OBRABOTKAMETALLOV Vol. 23 No. 3 2021 MATERIAL SCIENCE EQUIPMENT. INSTRUMENTS 7 5 s = 1) and f₁₂ and f₁₃ (n = 2, s = 1) differ by the presence of a nodal circle and a different number of nodal diameters. The pairs f₆ and f₇ (n = 2, s= 0), f₁₅ and f₁₆ (n = 3, s = 0), f₁₉ and f₂₀ (n = 1, s = 1), and mode f₁₄ (n = 0, s = 1) belong to the class of radial modes. These are characterized by tension-compression stresses, in which oscillations of microvolumes occur in the plane of the grinding wheel. The peculiarity of the pairwise manifestation of modes with nodal diameters (n ≠ 0) is emphasized. These modes are called multiple modes since they are vibration modes with close (or coinciding) natural frequencies, the same mode set but different orientations of nodal lines. Multiple modes appear in pairs and are characterized by the relative displacement of nodal diameters by some angle. Such modes occur in systems with a high degree of symmetry (e.g., circular discs, spherical shells, square plates). Their existence has been confirmed by both experimental studies and analytical calculations [23–26]. The smallest number of nodal lines, whether nodal diameters or nodal circles, are characteristic of the lowest modes, i.e., modes formed at the lowest frequencies characteristic of the “grinding wheel” system. As the number of nodal lines manifested in the vibrational motion of a particular mode increases, the frequency at which this mode occurs also increases. It is well known that the lowest modes are of primary importance in the overall dynamics of the vibrational process of an elastic solid. To describe the contribution of each mode, coefficients of modal participation and modal mass have been introduced. These coefficients will be discussed in more detail in “Modal Participation Coefficients”. Modal participation coefficients The participation coefficient indicates the relative contribution of each mode to the displacement or rotation of the system when excited in a specific direction and manner. Since no rotational modes or angular vibrations of the grinding wheel were identified in the computer simulations, the participation coefficients for rotational directions are not considered in this study. Participation coefficients are calculated when it is necessary to determine the parameters of an external load that could potentially cause undesirable resonance in the system [27]. Such calculations make it possible to assess the significance of each mode participating in the vibration process. These modes are characterized by high vibration energies and sensitivity to specific types of loads. After identifying a significant mode in strength calculations, either the system’s operating modes should be changed or the design modernized to avoid undesirable consequences. Fig. 4 shows the graph of participation coefficients of grinding wheel No. 3 – GW 1 600×50×305 25A F60 L 7 V 50 2 class GOST R 52781-2007 – plotted along three coordinate axes. It demonstrates that the most significant eigenmodes of vibration of the grinding wheel are modes f₁ and f₂, which are most pronounced in the X and Y directions. Regarding the Z axis, the largest contribution in this direction is made by the “umbrella” mode f₃. This mode will be used for acoustic monitoring of the grinding process. When applying boundary conditions, the displacement of the grinding wheel model is restricted – it is rigidly fixed along the seat diameter on the machine spindle. Additionally, a prestressing condition distributed over the volume of the grinding wheel is imposed, resulting from the centrifugal forces during rotation at a speed of 1,590 RPM. Fig. 4. Participation factors of the natural vibration modes of the grinding wheel along the coordinate axes
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