A parametric transmissibility-based method for identifying in-process FRFs of milling spindle–tool systems under measurement noise

Existing transmissibility-based operational modal analysis (TOMA) methods estimate in-process frequency response functions (FRFs) of milling spindle–tool systems directly from cutting responses, yet their accuracy is limited by the absence of explicit measurement noise modeling, resulting in biased estimates. This study introduces a parametric TOMA method tailored for identifying in-process FRFs in milling systems. The method explicitly addresses measurement noise by acquiring acceleration responses at multiple locations on non-rotating spindle components under actual cutting excitation. To counteract noise—modeled as white noise—a Frisch scheme integrated with high-order Yule–Walker (HOYW) equations is developed. By leveraging statistical independence between noise and the noise-free system response, the approach effectively isolates and suppresses noise while extracting true system dynamics. The transmissibility function is formulated via a polynomial matrix from a left matrix fraction description, enabling system pole identification through singular value decomposition and companion matrix techniques. Subsequently, in-process FRFs are reconstructed using the identified poles and residues obtained from impact testing. Milling tests demonstrate close agreement between identified and measured FRFs, enabling accurate stability lobe diagram (SLD) prediction and validating the method’s effectiveness and practicality.