Improved multi-order Vold-Kalman filter for order tracking analysis using split cosine and sine terms

Order tracking analysis is an effective tool for the analysis of non-stationary vibration and acoustical signals for the purposes of machinery fault diagnosis and operational modal analysis. The second-generation Vold-Kalman filter is widely used for order tracking analysis; however, the formulation of the data equation is based on incomplete complex kernels, which introduce an imaginary part error in the cost function. In this work, the data equation of the second generation Vold-Kalman filter is re-formulated using separated cosine and sine kernels that necessarily provide a real-valued sum. Doing so will minimize the error in the data equation and offer better optimization scheme to reduce the error in the structural equation and; hence, provide smooth envelopes. Unlike Vold-Kalman method which requires high filter weighting factors to suppress envelope artifacts, the proposed method can achieve high accuracy even when using small weighting factor as deduced from the simulation and experimental investigations. The proposed method can be used to measure the envelopes of close and crossing orders, as well as orders experiencing resonance conditions or orders with high slew rate due to lower filter decay time.