Monitoring varying speed machinery vibrations—I. The use of non-stationary recursive filters

Monitoring varying speed machinery vibrations—I. The use of non-stationary recursive filters

Mechanical Systems and Signal Processing (1995) 9(6), 635–645 MONITORING VARYING SPEED MACHINERY VIBRATIONS—I. THE USE OF NON-STATIONARY RECURSIVE FI...

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Mechanical Systems and Signal Processing (1995) 9(6), 635–645

MONITORING VARYING SPEED MACHINERY VIBRATIONS—I. THE USE OF NON-STATIONARY RECURSIVE FILTERS E. L, J. Z  J. M Centre for Machine Condition Monitoring, Monash University, Australia (Received 1 September 1994, accepted 1 May 1995) Varying speed machinery monitoring is fraught with difficulties due to non-stationary machine dynamics and vibrations. Most conventional signal processing methods based on digital sampling carried out in equal time intervals become inappropriate when monitoring the vibrations of varying speed machinery. Parts I and II of this paper present a new data acquisition technique which pre-processes varying speed signals so that conventional signal processing methods can be applied. It is based on the application of recursive filtering and angle domain analysis to non-stationary vibration analysis. The present paper, Part I, is primarily concerned with the recursive filtering technique. It is shown that the recursive filtering can be used as an effective and reliable method of pre-conditioning the original vibration signals. The theoretical background and some basic properties of the non-stationary recursive filtering approach are described. The approach is implemented and validated using computer simulation and experiments. 7 1995 Academic Press Limited

1. INTRODUCTION

The problem of variability in the operation of machines (due to change of loading, rotational speed, etc.) is of great concern when using current condition monitoring techniques. Conventional signal processing methods, which were developed for constant speed machinery monitoring, are based on digital sampling carried out in equal time intervals. If the machine operates under varying speed or load, its dynamic and vibrations become non-stationary. Fixed time sampling cannot cope with the varying rotational frequency of the machine, resulting in increasing leakage error and spectral smearing. Therefore most of the conventional methods for signal processing become inappropriate when monitoring the vibrations of varying speed machinery. Non-stationary vibrations signals from varying speed machinery may include more abundant information about its condition. Some phenomena, which are usually not obvious at constant speed operation, may become more apparent under varying speed conditions. For example, the detection of resonances can be effectively made during run-up or coast down conditions. Therefore in the last decade vibration analysis and condition monitoring techniques for varying speed machinery have attracted a lot of attention of scientists and engineers. Some progress has been made in the theoretical analysis, the signal processing methodology, measurements and practical applications of varying speed machinery monitoring. This study is concerned with a new data acquisition approach which pre-processes non-stationary signals so that conventional signal processing methods can be applied for 635 0888–3270/95/060635+11 $12.00/0

7 1995 Academic Press Limited

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varying speed machinery monitoring. It uses techniques based on recursive filtering and the angle domain analysis.

2. THE NON-STATIONARY RECURSIVE FILTERING APPROACH

Non-stationary vibration analysis in itself is not new. Adaptive noise cancelling techniques have been applied in the form of standard Kalman filters to machinery vibration monitoring with reported success [1, 2]. It was shown that the adaptive filtering approach allows one to track a harmonic component with varying magnitude from signals buried in noise. The main disadvantage of this technique is that it can only track a fixed frequency component and cannot deal with a varying frequency signal. To overcome this limitation a non-stationary recursive filtering approach (NSRFA) has been chosen for the present study. Recursive filtering is a time domain filtering approach which has a strong ability to extract desired waveforms from complex vibration signals varying both in magnitude and frequency. The distinct feature of the NSRFA is that it employs a mathematical model of the actual vibration signal derived from parametric analysis [3]. There are other features as well, such as its ability to provide strong noise cancelling and tracking complex signals. These and other matters are discussed in the following sections. 2.1.  In order to filter a desired waveform (target signal with a known structure) buried in noise and other extraneous signal components, a time series filter model (ARX or ARMAX) can be adopted [3]. Parameters of this model can be time varying and identified with a control signal as an input to the model and a measured vibration signal as its output. Then the predicted value of the output vector is the desired part of the measured signal. An ARX model of the following type: y(t)=B(q−1 )u(t)+v(t)

(1)

was employed for filtering. It is known as the finite impulse response (FIR) model. The FIR model is a linear regressive model. It requires only linear operations in the parameter identification algorithm [3]. For varying speed machinery monitoring a constant property filter cannot be used due to non-stationary vibration signals. The time series model should be time varying, and the model parameters need to be modified. A recursive algorithm was chosen to speed up calculations. This approach was called the non-stationary recursive filtering approach. A block diagram of the recursive filter is shown in Fig. 1. Here s(t) is the desired target signal, y(t) is the measured signal, e(t) is the noise, f(t) is the useless (extraneous) part of y(t) and u(t) is the control signal which describes the structure of the target (output) signal and thereby drives the filter to obtain the desired part of the measured signal. H(q−1 ) and T(q−1 ) are unknown transfer functions which vary with time and should be recursively identified. Thus the object of recursive filtering is to filter out the noise e(t) and the extraneous components f(t) from the measured signal y(t), obtaining the desired target signal s(t) under the control signal u(t). The difference equation of the system is given by: y(t)=H(q−1 )s(t−1)+e(t)+f(t) s(t)=T(q−1 )u(t).

(2)

   —

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Suppose that s(t), e(t) and f(t) are mutually independent, and T(q−1 ) is taken as a constant C (s(t) is linearly proportional to u(t)). Then y(t)=CH(q−1 )u(t−1)+e(t)+f(t).

(3)

In accordance with the output y(t) and the input u(t), the CH(q −1 ) parameters can be recursively identified with a common recursive identification algorithm. To identify CH(q−1 ) the following objective function was minimised: W(U)+E[y(t)−CH(q−1 )u(t−1)]2.

(4)

Then the predicted value of the target signal s(t) can be estimated as: sˆ (t)=H (q−1 )s(t−1)=C H (q−1 )u(t−1).

(5)

2.2.   In order to investigate the internal properties of the non-stationary recursive filtering approach, a computer simulation was performed using the MATLAB software package. Vibration signals with various forms ranging from simple sine waveforms to noise contaminated non-stationary swept sine waveforms were examined. Special attention was given to non-stationary signals consisting of a number of harmonic components, which are typical for rotating machinery elements (gears, bearings, fans, etc.). The main objective of the computer investigation was to select a number of appropriate recursive algorithms and mathematical models from the literature [3]. Initially, it was expected that some formal criteria for quantitative divergence assessment between control and target signals could be used. However, after preliminary simulations it was found that as a first pass a qualitative assessment of the results, based on visual comparison, was sufficient. As a rule, unsatisfactory parametric models or choice of model orders as well as an improper selection of the recursive algorithm, would significantly worsen matching of the control and target waveforms. A number of influencing factors and recommendations were derived from this preliminary investigation and are enumerated below. These recommendations must not be considered

Figure 1. Block diagram of the NSRF model.

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as a general solution to the selection of the NSRFA parameters, but nevertheless can be extended to the majority of vibration signals collected from rotating machinery. 2.2.1. Recursive algorithms To perform recursive identification a number of approaches are available. These are the Kalman Filter, the Forgetting Factor, the Unnormalized Gradient and Normalized Gradient approaches [3, 4]. In this investigation, these methods were evaluated and the Kalman Filter was selected as the most effective approach. 2.2.2. Model order By definition, an ARX model is presented as: A(q−1 )y(t)=B(q−1 )u(t−nk)+v(t).

(6)

Therefore a set of order parameters [na, nb, nk] fully describes the ARX model. After several tests it was found that: , The na parameter (the order of A(q−1 ) or auto regression) must be 0, implying that auto regression is not necessary. The result will become worse in any case if na$0. , The nb parameter (the order of B(q−1 ) or input regression) is related to a signal frequency to be tracked. For low frequency signals, as a rule, good accuracy can be obtained with nb=1. Higher frequency signals need higher orders, and nb=6 is usually sufficient. In some cases, nb may be increased to 10. , The nk parameter reflects the number of delays between the input and output. Usually delay nk=1 is appropriate. 2.2.3. Initial conditions and data length In the most general form, the set of initial parameters for a recursive algorithm is presented as [th(0), P(0), y(0), u(0)], where th(0) is the initial model parameter, P(0) is the initial covariance of parameters and y(0) and u(0) are initial values of the signals y(t) and u(t). If the set of initial parameters is chosen to be close to the real ones, the results would be greatly improved at the initial stage of the recursive identification. It has been found that usually the initial stage affects only the first one or two periods of the time waveform. If an appropriate set of initial parameters can be given, the target signal would be tracked with a short data length. In some cases, too long data length can introduce an accumulation of numerical errors and leads to distortions in the time waveform. According to the situation, it may be appropriate to divide the data into shorter data lengths. 2.3.      A number of simulations were performed to investigate the characteristics of the non-stationary recursive filtering approach. The following features were identified: , Strong noise cancelling capability. The non-stationary recursive filter approach has prominent filtering capability. It is possible to filter the target components of the measured signal without difficulty, even if the magnitudes of the extraneous component are several times greater than the desired signal. , Tracking of signals with complex structure. Both simple waveforms and complex signals can be tracked without a greater effort (e.g. a compound waveform with several frequency components, a periodic signal with varying amplitude, a modulated waveform, a repeated pulse signal, etc.). The only prerequisite is that the structure of the target signal be known previously.

   —

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, Linearity. In NSRFA, a linear relationship is maintained between the magnitudes of the target waveform and the control waveform. This is a useful property for real world applications where the magnitude of the target signal component is usually not known previously. 3. NUMERICAL IMPLEMENTATION

In order to demonstrate some of the useful properties of the NSRFA mentioned above, consider tracking of complex time waveforms using computer simulation. Figure 2 presents detection of a repeated pulse waveform with an amplitude modulation from a measured signal y(t) with a complex structure shown in Fig. 2(a). The measured signal was defined analytically as y(t)=2 sin (2p · 150t)(w1(t)+w2(t)+w3(t)+w4(t)+w5(t)) +2 sin (2p · 150t)+rand(t),

(7)

where rand(t) is white noise with zero mean value and w1(t)=cos (2p · 5t)+2 cos (2p · 10t), w2(t)=3 cos (2p · 15t)+4 cos (2p · 20t), w3(t)=5 cos (2p · 25t)+4 cos (2p · 30t), w4(t)=3 cos (2p · 35t)+2 cos (2p · 40t), w5(t)=cos (2p · 45t).

(8)

To obtain the desired target signal s(t), the control signal u(t) shown in Fig. 2(b) was chosen as u(t)=2 sin (2p · 150t)(w1(t)+w2(t)+w3(t)).

(9)

The model parameters [na, nb, nk] were defined as [0, 6, 1], the data length L=1 sec and the sampling frequency Fs=1000 Hz. The resulting tracked target signal s(t) is shown in Fig. 2(c). The next example (Fig. 3) shows tracking of an amplitude modulated swept waveform from a noisy signal. The measured signal y(t) [Fig. 3(a)] had the following structure: y(t)=w(t)+2 sin (2p · 50t)+2rand(t),

(10)

w(t)=(1+sin (2p · 0.5t)) · sin (2p · 20t 2 ).

(11)

where

The control and target signals were defined as u(t)=w(t),

(12)

s(t)=w(t).

(13)

The model parameters, the data length and the sampling frequency were chosen as in the previous example. The tracked target signal s(t) is shown in Fig. 3(c). The presented examples clearly demonstrate the NSRFA capability to track complex target signals varying both in frequency and in amplitude even when superposed with significant white noise. It should be noted that even with the relatively small magnitudes of the target signals, good results were obtained.

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Figure 2. Tracking of a repeated wave form with amplitude modulation. (a) Noisy signal; (b) control signal; (c) target signal.

4. EXPERIMENTS

Experimental testing and validation is considered an essential part of the development of any new signal processing approach for machine monitoring. Although a variety of well-established experimental procedures is now available, some specific requirements should be satisfied for adequate assessment of the NSRFA applicability due to the complexity of non-stationary conditions.

   —

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4.1.      It is well-known that the key to a successful application of a recursive identification algorithm is a precise knowledge of the structure of the signal to be tracked. In fault detection, the definition of the desired fault signal structure requires accumulation of previous experience. In some cases, typical characteristic waveforms of different kinds of faults may be assumed to be the control signals.

Figure 3. Tracking of a non-stationary modulated swept sine wave form. (a) Noisy signal; (b) control signal; (c) target signal.

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For varying speed machinery monitoring the definition of the control signal structure becomes more complex due to non-stationary vibration properties. An effective way to overcome this difficulty is to obtain the knowledge of the target signal structure from real measurements. The present non-stationary recursive filtering approach is suggested for signal preconditioning as a part of the combined data acquisition technique based on simultaneous application of recursive filtering and the angle domain analysis. To perform angle domain conversion, an optical encoder, producing a series of pulses and thereby reflecting machine shaft speed variations, is used. Since these pulses provide useful information about the actual rotational frequency of the machine, they can also be used for defining the target signal structure, corresponding to the actual (constant or varying) operating conditions of the machine. Estimation of rotational speed can be done using measured values of the rotational period duration with subsequent curve fitting. For example, polynomial fitting gives a polynomial representation of rotational speed which fits the data obtained from the encoder signal and can be used for analytical description of the shaft rotation at every instant of time. Then the structure of the control signal can be defined as a function of the rotational speed. 4.2.     The NSRF approach was applied to monitoring the vibrations of a single stage spur gear box consisting of a 24 teeth and a 32 teeth gear. The gear test rig was designed to operate at constant or varying speeds. An optical encoder was mounted on the output shaft of the gearbox and produced impulses indicating actual vibrations of the rotational speed. Vibration signals were measured under various speed variation rates. The first set of tests involved gathering vibration and encoder signal from the gear operating under constant load with different constant speeds. The second test set was conducted to investigate vibration of the gear under varying speed. Acceleration and deceleration times were varied from 10 to 60 sec with a linear pattern using an AC motor speed controller (Toshiba TOSVERT VF-A3). Data from two B&K Type 4371 vibration accelerometers and a Hewlett Packard optical encoder HEDS-5010 were recorded into a data recorder. The samples were subsequently converted into MATLAB files using a Tektronix 2630 Fourier Analyser for further analysis within the MATLAB software environment. 4.3.  The NSRF approach was implemented for tracking constant (Fig. 4) and varying (Fig. 5) gear mesh frequencies. It was found that to implement a reliable recursive filtering procedure, an adequate definition of the structure of the control signal is required. This requirement is essential for varying speed conditions with significant variations of shaft rotation. Good results were achieved when the number of discrete points representing vibration signals was at least five to ten per period of the target component. For example, if the gear mesh frequency is 1000 Hz, a time sample with a duration of 1 sec comprises 1000 gear mesh periods and should consist of 10 000 discrete points to satisfy the above mentioned requirements. On the other hand, working with large data blocks is computationally expensive and some additional hardware is required. Therefore data blocks consisting of 4 K (4096) discrete points with 0.32 sec total duration were used for signal proceessing and analysis. Figures 4(a) and 5(a) represent signals from the index channel of the optical encoder (one pulse per rotational period). The encoder signals consist of 4096 points an have a total duration of about 0.32 sec. As each data block in Figs 4(a) and 5(a) comprises about five

   —

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Figure 4. Tracking of the constant gear mesh frequency. (a) Encoder signal; (b) rotational speed, w, actual values; ——, approximation; (c) measured signal; (d) target signal; (e) – – –, control signal; ——, target signal; (f) – – –, control signal; ——, target signal.

rotational periods, only five actual (measured) values of rotational frequency are available for the block of data. To obtain approximate values of rotational speed for every data point, polynomial curve fitting was used [Figs 4(b) and 5(b)]. It was found that linear approximate was sufficient for this study. Polynomial coefficients were determined for each data block and analytical descriptions of the rotational speed were obtained. The figures clearly show that the rotational frequency remained nearly constant (218 Hz) for the first example and varied

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from 18.3 to 17.8 Hz for the second case. The measured vibration signals are shown in Figs 4(c) and 5(c). The structures of these signals are very similar except for variations in amplitude. The modulated gearmesh frequencies tracked by the NSRFA are presented in Figs 4(d) and 5(d). The comparison of tracked waveforms clearly shows a dependence of the signal

Figure 5. Tracking of the varying gear mesh frequency. (a) Encoder signal; (b) rotational speed, w, actual values; ——, approximation; (c) measured signal; (d) target signal; (e) – – –, control signal; ——, target signal; (f) – – –, control signal; ——, target signal.

   —

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structural on rotational speed. The amplitude of the gearmesh remains constant while the rotational speed is constant and decreases when the gear slows down. The final Figs 4(e, f) and 5(e, f) demonstrate the comparison of target and control signals. For analysis of the fine signal structure, short signal segments (0.03 sec) are depicted. For each case the fragments were taken at the beginning (0–0.03 sec) and in the middle (0.12–0.15 sec) of the analysed data blocks. The results confirm the above-mentioned feature of NSRFA to track complex waveforms accurately with varying frequencies. The linear relationship between the target and control waveforms is clearly visible in Fig. 5(e). It is important to note that the definition of the control signal was based only on the frequency dependence on time. Its amplitude was modulated, but remained constant in terms of the overall value. However the amplitude of the tracked target signal clearly reduces with time corresponding to a real decrease in rotational frequency. 5. CONCLUDING COMMENTS

This paper has investigated the non-stationary recursive filtering approach for tracking a desired signal component from vibration signals with a complex structure. The technique can be extremely useful for monitoring varying speed machinery. The approach is relatively straightforward and is used as a preconditioner to currently used methods, mainly, spectral analysis. REFERENCES 1. H. V and L. L 1993 LSM International Technical Report. High resolution order tracking at extreme slew rates using Kalman tracking filters. 2. N. K 1994 Hewlett Packard Realtime Update. Spring 1994, 1–4. The fine art of order tracking. 3. L. L 1985 Theory and Practice of Recursive Identification. Cambridge, MA: The MIT Press. 4. L. L 1987 System Identification—Theory for the User. Englewood Cliffs, NJ: Prentice-Hall.