This paper reports on the application of the correlation dimension in large rotating machinery fault diagnosis. The Grassberger-Procaccia algorithm and its modified version are introduced. Some important influencing factors relating directly to the computational precision of the correlation dimension are discussed. Industrial vibration signals measured from large rotating machinery with different faults are researched using the above-mentioned methods. The results show that the correlation dimension can provide some intrinsic information on an underlying dynamic system and can be used to classify different faults intelligently.
ChenC. L.YaoH. T.Chaos in the imbalance response of a flexible rotor supported by oil film bearings with nonlinear suspension. Nonlinear Dynamics, 1998, 16, 71–90.
2.
GoldmanP.MuszynskaA.Dynamic effects in mechanical structures with gaps and impacting: Order and chaos. J. Vibr. Acoust., 1994, 116, 541–547.
3.
MuszynskaA.GoldmanP.Chaotic responses of unbalanced rotor/bearing/stator systems with looseness or rubs. Chaos, Solitons and Fractals, 1995, 5(9), 1683–1704.
4.
ChuF. L.ZhangZ. S.Periodic, quasi-periodic and chaotic vibrations of a rub-impact system supported on oil film bearings. Int. J. Engng Sci., 1997, 35(11), 963–973.
5.
MullerP. C.BajkowskiJ.SoffkerD.Chaotic motions and fault detection in a cracked rotor. Nonlinear Dynamics, 1994, 5, 233–254.
6.
LoganD.MathewJ.Using the correlation dimension for vibration fault diagnosis of rolling element bearing—1. Basic concepts. Mech. Syst. and signal processing, 1996, 10(3), 241–250.
7.
LoganD.MathewJ.Using the correlation dimension for vibration fault diagnosis of rolling element bearing—2. Selection of experimental parameters. Mech. Syst. and signal processing, 1996, 10(3), 251–264.
8.
GrassbergerP.ProcacciaI.Characterization of strange attractors. Phys. Rev. Lett. A, 1983, 50(5), 346–349.
9.
PackardN. H.CrutchfieldJ. P.FarmerJ. D.ShowR. S.Geometry from a time series. Phys. Rev. Lett., 1980, 45, 712–716.
10.
TakensF.Detecting strange attractors in turbulence. Lecture Notes in Mathematics (Eds RandD. A.YoungL. S.), 1981, pp. 336–342 (Springer, Berlin).
11.
CasdagliM.EubankS.FarmerJ. D.GibsonJ.State space reconstruction in the presence of noise. Physica D, 1991, 51, 52–60.
12.
AlbanoM.MuenchJ.SchwartzC.Singular-value decomposition and the Grassberger-Procaccia algorithm. Phys. Rev. A, 1988, 38(6), 3107–3026.
13.
FraserA. M.SwinneyH. L.Independent coordinates for strange attractors from mutual information. Phys. Rev. A, 1986, 33(2), 1134–1140.
14.
BuzugT.PfisterG.Comparison of algorithms calculating optimal parameters for delay time coordinates. Physica D, 1992, 58, 127–140.
15.
LibertW.PawelzikK.SchusterH. G.Optimal embeddings of chaotic attractors from topological considerations. Europhys. Lett., 1991, 14, 521–527.
16.
AlbanoA. M.PassamanteA.FarrellM. E.Using higher-order correlations to define an embedding window. Physica D, 1991, 54, 85–97.
17.
TheilerJ.Efficient algorithm for estimating the correlation dimension from a set of discrete points. Phys. Rev. A, 1987, 36, 4456–4462.
18.
BinghamS.KotM.Multidimensional trees, range searching, and a correlation dimension algorithm of reduced complexity. Phys. Lett. A, 1989, 140, 327–330.
19.
GrassbergerP.An optimized box-assisted algorithm for fractal dimension. Phys. Lett. A, 1990, 148, 63–68.
20.
KantzH.SchreiberT.Dimension estimates and physiological data. Chaos, 1995, 5(1), 143–154.
21.
TheilerJ.Spurious dimension from correlation algorithms applied to limited time-series data. Phys. Rev. A, 1986, 34(3), 2427–2432.
22.
ProvenzaleA.SmithL. A.VioR.MuranteG.Distinguishing between low-dimensional dynamics and randomness in measured time series. Physica D, 1992, 58, 31–49.
23.
FraedrichK.WangR. S.Estimating the correlation dimension of an attractor from noisy and small data-sets based on re-embedding. Physica D, 1993, 65, 373–398.
24.
AlbanoM.MuenchJ.SchwartzC.Singular-value decomposition and the Grassberger-Procaccia algorithm. Phys. Rev. A, 1988, 38(6), 3107–3026.
25.
SmithL. A.Intrinsic limits on dimension calculations. Phys. Lett. A, 1998, 133, 283–288.
26.
TheilerJ.Estimating fractal dimension. J. Opt. Soc. Am., 1990, A7, 1055–1073.
27.
EckmannJ. P.RuelleD.Fundamental limitations for estimating dimensions and Lyapunov exponents in dynamical system. Physica D, 1992, 35, 357–359.
28.
GalkaA.MaassT.PfisterG.Estimating the dimension of high-dimensional attractors: A comparison between two algorithms. Physica D, 1998, 121, 237–251.
29.
LorenzE. N.Deterministic non-periodic flow. J. Atmos. Sci., 1963, 26, 130–136.
30.
AbarbanelH. D. L.BrownR.SidorowichJ. J.TsimringL. S.The analysis of observed chaotic data in physical system. Rev. Mod. Phys., 1993, 65(4), 1331–1390.
QuL. S.XieA. L.LiX.Study and performance evaluation of some nonlinear diagnostic methods for large rotating machinery. Mech. Mach. Theory, 1993, 28(5), 699–713.
