Recovering cutting tool modal parameters from fractionally uncorrelated and potentially aliased signals

Abstract

Modal parameters characterize how tools vibrate. Correct evaluation of modal parameters depends on how signals are sampled. Since tools can vibrate with frequencies of up to several kHz, respecting the Nyquist criterion requires sampling potentially at tens of kHz. This is easy enough with modern data acquisition systems. However, if/when using modal parameters to monitor condition of tools, transmitting, storing, and processing large data sets becomes difficult. Moreover, when extracting modal parameters using newer vision-based methods, it may not always be possible to acquire high resolution images at rates that avoid aliasing. This paper present solutions to address such cases by recovering modal parameters from signals sampled potentially below the Nyquist limit. No a priori knowledge of the system order is assumed, and folding properties of signals are leveraged to recover parameters from fractionally uncorrelated signals using notions of set theory. To aid recovery, we suggest formal procedures to group candidates of likely modes together and resolve the case of modes being potentially confounded. Special spatiotemporal decoupling properties inherent to modal analysis are leveraged to recover eigenvectors from potentially aliased signals. Recovery is illustrated using the eigensystem realization algorithm. Numerical experiments with systems of different orders, of signals with noise, and of systems in which likely modes can be confounded with other likely candidates are designed to test robustness of the method. Those findings guide experiments with measured accelerations and video of an end mill. Results confirm that parameters recovered using proposed methods agree with those extracted from accelerations and videos sampled properly.