CCU Course - Fundamental Machining Theory Part 1-Mechanics

Original Author: 楊志慶 Juhchin Yang Version 2.044

2021-10-15


3D Cutting Force Theory


Recap of Previous Learning 2D Merchant’s Model 3D Single Point Insert Orthogonal/ Normal/ Effective Rake Angles Chip Flow Angle


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


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


=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)


Fs

Fn

Fk

Fz n

View from Fm (same as Fp)

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)

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


= 𝑭𝒌𝑭𝒎𝑭𝒔

or𝐶𝑜𝑠 𝛼 0 𝑆𝑖𝑛 𝛼

0 1 0𝑆𝑖𝑛 𝛼 0 𝐶𝑜𝑠 𝛼


View from Fk (same as N)

Fl

Fz

Fc

Fs

FnFp

Fz

Fk

Fm

P

N


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


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


FkFmFs

= 𝟏 𝟎𝟎 𝑺𝒊𝒏 𝜼𝟎 𝑪𝒐𝒔 𝜼

𝑵

𝑷


PFm

Fs

Rake Face


= 𝑪𝒐𝒔 𝝀 𝑺𝒊𝒏 𝝀 𝟎𝑺𝒊𝒏 𝝀 𝑪𝒐𝒔 𝝀 𝟎

𝟎 𝟎 𝟏



= 𝑪𝒐𝒔 𝜶 𝟎 𝑺𝒊𝒏 𝜶

𝟎 𝟏 𝟎𝑺𝒊𝒏 𝜶 𝟎 𝑪𝒐𝒔 𝜶

𝑭𝒌𝑭𝒎𝑭𝒔

Fl

Fc

Fn

Fp

s


𝑪𝒐𝒔 𝝀 𝑺𝒊𝒏 𝝀 𝟎𝑺𝒊𝒏 𝝀 𝑪𝒐𝒔 𝝀 𝟎

𝟎 𝟎 𝟏

𝑪𝒐𝒔 𝜶 𝟎 𝑺𝒊𝒏 𝜶𝟎 𝟏 𝟎

𝑺𝒊𝒏 𝜶 𝟎 𝑪𝒐𝒔 𝜶


𝑵

𝑷

View from FzView from Fm (same as Fp)

Fs

Fn

Fk

Fz n


Fl

Fz

Fc

Fs

FnFp

Fz

Fk

Fm

P

N




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


𝑪𝒐𝒔 𝝀 𝑺𝒊𝒏 𝝀 𝟎𝑺𝒊𝒏 𝝀 𝑪𝒐𝒔 𝝀 𝟎

𝟎 𝟎 𝟏

𝑪𝒐𝒔 𝜶 𝟎 𝑺𝒊𝒏 𝜶𝟎 𝟏 𝟎



𝑵

𝑷

𝑪𝒐𝒔 𝝀 𝑪𝒐𝒔 𝜶 𝑺𝒊𝒏 𝝀 𝑪𝒐𝒔 𝝀 𝑪𝒐𝒔 𝜶𝑺𝒊𝒏 𝝀 𝑪𝒐𝒔 𝜶 𝑪𝒐𝒔 𝝀 𝑺𝒊𝒏 𝝀 𝑺𝒊𝒏 𝜶



𝑵

𝑷


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


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


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


Comparison of Cutting Tool Angle

Tips: Identify Main Cutting Edge Identify Feed Direction Identify Chip Flow Direction Identify Axial/Radial/Tangential Forces


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 (‐)


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


(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


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



𝑲𝒏


(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



𝑲𝒏


(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


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


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


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



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



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


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


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

Documents

CCU Course - Fundamental Machining Theory Part 1-Mechanics