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Advances in Mechanical Engineering

Volume 2013 (2013), Article ID 218576, 7 pages

http://dx.doi.org/10.1155/2013/218576

## Synthetic Amplitude Spectrum and Its Extensions for Analyzing the Two Perpendicular Directional Vibration Displacement Signals of a Rotating Rotor

School of Electronic Science and Engineering, Nanjing University, Nanjing 210093, China

Received 28 July 2012; Revised 13 January 2013; Accepted 14 January 2013

Academic Editor: A. Seshadri Sekhar

Copyright © 2013 Liang Yonggang 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

Classical Amplitude Spectrum analysis and Full Amplitude Spectrum analysis exhibit deficiencies in analyzing the two perpendicular directional vibration displacement signals of a rotating rotor. The shape of Classical Amplitude Spectrum is influenced by the installing position of its sensor. Neither Classical Amplitude Spectrum nor Full Amplitude Spectrum can indicate the actual radial rotor vibration amplitude on every frequency. Therefore, the previous two methods are not convenient to be used in rotating machine diagnoses. To solve these problems, this paper proposes a new rotor vibration analyzing tool here called Synthetic Amplitude Spectrum (SAS). The paper discusses the principle of SAS analysis, provides the specific making process of SAS, and applies it to two other current important analyzing methods in rotating machine diagnoses, resulting in two SAS extensions. The two extensions include a short-time SAS array tool for rotor vibration time-frequency analysis and a SAS waterfall plot tool for analyzing rotor vibration during machine startup or shutdown. The experiments and theoretical analysis showed that SAS and its two extension methods are *not* influenced by the installation position of the two sensors, and each amplitude of the spectrums can represent the actual radial rotor vibration amplitude on the frequency.

#### 1. Introduction

Large rotating rotors found in large motors and centrifugal compressors, and so on, are usually supported on various forms of sliding bearings. They often have two vibration displacement sensors placed at each sliding bearing with angle of 90 degrees between each other, as in Figure 1, called Sensor and Sensor , respectively. The two displacement sensors are set up to measure the running condition of the rotor so that the conclusion on whether the whole machine is healthy can be drawn. The two discretized displacement signals from sensors and , here called and signal sequences, respectively, are always acquired synchronously. The most basic tools used in analyzing the and signal sequences are amplitude spectrums.

The most well-known amplitude spectrum, here called Classical Amplitude Spectrum (CAS), is usually obtained as follows: process and signal sequences using FFT, respectively, get two corresponding Fourier series of them, and then take the amplitudes and the frequencies of the two series to plot their respective amplitude spectrums. The shortcoming of Classical Amplitude Spectrum is that if the installation mode of two sensors changes from Figure 1(a) to Figure 1(b), the obtained spectrums will show large differences even for same rotor and bearing conditions. Therefore, it is not easy for us to establish the relationship between the spectrum characteristics and the machine conditions.

Full Amplitude Spectrum (FAS) is another amplitude spectrum [1–3], and it can be obtained as follows: construct a complex signal sequence , letting , make FFT of , and then calculate its Fourier series and plot the Fourier amplitude spectrum. Since Full Amplitude Spectrum contains the information of both signals and , it is not influenced by the installation mode of sensors and anymore. Thus, it is more convenient than Classical Amplitude Spectrum to establish the relationship between the spectrum characteristics and the machine conditions. However, in the Full Amplitude Spectrum, each original actual frequency component is broken down into a negative component and a positive component, and the corresponding two amplitudes are more abstract and difficult for users to understand.

Moreover, neither Classical Amplitude Spectrum nor Full Amplitude Spectrum can indicate the actual radial rotor vibration amplitude on every frequency theoretically.

The motive of this paper is to propose a new amplitude spectrum, called Synthetic Amplitude Spectrum (SAS), so as to be able to overcome the shortcomings of both Classical Amplitude Spectrum and Full Amplitude Spectrum. Furthermore, if possible, apply the SAS to two other current important analyzing methods, short-time Fourier transform time-frequency analysis, and waterfall plot analysis in rotating machine diagnoses, so as to result in two SAS extensions.

In order to verify the shortcomings of the existing amplitude spectrum methods and the virtues of the proposed methods, an appropriate testing is needed. The following sections of the paper are arranged as follows. First, how the testing data were acquired is described in Section 2. Then, the deficiency of the two existing amplitude spectrums is tested in Section 3, and on the basis of these, in Section 4, the Synthetic Amplitude Spectrum is proposed and tested, which is also compared with Holospectrum. In Section 5, some extensions of SAS are given. Section 6 is the conclusion.

#### 2. On Acquisition of the Testing Data

A gas turbine set in a factory (named Shijiazhuang Oil Refinery) had been under monitoring when we went to acquire the testing data. The monitoring system was Bently 3300 system. The two displacement sensors and at each bearing were configured as mode 1 shown in Figure 1(a). We used an NI card 6036E to set up an easy data acquisition system; the card was connected to a notepad computer through PCMCIA port, and in the opposite direction, it was connected to two output terminals at the front panel of 3300 system, which output two modified displacement signals from sensors and at a bearing. The whole testing data acquisition system is shown in Figure 2. We set up 2 kHz as the sampling rate and took 1024 points synchronously for each channel every time.

Since the following testing also needs the synchronous data under the sensors configuration of mode 2, but we had no means to actually configure the sensors in both mode 1 and mode 2 simultaneously, especially on the productive gas turbine set, we thought out a substitute scheme. The scheme was that we can calculate the synchronous data in sensors of mode 2 from the actually detected data in sensors of mode 1, which is illustrated in Figure 3.

By the way, from Figure 3, it is easy to know that the rotor vibration amplitude in any direction on any frequency cannot fully represent the rotor vibration amplitude in radial direction, which is the real rotor vibration amplitude on the frequency.

#### 3. Deficiency Verification of the Two Existing Amplitude Spectrums

As mentioned earlier, we acquired a set of rotor vibration displacement data from a bearing of the gas turbine set, with the help of Monitor 3300, the sampling rate being 2 kHz and the sample length being 1024 points for each channel. The acquired signal sequences and were corresponding to the sensors-installation mode 1 as we know.

We also calculated the rotor vibration displacement signal sequences and in sensors of mode 2, using the previous method, and the obtained data were called signal sequences and in mode 2.

With the two sets of the data for two sensors-position modes, we made 4 Classical Amplitude Spectrums, 2 CASs for each mode, as in Figure 4. In Figure 4, the unit for each frequency axis is order, which means that they are all order spectrums, and the horizontal coordinate values are all divided by the basic frequency (rotating frequency). By estimating accurately [4, 5], the basic frequency was 135.4294 Hz. For either signal sequence or in any sensors-position mode, the estimated basic frequency was of the same value.

From Figure 4, it can be clearly seen that for different sensors-installation modes, the Classical Amplitude Spectrums are different, although the signals are from the same machine-running condition. The signal amplitudes on basic frequency in two sensors-position modes were 12.8 and 22.1, respectively, as in Figures 4(a) and 4(c). The signal amplitudes on basic frequency in two sensors-position modes were 25.7 and 18.4, respectively, as in Figures 4(b) and 4(d). In Figure 4, each of the estimated amplitudes represents the rotor vibration amplitude of the component towards the detecting direction but not representing radial rotor vibration amplitude on the frequency.

With the previously detected data and in sensors of mode 1, as we know, we constructed a complex signal sequence and made a Full Amplitude Spectrum as mentioned earlier. We also constructed another complex signal sequence and made its Full Amplitude Spectrum, with the previously calculated data and in sensors of mode 2. We found that the two made Full Amplitude Spectrums are the same absolutely, as shown in Figure 5.

The Full Amplitude Spectrum independent from the sensors position brings us an ease to use it, as mentioned earlier. However, in Figure 5, each frequency component is divided into a negative subcomponent and a positive subcomponent, such as the basic frequency component is divided into a negative component of 38.3 and a positive component of 13.5, which is difficult for users to understand and so not convenient to be used.

#### 4. Proposed Synthetic Amplitude Spectrum

##### 4.1. Principle and Making Steps

A signal, especially a periodic one, can be Fourier decomposed into harmonic components. Both the signal sequences and for any sensors-installation mode can be regarded as nearly periodic signal sequences. Suppose that the two decomposed components on the frequency , from periodic sequences and , respectively, are

Combining these components, we can get the rotor-center orbit on the frequency : which is a synthetic ellipse; see Figure 6.

From (2), we can also have Let ; then,

Therefore, when is equal to , will be the square of the synthetic ellipse semimajor or semiminor. Then, calculate in the angle of , using . By comparing, we can determine the elliptic semi-major shown in Figure 6.

The ellipse in Figure 6 fully reflects the real orbit of the rotor vibration component on the frequency, and the length of its semi-major represents the amplitude of the radial rotor vibration on the frequency [4, 5].

Based on the previous analysis, steps of making SAS can be described in the following six steps.(1)Take two synchronously acquired signal sequences and from sensors and .(2)Process two signal sequences and using FFT, respectively.(3)Calculate their Fourier series respectively.(4)Combine each pair of two sine-wave components on same frequency from the previous Fourier series, thus resulting in a series of ellipses.(5)Calculate the semi-major for each ellipse (amplitude of radial rotor vibration for each frequency).(6)Plot the amplitude spectrum using frequency or its order as the horizontal axis and the corresponding semi-major (amplitude) as the vertical axis, that is, the so-called Synthetic Amplitude Spectrum (SAS). The frequency order refers to a relative frequency from division by a reference one, and the order is dimensionless.

##### 4.2. Test and Comparison

With the detected data and in Figures 4(a) and 4(b), which were acquired under the condition of sensors of mode 1, according to the previous making steps, we accomplished the Synthetic Amplitude Spectrum. We also accomplished another Synthetic Amplitude Spectrum with the previously calculated data and in Figures 4(c) and 4(d), which are corresponding to sensors of mode 2. We found that the two Synthetic Amplitude Spectrums are the same completely, as shown in Figure 7.

Not only the Synthetic Amplitude Spectrum is not influenced by and sensors position, it also can reflect the actual amplitude of radial rotor vibration on every frequency theoretically, unlike Full Amplitude Spectrum. In Figure 7, the amplitude 25.9 marked in the figure, larger than the corresponding four marked amplitudes 12.8, 22.1, 25.7, and 18.4 in Figure 4, is also different from two marked amplitudes 38.3 and 13.5 in Figure 5.

In 1989 [4], Qu put forward the Holospectrum proposal [4, 5]. It was also made with both and displacement signals. It calculated the accurate amplitude, frequency, and phase value of the main harmonics of the two vibration displacement signals and . But it only brought the main harmonics into a combination to form the Holospectrum. So, the Holospectrum is different from SAS, which does not exhibit the whole amplitude spectrum. Figure 8 was the Holospectrum made with the same data in Figure 4, here consisting of the main components of frequency orders 1, 2, 3, and 4.

#### 5. Extensions of Synthetic Amplitude Spectrum

##### 5.1. Short-Time SAS Array Method

Sometimes there is a need to observe whether the machine signal is steady in certain time periods. One of the most common methods to achieve this is to use short-time amplitude spectrum array analysis. The traditional short-time amplitude spectrum array analysis is based on one directional signal, that is, either or . The brief making steps are (1) first, divide the signal into a group of short-time signal sequence according to certain parameters of delay and window length ; (2) then, calculate their Classical Amplitude Spectrums and queue these CAS according to the short signals time order, and thus a short-time amplitude spectrum array emerges [5, 6].

Our idea is that to apply the Synthetic Amplitude Spectrum to substitute for Classical Amplitude Spectrum to make the short-time amplitude spectrum array. The brief steps can be as follows (1): divide a group of synchronous and signals into a series of short-time signal sequences according to certain parameters of delay and the window length , as shown in Figure 9; (2) then, calculate the Synthetic Amplitude Spectrums for each pair of short and signals and queue these SASs by time order, and thus a short-time SAS array emerges, which is one extension of SAS method.

We also took the detected data and in Figures 4(a) and 4(b) to make the short-time SAS array, according to the previous making steps. Figure 10(a) was the accomplished CAS array with the detected signal; Figure 10(b) was the accomplished CAS array with the detected signal; Figure 10(c) was the proposed short-time SAS array with both and signals.

##### 5.2. SAS Waterfall Plot

Similarly, we applied the Synthetic Amplitude Spectrum to substitute for Classical Amplitude Spectrum to make SAS waterfall diagram for analyzing rotor vibration during startup or shutdown process. The classical waterfall diagram using CAS is based on one signal from one sensor [3, 7], while the SAS waterfall diagram is based on two signals from corresponding two sensors.

To test the proposed SAS waterfall diagram method, we were able to acquire a set of data during the startup of another gas turbine machine. We started to record the data at the rotor speed of 676.404 rpm and stopped the recording at the rotor speed of 5381.45 rpm. Figures 11(a) and 11(b) were the classical waterfall plots made from sensor data or sensor data, respectively. Figure 11(c) was the proposed SAS waterfall plot made on basis of combining the two sensors’ signals. In Figure 11, the maximum amplitude 309.3 for SAS array methods was larger than the other two maximum amplitude values 177.7 and 300.3.

#### 6. Conclusion

When analyzing the two directional rotor vibration displacement signals, neither Classical Amplitude Spectrum nor Full Amplitude Spectrum can represent the amplitude of the actual radial rotor vibration on every frequency. Classical Amplitude Spectrum is also influenced by the installation position of the sensors. The proposed SAS method and its extensions can overcome these defects. Their spectrums are no longer influenced by the sensor-mounting position, and any amplitude of them can also represent the actual size of the radial rotor vibration on the corresponding frequency. They are convenient for users to analyze and establish the relationship between the spectrum characteristics and the running state of the rotating machine. So, they are of practical significance.

#### Acknowledgments

This paper is funded by the Natural Science Foundation of China (61271079) and the Supporting Plan of Jiangsu Province (BE2010720). The authors also want to thank the anonymous referees for their much contribution to this work.

#### References

- D. Southwick, “Plus and minus spectrum,”
*Orbit*, vol. 14, no. 2, pp. 16–20, 1993. - D. Southwick, “Using full spectrum plots,”
*Orbit*, vol. 14, no. 4, pp. 12–16, 1993. - D. Southwick, “Using full spectrum plots—part 2,”
*Orbit*, vol. 15, no. 2, pp. 10–16, 1994. - L. Qu, X. Liu, G. Peyronne, and Y. Chen, “The holospectrum: a new method for rotor surveillance and diagnosis,”
*Mechanical Systems and Signal Processing*, vol. 3, no. 3, pp. 255–267, 1989. View at Publisher · View at Google Scholar · View at Scopus - D. Shi, L. Qu, and M. Bao, “Instantaneous purified orbit: a new tool for analysis of nonstationary vibration of rotor system,”
*International Journal of Rotating Machinery*, vol. 7, no. 2, pp. 105–115, 2001. View at Scopus - B. S. Kim, S. H. Lee, M. G. Lee, J. Ni, J. Y. Song, and C. W. Lee, “A comparative study on damage detection in speed-up and coast-down process of grinding spindle-typed rotor-bearing system,”
*Journal of Materials Processing Technology*, vol. 187-188, pp. 30–36, 2007. View at Publisher · View at Google Scholar · View at Scopus - M. F. Abdul Azeez and A. F. Vakakis, “Numerical and experimental analysis of a continuous overhung rotor undergoing vibro-impacts,”
*International Journal of Non-Linear Mechanics*, vol. 34, no. 3, pp. 415–435, 1999. View at Scopus