MECHANICS OF METAL CUTTING
(feed
force)
Radial
force
Tool feed
direction
Main
Cutting
force
Topics to be covered





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Tool terminologies and geometry
Orthogonal Vs Oblique cutting
Turning Forces
Velocity diagram
Merchants Circle
Power & Energies
Need for calculating forces, velocities and
angles during machining??
• We need to determine the cutting forces in turning for
Estimation of cutting power consumption, which also enables
selection of the power source (e.g. motors) during design of the
machine tools.
• Structural design of the machine – fixture – tool system.
•
Evaluation of role of the various machining parameters (tool
material and geometry) on cutting forces to make machining
process more efficient and economical.
• Condition monitoring of the cutting tools and machine tools.
Heat Generation Zones
30% (Dependent on m)
(Dependent on f) 60%
Chip
Tool
Workpiece
10%
(Dependent on sharpness
of tool)
https://www.youtube.com/watch?v=bUrp8JMRwx4
Tool Terminology
Side Rake
(SR), +
End Cutting
edge angle
(ECEA)
Facing
Cutting
edge
Nose
Radius
Clearance or end
relief angle
Back
Rake
(BR),+
Turning
Cutting
edge
Side relief
angle
Side cutting
edge angle
(SCEA)
Cutting Geometry
Cutting Geometry
n = Revolutions
per minute
D
MRR  vfd
𝑉 = 𝜋𝐷n
Metal Cutting is the process of removing unwanted material from the workpiece
in the form of chips
 Cutting Edge is normal to tool feed.
 Here only two force components are
considered i.e. cutting force and thrust
force. Hence known as two dimensional
cutting.
 Shear force acts on smaller area.
 Cutting Edge is inclined at an acute
angle to tool feed.
 Here three force components are
considered i.e. cutting force, radial force
and thrust force. Hence known as three
dimensional cutting.
 Shear force acts on larger area.
Assumptions
(Orthogonal Cutting Model)
 The cutting edge is a straight line extending perpendicular
to the direction of motion, and it generates a plane surface
as the work moves past it.
 The tool is perfectly sharp (no contact along the clearance
face).
 The shearing surface is a plane extending upward from
the cutting edge.
 The chip does not flow to either side
 The depth of cut/chip thickness is constant uniform
relative to velocity between work and tool
 Continuous chip, no built-up-edge (BUE)
TERMINOLOGY
 α : Rake angle
F: Frictional Force
 b : Frictional angle
 N: Normal Frictional Force
 ϕ : Shear angle
 V: Cutting velocity
 Ft : Thrust Force
Vc: Chip velocity
Fc: Cutting Force
Vs: Shear velocity
Fs: Shear Force
 Fn: Normal Shear Force
‘Facing’ Forces For Orthogonal Model
DIRECTION OF ROTATION
Velocity of
Tool relative to
workpiece V
F C Tangential Force
WORKPIECE
'Cutting' Force
Fr Radial Force
‘Thrust’ Force
FL
Longitudinal Force
CUTTING TOOL
DIRECTION OF FEED
Note: For the 2D Orthogonal Mechanistic
Model we will ignore the Longitudinal
component
End view
Orthogonal Cutting Model
(Simple 2D mechanistic model)
tc
Chip thickness
Velocity V
Rake
Angle
-
Chip
+
a
tool
Tool
depth of cut
t0
Shear Angle
f
Clearance Angle
Workpiece
Mechanism: Chips produced by the shearing process along the shear plane
Orthogonal Cutting
r
to
ls sin f

tc ls cos(f  a )
tan f 
 
r cos a
1  r sin a
AC AD  DC

 tan( f  a )  cot f
BD
BD
Cutting Ratio
(or chip thickness ratio)
Chip
fa)
B
to
tool
tc
f
A
Workpiece
to
tc
As Sinf =
and Cosf-a) =
AB
AB
t0
sinf
Chip thickness ratio (r) = =
tc cos(fa)
Experimental Determination of
Cutting Ratio
Shear angle f may be obtained
either from photo-micrographs
or assume volume continuity
(no chip density change):
Lc
wc
tc
t0
w0
L0
Since t 0w 0L 0 = t cw cL c and w 0=w c (exp. evidence)
Cutting ratio , r =
t0 Lc
=
tc L0
i.e. Measure length of chips which is easier than measuring thickness
Shear Plane Length
and Angle f
Chip
tool
B
to
fa)
tc
f
A
Workpiece
Shear plane length AB =
t0
sinf
-1 rcosa
Shear pl ane angl e (f) = Tan
1-rsi na
or make an assumption, such as f adjusts to minimize
0
f = 45 + a/2 - b/2 (Merchant)
cutting force:
Shear Velocity
(Chip relative
to workpiece)
Vc = Chip Velocity
(Chip relative to tool)
Velocities
(2D Orthogonal
Model)
V = Cutting Velocity
V
s
Chip
Tool
(Tool relative to
workpiece)
Workpiece
Velocity Diagram
Vc
From mass continuity: Vt o = V ct c
si nf
V c = Vr and V c = V
cos(fa)
From the Velocity diagram:
c osa
Vs = V
cos(fa)
Vs
a
fa
90  f
V
f
Cutting Forces
(2D Orthogonal Cutting)
Chip
Tool
R
F
f
Fs
Fn
R
Workpiece
N
Fc
Ft
R
Dynamometer
Free Body Diagram
Cutting Forces
(2D Orthogonal Cutting)
 Fs , Resistance to shear of the metal in forming the chip. It
acts along the shear plane.
 Fn , ‘Backing up’ force on the chip provided by the
workpiece. Acts normal to the shear plane.
 N, It is at the tool chip interface normal to the cutting face of
the tool and is provided by the tool.
 F, It is the frictional resistance of the tool acting on the chip.
It acts downward against the motion of the chip as it glides
upwards along the tool face.
Fs
α
Fn
Fc
Ft
ϕ
β
β-α
V
ϕ
α
R
F
N
Knowing Fc , Ft , α and ϕ, all other component forces can be calculated.
Force Circle Diagram
(Merchants Circle)
a
Fs
Tool
f
Fc
ba
F
n
F
t
a
R
f
b
N
ba
F
a
Force Circle Diagram
(Merchants Circle)
Cutting Forces
•
Forces considered in orthogonal cutting include
–
•
Cutting force,Fc acts in the direction of the cutting
speed V, and supplies the energy required for cutting
–
•
•
•
Cutting, friction (tool face), and shear forces
Ratio of Fc to cross-sectional area being cut (i.e. product of
width and depth of cut, t0) is called: specific cutting force
Thrust force,Ft acts in a direction normal to the cutting
force
These two forces produces the resultant force, R
On tool face, resultant force can be resolved into:
–
–
Friction force, F along the tool-chip interface
Normal force, N to  to friction force
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Cutting Forces
•
It can also be shown that (b is friction angle)
F  R sin b  N  R cos b
•
•
•
Resultant force, R is balanced by an equal and
opposite force along the shear plane
It is resolved into shear force, Fs and normal force, Fn
Thus,
Fs  Fc cos f  Ft sin f
Fn  Fc sin f  Ft cos f
•
The magnitude of coefficient of friction, m is
m
F Ft  Fc tan a

N Fc  Ft tan a
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Cutting Forces
•
The tool holder, work-holding devices, and machine tool
must be stiff to support thrust force with minimal
deflections
–
–
If Ft is too high ⇒ tool will be pushed away from workpiece
this will reduce depth of cut and dimensional accuracy
•
The effect of rake angle and friction angle on the direction
of thrust force is
Ft  R sin b  a )
•
Magnitude of the cutting force, Fc is always positive as the
force that supplies the work required in cutting
However, Ft can be +ve or –ve; i.e. Ft can be upward with
a) high rake angle, b) low tool-chip friction, or c) both
•
26
Forces from Merchant's Circle
Friction Force F = Fcsina + Ftcosa
Normal Force N = Fccosa - Ftsina
m = F/N and m = tanb typically 0.5 - 2.0)
Shear Force Fs = Fccosf - Ftsinf
Force Normal to Shear plane F n = F csinf + F tcosf
Stresses
On the Shear plane:
Fn
Fnsinf
Normal Stress = s = Normal Force / Area =
=
AB w
tow
Fs
Fssinf
Shear Stress = s = Shear Force / Area =
=
AB w
tow
On the tool rake face:
 = Normal Force / Area =
N
(often assume tc = contact length)
tc w
 = Shear Force / Area =
F
tc w
Power
•Power (or energy consumed per unit time) is the product of force and
velocity. Power at the cutting spindle:
Cutting Power Pc = FcV
•Power is dissipated mainly in the shear zone and on the rake face:
Power for Shearing Ps = FsVs
Friction Power Pf = FVc
•Actual Motor Power requirements will depend on machine efficiency
E (%):
Pc
Motor Power Required =
x 100
E
Material Removal Rate (MRR)
Material Removal Rate (MRR) =
Volume Removed
Time
Volume Removed = Lwto
Time to move a distance L = L/V
Lwto
Therefore, MRR =
= Vwto
L/V
MRR = Cutting velocity x width of cut x depth of cut
Specific Cutting Energy
(or Unit Power)
Energy required to remove a unit volume of material (often quoted as
a function of workpiece material, tool and process:
Ut =
Energy
Energy per unit time
=
Volume Removed Volume Removed per unit time
Cutting Power (Pc)
FcV
Fc
Ut =
=
=
Material Removal Rate (MRR) Vwto wto
Ut = Us + Uf +Uc +Um
Specific Energy for shearing Us =
FsVs
Vwto
FVc
Fr
F
Specific Energy for friction Uf =
=
=
=
Vwto
wto wtc
Cutting Forces and Power measurement
Measuring Cutting Forces and Power
• Cutting forces can be measured using a force
transducer, a dynamometer or a load cell mounted
on the cutting-tool holder
• It is also possible to calculate the cutting force from the
power consumption during cutting (provided
mechanical efficiency of the tool can be determined)
• The specific energy (u) in cutting can be used to
calculate cutting forces
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Cutting Forces and Power
Power




Prediction of forces is
based largely on
experimental data (right)
Wide ranges of values
is due to differences in
material strengths
Sharpness of the tool tip
also influences forces
and power
Duller tools require
higher forces and power
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