Thermally affected stiffness matrix of angular contact ball bearings in a high-speed spindle system

Abstract

Bearing stiffness directly affects the dynamic characteristics in a high-speed spindle system and plays an important role in terms of manufacturing quality. We developed a new approach for predicting the thermal behavior of a high-speed spindle, calculated the thermal expansion, and generated a bearing stiffness matrix for angular contact ball bearings. The heat convection of spindle housing in air, the balls in lubricant, the spindle shaft in quiescent air, and the bearing inner ring surfaces were determined. Heat sources such as bearing friction, and the heat contributed by the built-in motor, were simulated using an analysis systems (ANSYS) steady-state thermal model. The results were imported into a static ANSYS structural model. Ball thermal expansion was calculated based on changes in the coordinates of nodal points on the ball surface. Finally, a thermally affected bearing stiffness matrix was generated by applying the Newton–Raphson technique. Decreases in the bearing radial, axial, angular, and coupling stiffness values as rotational spindle speed increased were calculated. Also, the stiffness coefficients at a specific speed increased significantly caused by the thermal effects. Finally, for validation, the bearing stiffness was compared to that calculated using an earlier thermal network approach.

Development of a Digital Model for Predicting the Variation in Bearing Preload and Dynamic Characteristics of a Milling Spindle under Thermal Effects

“…Such changes after the low-speed operation are relatively small, but this shows that the thermal temperature rise under high-speed operation affects the dynamic characteristics of the spindle tool. As found, when the spindle temperature increases, the natural vibration frequency decreases, and the dynamic deflection increases, which implies a softening of the spindle system with increasing speed [24,25]. In addition, as shown in Figure 10, the bearing preload positively affects the modal frequency and modal stiffness with a correlation coefficient (R > 0.9), in which the bearing preload was predicted by thermal-mechanical model, reported in Section 6.…”

“…They conducted a Finite Element modeling approach to further explore the impact of bearing preload on the rigidity of the spindle shaft as the temperature rises. In general, the thermal effects on the stiffness of the spindle bearing system have been extensively investigated using thermal-mechanical analysis based on analytical approaches and the Finite Element method [24,25]. For instance, Zahedi et al [24] introduced a simplified thermal-mechanical coupling model for the spindle-bearing system, simplifying the spindle housing and shaft as six-degree-of-freedom Timoshenko beam elements.…”

“…This model was used to analyze the deformation and temperature increase in the spindle at various speeds, revealing a decrease in bearing stiffness and modal frequency with higher speeds. Truong et al [25] studied the impact of thermal effects on bearing stiffness using the Newton-Raphson technique, incorporating thermal radial expansion and centrifugal expansion of rolling balls into their analytical model through Finite Element modeling. The results indicated a significant reduction in all dominant bearing stiffness coefficients at high speeds compared to static stiffness values.…”

Dynamic stiffness and vibration analysis model of angular contact ball bearing considering vibration and friction state variation

“…The result makes clear that external load has a direct effect on bearing stiffness, such as the radial load, axial load, moment load and hybrid load [17][18][19][20][21][22][23]. Some factors, such as centrifugal force [9,24], gyroscopic moment [9,24], centrifugal expansion of ring [9,25], thermal effect [9,21,26,27], number of rolling element [23] and defects JOURNAL OF VIBROENGINEERING [22,28,29], make the bearing stiffness present non-linear and complexity. As the preload increases, nonlinear of the axial stiffness add up, the increment of radial load employs the radial stiffness to be an uneven change [9].…”

“…As the preload increases, nonlinear of the axial stiffness add up, the increment of radial load employs the radial stiffness to be an uneven change [9]. However, the enlargement of rotational speed has radial stiffness, axial stiffness, angular stiffness and coupling stiffness decline, thermal effect cause stiffness notable enlargement [26]. With the magnification of defect size, bearing even radial stiffness along radial and axial loading direction decrease, but increase along unloading direction [22].…”