388 E. Di Lorenzo et al. the experimental modal model of a structure with rotating parts by applying a two-steps procedure. First of all, the so-called orders are extracted and then the OMA procedure can be applied on the orders instead than on the spectra. Order domain analysis relates the vibration signal to the rotating speed of the system. In this way, vibration components that are proportional to multiples of this speed can be easily identified. The next sections will describe the implemented techniques and afterwards the gear test rig and its instrumentation will be introduced. Finally, the results will be discussed. This analysis will be mainly focusing on the test rig behavior and on how the whole structure is interacting with the gears. In the future, the rotating accelerometers mounted on the gears will be included in the analysis and their signals will be processed in order to have some more insights into gear dynamics. 36.2 Order Tracking Techniques Order tracking is the estimation of the amplitude and phase of the response of a machine to a referenced rotating components that is allowed to vary in amplitude and frequency over the time. Regardless of the method used for performing the order tracking step, the most important channel is the tachometer signal which can be measured directly or generated virtually by using a response channel. Since the response amplitude and phase estimation of an order is referenced to a rotating component, it is obvious that the accuracy of the estimation is strictly dependent on the accuracy of the tachometer signal. Several techniques for extracting the orders can be found in the literature [3]. The Fourier transform methods are based on time domain data sampled with a constant time interval t. The two main assumptions of these methods are that the data has constant frequency across the length of the data block and that also amplitude and phase are constant across the same data block. The effect of variations will be reflected in the presence of leakage into the transform. The evolution of these methods into the first computed order tracking method was developed and patented by Potter at Hewlett Packard [4]. This method is based on the resampling of the data from the time domain to the angle domain which means from data with constant t to data with constant angular spacing . The Fourier transform is applied to the angle domain data to transform it into the order domain data. Orders that sweep through several frequencies in the frequency domain will be constant in the order domain allowing a leakage free amplitude and phase estimation. This is true only if the estimation is performed over an integral number of revolutions. Some more methods for extracting orders have been implemented. In this paper two methods have been used and a comparison with the classical techniques have been performed. The first method was proposed by Blough [5] and it has been named Time Variant Discrete Fourier Transform (TVDFT) method. It is defined as a discrete Fourier transform with a kernel that allows the frequency to vary in time, but amplitude of the data is assumed to remain constant across the data block. Also TVDFT allows a leakage free estimates of orders, but there is no need to resample the data from time to angle domain which allows the technique to be much less computational demanding and suitable for real-time processing. Borghesani [6] proposed the velocity synchronous discrete Fourier transform (VSDFT) for order tracking in the field of rotating machinery and compared its performances with the ones of TVDFT underlining advantages and drawbacks of the two methods. A method developed as a post-processing technique has been named Vold-Kalman filter since it was developed by Vold and Leuridan as an extension of the classical Kalman filter [7]. It requires information about “future” data points when estimating an order and it allows extracting close and crossing orders. Another method developed for post-processing purposes is the Gabor transform which has been compared to the Vold-Kalman filter and to other techniques as well [8]. 36.2.1 Angle Domain Computed Order Tracking (AD) A very well-known order tracking method which is widely used in commercial software is based on the angular resampling procedure. Data are acquired with a uniform t and then resampled to the angle domain through the use of an adaptive digital resampling algorithm. The final result of the technique is that uniform t data become uniformly spaced angle data. Estimates of amplitude and phase of the orders are obtained by processing these data by means of a Discrete Fourier Transform (DFT) instead of a FFT for computational flexibility in performing the transform without being restricted to a power of two samples. In order to perform the transformation from time domain data to angle domain data, a reference signal has to be identified to define the instant of time in which the uniform angular intervals have been spaced. Typically, this signal is considered to be the tachometer signal measured on a reference shaft of the operating machine. The kernels of the Fourier transform are reformulated as shown in Eq. (36.1) where om is the order which is being analyzed.
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