972006 The Time Variant Discrete Fourier Transform as an Order Tracking Method Jason R. Blough and David L. Brown Structural Dynamics Research Laboratory University of Cincinnati Håvard Vold Vold Solutions ABSTRACT 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. Both analytical and experimental data are used to establish the behavioral characteristics of this new method. INTRODUCTION Traditionally two basic methods have been employed to digitally track orders which result from rotating components in noise and vibration problems. These two basic methods are the FFT based methods and the re-sampling based methods. Recently, a Kalman filtering based method has been introduced to track orders in noise and vibration data. This Kalman filtering based method is not presented in this paper because it is a very different approach to estimating the amplitude and phase of orders. This filtering approach does have many advantages which the two traditional digital methods do not, however, it also has several disadvantages such as the determination of the harmonic confidence factor, and computational load and complexity [ref. 1]. A new method which is presented in this paper is a method which, like the FFT, is based on acquiring constant delta-t spaced time data. This approach however, also makes use of angular position information similar to the re-sampling based methods. The method requires a very accurate tachometer signal, as do all of the order tracking methods. The time variant discrete Fourier transform (TVDFT) method which is presented is based upon a discrete Fourier transform which has a kernel whose frequency is not constant. The frequency of this kernel varies with the frequency of the order of interest. The bandwidth of this technique may be either a constant frequency or a constant order width. Through a post-processing calculation with an orthogonality compensation matrix (OCM), the TVDFT may be extended to separate contributions of closely spaced or crossing orders. This analysis is not possible using either an FFT technique or a re-sampling based technique. The limitations of the orthogonality compensation matrix are discussed and are currently a topic of further research. ORDER TRACKING THEORY FAST FOURIER TRANSFORM BASED ORDER TRACKING - Fast Fourier transform (FFT) based order tracking techniques are probably the most commonly used order tracking methods. These methods are all based upon the FFT and require time domain data sampled with a constant delta-t. These methods require an accurate tachometer signal to allow the computation of the exact frequency which the order of interest possesses at each instant in time. FFT techniques follow Shannon’s sampling theorem [ref. 2]. This theorem states the basic relationships between the sampling rate of the data and the frequency range over which the FFT is performed.