International Journal of Rotating Machinery

Volume 2015, Article ID 809785, 12 pages

http://dx.doi.org/10.1155/2015/809785

## Rotor-System Log-Decrement Identification Using Short-Time Fourier-Transform Filter

^{1}Beijing Key Laboratory of Health Monitoring and Self-Recovery for High-End Mechanical Equipment, Beijing University of Chemical Technology, Beijing 100029, China^{2}Liaoning Key Lab of Advanced Test Technology for Aerospace Propulsion System, Shenyang Aerospace University, Shenyang 110136, China

Received 27 June 2015; Accepted 12 October 2015

Academic Editor: Robert C. Hendricks

Copyright © 2015 Qihang Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

With the increase of the centrifugal compressor capability, such as large scale LNG and CO_{2} reinjection, the stability margin evaluation is crucial to assure the compressor work in the designed operating conditions in field. Improving the precision of parameter identification of stability is essential and necessary as well. Based on the time-varying characteristics of response vibration during the sine-swept process, a short-time Fourier transform (STFT) filter was introduced to increase the signal-noise ratio and improve the accuracy of the estimated stability parameters. A finite element model was established to simulate the sine-swept process, and the simulated vibration signals were used to study the filtering effect and demonstrate the feasibility to identify the stability parameters by using Multiple-Input and Multiple-Output system identification method that combines the prediction error method and instrumental variable method. Simulation results show that the identification method with STFT filter improves the estimated accuracy much well and makes the curves of frequency response function clearer. Experiment was carried out on a test rig as well, which indicates the identification method is feasible in stability identification, and the results of experiment indicate that STFT filter works very well.

#### 1. Introduction

With the ever increasing capacity in today’s ethylene, refinery, LNG, and high pressure CO_{2} reinjection, it is critically important to optimize the design of rotors, impellers, and journal bearings to improve rotor dynamic stability in field. Definitely, it is crucial to ensure that centrifugal compressors are rotor dynamically stable under operating conditions before they are delivered.

Rotor stability identification methods for rotor-bearing systems have been investigated by many researchers, and they estimated rotor stability during the shop test based on both frequency and time domain signal analysis [1, 2]. Takahashi et al. [1], Kocur and Cloud [2], and Pettinato et al. [3] did a plenty of work on shop rotor dynamic testing by using the electromagnetic bearing as a shaker to excite the rotor system. Cloud [4] developed the Multiple Output Backward Autoregressive (MOBAR) method for time domain signals and carried out significant work on the rotor stability investigation. A stability identification method combining the PEM and IV method to estimate the frequency response function (FRF) in frequency domain was developed by Wang et al. [5], and numerical simulation results show that the method can identify the damping ratio of the first forward and backward modes with high accuracy, even in a severe noise environment; furthermore, two example centrifugal compressors (nine-stage straight-through and six-stage back-to-back) were employed to verify the feasibility of identification method in industrial configurations as well. With this method, Wang et al. [6, 7] carried out experiments to investigate two sets of bearings with preloads 0.1 and 0.3 under different rotating speeds, oil inlet temperatures, and pressure conditions; Wang et al. [8] studied the effect of specific load on rotor stability as well. Recently, a new method, Operational Modal Analysis (OMA) method, which was used in structure modal analysis successfully, is very popular for its reputation of being convenient to use and having no need of any artificial excitation to the rotor system. Guglielmo et al. [9] and Carden and Morosi [10] used the method to evaluate the stability of rotating machinery. The only drawback is that OMA requires the rotor running at steady state condition and a sufficient aerodynamic excitation force acting on the compressor as pointed out by Guglielmo et al. [9].

The key of identification method is of high antinoise ability and identification accuracy, and improving quality of the raw vibration data is crucial to the reliability of the identified results. Also, Kocur and Cloud [2] pointed out that the estimation process faced the challenges that data contain additional response from the significant presence of the immeasurable, internal excitations within the machine such as unbalance, misalignment, aerodynamics when dealing with the measurement data. The traditional filtering method, like band-pass filter, can retain the signal just in interest frequency range and decay other frequency component signals, but the noise signal mixing in the interest frequency band cannot be removed. Particularly, for the sine-swept response signal, its sinusoidal frequency changes over time, so a filter with a fixed bandwidth cannot eliminate the effect of noise in the interest frequency band. Focusing on the time-varying characteristic of the response vibration, a time-frequency filter, such as short-time Fourier transform (STFT) [11], was studied to eliminate other uncorrelated interference and retain the useful signal component. To some extent, STFT is essentially the technique of using Compact Harmonics Wavelets (CHW) for time-frequency analysis and system identification; many researchers vested effort to the development of CHW theory and its applications. Coifman and Guido [12] did some pioneer works about CHW. Daubechies [13] provided lectures about the wavelets and its applications in signal processing and other fields, in which the comparison between STFT and wavelets was mentioned. Farge et al. [14] applied CHW ideas into the computation of two-dimensional turbulent flows, where the wavelet packet best-basis seems to distinguish the low-dimensional dynamically active part of the flow from the high-dimensional passive components, providing a hope of drastically reducing the number of degrees of freedom necessary to the computation. Le [15] did a practicable framework to represent the wavelets transform of linear dynamics delivering a formula for projecting linear time-invariant dynamics and stochastic processes on the wavelet spaces of a multiresolution analysis; applications of this formula to system and parameter identification for control and modeling of mechanical systems were demonstrated. Latest, Yao et al. [16] used the STFT to estimate the linear frequency modulation signal parameters and the experimental results verified its validity and feasibility. Liu et al. [17] developed a variable window based on STFT to study the rotating machinery vibrating signal and its relationship to rotating speed in the process of startup.

In this paper, the STFT filter was introduced to increase the signal-noise ratio and improve the accuracy of the estimated stability parameters. Numerical and experimental study verified its feasibility and effect in identification process, and using the STFT filter made the FRFs clearer and the estimated results more reliable. An improved Multiple-Input and Multiple-Output (MIMO) system identification was adapted to estimate the modal parameters. Studying a stable and accurate identification method provides possibility and quantitative criteria to study the influence of bearing, seal, load, or other operation parameters on the rotor system.

#### 2. STFT Theory

The STFT is used to observe and analyze the signal whose sinusoidal frequency changes over time. For a given signal , it can be intercepted to series by a window function that is slid along the time axis and is nonzero for only a short period of time, and then the intercepted signal series can be transferred into complex form that contains the phase and magnitude information of the signal over time and frequency by the Fourier transform, resulting in a two-dimensional representation of the signal. The transform for the continuous-time case is written as

Making , then is essentially the Fourier transform of . In practice, signal that we obtained is discrete, so it can be broken up into chunks. Fast Fourier transform (FFT) is used to transform the chunks to form a complex matrix, which records magnitude and phase for each point in time and frequency. Mathematically, this can be expressed as where and is discrete because of using FFT in computation.

Like Fourier transform, the STFT is also invertible and the most widely accepted way of inverting the STFT is by using the overlap-add (OLA) method. Selesnick [11] provided a signal reconstruction method to obtain the inverse STFT and chose a half-cycle sine window for analysis.

Window function is expressed aswhere is the length of window and “” can be taken as the scaling factor that determines frequency resolution of each intercepted signal.

And the th intercepted signal can be written as

Then the STFT is , and its inverse form is .

For the original signal , it can be reconstructed to be as

Evidently, once , then ; that is, the signal is a perfect reconstruction. Hence, the condition can be simplified as [11]

The half-cycle sine window in (3) satisfies this condition [11]. Furthermore, many other windows, like rectangular window, can also be designed to do the STFT and inverse STFT. In this paper, a half-cycle sine window was chosen for its less spectral leakage when doing FFT.

The length of the window () determines the scaling factor “” that affects the resolution of resulting FFT. In this paper, the length of time of each intercepted signal was set to be 1 s because the step of increasing frequency was 1 Hz/s, and the resolution was 1 Hz which is enough for analysis and filtering the noise. Because of the linear characteristic of the frequency, the scaling factor was chosen to be constant when decomposing and reconstructing the whole signal.

It is worth pointing out that the STFT idea is essentially the forward CHW using windowed sinusoidal bases for analysis, and that the inverse FFT on such transformed signals is just the typical inverse wavelet transform; thus, many other methods based on CHW can complete the work of STFT as well. But STFT is very simple to use without changing the scaling factor “”.

#### 3. Sine-Swept Response Signal

Logarithm decrement of the rotor system is usually estimated using frequency method by applying the sine sweep force on the rotating rotor system. Because the rotor system is assumed to be a linear system, the frequency of response displacement is theoretically corresponding or equal to that of the exciting force during the test process. Other frequencies appearing in the signal are not needed for the modal identification, and they affect the results that are identified with FRF method.

In practice, the exciting force is not a strict chirp signal, and it is accompanied with little white noise, which can excite the vibration near the natural frequency of rotor system, just like the right rectangular portion in the waterfall graph (see Figure 1(a)). Although the vibration is little in a certain time slice, it can accumulate together to be big enough to affect the identification when doing FFT for the whole time signal. Besides, some interferential and irrelevant frequency vibrations that are not needed are excited as well.