page 20

where,

Fc = cutting force (measured) Ft = tangential force (measure) R = resultant of Fc and Ft

Fc

Ft

R

CALCULATE

R

F

N

where,

F = friction force between tool and chip

N = normal force between tool and chip

5.2.1 Force Calculations

5.2.1.1 - Force Calculations

• The forces and angles involved in cutting are drawn below,

page 21

	α
t2
the shear plane
	tool
Fs	φ
t1
Fc
Fn
R
Ft
τ	F
N

Fs = shear force

Fn = force normal to shear plane

α = tool rake angle (positive as shown) φ = shear angle

τ= friction angle

•Having seen the vector based determination of the cutting forces, we can now look at equivalent calculations

page 22

F	= tan τ = µ
---
N

where,

µ = the coefficient of friction

rc

=

t1

---

t2

where,

rc

=

the cutting ratio

(φ -α

)

α

t1

h

t2

tool

φ

t1

= h sin φ

t2 = h cos ( φ –α

)

rc

=

t1

=

h sin φ

=

sin φ

---

h-----------------------------cos ( φ –α

)

------------------------------------------------------cos φ cosα

+ sin φ sinα

t2

rc cos φ

cosα

+ rc sin φ sinα

=

sin φ

r-----------------------------c cos φ

cosα

+ r---------------------------c sin φ

sinα

=

1

sin φ

sin φ

rc cos α

1 – rc sin α

---------------- =

tan φ

tan φ

rc cos α

=

------------------------

1 – rc sin α

page 23

And, by trigonometry,
	F =	Ft cos α	+ Fc sin α	Fs	=	Fc cos φ	– Ft sin φ
	N =	Fc cos α	– Ft sin α	Fn	=	Fc sin φ	+ Ft cos φ

• The velocities are also important, and can be calculated for later use in power calculations. The Velocity diagram below can also be drawn to find cutting velocities.

(90°+α -φ )	(φ -α )		α		**Note: graphical solutions
							are possible
Vf	Vs		tool

Vc φ (90°-α )

where,

Vc = cutting velocity (ft./min.) - as set or measured on the machine Vs = shearing velocity

Vf = frictional velocity

Using the sine rule,
				Vs		=			Vc
		------------------------------sin ( 90° – α			)		----------------------------------------sin ( 90° + α				– φ )
			Vs	Vc sin ( 90°				– α	)	=	V c cos α
				= ----------------------------------------	( 90°		+ α	–φ	)		--------------------------cos ( φ – α )
				sin
Also,
		Vf	=	Vc sin φ
				--------------------------cos ( φ	–α	)

• A final note of interest to readers not completely familiar with vectors, the forces Fc and Ft, are used to find R, from that two other sets of equivalent forces are found.,