Study on Fractal Box Dimension and Its Application to Fault Diagnosis of Hydraulic Pump

Transcription

1 Advanced Materials Research Online: ISSN: , Vols , pp doi:1.428/ 212 Trans Tech Publications, Switzerland Study on Fractal Box Dimension and Its Application to Fault Diagnosis of Hydraulic Pump Xiliang Liu 1,a, Guiming Chen 1,b and Fangxi Li 1,c 1 Xi an Research Institute of Hi-tech, Xi an, 7125, China a liuxiliang111@yahoo.com.cn, b chenguiming@yahoo.com, c lifangxi@yahoo.com.cn Keywords: fractal, box ension, hydraulic pump, fault diagnosis Abstract: Fractal box ension is a new method to represent self-similarity and complexity of non-stationary mechanical fault signal. In this paper, we first present the basic conception of fractal and box ension. Then we propose the algorithm to calculate the box ension of discrete signal. On the basis of wavelet packet decomposition, we quantitatively analyze the vibration signal under different conditions with fault characteristic by way of box ension. Experimental results show that the box ensions are different in evidence because of the difference of fault mechanism. As a successful application, fractal box ension can be used to recognize the fault pattern of hydraulic pump effectively. Introduction The hydraulic pump is a core part of a hydraulic system, which can convert mechanical energy into hydraulic energy, and provide pressure for the whole system. Its running state influences the systemic reliability. Therefore, how to monitor its working state is an important topic for fault diagnosis of mechanical equipment. In recent years, several researchers begin to introduce fractal a new kind of mathematical tool into signal processing, and fractal ensions become important characteristic parameters that describe chaos phenomenon of complicated system. Among them, box ension adopts the thought of box filling to link with nonlinear problem. For the state changing, it can be used to describe statistical self-similarity of fractal boundary quantitatively. Thus, the box ension can give quantitative description to realize the fault diagnosis for complicated mechanical equipment. This paper applies the box ension to characteristic extraction of hydraulic pump fault signal, and describes the normal and abnormal characteristics of vibration signal in a simple and brief way. Then, it provides a new practical method for the classification of fault pattern. Fractal In a brief, fractal is a similar set of part and whole in a certain way. Generally, it has precise structure, non-regularity and unlimited self-similarity structure. Invariant scale and self-regularity are its two important characteristics[1]. The word fractal originally means the irregular, fractional, and fragmented object. In general, fractal is taken for the state of fragment accumulation. It is the generalization of graph, conformation and phenomenon with no characteristic length, the overly unsmooth and irregular set and function which traditional and analytic means cannot describe. It is a newborn subject which researches most disorder, unstable, non-stationary and non-equilibrium and random objects in nature[2]. Although there is no accurate and satisfactory definition, it has gone deep into the wide range of areas so far. This study makes use of fractal box ension to analyze and discuss the fault of hydraulic pump, just from mechanical application provide a simple and feasible method. All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications, (# , Pennsylvania State University, University Park, USA-18/9/16,14::5)

2 Advanced Materials Research Vols The box ension and algorithm The box ension definition Traditional and classical Euclidian Geometry and mathematical analysis are not suitable for fractal, whose main tool is all kinds of ensions. General geometry object has integral ension, to the contrary, fractal has non-integral ension. Fractal ension is an epoch-making development since it extends empirical ension to nonintegral value. It is an important parameter describing fractal characteristic quantitatively, not only defined by value, but also calculated precisely by use of all sorts of software through experimental method[3,4]. Similar ension is defined as D s =ln b / ln a, where a is what the graph is reduced 1/a, b is what b analogues form the graphics whole. However, similar ension s application is very limited, only to regular fractal owning rigorous self-similarity. Nevertheless, the box ension is different entirely, if F is nonempty bounded subset of R n, N(F,δ ) denotes minimal units covering F set with maximal diameterδ, then the upper and lower box ensions can be defined respectively as B F B F δ (1) (2) δ If upper box ension equals lower ension, the box ension of F can be computed from B F δ (3) The box ension algorithm of discrete signal In engineering practice, we often collect some discrete signals, i.e. vibration, pressure, temperature, et al. Box ension, as one of fractal ensions, its value and variety can reflect the irregularity and complexity of signals. Let discrete signal x(j) X, where X is a closed set in n ensions Euclidian space R n. R n can be divided into smallδ grid, and N δ is the grid number of set X with widthδ in discrete space. Because the maximal resolution of x(j) is sampling intervalδ t, so Eq. (3) cannot calculate whenδ. Above all, it adopts approximate method when calculating: regardδ grid as minimal grid, then enlarge to + grid gradually ( k Z ). Given the grid number N with width in X set, it can be abtained from Eq. (4) and (5) P N / k { + k+ 1} min{ xk( j 1) + 1, xk( j 1) + 2,, xk( j 1) + k+ 1} ( ) = max xk( j 1) + 1, xk( j 1) + 2,, xk( j 1) j= 1 where N denotes sampling points. Then grid number N is where N >1 is necessary condition. ( j = 1,2,, N / k; k = 1,2,, K K < N ) (4), ( ) /( k ) + 1 N k = P δ (5) δ It can use correlation coefficient test method, triple line fitting method or genetic optimization algorithm to confirm a better linear line as signal scale-invariant region. If the region s threshold and terminal are k 1, k 2, respectively, in the internal [lg k δ, lg N ] it should satisfy linear regression model lg N = a lg + b (k 1 k k 2 ) (6)

3 1528 Mechatronics and Intelligent Materials II The slope rate of linear regression line is confirmed by least square method, that is So the box ension d B is a = ( k 2 k1+ 1) lg k lg N lg k ( k k + 1) lg 2 k ( lg k) lg N (7) d B = a (8) The fractal box ension of 1 ension discrete vibration signal is between 1 and 2. The more irregular the signal is, the greater the fractal box ension is. Hence, it can judge the signal s complexity by way of ensions[5-7]. Application to fault diagnosis of hydraulic pump The gear pump is a common hydraulic pump. Because of its advantages of small volume, light weight, high reliability, and not sensitive to hydraulic oil pollution, it is applied in various hydraulic machinery far and wide. In running processing the gear pump produces vibration certainly, which will intensify, especially partial fault happening. The fault signal s outstanding performance is non-stationary, non-gaussian and non-linear, and this kind of instantaneous signal lasts for a short time, often submerged by normal signal. Since it is difficult to extract useful characteristics from spectrum analysis based on traditional Fourier transform, this paper analyze the vibration signal by means of box ensions. We carry out experiments towards the CB-KP63 hydraulic gear pump on the test bench, in which we set three kinds of fault bearing abrasion, gear abrasion and sideboard abrasion. Acceleration sensors measure the pump radial vibration under the condition of 2k Hz sampling frequency and sampling points. The original time-domain waveform of vibration signal shows in Fig.1. Fig.1 Original time-domain waveform of vibration signal

4 Advanced Materials Research Vols Due to a great deal of random noise mixing up with vibration signal in experiment, the signals need to preprocess before calculating box ension. We adopt the method of wavelet packet decomposition, after a large number of analysis and comparison, choose db3 wavelet function to decompose four wavelet packets. The coefficient distribution of the fourth wavelet packet (4,1) shows in Fig.2. Fig.2 Wavelet packet decomposition coefficients Based on box ension algorithm of the discrete signal, according to the coefficient in Fig.2, we make use of triple-line fitting method minimize the residual sum of squares of fitting lines, choose a invariant-scale region with better linearity, that is Fig.3 Box ension fitting line

5 153 Mechatronics and Intelligent Materials II The slope rate of fitting line in Fig.3 is the fractal box ensions in all kinds of state of the gear pump. The concrete results show in table 1. Table 1. Results of fractal box ension fault pattern normal state bearing abrasion gear abrasion sideboard abrasion fractal box ension From table 1, we can find: 1. The box ension of sideboard abrasion is the largest, because the pump creates a large amount of irregular vibration signal when sideboard abrasion happens. 2. The box ensions of bearing abrasion and gear abrasion are larger, because vibration signal mixes up with some periodic impact signal when these faults happen. 3. The box ension of normal state is the least, because the vibration signal is regular when gear pump works well. The calculation results above also coincide with the diagnosis rule The more irregular the signal is, the greater the fractal box ension is. Summary In this paper, we apply the fractal box ension to fault pattern recognition of hydraulic pump. According to comparing the variety of fractal box ension in different states, it can monitor fault s occurrence and development. What s more, fractal box ensions have obvious separability under different patterns, which make it easy to recognize the pump fault pattern. As a result, fractal box ension can be used to recognize the fault pattern, at last, provide an exact and reliable quantitative method for fault diagnosis of hydraulic pump. References [1] Li Shuigen, Fractal(Higher Education Press, Beijing 24), in Chinese. [2] He Zhengjia, Zi Yanyang and Meng Qingfeng. Theory and Application of Fault Diagnosis in Mechanical Equipment Nonstationary Signal(Higher Education Press, Beijing 21), in Chinese. [3] Shen Xiaohua, Zhou Lejun, LI Hongsheng et al.: Acta Geologica Sinica Vol. 2(21), p [4] Kuo-Kai Shyu, Yu-Te Wu and Tzong-Rong Chen. Nonlinear Dynamics Vol. 3(21), p [5] I. M. Zhuravel, L. M. Svirs ka. Materials Science Vol. 3(21), p [6] André R. Backes, Odemir M. Bruno. Image and Signal Processing Vol. 6134(21), p [7] João B. Florindo, André R. Backes and Odemir M. Bruno. Image and Signal Processing Vol. 6134(21), p