2. Brief introduction to Merchant’s Circle.
Assumptions for Merchant’s Circle Diagram.
Construction of Merchant’s Circle.
Solutions of Merchant’s Circle.
Advantages of Merchant’s Circle.
Need for the analysis of cutting forces.
Limitations of Merchant’s Circle.
Conclusion
3. Merchant’s Circle Diagram is
constructed to ease the analysis of
cutting forces acting during
orthogonal (Two Dimensional)
cutting of work piece.
Ernst and Merchant do this
scientific analysis for the first time
in 1941 and gives the following
relation in 1944
It is convenient to determine
various force and angles.
4. 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 only 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.
5. α : Rack angle
Fc: Cutting Force
λ : Frictional angle
Fs: Shear Force
ϕ : Shear angle
F: Frictional Force
Ft : Thrust Force
N: Normal Frictional Force
Fn: Normal Shear Force
V: Feed velocity
Back Rake Angle
Side Rake Angle
Fs
Fn
Fc N φ
Ft
λ
V
R
Front View
F
P
N
F
Normal FrictionForce
Normal Shear Force
FrictionalForce
FrictionForce
RAKE ANGLE
Shear Angle
CuttingForce
ThrustAngle
Resisting the alongnormal metal in
Force on angle madebetweenshear
It is force Angle: provided to tool
act acted angle chip theby
at chip is at angle
Thisis Rake theshear thetheinterface
Backthethe toactsactedbyvelocitythe
ResistanceforcetoolItof the the of
normal force chip. the to and
workpiece face Frictionalalong
workpiece. the of It face ofForce &
resultanttheActsdirectiontoresistshear
cutting tool
plane withthethe normal oftheof the
betweento,ofinterface velocitythe tool
tool
cutting
forming the or the acts the
and is
motion
plane.plane.
Normalof in a by the tool. Normal
Force,
and
travel.provided
measured tool. plane perpendicular
tool.
shear
Reaction. force edge
to the side cuttingincreases as speed
Cutting
Side Rake Angle: It is the as rake
increases and decreases angle
-1
λ = decreases
tan μ
between the face of the tool and
angle
μ: coefficient of friction
measured in a plane perpendicular
to the base
6. Tool edge is sharp.
The work material undergoes deformation across a
thin shear plane.
There is uniform distribution of normal and shear
stress on shear plane.
The work material is rigid and perfectly plastic.
The shear angle ϕ adjusts itself to minimum work.
The friction angle λ remains constant and is
independent of ϕ.
The chip width remains constant.
The chip does not flow to side, or there is no side
spread.
8. 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 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.
9. Knowing Fc , Ft , α and ϕ, all other component forces
can be calculated as:
The coefficient of friction will be then given as :
Fs
α
Fn
On Shear plane,
Fc
Ft φ
λ
Now,
λ-α
α
R
F
N
V
φ
10. Let ϕ be the shear angle
Where,
Fs
Now shear plane angle
α
Fn
Fc
The average stresses on the
shear plane area are:
Ft
φ
λ-α
α
R
λ
F
N
V
φ
11. Now the shear force can be written as:
Fs
and
α
Fn
Fc
Ft
φ
λ
Assuming that λ is independent of ϕ ,
for max. shear stress
λ-α
α
R
F
N
V
φ
12. Analysis of cutting forces is helpful as:-
Design of stiffness etc. for the machine tolerance.
Whether work piece can withstand the cutting force
can be predicted.
In study of behavior and machinability
characterization of the work piece.
Estimation of cutting power consumption, which
also enables selection of the power source(s) during
design of the machine tool.
Condition monitoring of the cutting tools and
machine tool.
13. Proper use of MCD enables the followings :-
Easy, quick and reasonably accurate determination
of several other forces from a few forces involved in
machining.
Friction at chip-tool interface and dynamic yield
shear strength can be easily determined.
Equations relating the different forces are easily
developed.
14. Some limitations of use of MCD are :-
Merchant’s Circle Diagram (MCD) is valid only for
orthogonal cutting.
By the ratio, F/N, the MCD gives apparent (not
actual) coefficient of friction.
It is based on single shear plane theory.
15. Following conclusions/results are drawn from MCD :-
Shear angle is given by
For practical purpose, the following values of ϕ has
been suggested:
ϕ = α for α>15o
ϕ = 15o for α<15o