For establishing the relationship between measurable and actual forces Merchant’s circle diagram will be used.

- Merchant circle diagram is used to analyze the forces acting in metal cutting.
- The analysis of three forces system, which balance each other for cutting to occur. Each system is a triangle of forces.

### Assumptions made in drawing Merchant’s circle:

- Shear surface is a plane extending upwards from the cutting edge.
- The tool is perfectly sharp and there is no contact along the clearance force.
- The cutting edge is a straight line extending perpendicular to the direction of motion and generates a plane surface as the work moves past it.
- The chip doesn’t flow to either side, that is chip width is constant.
- The depth of cut remains constant.
- Width of the too, is greater than that of the work.
- Work moves with uniform velocity relative tool tip.
- No built up edge is formed.

**The three triangles of forces in merchant’s circle diagram are**

- A triangle of forces for the cutting forces,
- A triangle of forces for the shear forces,
- A triangle of forces for the frictional forces.

Let F = Frictional force

N = Normal to frictional force

F_{s} = Shear force

F_{sn} = Normal to shear force

F_{c} = Cutting force or tangential component of force

F_{t} =Thrust force or feed force

β = Friction angle

μ = Coefficient of friction = tanβ

F_{c} and F_{t} are along and normal to the direction of velocity.

Let R = resultant force

Then resultant force is given by the formula

_{c}

^{2}+ F

_{t}

^{2})

^{0.5 }

R = Diameter of merchant’s circle

F_{t}, F_{c} forces are defined based on actual machining conditions

- From the above merchant’s circle diagram it is found that there are three right angled triangles are present and all the three right angled triangle possessing common hypotenuse (largest side opposite to right angle in a right angled triangle).
- Merchant’s circle is used for establishing relationship between measurable and actual forces.

The above two forces can be measurable by using dynamometer or spring balance but these forces can’t be used in the analysis of machining. Hence to correlate the actual and measurable force the merchant’s circle will be used.

By using a dynamometer, the measurable forces can be measured and by using the merchant’s circle, the actual forces can be calculated. Using this actual force the machining can be analyzed.

From the above circle, it is found that there are three right-angled triangles are present and all will have common hypotenuse. Using this principle the forces can be related as

The resultant force, R = hypotenuse

using the above equations if the force F_{t} and F_{c} are known, the friction angle can be determined, the rake angle is already known from the tool designation and shear angle is known from the chip thickness so that the actual forces F_{s}, F_{sn}, F, N can be determined.

## All Comments

data still is incomplete

You can generate more relationships from figure 1 and figure 2