# Machining Forces and Merchant’s Circle Diagram (MCD) | Mechanics of Machining

## Benefit of knowing and purpose of determining cutting forces

The aspects of the cutting forces concerned :
• Magnitude of the cutting forces and their components
• Directions and locations of action of those forces
• Pattern of the forces : static and / or dynamic.

Knowing or determination of the cutting forces facilitate or are required for :
• Estimation of cutting power consumption, which also enables selection of the power source(s) during design of the machine tools
• Structural design of the machine – fixture – tool system
• Evaluation of role of the various machining parameters ( process – $V_C$, $S_o$, t, tool – material and geometry, environment – cutting fluid) on cutting forces
• Study of behaviour and machinability characterisation of the work materials
• Condition monitoring of the cutting tools and machine tools.

## Cutting force components and their significances

The single point cutting tools being used for turning, shaping, planing, slotting, boring etc. are characterised by having only one cutting force during machining. But that force is resolved into two or three components for ease of analysis and exploitation. Fig. ‘Cutting force R resolved into $P_X$, $P_Y$ and $P_Z$‘ visualises how the single cutting force in turning is resolved into three components along the three orthogonal directions; X, Y and Z.

##### figure: Cutting force R resolved into $P_X$, $P_Y$ and $P_Z$

The resolution of the force components in turning can be more conveniently understood from their display in 2-D as shown in Fig. ‘Turning force resolved into $P_Z$, $P_X$ and $P_Y$‘.

##### figure: Turning force resolved into $P_Z$, $P_X$ and $P_Y$

The resultant cutting force, R is resolved as,

In Fig. ‘Cutting force R resolved into $P_X$, $P_Y$ and $P_Z$‘ and Fig. ‘Turning force resolved into $P_Z$, $P_X$ and $P_Y$‘ the force components are shown to be acting on the tool. A similar set of forces also act on the job at the cutting point but in opposite directions as indicated by $P_Z$‘, $P_X$$_Y$‘, $P_X$‘ and $P_Y$‘ in Fig. ‘Turning force resolved into $P_Z$, $P_X$ and $P_Y$‘.

### Significance of $P_Z$, $P_X$ and $P_Y$

$P_Z$ : called the main or major component as it is the largest in magnitude.
It is also called power component as it being acting along and being
multiplied by $V_C$ decides cutting power ($P_Z$.$V_C$) consumption.
$P_y$ : may not be that large in magnitude but is responsible for causing
dimensional inaccuracy and vibration.
$P_X$ : It, even if larger than $P_Y$, is least harmful and hence least significant.

### Cutting forces in drilling

In a drill there are two main cutting edges and a small chisel edge at the centre as shown in Fig. ‘Cutting forces in drilling’.

The force components that develop (Fig. Cutting forces in drilling) during drilling operation are :
• a pair of tangential forces, $P_T$$_1$ and $P_T$$_2$ (equivalent to $P_Z$ in turning) at the main cutting edges
• axial forces $P_X$$_1$ and $P_X$$_2$ acting in the same direction
• a pair of identical radial force components, $P_Y$$_1$ and $P_Y$$_2$
• one additional axial force, $P_X$$_e$ at the chisel edge which also removes material at the centre and under more stringent condition.

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$P_T$$_1$ and $P_T$$_2$ produce the torque, T and causes power consumption $P_C$ as,
T = $P_T$ x ½ (D)                                                                                                                                         (8.3)
and $P_C$= 2πTN                                                                                                                                     (8.4)
where, D = diameter of the drill
and N = speed of the drill in rpm.
The total axial force $P_X$$_T$ which is normally very large in drilling, is provided by
$P_X$$_T$ = $P_X$$_1$ + $P_X$$_2$$P_X$$_e$                                                                                                                             (8.5)
But there is no radial or transverse force as $P_Y$$_1$ and $P_Y$$_2$, being in opposite direction, nullify each other if the tool geometry is perfectly symmetrical.

### Cutting forces in milling

The cutting forces (components) developed in milling with straight fluted slab milling cutter under single tooth engagement are shown in Fig. ‘Cutting forces developed in plain milling’.

The forces provided by a single tooth at its angular position, ψI are :
• Tangential force $P_T$$_i$ (equivalent to $P_Z$ in turning)
• Radial or transverse force, $P_R$$_i$ (equivalent to $P_X$$_Y$ in turning)
• R is the resultant of $P_T$ and $P_R$
• R is again resolved into $P_Z$ and $P_Y$ as indicated in Fig. ‘Cutting forces developed in plain milling’ when Z and Y are the major axes of the milling machine.

Those forces have the following significance:
o $P_T$ governs the torque, T on the cutter or the milling arbour as
T = $P_T$ x D/2                                                                                                                 (8.5)
and also the power consumption, $P_C$ as
$P_C$ = 2πTN                                                                                                                    (8.6)
where, N = rpm of the cutter.

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The other forces, $P_R$, $P_Z$, $P_Y$etc are useful for design of the
Machine – Fixture – Tool system.
In case of multitooth engagement;

Total torque will be D/2.Σ$P_T$$_i$ and total force in Z and Y direction will be Σ$P_Z$ and Σ$P_Y$ respectively.

One additional force i.e. axial force will also develop while milling by helical fluted cutter

## Merchant’s Circle Diagram and its use

In orthogonal cutting when the chip flows along the orthogonal plane, $π_O$, the cutting force (resultant) and its components $P_Z$ and $P_X$$_Y$ remain in the orthogonal plane. Fig. ‘Development of Merchant’s Circle Diagram’ is schematically showing the forces acting on a piece of continuous chip coming out from the shear zone at a constant speed. That chip is apparently in a state of equilibrium.

##### figure: Development of Merchant’s Circle Diagram

The circle(s) drawn taking R or $R_1$ as diameter is called Merchant’s Circle which contains all the force components concerned as intercepts. The two circles with their forces are combined into one circle having all the forces contained in that as shown by the diagram called Merchant’s Circle Diagram (MCD) in Fig. ‘Merchant’ s Circle Diagram with cutting forces’.

##### figure: Merchant’ s Circle Diagram with cutting forces

The significance of the forces displayed in the Merchant’s Circle Diagram are:

$P_S$ – the shear force essentially required to produce or separate the chip from the parent body by shear
$P_n$ – inherently exists along with $P_S$
F – friction force at the chip tool interface
N – force acting normal to the rake surface
$P_Z$ – main force or power component acting in the direction of cutting velocity

The magnitude of $P_s$ provides the yield shear strength of the work material under the cutting condition.

The values of F and the ratio of F and N indicate the nature and degree of interaction like friction at the chip-tool interface. The force components $P_X$, $P_Y$ and $P_Z$ are generally obtained by direct measurement. Again $P_Z$ helps in determining cutting power and specific energy requirement. The force components are also required to design the cutting tool and the machine tool.

## Advantageous use of Merchant’s Circle Diagram (MCD)

Proper use of MCD enables the following:
• Easy, quick and reasonably accurate determination of several other forces from a few known 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

### Some limitations of use of MCD

• Merchant’s Circle Diagram is valid only for orthogonal cutting
• by the ratio, F/N MCD gives apparent (not actual) coefficient of friction
• It is based on a single shear plane theory

The advantageous of constructing and using MCD has been illustrated  as by an example as follows;

Suppose, in a simple straight turning under orthogonal cutting condition with given speed, feed, depth of cut and tool geometry, the only two force components $P_Z$ and $P_X$ are known by experiment i.e., direct measurement, then how can one determine the other relevant forces and machining characteristics easily and quickly without going into much equations and calculations but simply constructing a circle-diagram. This can be done by taking the following sequential steps :

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• Determine $P_X$$_Y$ from $P_X$ = $P_X$$_Y$sinΦ, where  $P_X$ and Φ are known

• Draw the tool and the chip in orthogonal plane with the given value of $γ_o$ as shown in Fig. ‘Cutting forces developed in plain milling (with single tooth engagement)’ Choose a suitable scale (

• Choose a suitable scale (e.g. 100 N = 1 cm) for presenting $P_X$$_Y$ in cm

• Draw $P_Z$ and $P_X$$_Y$ along normal to  as indicated in Fig. ‘Merchant’ s Circle Diagram with cutting forces’

• Draw the cutting force R as the resultant of $P_Z$ and $P_X$$_Y$

• Draw the circle (Merchant’s circle) taking R as diameter

• Get F and N as intercepts in the circle by extending the tool rake surface and joining tips of F and R

• Divide the intercepts of F and N by the scale and get the values of F and N

• For determining $P_s$ (and $P_n$) the value of the shear angle $β_o$ has to be evaluated from

• Draw the shear plane with the value of $β_o$ and then $P_s$ and $P_n$ as intercepts shown in Fig. Merchant’ s Circle Diagram with cutting forces.

• Get the values of $P_s$ and $P_n$ by dividing their corresponding lengths by the scale

• Get the value of apparent coefficient of friction, µ$_a$ at the chip tool
interface simply from the ratio, µ$_a$ = $N/F$

• Get the friction angle, η, if desired, either from tanη = µ$_a$ or directly from the MCD drawn as indicated in Fig. Merchant’ s Circle Diagram with cutting forces.

• Determine dynamic yield shear strength (τs) of the work material under the cutting condition using the simple expression

where, A$_s$ = shear area as indicated in Fig. Shear area in orthogonal turning

### Evaluation of cutting power consumption and specific energy requirement

Cutting power consumption is a quite important issue and it should always be tried to be reduced but without sacrificing MRR.

Cutting power consumption, P$_C$ can be determined from, PC = P$_Z$.V$_C$ + P$_X$.V$_f$

where, V$_f$ = feed velocity
= Ns$_o$/1000 m/min [N=rpm]

Since both P$_X$ and V$_f$, specially V[/latex]_f[/latex] are very small, P$_Z$.V$_C$ can be neglected and then P$_C$ ≅ P$_Z$.V$_C$

Specific energy requirement, which means amount of energy required to remove unit volume of material, is an important machinability characteristics of the work material. Specific energy requirement, U$_s$, which should be tried to be reduced as far as possible, depends not only on the work material but also the process of the machining, such as turning, drilling, grinding etc. and the machining condition, i.e., V$_C$, S$_o$, tool material and geometry and cutting fluid application.

Compared to turning, drilling requires higher specific energy for the same work-tool materials and grinding requires very large amount of specific energy for adverse cutting edge geometry (large negative rake).

Specific energy, U$_s$ is determined from