝑪𝒐𝒔 𝜶𝒂 𝑺𝒊𝒏 𝜶𝒏 𝑪𝒐𝒔 𝝓 𝑪𝒐𝒔 𝜶𝒏
𝑺𝒊𝒏 𝜶𝒂 𝑺𝒊𝒏 𝝓 𝑪𝒐𝒔 𝜶𝒂
𝑲𝒏
𝑲𝒇
(A)chip
𝑻𝒂𝒏 n 𝑻𝒂𝒏(r)𝑪𝒐𝒔(L) + 𝑻𝒂𝒏 a 𝑺𝒊𝒏 L 𝑪𝒐𝒔(𝝀)
𝑻𝒂𝒏 𝝀 𝑻𝒂𝒏(a)𝑪𝒐𝒔(L) - 𝑻𝒂𝒏 r 𝑺𝒊𝒏 L
Known: a Back Rake Angle (Axial Rake Angle in Milling a ) r Side Rake Angle (Radial Rake Angle in Milling r ) L Lead Angle
Need to Determine: • n Normal Rake Angle• Inclination Angle• 𝝓 = Effective Lead Angle • (A)chip : Uncut Chip Area (Chip Load)• Kn , Kfr : Normal and Frictional Cutting Pressure• Kf = Kn Kfr
L
Kf =Kn Kfr
Original Author: 楊志慶 Juhchin Yang Version 2.061
Definition of Chip Load (A)chip
With Nose RadiusWhen d 𝒓 𝟏 𝒔𝒊𝒏 L :
(A)chip = d x fFeed
d (Depth of Cut)
F
d (Depth of Cut)
r (Nose Radius)
L1
d (Depth of Cut)
L
When d 𝒓 𝟏 𝒔𝒊𝒏 L :
Without Nose Radius
Feed
Depth of Cut
Axis of Rotation
𝝓Fa
FrR
(A)chip
Feed
Axis of Rotation
L1
tavg
Original Author: 楊志慶 Juhchin Yang Version 2.062
𝑯𝑲 ∗ 𝑯𝑱𝟏𝟐𝑮𝑱 ∗ 𝑮𝑷
= 𝐇𝐏𝐉𝐊 𝐆𝐉𝐏
Calculation of Chip Area (Chip Load)
AreaC (Trapezoid GHJK):
When d L
d (Depth of Cut)
L
L : Lead Angle
E
𝑵
r (Nose Radius)
𝑮
𝑱
𝑲𝑯
𝒇𝟐
𝒇𝟐
𝑶
𝑮𝑯 𝑱𝑲 𝑮𝑱𝟐
r 𝑷
Achip = AreaC + AreaE
C
𝒓 sin L L
𝒇 𝒅 𝒓 𝟏 sin L 𝟏𝟒𝒇
𝟐 sin 𝟐L
Original Author: 楊志慶 Juhchin Yang Version 2.063
Calculation of Chip Area (Chip Load)
AreaE : Use Integration Skills
When d L
Achip = AreaC + AreaE
From Cosine Rule:𝑵𝑻𝟐 𝑵𝑶𝟐 + 𝑶𝑻𝟐 𝟐𝑵𝑶 𝑶𝑻 𝒄𝒐𝒔 𝜽
𝒅𝑨 𝑬 = 𝒅𝑨 𝑬𝜽𝟐𝜽𝟏
= 𝒅𝒔 𝒕 𝜽 𝑬𝜽𝟐𝜽𝟏
= 𝒓𝒕 𝜽 𝒅𝜽 𝑬𝜽𝟐𝜽𝟏
𝑶𝑻𝟐 𝑵𝑻𝟐 𝑵𝑶𝟐 𝟐𝑵𝑶𝑶𝑻 𝒄𝒐𝒔 𝜽
𝒓 𝒕 𝜽 𝟐 𝒓𝟐 𝒇𝟐 𝟐𝒇 𝒓 𝒕 𝜽 𝒄 𝒐𝒔 𝜽
𝒕 𝜽 𝒓 𝒇𝒄𝒐𝒔 𝜽 𝒓𝟐 𝒇𝟐𝒔𝒊𝒏𝟐 𝜽
t()
𝑻
ds
𝜽
AreaE = 𝒓 𝒓 𝒇𝒄𝒐𝒔 𝜽 𝒓𝟐 𝒇𝟐𝒔𝒊𝒏𝟐 𝜽 𝒅𝜽𝜽𝟐𝜽𝟏
Where 𝜽𝟏 𝒄𝒐𝒔 𝟏 𝒇𝟐𝒓
; 𝜽𝟐 𝝅 L Where ON = feed ; OT= r - t() ; NT = r
Original Author: 楊志慶 Juhchin Yang Version 2.064
Calculation of Chip Area (Chip Load)
Region D : Considering ONT, Cosine Rule: 𝑵𝑻𝟐 𝑵𝑶𝟐 + 𝑶𝑻𝟐 𝟐𝑵𝑶 𝑶𝑻 𝒄𝒐𝒔 𝜽
AreaD 𝒅𝑨 𝑫 = 𝒅𝑨 𝑫𝜽𝟐𝜽𝟏
= 𝒓𝒕 𝜽 𝒅𝜽 𝑫𝜽𝟐𝜽𝟏
𝑶𝑻𝟐 𝑵𝑻𝟐 𝑵𝑶𝟐 𝟐𝑵𝑶 𝑶𝑻 𝒄𝒐𝒔 𝜽𝒓 𝒕 𝜽 𝟐 𝒓𝟐 𝒇𝟐 𝟐𝒇 𝒓 𝒕 𝜽 𝒄 𝒐𝒔 𝜽
𝒕 𝜽 𝒓 𝒇𝒄𝒐𝒔 𝜽 𝒓𝟐 𝒇𝟐𝒔𝒊𝒏𝟐 𝜽 r (Nose
Radius)
d (Depth of Cut)
dsBD
t()
𝑶
𝑻
𝑵
𝒅𝑨 ds t r d t Where NO = feed ; OT= r - t() ; NT = r
r 𝒅
Region B :Considering MNS, Cosine Rule:𝒕 𝜽 𝒓 𝑶𝑺 𝒓 𝒓 𝒅 𝒄𝒔𝒄 𝝅 𝜽
AreaB 𝒅𝑨 𝑩 = 𝒅𝑨 𝑩𝜽𝟑𝜽𝟐
= 𝒓𝒕 𝜽 𝒅𝜽 𝑩𝜽𝟑𝜽𝟐
𝜽𝟏 𝒄𝒐𝒔 𝟏 𝒇𝟐𝒓 ; 𝜽𝟐 𝝅 𝒕𝒂𝒏 𝟏 𝒓 𝒅
𝟐𝒓𝒅 𝒅𝟐 𝒇; 𝜽𝟑 𝝅 𝒔𝒊𝒏 𝟏 𝒓 𝒅
𝒓
When d L
Achip = AreaB + AreaD
𝑺
r
𝑴
Original Author: 楊志慶 Juhchin Yang Version 2.065
Summary of Chip Load (A)chip
With Nose Radius: When d 𝒓 𝟏 𝒔𝒊𝒏 L :
(A)chip = d x f
Feed
d (Depth of Cut)
F
When d 𝒓 𝟏 𝒔𝒊𝒏 L :
Without Nose Radius :
FeedAxis of Rotation
(A)chip = AreaC + AreaE
(A)chip = AreaB + AreaD
AreaE = 𝒓 𝒓 𝒇𝒄𝒐𝒔 𝜽 𝒓𝟐 𝒇𝟐𝒔𝒊𝒏𝟐 𝜽 𝒅𝜽𝜽𝟐𝜽𝟏
AreaC 𝒇 𝒅 𝒓 𝟏 sin L 𝟏𝟒𝒇
𝟐 sin 𝟐L
AreaD 𝒅𝑨 𝑫 = 𝒅𝑨 𝑫𝜽𝟐𝜽𝟏
= 𝒓𝒕 𝜽 𝒅𝜽 𝑫𝜽𝟐𝜽𝟏
AreaB 𝒅𝑨 𝑩 = 𝒅𝑨 𝑩𝜽𝟑𝜽𝟐
= 𝒓𝒕 𝜽 𝒅𝜽 𝑩𝜽𝟑𝜽𝟐
𝜽𝟏 𝒄𝒐𝒔 𝟏 𝒇𝟐𝒓 ; 𝜽𝟐 𝝅 𝒕𝒂𝒏 𝟏 𝒓 𝒅
𝟐𝒓𝒅 𝒅𝟐 𝒇; 𝜽𝟑 𝝅 𝒔𝒊𝒏 𝟏 𝒓 𝒅
𝒓
Where 𝜽𝟏 𝒄𝒐𝒔 𝟏 𝒇𝟐𝒓
; 𝜽𝟐 𝝅 L
Original Author: 楊志慶 Juhchin Yang Version 2.066
Homework # 3 Due by 2021/11/051) Derive the equations for Chip Area Calculation and show your work Step-by-Step2) Program your equations to calculate the Chip Area for the condition as listed in
the attached excel file. Please approach to 4th digit after dismal point.3) Compare your results with simple approach (feed * Depth of Cut)4) Conclude your findings and comments