Question

The following figure shows the Merchant's Circle which represent the forces in an orthogonal cutting system. Using the Merchant's circle, prove the following relationships between the unknown cutting forces (F,, Fv). (F,N) and known forces (Fe, Ft) F-E, sin α + Ft cos α N = Fc cos α-Ft sin α E, = E, cos φ-E, sin φ FN-E, sin φ + Ft cos φ F.

Answer 1

Here, F_o = Horizontal force N = Vertical force F_s, N_s - force along and perpendicular to shear plane We have to prove F = Fc Sin alpha + Ft cos alpha N = Fc cos alpha - Ft Sin alpha F_s = fc cos phi - Ft Sin phi Draw the triangle ACD from merchant circle From figure t_s = AG = AF - FG = AF - DH = F_c cos phi - CD Sin phi F_s = F_c cos phi - F_t Sin phi (proved) F_N = CG = CH + HG = CD cos phi + DF = F_t cos phi + AD Sin phi F_N = F_t cos phi + F_c sin phi = F_c sin phi + F_t cos phi (proved) Now draw the tool plane forces from merchant circle F = AB = CJ + JI = CJ + JD F = CD cos alpha + AC Sin alpha F = F_t cos alpha + F_c Sin alpha F = F_c sin alpha + F_t cos alpha (proved) N = CB = AJ - IJ = AD cos alpha - CD sin alpha N = F_c cos alpha - F_t sin alpha (proved)