Pre-Filtered Numerical Integration for Machine Tool Vibration Measurements

Accurate knowledge of spindle displacement is essential for assessing the dynamic behavior and machining performance of machine tools, as relative tool–workpiece motion directly affects surface quality and dimensional accuracy. In practice, however, spindle vibrations are commonly measured using accelerometers. Therefore, the displacement has to be calculated from accelerometer data. A major challenge in deriving displacement from acceleration measurements is the accumulation of low-frequency noise and offsets during numerical integration, which leads to unrealistic drift and unphysical displacement signals. As a result, direct double integration of raw acceleration data is not suitable for reliable machine tool vibration analysis with respect to displacements. This paper addresses this problem by applying pre-filtered numerical integration of acceleration signals, following the methodology proposed by Hofmann (2013). High-pass Butterworth filtering with varying cutoff frequencies is applied prior to double integration to suppress low-frequency disturbances while preserving relevant dynamic content. The proposed approach is experimentally validated on a milling machine spindle using broadband impulse hammer excitation. Reconstructed displacement signals are directly compared with reference measurements obtained from a laser interferometer in both the time and frequency domains. The results show that a high-pass cutoff frequency of approximately 10~Hz provides the best agreement with the reference data, enabling physically meaningful spindle displacement reconstruction from accelerometer measurements.