Leakage fault diagnosis method of aircraft landing gear hydraulic cylinder based on wavelet packet

The main executive element of the retractable landing gear is the hydraulic cylinder, and its common fault is caused by the leakage of the hydraulic cylinder. In this study, the mathematical model of the energy entropy of the wavelet packet is constructed according to the fault characteristic information, and the hydraulic system is modelled by the Amesim software. Through the simulation calculation of the leakage and leakage of the hydraulic cylinder, the fault characteristic parameters such as displacement and pressure are extracted. Wavelet packet transform is used to obtain the energy entropy of the wavelet packet, and a mapping relationship between the leakage of the hydraulic cylinder and the energy entropy is established. The research shows that the leakage fault diagnosis method based on wavelet packet can effectively predict the failure of the hydraulic cylinder of the landing gear, and it is effective and practical for guiding the fault prediction and intelligent maintenance of the hydraulic system of the aircraft.


Introduction
The landing gear performance has an important influence on the normal landing of the aircraft, which is the key to ensure the safety of the aircraft and the lives of the passengers.The hydraulic cylinder is the main executive part of the landing gear, and its failure will directly affect the landing gear performance.From the use of current aircraft, the failure rate of hydraulic system is high, but the diagnosis and research of the fault can only rely on experience and test in ground [1].With the development of computer simulation, it is a tendency to analyse the landing gear retracting system by simulation.AMESim is a hydraulic, mechanical system modelling, simulation and dynamic analysis software based on bond graphs.Boeing and Airbus are using the software in different degrees in the design and development of the system [2].At present, fault diagnosis based on transfer function and fault diagnosis based on artificial intelligence are the main methods of fault diagnosis of hydraulic transmission system.Fault diagnosis based on transfer function needs to have an accurate mathematical model of system control process, based on artificial intelligence and expert system for fault diagnosis requires a lot of knowledge, it is widely used in hydraulic transmission systems [3].A diagnosis method based on wavelet packet energy entropy can multi-scale refinement, and the extracted signal can adapt to timefrequency analysis requirements to focus on any details of the signal, so the method can effectively diagnose the leakage of the complex non-linear system with distributed parameters, such as aircraft hydraulic systems [4].

Wavelet packet analysis theory
The wavelet packet analysis method can decompose the signal comprehensively in time and frequency, and further decompose the high-frequency part without subdivision of multiresolution analysis, so that ω n satisfies the following recursion double scale equation: In the formula, h n is the scale coefficient and g n is the wavelet coefficient.It is called the set of functions ω n t as a wavelet packet determined by ω 0 t = φ(t).Wavelet packets are a set of functions that have a certain connection with scaling functions and wavelet functions.Wavelet packet decomposition which has scale parameters, translation parameters and frequency parameters is a fine decomposition method that use ω 2n t and ω 2n + 1 t to filter the two continuous spatial filtering into two sub-bands of relatively low frequency and relatively high frequency, and it has better timefrequency characteristics.According to the wavelet packet coefficient d k j + 1, n of the previous frequency band the wavelet packet coefficients of two subbands can be rapidly decomposed [5].
Wavelet packet reconstruction algorithm is In the formula (2), d l j, 2n and d l j, 2n + 1 are the l decomposition sequences of the original signal after the original signal is decomposed by the j + 1 layer wavelet packet.

Energy entropy of wavelet packet
When the hydraulic cylinder of the landing gear system is leakage, it will have a great influence on the displacement of the hydraulic cylinder and the energy distribution in the frequency band, such as the flow rate of the inlet and outlet, so the signal energy that decomposes the frequency band can be used as the fault feature information.The result of wavelet packet decomposition expressed in terms of energy is called wavelet packet energy spectrum.By means of the (4) Parseval identities, it is known that there is an equivalence between the energy of the wavelet packet transform and the energy of the original signal Therefore, the sum of squares of signals in each frequency band can be selected as the energy spectrum of wavelet packets after wavelet packet decomposition.The result of wavelet decomposition is expressed by d i, j (k), and the energy of signal in each frequency band is In the formula, N represents the length of the original signal, and all of the E i, j constitute the energy spectrum of the wavelet packet When energy is large, normalisation can be done as Define The relative wavelet packet energy entropy is calculated as Wavelet energy entropy is defined as the degree of order or disorder of signals, so it can provide useful information related to signal potential dynamic processes [6].
The wavelet packet analysis method can adaptively select the corresponding frequency band according to the characteristics of the analysed signal to match the signal spectrum, thus the timefrequency resolution can be improved, and the fault features can be extracted more effectively, which is more valuable to the fault diagnosis [7].

Analysis and modelling of hydraulic system
Taking the common aircraft landing gear retractable hydraulic cylinder as the research object, when working, the pressure oil from the oil supply system enters the lock cylinder, hydraulic lock, and actuating cylinder in order to retract and lower the landing gear.Under the sketch model of AMESim, based on the working principle of the hydraulic cylinder and using the hydraulic library and the design library of the hydraulic components, the simulation model of the working loop of the landing gear hydraulic cylinder, as shown in Fig. 1, is set up.The simulation model mainly consists of (i) hydraulic energy parts, (ii) electromagnetic reversing valves, (iii) one-way valves, (iv) open locking actuators, (v) liquid control one-way valves, (vi) one-way valve, (vii) expansion valve, and (viii) storage hydraulic cylinder.The hydraulic cylinder simulation model is shown in Fig. 2 by adding internal and external leakage modules.
In AMESim, a component icon may have multiple mathematical models, so a mathematical submodel of the component is selected and the submodel is closed under the Submodel model mode.This paper mainly simulates the internal and external leakage of the hydraulic cylinder, so the parameters of the pipeline length and the viscosity, compressibility and quality of the oil in the pipeline are not considered.In order to simulate the working condition of the landing gear's hydraulic cylinder, a priority model is adopted to set the main parameters according to the actual situation in the parameter mode.The specific setting is shown in Table 1.The equivalent mass of the moving parts of the main landing gear is estimated to be 3.8 kg, and the approximate load of the hydraulic cylinder is 10,000 N.

Simulation model verification
In order to verify the correctness of the simulation model, simulate the working process of the hydraulic cylinder, set the simulation time 80 s, the sampling interval is 0.1 s, the output signal of the electromagnetic reversing valve is the middle 10 s, the left (down) 25 s, the middle 10 s, the right position 15 s, the middle 10 s.As shown in Fig. 3, when the system runs 35 s, the hydraulic cylinder is completely put down and the electromagnetic reversing valve is shifted to the right position in t = 45 s.After the system runs 15 s, the hydraulic cylinder is fully recovered and the landing gear is completely recovered.The displacement curve and the actual working condition of the aircraft landing gear show that it is feasible to use AMESim to set up the working process of the hydraulic cylinder of the aircraft landing gear to simulate the working process of the hydraulic cylinder.

Simulation analysis of internal leakage: Internal leakage is crack leakage and the mathematical model is
In the equation ΔP = P1 − P2, μ is the fluid dynamic viscosity, L c is the gap length, r c = (d c /2), d c is the gap width, d p is the cylinder body diameter, ecc is the centrifugal rate, V + is the external speed, and V − is the internal velocity.
According to the mathematical model, the leakage rate of the hydraulic cylinder is inversely proportional to the length of the gap, and is directly proportional to the width of the gap.In this paper, the inner leakage is changed by changing the gap diameter (Table 2).
As shown in Fig. 4, with the increase of the diameter of the internal leakage gap, the leakage of the hydraulic cylinder becomes larger, and the acting time of the landing gear is longer.Under the condition of internal leakage, the effect of failure on the lowering condition is more intense than that under the retracting condition.

Simulation analysis of external leakage:
External leakage is crack leakage and in single piston rod hydraulic cylinder, external leakage mainly occurs between piston rod and cylinder body in the rod cavity, its mathematical model In the equation P mid = (P a + P b /2), in aircraft landing gear retracting system, P a = P, P b = 0, P is oil pressure, μ is the fluid dynamic viscosity, L is the gap length, δ is the gap width, d p is the cylinder body diameter, e is the centrifugal rate, ρ(x) is the density under pressure x.
In this paper, the external leakage is also changed by changing the gap diameter (Table 3).As can be seen from Fig. 5, with the increase of the diameter of the external leakage clearance, the leakage amount of the hydraulic cylinder becomes larger, and the action time of the landing gear in lowering and retracting is longer.Under the condition of external leakage, the failure has less influence on the landing gear retracting condition of aircraft, and more influence on the retracting condition than on the landing condition.

Wavelet packet transform energy entropy of the internal and external leakage of hydraulic cylinders
This paper uses the wavelet analysis tool in MATLAB to analyse the displacement energy entropy of the internal and external leakage of hydraulic cylinders.The choice of wavelet function is an important problem.At present, there is no unified theoretical standard.The wavelet coefficient of wavelet transform provides the basis for the selection of wavelet function.The wavelet coefficients represent the similarity between the wavelet and the analysed signal after wavelet transform.If the wavelet coefficient is large after wavelet transform, it means that the similarity is large.In practical applications, wavelets are selected based on experience for different purposes of signal processing.This paper selects Daubechies 8 wavelet as wavelet basis function [8].

Analysis of internal leakage wavelet packet
The leakage displacement signal in the hydraulic cylinder is decomposed by four layers wavelet packet.In this study, the displacement signal obtained by leaking inside and outside of AMESim is decomposed by four layers wavelet packets.The adjustment frequency sets 20 Hz and the collected signal is further subdivided into 16 frequency bands.The four-layer wavelet packet is used to resolve the displacement signal of the piston rod.The energy spectrum of the wavelet packet is obtained as shown in Fig. 6.The energy entropy of the wavelet packet calculated based on the above formula is shown in Table 4.
It can be seen from Fig. 6 that the energy value of the 1 gradually becomes smaller as the diameter of the inner leakage gap becomes larger.This feature can reveal the influence of internal leakage on the energy of the system from the energy viewpoint.As the internal leakage of the hydraulic cylinder increases, the disordered movement of the hydraulic oil increases, the pressure rise rate decreases, and the system energy loss increases.From the frequency of the signal, the most influential signal is the lowfrequency band.The energy value of 3 becomes large as the diameter of the inner leakage gap becomes large.It can be seen from Fig. 7 that the energy entropy of the wavelet packet of the internal leakage displacement signal decreases with the increasing of the leakage rate.

Transform of external leakage wavelet packet
The external leakage is also extracted from the displacement curve in the AMESim simulation for four layers wavelet packet decomposition, and the wavelet packet energy spectrum and energy entropy are obtained through solutions as shown in Figs. 8 and 9.
It can be seen from Fig. 8 that the energy value of the external leakage 1 is the same trend as the energy value of the internal leakage 1, and the diameter of the external leakage gap becomes large, the energy values become small, which confirms the effect of leakage on system energy (Table 5).As the leakage increases, the system energy decreases.It can be seen from Fig. 9 that the energy entropy of the wavelet packet of the internal leakage displacement signal gradually increases with the increasing of the leakage rate.

Conclusion
(i) Using the leakage module in the model of the hydraulic cylinder working loop of the aircraft landing gear collection system built by   AMESim, it can effectively simulate the leakage of the hydraulic cylinder inside and outside, and the simulation of the hydraulic model is basically consistent.Therefore, the model of the aircraft hydraulic system can be built with AMESim, and the dynamic simulation of the leakage fault can be used.
(ii) In the case of internal leakage in the hydraulic cylinder, the energy value of the 1 of the wavelet packet energy decreases with the increase of the diameter of the internal leakage gap, that is, the leakage of the hydraulic cylinder becomes larger.The wavelet packet energy entropy of the internal leakage displacement signal decreases with the increase of leakage.
(iii) Under the condition of the external leakage of the hydraulic cylinder, the energy of the 1 of the wavelet packet energy decreases with the increase of the diameter of the internal leakage gap, that is, the leakage of the hydraulic cylinder becomes larger.The wavelet packet energy entropy of the internal leakage displacement signal increases with the increase of leakage.
Therefore, the energy entropy of the wavelet packet transform which is obtained by MATLAB can take the relationship between the energy entropy and the energy entropy of the wavelet packet, and the fault diagnosis based on the energy entropy of the wavelet packet is used to provide a practical method for the performance detection of the aircraft's landing gear.

Fig. 1 Fig. 2
Fig. 1 Hydraulic circuit model of the landing gear system

Fig. 4
Fig. 4 Displacement of cylinder in internal leakage

Fig. 7 Fig. 8
Fig. 7 Energy entropy of the internal leakage signal wavelet packet

Table 1
Parameter setting of simulation model Submodel Parameter name Parameter values

Table 2
Inner leakage changed by changing the gap diameter Simulation, data and the curve of Fig.5can be taken.

Table 3 External
Simulation, data and the curve of Fig. 5 can be taken.J. Eng., 2019, Vol.2019 Iss. 13, pp.427-431 This is an open access article published by the IET under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0/)

Table 4 Energy
entropy of the internal leakage signal