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Experimentalists are familiar with the aliasing that happens in data acquisition when the sampling rate is less than twice the highest frequency of energy in the signal to be sampled. Much effort has been made over the years using a combination of analog and digital filters to make sure that the higher frequencies are filtered out to avoid or minimize the effect of this aliasing. Much less talked about is the aliasing that occurs in modal parameter estimation, or curvefitting, when the residual effects of out of band modes violate the assumptions of the finite dimensional parametric model that the experimentalist uses to curvefit the acquired digitized data. While the out of band energy has been filtered out of the now band limited data, the tails, sometimes called residual flexibility and inertial restraint of the out of band modes are still present in the data.

Vold and Leuridan [1] introduced in 1993 an algorithm for high resolution, slew rate independent order tracking based on the concepts of Kalman filters [5, 6]. The algorithm has been highly successful as implemented in a commercial software system in solving data analysis problems previously intractable with other analysis methods. At the same time certain deficiencies have surfaced, prompting the development of an improved formulation. This paper presents for the first time the second generation algorithm and its theoretical foundations. The new algorithm allows for the simultaneous estimation of multiple orders, effectively decoupling close and crossing orders. This is especially important for acoustics applications, where order crossings cause transient beating events. The algorithm now allows for a much wider range of filter shapes, such that signals with sideband modulations are processed with high fidelity.

Present order tracking methods for solving noise and vibration problems are reviewed, both FFT and re-sampling based order tracking methods. The time variant discrete Fourier transform (TVDFT) is developed as an alternative order tracking method. This method contains many advantages which the current order tracking methods do not possess. This method has the advantage of being very computationally efficient as well as the ability to minimize leakage errors. The basic TVDFT method may also be extended to a more complex method through the use of an orthogonality compensation matrix (OCM) which can separate closely spaced orders as well as separate the contributions of crossing orders. The basic TVDFT is a combination of the FFT and the re-sampling based methods. This method can be formulated in several different manners, one of which will give results matching the re-sampling based methods very closely.

The use of Kalman filter methods for high-performance order tracking of noise and vibration signals was introduced in 1993. Based on experience with that original formulation, further work has produced significant enhancements which greatly extend the ability of these methods to deal with several practical issues of concern in vehicle testing. This paper reports on advances in the areas of: RPM estimation accuracy, even for fast-changing events such as gear shifts; Higher order Kalman filters, with improved shapes for extracting modulated orders; Decoupling of close and even crossing orders by use of multiple RPM references; Significant speed improvement over the original algorithm. Besides obtaining the magnitude and phase of selected orders as a function of time or RPM, the harmonic content may be extracted as time-histories, with no phase or leakage distortion.

Kalman filters have been employed very successfully in control and guidance systems since the sixties, with particular application to avionics and navigation. These filters can track accurately signals of known structure among noise and other signal components with different structure. This paper investigates the application of nonstationary Kalman filters to track harmonic components. This approach enables the analysis of harmonic components in signals even when the rate of change of frequency, or slew rate, is high. A Kalman filter formulation to track harmonic components in data records sampled with constant frequency is first reviewed. Particular attention is paid to the characteristics of the filters in terms of filter resolution and harmonic separation, as well as ability to handle higher slew rates. The applicability of the method is demonstrated for analyzing driveline harshness problem.

The analysis of the periodic components in noise and vibration signals measured on rotating equipment, like car power trains, must more and more be done under rapid changes of an axle, or reference RPM. Normal tracking filters (analog, or digital implementations) have limited resolution in such situations; wavelet methods, even when applied after resampling the data to be proportional to an axle RPM, must compromise between time and frequency resolution. The authors propose the application of nonstationary Kalman filters for the tracking of periodic components in such noise and vibration signals. These filters are designed to track accurately signals with a known structure among noise and signal components of different, ‘unknown’, structure.

The most damaging events controlling the low cycle fatigue life of a mechanical component are typically characterized by a relatively few, large amplitude, low frequency loadings where, for this paper, signals below 60 to 100Hz are considered low frequency. These are usually accompanied by a large number of small amplitude, high frequency loadings that may contribute little or no damage and often are merely a measure of system noise. Over sampling by factors of five to ten relative to the sampling theorem is often recommended to achieve the time domain resolution necessary for an accurate capture of peaks and valleys used in component fatigue evaluation. This over sampling limits the elapsed time capacity of digital recording devices used for field tests, slows data transfer rates, and expands database storage requirements.