Study on the extraction method of the friction-induced vibration signal of rotary compressors

2.1. Rotary compressor structure

At present, rotary compressors are commonly used in household air conditioners. The rotary compressor has the characteristics of small size, simple structure, stable operation, etc., which can reliably work under the most working conditions. Fig. 1 shows a rotary compressor structure. It mainly consists of a motor, a main bearing, a crankshaft, a cylinder, a roller, a vane, a sub bearing, and an accumulator, etc. There are several friction pairs in the rotary compressor, which include those friction pairs between the crankshaft and bearings, and between the roller and vane.

Fig. 1Structure diagram of a rotary compressor

2.2. Test details

A series of tests on the friction-induced vibration and noise of rotary compressors was performed in an anechoic room. The rotary compressor was located in the center of the anechoic room as shown in Fig. 2. The compressor was fixed on three elastic pads, and these three elastic pads were located freely on the ground. Three 3-D accelerometers were fixed on the different positions of the rotary compressor. Three sound sensors were fixed on the auxiliary frame. All vibration and sound signals were simultaneously collected with a data-acquisition system at a sampling frequency of 25600 Hz. All vibration and sound signals were input to the data acquisition system by using AC coupling. The data-acquisition system was equipped with anti-alias filters to ensure the correctness of the measured signals.

Two different rotary compressors were tested. Each compressor was tested at electric current frequencies of 30, 40, 50, 60, 70, 80, and 90 Hz, which corresponded to rotational speeds of 1800, 2400, 3000, 3600, 4200, 4800, and 5400 rpm, respectively. At each rotational speed of 1800-5400 rpm, the compressors were run for about twenty minutes, then the vibration and noise signals of the compressors were collected.

After all tests were carried out, the compressors were disassembled to check which one of the compressor parts has been worn.

Fig. 2Measurement environment of the friction-induced vibration and noise of rotary compressors

a) Overview of the test site

b) Measurement positions of the compressor vibration

2.3. Extraction method of the frictional vibration signal

For high-speed rotating machinery, the rotation conditions are more complicated. Especially when the rotating parts have abnormal phenomena such as eccentricity, bearing defects, oil film turbulence, friction, trapping flow, etc., the vibration of the rotating machinery is more complicated, including many vibration modulation components. At this time, it is difficult to extract the characteristic signals of rotating machinery faults by a simple frequency analysis.

There are several extraction methods of the characteristic signals of rotating machinery faults [13-17]. They are divided into two categories: those are based on the time-domain analysis, including dimensional parameters such as maximum value, mean value, variance, extreme value, peak-to-peak value, etc., as well as dimensionless evaluation parameters such as kurtosis, margin factor, coefficient of variation, and smoothing factor. Those methods are based on the frequency-domain analysis, including amplitude spectrum and power spectrum analysis, cepstrum analysis, envelope spectrum analysis, etc. More recently, the wavelet transform and t are often applied to extract the characteristic signals of rotating machinery faults.

The authors have made many endeavors to extract the characteristic signals of the frictional vibration of the rotary compressors, and have found that using a combination of the harmonic wavelet transform and the envelope spectrum analysis, the characteristic signals of the frictional vibration of the rotary compressors could be extracted.

Newland [18] proposed a harmonic wavelet packet transform in 1993, which was based on the harmonic wavelet family. The harmonic wavelet packet transform is actually a completely ideal bandpass filter, which consists of a complex wavelet with a box-like spectrum. The harmonic wavelet family’s expression in the frequency domain is presented as follows:

where, $m$, $n$ are positive real numbers, and $m<n$. $m$, $n$ determine the frequency band range of the harmonic wavelet. Taking the inverse Fourier transform of Eq. (1), the time domain expression is obtained as follows:

The harmonic wavelet packet transform is the natural extension of the harmonic wavelet transform after translation and expansion. It not only inherits the excellent orthogonal and filtering characteristics of the harmonic wavelet transform, but also breaks through the limitations of the harmonic wavelet in the frequency band division, and realizes the equal division of high and low frequency parts of signals. The frequency domain distribution of the harmonic wavelet packet decomposition is shown in Fig. 3, where $k$ is the number of decomposition levels; $w_k^i$ is the wavelet coefficient of the $i$-th band after $k$-level harmonic wavelet packet decomposition; $f_m$ is the Nyquist frequency; $B$ is the bandwidth of each subband; $B$ and the upper and lower limits $m$ and $N$ of the analysis frequency of each frequency band are respectively determined by Eq. (3) and Eq. (4):

Fig. 3Frequency domain distribution of the harmonic wavelet packet decomposition

More knowledge of the harmonic wavelet packet transform can be found in Newland’s publication [19]. A numerical example was presented so that readers can roughly know the harmonic wavelet packet transform. Assuming that a signal consists of three waveforms with different frequencies (30, 150, and 2000 Hz, respectively) and amplitudes (3, 100, and 26, respectively) and a random signal as follows:

where $t$ is time. The sampling frequency is assumed to equal to 10000 Hz. Fig. 4 shows a comparison between the original signal components and the extracted signal components using the harmonic wavelet packet transform. From Fig. 4, it is seen that the extracted signal components are in good agreement with the original signal components.

The envelope spectrum of vibration signals removes the high-frequency sinusoidal components and focuses on the lower-frequency repetition behavior [20]. This is why the envelope spectrum is a useful tool for the defect analysis of rotational machinery. In the bearing fault diagnosis, for example, the envelope spectrum of vibration signals is generally used to identify bearing faults. Software MATLAB has provided the analysis capability of the envelope spectrum.

Fig. 4Comparison between the original and the extracted signal components

3.1. Vibration signal analysis

In the present test work, compressor A and compressor B were tested to measure vibrations of their shells. Fig. 5 shows time history records of two compressor vibrations in $x$-direction. From Fig. 5, it is seen that there is not an obvious difference between the vibrations of compressor A and compressor B in amplitude. In comparison with Fig. 5(a) and 5(c), it is found that the amplitude of the vibration increased with increasing electric current frequency.

Fig. 5Time history records of the two compressors vibrations in x-direction

a) Electric current frequency $f=\mathrm{}$30 Hz

b) Details of the signal shown in Fig. 5(a)

c)$f=\mathrm{}$90 Hz

d) Details of the signal shown in Fig. 5(c)

Fig. 6Amplitude spectrum analysis of the vibrations of compressors A and B in x-direction

a)$f=\mathrm{}$30 Hz

b)$f=\mathrm{}$90 Hz

Fig. 6 shows an amplitude spectrum analysis of compressors A and B. From Fig. 6(a), it is obviously found that about 9819.23 Hz and 10000.00 Hz vibration components of compressor B are stronger than those of compressor A. From Fig. 6(b), it is seen that the 10000.00 Hz vibration component of compressors B and A is still the strongest.

It needs to be mentioned that the characteristics of the vibrations in $y$- and $z$-directions were similar to those of the vibrations in $x$-direction. Therefore, they were not presented in the paper.

3.2. Cepstrum analysis of vibration signals

Fig. 7 shows a cepstrum analysis of the vibration signals. It is found that there is a characteristic frequency which is corresponding to the rotational frequency of the compressor crankshaft, and there are several lower harmonic frequencies. Form Fig. 7(a), one can see that at the rotational frequency of 30 Hz, the characteristic frequency of the vibration is 1/0.0333 = 30 Hz. Low harmonic frequencies are 1/0.0666 = 15 Hz, 1/0.0999 =10 Hz, and 1/0.01332 = 7.5 Hz, respectively. Form Fig. 7(b), one can see that at the rotational frequency of 90 Hz, the characteristic frequency of the vibration is 1/0.0111 = 90 Hz. Low harmonic frequencies are 1/0.0222 = 45 Hz, 1/0.0333 = 30 Hz, and 1/0.0444 = 22.5 Hz, respectively.

Fig. 7Cepstrum analysis of the vibrations of compressors A and B in x-direction

a)$f=$30 Hz

b)$f=$90 Hz

3.3. Harmonic wavelet packet transform of the vibration signals

As mentioned in Section 2.3, the envelope spectrum analysis of vibration signals is a useful tool for the defect analysis of rotational machinery. In the present work, the envelope spectrum analysis is also applied to identify the friction-induced vibration of rotary compressors. Since the vibration of the rotational machinery is more complicated, including many vibration modulation components due to complicated work conditions such as eccentricity, bearing defects, oil film turbulence, friction, trapping flow, etc. In order to identify the friction-induced vibration signal, it is necessary to firstly remove those signals that are not related to the friction-induced vibrations. The harmonic wavelet packet transform of the vibration signals was used to identify the friction-induced vibration components of rotary compressors. After the harmonic wavelet packet transform, a series of vibration components was obtained. If the characteristic frequency of the envelope spectrum of a vibration component is equal to the rotational frequency of the compressor crankshaft, the corresponding vibration component is thought to be a friction-induced vibration component of rotary compressors. From Fig. 6, it is seen that the frequency components of compressor vibrations are very rich. Therefore, an 11-layer harmonic wavelet transform was used to decompose the measured compressor vibration signals into a series of components with 2048 frequency bands at a bandwidth of 6.25 Hz. 489 vibration components were obtained, whose frequency bands did not overlap with each other. The harmonic wavelet packet transform was an accurate transform, and the sum of the transformed vibration components was the same as the original signal. Fig. 8 show characteristics of four vibration components in time and frequency domains among these 489 vibration components. From Fig. 8(a) and Fig. 8(b), it is found that vibration components of frequencies 30 Hz and 60 Hz were excited by the rotational frequency of the compressor crankshaft. From Fig. 8(c) and Fig. 8(d), it is found that vibration components of dominant frequencies 819.62 Hz and 7181.03 Hz were excited by the rotational frequency of the compressor crankshaft. It is clearly found that there are two dominant frequencies in each vibration component shown in Fig. 8(c) and Fig. 8(d), suggesting that the vibration components of dominant frequencies 819.62 Hz and 7181.03 Hz have been modulated. Fig. 9 shows an envelope spectrum analysis of the vibration components of dominant frequencies 819.62 Hz and 7181.03 Hz. It is seen that the characteristic frequencies of the envelope spectrums are all equal to the rotational frequency of the compressor crankshaft, suggesting that the vibration components of dominant frequencies 819.62 Hz and 7181.03 Hz are the friction-induced vibration of the compressor.

Fig. 8Extracted four vibration components of the compressor B at the rotational frequency of 30 Hz from the harmonic wavelet packet transform

a)

b)

c)

d)

Fig. 9Envelope spectrum analysis of the vibration components of dominant frequencies 819.62 Hz and 7181.03 Hz

a)

b)

3.4. Extraction of the friction-induced vibration signals

From Fig. 6, it is seen that the vibration of rotary compressors consists of many different-frequency vibration components. Some vibration components are excited by friction, but some are excited by other factors. In the present work, the friction-induced vibration signals are considered to be the sum of all vibration components whose characteristic frequencies are equal to the rotational frequency of the compressor crankshaft.

Fig. 10Extracted friction-induced vibration signals

a)$f=$30 Hz for compressor A

b)$f=$30 Hz for compressor B

c)$f=$60 Hz for compressor A

d)$f=$60 Hz for compressor B

Fig. 10 shows four friction-induced vibration signals and their power spectrum analyses. Comparing Fig. 10(a)-10(b) with Fig. 5 and Fig. 6, it is seen that the amplitudes of the friction-induced vibration signals are much less than those of the overall vibrations of the compressor, and the number of the friction-induced vibration frequencies is less than that of the overall vibration frequencies of the compressor.

The authors have found that the number of the friction-induced vibration components obtained through the harmonic wavelet packet transform decreases with increasing rotational frequency of the compressor crankshaft. When the rotational frequency of the compressor crankshaft is equal to 80 Hz or 90 Hz, no friction-induced vibration component was detected. This phenomenon is attributed to the relation between the friction coefficient and sliding speed (Stribeck curve) as shown in Fig. 11. From Fig. 11, it is seen that when the rotational frequency of the compressor crankshaft is low, the compressor crankshaft and bearings work under the boundary lubrication. In this case, the friction between the compressor crankshaft and bearing is the largest, suggesting the compressor crankshaft is probably pressed into contact with the bearing, which induces friction-induced vibration of the compressor. When the rotational frequency of the compressor crankshaft is higher, the compressor crankshaft and bearings work under the mixed lubrication or hydrodynamic lubrication. In these cases, the friction between the compressor crankshaft and bearing is lower, suggesting the compressor crankshaft is not probably pressed into contact with the bearing, which is not able to induce friction-induced vibration of the compressor.

Fig. 11Stribeck curve

3.5. Noise signal analysis

In the test, noise signals of the rotary compressors were also measured. Fig. 12 shows time history recorders of both compressor A and compressor B and their spectrum analysis. It is seen that two dominant frequencies of the noise signal of compressor A are 960.8 Hz and 10360 Hz, respectively. Two dominant frequencies of the noise signal of compressor B are 1080 Hz and 9640 Hz, respectively. Using the same method as in Section 3.4, friction-induced noise components were extracted as shown in Fig. 13. It is seen that the dominant frequencies of the friction-induced noise components of both compressor A and compressor B are close to those of the friction-induced vibrations shown in Fig. 10(c) and Fig. 10(d).