Development of equations for estimation of cutting forces

The two basic methods of determination of cutting forces and their characteristics are :

(a) Analytical method : enables estimation of cutting forces characteristics : –
• easy, quick and inexpensive
• very approximate and average
• effect of several factors like cutting velocity, cutting fluid action etc. are not revealed
• unable to depict the dynamic characteristics of the forces.

(b) Experimental methods : direct measurement characteristics : –
• quite accurate and provides true picture
• can reveal effect of variation of any parameter on the forces
• depicts both static and dynamic parts of the forces
• needs measuring facilities, expertise and hence expensive.

The equations for analytical estimation of the salient cutting force components are conveniently developed using Merchant’s Circle Diagram (MCD) when it is orthogonal cutting by any single point cutting tool like, in turning, shaping, planning, boring etc.

Development of mathematical expressions for cutting forces under orthogonal turning.

Tangential or main component, P_Z

This can be very conveniently done by using Merchant’s Circle Diagram, MCD, as shown in Fig. Forces involved in machining and contained in Merchant’s Circle.

Fifure: Forces involved in machining and contained in Merchant’s Circle

From the diagram in Fig. Forces involved in machining and contained in Merchant’s Circle,

For brittle work materials, like grey cast iron, usually, 2β_o + η- γ_o = 90º and τs remains almost unchanged.
Then for turning brittle material,

It is difficult to measure chip thickness and evaluate the values of ζ while machining brittle materials and the value of τs is roughly estimated from

τ_s = 0.175 BHN                                                                                         (9.8)

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where, BHN = Brinnel Hardness number.

But most of the engineering materials are ductile in nature and even some semi-brittle materials behave ductile under the cutting condition.

The angle relationship reasonably accurately applicable for ductile metals is

Friction force, F, normal force, N and apparent coefficient of friction μ_a

Again from the MCD in Fig. Forces involved in machining and contained in Merchant’s Circle

Therefore, if P_Z and P_X_Y are known or determined either analytically or experimentally the values of F, N and μa can be determined using equations only.

• Shear force P_s and P_n

Again from the MCD in Fig. Forces involved in machining and contained in Merchant’s Circle

• Cutting forces in turning under oblique cutting

In orthogonal cutting, the chip flows along the orthogonal plane, πo and all the forces concerned, i.e., P_Z, P_X_Y, F, N, P_S and P_n are situated in πo and contained in the MCD. But in oblique cutting the chip flow is deviated from the orthogonal plane and a force develops along the cutting edge and hence MCD (drawn in π_o) is not applicable. However, since it is a single point tool, only one force will really develop which will have one component along the cutting edge in oblique cutting.

Fig. Resolving the cutting force in oblique cutting (turning) shows how the only cutting force, R can be resolved into

Either, P_X, P_Y and P_Z; which are useful for the purpose of measurement and Design of the M – F – T system

or, P_l, P_m and P_n; which are useful for the purpose of design and stress analysis of the tool and determination of chip-tool interaction in oblique cutting when the chip does not flow along π_o.

Figure: Resolving the cutting force in oblique cutting (turning)

For convenience of analysis, the set of force components are shown again in Fig. ‘Resolved components of the cutting force in oblique cutting’ where the cutting force R is resolved into two components R_C and R_r as

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Figure: Resolved components of the cutting force in oblique cutting

From equations 9.28 to 9.31, the following three expressions are attained.

The equation 9.35 is very important and useful for evaluating the force components P_l, P_m and P_n from the measured or known force components P_X, P_Y and P_Z in case of oblique cutting.

By inversion of the Eqn. 9.35, another similar matrix form can be developed which will enable evaluation of P_X, P_Y and P_Z, if required, from P_l, P_m and P_n if known other way.

Under oblique cutting, the coefficient of friction, μa is to be determined from

where, F′ and N′ are to be determined from the values of P_n and P_m as,

Analytical Estimation of cutting forces in drilling and milling

(a) Cutting forces in drilling.

In drilling ductile metals by twist drills, the thrust force, PX and torque, T can be evaluated using the following equations (Shaw and Oxford):

Where, K_x_1, K_x_2 and K_t are constants depending upon the work material. H_B is Brinnel Hardness and d is drill diameter (mm).

As for example, for steels of H_B ≤ 250 and d_c/d = 0.18 [ d_c = chisel edge diameter, mm ]

Eqn. 9.39 and 9.40 become

The drilling torque and thrust can also be roughly evaluated using following simpler equations:

Table “Constant and exponents in drilling” typically shows the approximate values of the constants C_1 and C_2 and the exponents x, y, x′ and y′ for some common engineering materials (Fe-based):

Table “Constant and exponents in drilling”

Table 9.1 Constant and exponents in drilling. Work material C1 C2 x y x′ y′
Plain carbon and low alloy steels 35 ~ 55 85 ~ 160 2.0 0.6 ~ 0.8 1.0 0.7
Cast iron

BHN 150 ~ 190

20 ~ 23 50 1.9 0.8 1.0 0.8

(b) Cutting forces in Plain milling

In plain or slab milling, the average tangential force, PTavg, torque, T and cutting power, PC can be roughly determined irrespective of number of teeth engaged and helix angle, by using the following expressions :



There are several other equations available (developed by researchers) for evaluating milling forces approximately under given cutting conditions.

Needs and Purposes of Measurement of Cutting Forces

In machining industries and R & D sections the cutting forces are desired and required to be measured (by experiments)

• for determining the cutting forces accurately, precisely and reliably (unlike analytical method)
• for determining the magnitude of the cutting forces directly when equations are not available or adequate
• to experimentally verify mathematical models
• to explore and evaluate role or effects of variation of any parameters, involved in machining, on cutting forces, friction and cutting power consumption which cannot be done analytically
• to study the machinability characteristics of any work – tool pair
• to determine and study the shear or fracture strength of the work material under the various machining conditions
• to directly assess the relative performance of any new work material, tool geometry, cutting fluid application and special technique in respect of cutting forces and power consumption
• to predict the cutting tool condition (wear, chipping, fracturing, plastic deformation etc.) from the on-line measured cutting forces.

General methods of measurement of cutting forces

(a) Indirectly
• from cutting power consumption
• by calorimetric method
o inaccurate
o average only
o limited application possibility

(b) Directly: Using tool force dynamometer(s)
o accurate
o precise / detail
o versatile
o more reliable

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