390 E. Di Lorenzo et al. orders generated from the machine and other random noise present in the data. The random noise and non-tracked orders are combined into the signal .n/. y.n/ Dx.n/ C .n/ (36.4) A weighted solution can be obtained by introducing the Harmonic Confidence Factor (HCF) r. The value of this parameter is what determines the tracking characteristics of the filter. It is calculated, as shown in Eq. (36.5) as the ratio between the standard deviations of the structure and data equations. r.n/ D s".n/ s .n/ (36.5) The choice of a large value for the weighting factor r leads to a highly selective filtration in the frequency domain, while by choosingr small the resolution in frequency is very small, but a fast convergence in amplitude can be obtained. Applying the ratio as a weighting function and combining together the two equations, the set of linear equations shown in Eq. (36.6) is obtained. 1 2cos.! t/ 1 0 0 r.n/ 2 4 x.n 2/ x.n 1/ x.n// 3 5D ".n/ r.n/y.n/ .n// (36.6) Applying Eq. (36.6) to all observed time points will give a global system of over determined equations for the desired waveformx.n/ which may be solved by using standard least squares techniques such as normal equations or the singular value decomposition. For the purpose of order tracking, the filtered waveform is most conveniently described in terms of amplitude and phase with respect to a reference channel such as the tachometer channel. 36.3 Order-Based Modal Analysis Operational Modal Analysis (OMA) algorithms, such as Operational Polymax [9], allows the identification of the modal parameters of a structure by taking into account only operational measurements. Previous studies have demonstrated that the classical OMA has some drawbacks when it has to be applied in the rotating machinery field. Some of the peaks in the overall spectrum are originated from order components that suddenly stop at the maximum rpm. These components are identified as poles, while they are not physically present in the system. They have been named “end-of-order” related poles. Order-Based Modal Analysis (OBMA) estimates the modal parameters of a structure during a run-up or a run-down test by applying the curve fitting algorithms to the extracted orders instead than to the overall spectra. It can be assumed that the measured responses are mainly caused by the rotational excitation. Run-up and run-down can be assimilated to multi-sine sweep excitation in the frequency band of interest. Several observations need to be taken into account for applying the same OMA algorithms to the orders: 1. Displacement orders are proportional to the squared rotation speed and acceleration orders are proportional to the forth power of the same rotation speed. The main difference is that in the classical modal analysis the acceleration FRFs are proportional to the squared frequency axis. 2. Complex upper and lower residuals, while in classical modal analysis they are real. 3. Complex participation factor both in classical modal analysis and in order-based modal analysis. Methods such as Operational Polymax are robust again these observations and they can be employed for estimating the modal parameters in case of rotating machineries by looking at the orders rather than at the spectra. Compared to other applications of order tracking, OBMA requires that the orders are calculated with respect to a reference signal ‚.t/ which is synchronous with the excitation coming from the rotating source. This signal has been chosen as a sine sweep with frequency equal to the instantaneous rotational speed!.t/ of the machine multiplied by the order om which is being extracted, as shown in Eq. (36.7). ‚.t/ Dcos Z T 0 om !.t/dt (36.7)
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