The stiffness of a spindle at high speeds has a significant effect on the quality of the cutting surface and the machining accuracy. However, the spindle stiffness is difficult to be detected directly when the spindle is rotating, and the measured stiffness values are usually coupled with other parts of the spindle–tool system (such as toolholder, spindle–toolholder joint, tool). This paper presents a comprehensive method to deal with the stiffness modeling, identification, and measuring of a rotating spindle. Based on the deflection equation and the principle of superposition, the stiffness model is derived, for a spindle–tool system including a spindle, a specially manufactured toolholder, and a spindle–toolholder joint. A three-step identification algorithm is proposed to decouple and identify the actual spindle stiffness value. First, when the spindle is static, the stiffness values of the shaft, toolholder, and joint are obtained by using the least-squares method. Second, when the spindle is rotating, the stiffness values of the rear bearings and front bearings are identified based on the spindle error analysis method. Third, the stiffness values of the spindle under different rotating conditions are calculated based on the identification results from the previous two steps. Furthermore, the stiffness model and identification algorithm are verified experimentally on an instrumented spindle. The static stiffness value of the same spindle is also measured and compared with the stiffness value under rotating conditions subsequently. This work is useful and can be utilized as a guide for spindle stiffness testing and spindle performance evaluation to spindle manufacturers.

中文翻译：

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旋转主轴的刚度建模、识别和测量

主轴在高速下的刚度对切削表面质量和加工精度有显着影响。然而，在主轴旋转时很难直接检测主轴刚度，并且测量的刚度值通常与主轴-刀具系统的其他部分（如刀架、主轴-刀架接头、刀具）耦合。本文提出了一种处理旋转主轴刚度建模、识别和测量的综合方法。基于挠度方程和叠加原理，推导出主轴-刀具系统的刚度模型，包括主轴、特制刀架和主轴-刀架接头。提出了一种三步识别算法来解耦和识别实际主轴刚度值。第一的，当主轴静止时，轴、刀架和关节的刚度值通过最小二乘法获得。其次，当主轴旋转时，根据主轴误差分析方法确定后轴承和前轴承的刚度值。第三，根据前两步的辨识结果，计算主轴在不同旋转条件下的刚度值。此外，刚度模型和识别算法在仪表主轴上进行了实验验证。还测量了同一主轴的静态刚度值，并与随后的旋转条件下的刚度值进行了比较。这项工作很有用，可以用作主轴刚度测试和主轴制造商主轴性能评估的指南。