Lecture-10 Force Relationships in Orthogonal Cutting (Merchant circle)

It is clear from Fig. 10.1 that a number of forces act on the chip during metal cutting. The relationships among these forces were established by Merchant (Merchant’s circle suggested by V.E. Merchant in 1945) with the following assumptions: 

1. Cutting velocity always remains constant. 

2. Cutting edge of the tool remains sharp throughout cutting and there is no contact between the workpiece and tool flank. 

3. There is no sideways flow of chip. 

4. Only continuous chip is produced. 

5. There is no built-up Edge. 

6. No consideration is made of the inertia force of the chip. 

7.The behaviour of chip is like that of a free body which is in the state of a stable equilibrium due to the action of two resultant forces which are equal opposite and collinear. 

However, there were a number of flaws and practical difficulties in these assumptions and that is why they were modified later. 

Figure 10.1 illustrates the forces acting on a chip in Orthogonal Cutting. The forces represented are the following: 

F_s = Metal resistance to shear in chips formation, acting along the shear plane, or Shear Force. 

F_n= Backing up force exerted by the workpiece on the chip, acting normal to the shear plane. 

N = Force exerted by the tool on the chip, acting normal to the tool face. 

F = μN = Frictional resistance of the tool against the chip flow, acting along the tool face, 'μ' being the coefficient of friction between the tool face and the- chip. 

μ = F/N 

Fig. 10.1 Forces acting on a Chip in Orthogonal Cutting. 

These forces are vectorially represented in the free-body diagram shown on the right-hand side in Fig. 10.1. It will be observed that forces Fs and Fn can be easily replaced by their resultant R and force's F and N by their resultant R'. Thus, all these forces are resolved to only two forces R and R'. For equilibrium, these forces R and R' should be equal, act opposite to each other and should be collinear, i.e., 

For the convenience in studying further relationship, the two Triangles of Forces of the above free body diagram have been combined together in Fig. 10.2, called the Merchant's Circle Diagram for Cutting Forces, in which the following new components figure: 

F_c = Horizontal cutting force exerted by the tool on the workpiece. 

F_t = Vertical or tangential force which helps in holding the tool in position and acts on the tool nose. 

These two forces can easily be found out with the help of Strain Gauges or Force Dynamometer. The angle α is a known quantity, being the rake angle of the tool. With the help of the equations given in Lecture 8.1, Eq. 7 & 8, the value of ' φ ' can also be determined. When all these four values, i.e., of Fc, Ft, α, and φ are known, all the other forces can be easily calculated with the help of geometry with reference to Fig. 10.2, as follows: 

Fig. 10.2 Forces acting in Orthogonal Cutting (Merchant's Circle Diagram). 

                                                                                    F = AQ + QB 

                                                                                       = AQ + DC  [·: QB = DC] 

I.e.                                                                               F = Fc sin α + Ft cos α  ...................................................... (1) 

and          

                                                                                    N = QD = PQ - PD 

or,                                                                               N = Fc cos α - Ft sin α ........................................................(2)  

Again,                                                                         F = AH-HK 

                                                                                     =AH-PE 

                                                                                    F = Fc cos φ - Ft sin φ …....................................................(3) 

                                                                                   Fn = CK = CE + EK 

                                                                                       = CE + PH [EK = PH] 

                                                                             Fn = Ft cos φ  + Fc sin φ …................................................................(4) 

From equations (1) and (2) , we have : 

.....................................(8) 

Also, by dividing the numerator and denominator both by cosα, we get: 

.................................................(9) 

From, right angled triangle ABC, we also have: 

..................................................................(10) 

Where, μ = Kinetic coefficient of friction between the upward sliding chip and tool face 

and, 

 = Angle of friction 

By substituting the values of CP, AP and angle PAC, we get: 

...............................................(11)