The Application of the Secant's Equation to the Sewing Machine Needle

The sewing needle of the industrial sewing machines is an essential element. Its' target is to penetrate the sewn fabric layers by a penetrating axial compressive force, which, coincides with the sewing needle geometrical axis. Author wise, it will cause stress on the needle's cross section. Practically, there is always a shift - eccentricity- between the effective action line of the force and the sewing needle geometrical axis. In the present work a mathematical approach has been carried out to study the mathematical relationship between the eccentricity ecr2 and the penetration force (Pa), taking into consideration the critical load (Pcr) (Euler load) of the sewing needle. This relationship is named the “Secant formula”, where it was computerized and graphed. It was found that; the limiting values for the ecr2 was 0.7 and for the ratio Pa/Pcr was 0.8 to make the needle to run in its design stress 538 MPa (steel). When the ratio Pa/Pcr was equal to unity, the sewing needle max stress σmax changed from 101 MPa to 58 GPa i.e. the working penetration force Pa must be far enough from the critical load by about 20%. The required value of Pa must be equal or less than 0.8 Pcr. This work focused on the static case.