DETERMINING THE FREQUENCY OF TRANSVERSE OSCILLATIONS OF AN ELASTICALLY FIXED DISK OF A CIRCULAR SAW

Abstract

The object of the study is a circular saw blade. An annular plate of constant thickness with a free outer contour and a fixed inner contour was taken as the calculation scheme of the saw blade. The real conditions for fixing the internal circuit correspond to the elastic fixing of the saw blade with clamping flanges on the shaft of the machine. For the accepted calculation scheme of the circular saw, the dynamic model was a fourth-order nonlinear differential equation of the transverse oscillations of the annular plate with the corresponding boundary conditions. The rotation of the circular saw was taken into account in the dynamic model due to the radial force in the middle surface of the ring plate. This force arises as a result of the action of centrifugal forces during the rotation of the saw blade. The solution to the fourth-order nonlinear differential equation was constructed using the Bubnov-Galyorkin numerical method. The boundary conditions for constructing the solution were as follows: the outer contour of the saw disk was considered free; the inner contour of the saw disk – elastically fixed with a certain stiffness coefficient. The solution was implemented in the Maple 15 mathematical environment in the form of a developed program. According to the obtained frequency equation, the values of cyclic and natural frequencies of transverse oscillations of circular saw disks of different thicknesses with the same radius of the inner contour and three values of the radii of the outer contour were determined: 150 mm, 200 m, and 250 mm. The effect of the rigidity of the internal contour fixing and the angular speed of rotation of the saw blade on the natural frequencies of transverse oscillations was studied. The study was performed for saw disks in the case of oscillations with one, two, and three nodal diameters. It was established that the rigidity of the internal contour of the saw blade has the greatest influence on the natural frequency of transverse oscillations with one nodal diameter