The Application of Gorman’s Eigen Values to The Industrial Sewing Machine’s Needle Vibration

The free vibration of the sewing needle is divided into lateral free vibrations and an axial free vibration. In this work a theoretical study that concerns the free lateral vibrations will be applied to the sewing needle by the use of Gorman’s Eigen values (Daniel, 1975) technique. The study will be divided into the following: needles with constant cross-section (with classical and non classical boundary conditions) and needles with variable cross-section (conical and stepped). For all the different shapes of needles (Gorman’s classifications) the linear natural frequency in stitches per min (SPM) will be calculated by the use of Gorman’s Eigen values via special tables and graphs. It was found that the linear fundamental natural frequencies of the following: clamped free sewing needle (CF) is 21,548 SMP, clamped simple sewing needle (CS) is 94,522 SMP while for free–free needle (FF) for n = 2 is 137,130 SMP. For each type the Eigen value β was selected due to the sewing needle boundary conditions. The ratio between the lowest (CF) linear natural frequency and the highest (FF) one is 16%. In this work the selected sewing needle material was steel with E = 206 GPa and specific weight 785,000 N/m3.