This study presented a method for modeling the nonlinear system of a planetary gearbox and the fault diagnosis of a crack in a planetary gear based on the Volterra series theory. First, the exponential Hilbert reproducing kernel and its fast optimization algorithm was proposed and deduced in theory, and the fast solution of the fourth-order kernel of the Volterra series was successfully solved. Second, the Volterra series model estimation was compared with the least squares estimation of the actual collected signals from the planetary gearbox and the time-domain output signal was estimated using a neural network. The accuracy and the superiority of the Volterra series model of the planetary gearbox were then verified. At the same time, the convergence and the memory length of the Volterra series were discussed. In order to further mine and extract fault feature information, coupling relationship between the generalized frequency response of higher order spectrum of the Volterra series model and fault frequency was also studied. This study attempted to reflect the fault state and fault degree of a crack in a planetary gear from different observation angles and dimensions. Finally, the real condition loading test of a gearbox's comprehensive fault test platform was carried out. The validity of the method of nonlinear system modeling and fault diagnosis of the planetary gearbox, based on the Volterra series theory, was verified, and a new solution has been provided for related research in this field.

References

References
1.
Al-Hamadani
,
H. A.
, and
Taylan King
,
M.
,
2017
, “
System Dynamic Modelling of Three Different Wind Turbine Gearbox Designs Under Transient Loading Conditions
,”
Int. J. Precis. Eng. Manuf.
,
18
(
11
), pp.
1659
1668
.
2.
Skrimpas Georgios
,
A.
,
Thomas
,
U.
, and
Christian
,
S.
,
2017
, “
Residual Signal Feature Extraction for Gearbox Planetary Stage Fault Detection
,”
Wind Energy
,
20
(
8
), pp.
1389
1404
.
3.
Martin-Christopher
,
N.
,
Julian William
,
G.
, and
Ralf
,
S.
,
2017
, “
Analysis of Time-Domain Signals of Piezoelectric Strain Sensors on Slow Spinning Planetary Gearboxes
,”
Mech. Syst. Signal Process.
,
72
, pp.
727
744
.
4.
Liang
,
X.
,
Zuo
,
M. J.
, and
Feng
,
Z.
,
2018
, “
Dynamic Modeling of Gearbox Faults
,”
Mech. Syst. Signal Process.
,
98
, pp.
852
876
.
5.
Feng
,
Z.
, and
Liang
,
M.
,
2016
, “
Complex Signal Analysis for Planetary Gearbox Fault Diagnosis Via Shift Invariant Dictionary Learning
,”
Measurement
,
90
, pp.
382
395
.
6.
Singh
,
A.
, and
Parey
,
A.
,
2019
, “
Gearbox Fault Diagnosis Under Non-Stationary Conditions With Independent Angular Re-Sampling Technique Applied to Vibration and Sound Emission Signals
,”
Appl. Acoust.
,
144
, pp.
11
22
.
7.
Zimroz
,
R.
, and
Bartkowiak
,
A.
,
2013
, “
Two Simple Multivariate Procedures for Monitoring Planetary Gearboxes in Non-Stationary Operating Conditions
,”
Mech. Syst. Signal Process.
,
38
(
1
), pp.
237
247
.
8.
Feng
,
Z.
,
Chen
,
X.
, and
Liang
,
M.
,
2016
, “
Joint Envelope and Frequency Order Spectrum Analysis Based on Iterative Generalized Demodulation for Planetary Gearbox Fault Diagnosis Under Nonstationary Conditions
,”
Mech. Syst. Signal Process.
,
77
, pp.
242
264
.
9.
Feng
,
K.
,
Wang
,
K.
,
Ni
,
Q.
, and
Zuo
,
M. J.
,
2017
, “
A Phase Angle Based Diagnostic Scheme to Planetary Gear Faults Diagnostics Under Non-Stationary Operational Conditions
,”
J. Sound Vib.
,
408
(
10
), pp.
190
209
.
10.
Randall
,
R. B.
,
2017
, “
A History of Cepstrum Analysis and Its Application to Mechanical Problems
,”
Mech. Syst. Signal Process.
,
97
, pp.
3
19
.
11.
Samuel
,
P. D.
, and
Pines
,
D. J.
,
2009
, “
Constrained Adaptive Lifting and the CAL4 Metric for Helicopter Transmission Diagnostics
,”
J. Sound Vib.
,
319
(
1–2
), pp.
698
718
.
12.
Teng
,
W.
,
Ding
,
X.
,
Zhang
,
X.
,
Liu
,
Y.
, and
Ma
,
Z.
,
2016
, “
Multi-Fault Detection and Failure Analysis of Wind Turbine Gearbox Using Complex Wavelet Transform
,”
Renewable Energy
,
93
, pp.
591
598
.
13.
Yang
,
W. X.
, and
Tse
,
P. W.
,
2003
, “
Development of an Advanced Noise Reduction Method for Vibration Analysis Based on Singular Value Decomposition
,”
NDTE Int.
,
36
, pp.
419
432
.
14.
Lei
,
Y.
,
Han
,
D.
,
Lin
,
J.
, and
He
,
Z.
,
2013
, “
Planetary Gearbox Fault Diagnosis Using an Adaptive Stochastic Resonance Method
,”
Mech. Syst. Signal Process.
,
38
(
1
), pp.
113
124
.
15.
Wang
,
Z.
,
Han
,
Z.
,
Gu
,
F.
,
Gu
,
J. X.
, and
Ning
,
S.
,
2015
, “
A Novel Procedure for Diagnosing Multiple Faults in Rotating Machinery
,”
ISA Trans.
,
55
, pp.
208
218
.
16.
Liu
,
B.
,
Riemenschneider
., and
Xu
,
S. Y.
,
2005
, “
Gearbox Fault Diagnosis Using Empirical Mode Decomposition and Hilbert Spectrum
,”
Mech. Syst. Signal Process.
,
20
(
3
), pp.
718
734
.
17.
Haitao
,
W.
,
Kun
,
W.
, and
Lichen
,
S.
,
2018
, “
Quantitative Extraction of Rolling Bearings' Inner Race Fault Level Based on Volterra Theory
,”
J. Vib. Shock
,
37
(
9
), pp.
173
179
.
18.
Wang
,
H.
,
Kang
,
Z.
,
Shi
,
L.
,
Wang
,
K.
, and
Zhang
,
X.
,
2018
, “
Research and Application of Volterra Series Theory in Rolling Bearing Fault State Feature Extraction
,”
J. Vibroeng.
,
20
(
1
), pp.
189
201
.
19.
Peng
,
Z. K.
,
Lang
,
Z. Q.
, and
Billings
,
S. A.
,
2007
, “
Resonances and Resonant Frequencies for a Class of Nonlinear Systems
,”
J. Sound Vib.
,
300
(
3–5
), pp.
993
1014
.
20.
Zhu
,
A.
,
Draxler
,
P. J.
,
Hsia
,
C.
,
Brazil
,
J. T.
,
Kimball
,
F. D.
, and
Asbeck
,
M. P.
,
2008
, “
Digital Predistortion for Envelope-Tracking Power Amplifiers Using Decomposed Piecewise Volterra Series
,”
IEEE Trans. Microwave Theory Tech.
,
56
(
10
), pp.
2237
2247
.
21.
Peng
,
Z. K.
,
Lang
,
Z. Q.
,
Billings
,
S. A.
, and
Lu, Y.
,
2007
, “
Analysis of Bilinear Oscillators Under Harmonic Loading Using Nonlinear Output Frequency Re-Sponse Functions
,”
Int. J. Mech. Sci.
,
49
(
11
), pp.
1213
1225
.
22.
Hélie
,
T.
,
2010
, “
Volterra Series and State Transformation for Real-Time Simulations of Audio Circuits Including Saturations: Application to the Moogladder Filter
,”
IEEE Trans. Audio Speech
,
18
, pp.
747
759
.
23.
Guo
,
L. Z.
,
Billings
,
S. A.
, and
Coca
,
D.
,
2010
, “
Identification of Partial Differential Equation Models for a Class of Multiscale Spatio-Temporal Dynamical Systems
,”
Int. J. Control
,
83
(
1
), pp.
40
48
.
24.
Irving
,
A. D.
,
2008
, “
Dynamical Hysteresis in Communications: A Volterra Functional Approach
,”
IET Signal Proc.
,
2
(
2
), pp.
75
86
.
25.
Hélie
,
T.
, and
Hasler
,
M.
,
2004
, “
Volterra Series for Solving Weakly Non-Linear Partial Differential Equations: Application to a Dissipative Burgers' Equa-Tion
,”
Int. J. Control
,
77
, pp.
1071
1082
.
26.
Yuan
,
F.
, and
Opal
,
A.
,
2001
, “
Distortion Analysis of Periodically Switched Nonlinear Circuits Using Time-Varying Volterra Series
,”
IEEE Trans. Circuits-I
,
48
(
6
), pp.
726
738
.
27.
Boyd
,
S.
, and
Chua
,
L. O.
,
1985
, “
Fading Memory and the Problem of Approximating Nonlinear Operators With Volterra Series
,”
IEEE Trans. Circuits Syst
,
32
(
11
), pp.
1150
1161
.
28.
Volterra
,
V.
,
1959
,
Theory of Functionals and of Integral and Integro-Differential Equations
,
Dover Publications
,
New York
.
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