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where d is the diameter of orifice, l is the length of orifice, l is thefluid dynamic viscosity; Dp is the pressure difference between theorifice.With the reference to the test requirements in GB/T15622-2005[9], the experimental principle of hydraulic cylinder internal andexternal leakage is shown in Fig. 1.The system pressure can be adjusted by the relief valve 8, andthe system flow can be changed by adjusting the input frequencyprovided by the variable frequency motor 4, the load force canbe changed by adjusting the opening pressure of the relief valve23 and 27. So, the system pressure, flow and load can be regulated.In this experimental system, different leakage levels of the hydrau-lic cylinder are simulated by the variable orifice selection. Anindustrial inverter 5 is used to adjust the rotating speed of the var-iable frequency motor 4.The size of the load cylinder 19 and the experimental hydrauliccylinder 16 are the same; the diameter of stroke is 330 mmand thediameter of piston rod is 45 mm. The maximum system pressure is10 MPa, and the maximum flow rate is 50 L/min. Two AK-4 typepressure sensors have been used to collect the pressure signal ofthe hydraulic cylinder chamber with and without rod respectively,and SGC-5 type line grating sensor has been used to collect the dis-placement signal of the hydraulic cylinder piston rod with sam-pling frequency of 1 kHz.In the experiment, variable orifice is applied to simulate differ-ent leakage level of hydraulic cylinder, which is shown in Table 1.Leakage of orifice can be calculated by the formula (2); system loadpressure is set in 5 MPa; the inverter output frequency is 20 Hz;gear pump output flow is 23.6 L/min.During the process of piston rod stretching and retracting, thechanges of pressure and piston position over time with no leakageand different levels of leakage are recorded by the pressure sensorand the position sensor.For the fault diagnosis of hydraulic cylinder, An [10], Sepehri[11], Goharrizi [12–13], Tang [14–15] had done some research.The data processing methods they used included the extendedKalman filter, wavelet analysis, Hilbert transform and wavelettransform. And they found wavelet analysis for fault diagnosis ofhydraulic cylinder was very effective, and the failure featuredecomposed by the wavelet decomposition was very sensitive to the impact of internal leakage, which was very beneficial for thediagnosis. So the paper chooses wavelet analysis to extract thehydraulic cylinder leakage fault feature. In the following content,the related theory of wavelet analysis will be reviewed firstly,and then the application of wavelet packet analysis is analyzed3. Relevant theory analysisWavelet analysis is a time-frequency analysis method whichcan be applied to handle the non-stationary signals effectively. Itdecomposes the signal into the superposition of a series of wave-lets functions [16]. Reconstruct the signal after the decomposition,and the signal can reach the purpose of de noising.3.1. Wavelet transformWavelet transform is classified as Continuous Wavelet Trans-form (CWT) and discrete wavelet transform (DWT). The CWT isa convolution of the input data sequence x(t) with a set of func-tions generated by the mother wavelet w(t).CWTða; bÞ¼ 1ffiffiffiapZ þ1 1xðtÞwt ba dt ð3Þwhere w(t) is the mother wavelet that can be scaled and shifted;b 2 R; a 2 R+; and a – 0; a is scale factor and b is shift factor.In most applications, the signal is dispersed by a scale factor of2m and by a shift factor of 2mn.
Then Eq. (3) can be defined as:DWTðm; nÞ¼ 2 m=2Z þ1 1xðtÞwð2 mt nÞdt ð4Þwhere m and n are integers.Fast wavelet transform is commonly used in practical calcula-tion of the discrete wavelet transform. In order to get the approx-imation signal, the original signal S is decomposed according to itsfrequency. As shown in Fig. 2(a), Aj is the approximation signal oforiginal signal S, LPF is low pass filter, and HPF is high pass filter.3.2. Wavelet packetIn application, the resolution ability of high frequency band sig-nal is expected to be improved. However, wavelet transform can-not meet the resolution requirement, so wavelet packet analysisis introduced to solve this problem [17]. As shown in Fig. 2(b),wavelet packet analysis can continue to decomposes the high fre-quency band signal further than the wavelet transform. So it canprovide an elaborate method which pides the original signal intomulti-layers in the full band frequency, improving the resolution.The original signals S in Fig. 2 is decomposed into layers, and Si,jis used to represent the jth band reconstructed signal at the ithdecomposition level. As shown in Fig. 2(b). Signal S is decomposedinto 2ifrequency bands at ith decomposition level. The original sig-nal S can be expressed as:S ¼ Si;0 þ Si;1 þ þ Si;j ð5Þ3.3. Wavelet packet energyFor a further analysis of the signal, the energy value of detailsignal of wavelet packet decomposition is presented. Define theenergy of Si,j as Ei,j: k¼1where xi,k is the amplitude of kth discrete points of the detail signalSi,j.Construct feature vectors based on energy. Feature vector T isconstructed as follows.T ¼½Ei;0; Ei;1; ; Ei;j ð7ÞWhen the energy is large, Ei,j is usually a large value, causingsome inconvenience in the analysis, so normalize the above featurevector as follows.E ¼X 2i 1j¼0Ei;j ð8ÞT0¼½Ei;0=E; Ei;1=E; ; Ei;j=E ð9Þwhere vector T0is the normalized feature vector.3.4. Wavelet energy entropyThe wavelet energy entropy is defined as a measure of thedegree of order/disorder of the signal, so it can provide usefulinformation about the underlying dynamical process associatedwith the signal [18]. For a random signal, if it is generated by acompletely unordered process, the amplitude and energy in eachfrequency band are approximately the same. Namely, the probabil-ity distribution of the signal is close to the disorder degree and theentropy is close to the theoretical maximum.Define:pj¼ Ei;jEð10ÞX 2i 1j¼0pj¼ 1 ð11ÞThen calculate the corresponding wavelet energy entropy as:WWE ¼ X 2i 1j¼0pjlog pjð12Þ3.5. Wavelet packet energy varianceSimilar to using ‘‘Entropy’’ to describe the distribution ofwaveletpacket energy, the ‘‘variance’’ is adopted to indicate the probabilitydistribution of the wavelet packet energy value quantitatively inthis paper. The wavelet packet energy variance is defined as:s2WE ¼ 12i 1X 2i 1j¼0ðEj EmeanÞ2ð13Þwhere s2is the variance; Ei is the energy values of ith band; Emean isthe average value of energy.On the basis of above theory, paper will discuss how to analyzethe test data to find the effective leakage fault feature extractionmethod. 4. Leakage characteristic parameters selection and sensitivityanalysis4.1. Leakage data collectionThe position and pressure signals of no-leakage and seven sizesof different levels of leakage are collected in above test-bed. Thetime domain waveforms of the collected data are shown in Fig. 3.Compared with the external leakage, the internal leakage ismore harmful and more difficult to be detected. Therefore, thispaper concentrates on the study of the internal leakage. Waveletanalysis tool inMatlab are applied to analyze the displacement sig-nals of piston rod and the pressure signals of inlet and outlet ofhydraulic cylinder.An important issue is the choice of the wavelet basis function.So far there is not a unified theory standard, but the wavelet coef-ficients of wavelet transform provides the basis for choosing thewavelet based function. Wavelet coefficients after wavelet trans-forming indicate the degree of similarity between wavelet andanalytical signal. If the wavelet coefficient after wavelet transformis large, it means degree of similarity is large. In practice, the wave-let is selected by experience according to different purposes of sig-nal processing.