Digital twin-assisted gearbox dynamic model updating toward fault diagnosis

Abstract

One of the core challenges of intelligent fault diagnosis is that the diagnosis model requires numerous labeled training datasets to achieve satisfactory performance. Generating training data using a virtual model is a potential solution for addressing such a problem, and the construction of a high-fidelity virtual model is fundamental and critical for data generation. In this study, a digital twin-assisted dynamic model updating method for fault diagnosis is thus proposed to improve the fidelity and reliability of a virtual model, which can enhance the generated data quality. First, a virtual model is established to mirror the vibration response of a physical entity using a dynamic modeling method. Second, the modeling method is validated through a frequency analysis of the generated signal. Then, based on the signal similarity indicator, a physical–virtual signal interaction method is proposed to dynamically update the virtual model in which parameter sensitivity analysis, surrogate technique, and optimization algorithm are applied to increase the efficiency during the model updating. Finally, the proposed method is successfully applied to the dynamic model updating of a single-stage helical gearbox; the virtual data generated by this model can be used for gear fault diagnosis.

Graphical abstract

Keywords

digital twin / gearbox / model construction / model updating / physical–virtual interaction

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Nomenclature

Abbreviations
FA	Firefly algorithm
FEM	Finite element method
FFT	Fast Fourier transform
LHS	Latin hypercube sampling
LPM	Lumped parameter method
LSM	Least squares method
PHM	Prognostics and health management
PRSM	Polynomial response surface model
RFCD	Rigid–flexible coupling dynamic
RFCM	Rigid–flexible coupling method
RSM	Response surface methodology
Variables
ai	Amplitude of the exciting force
{\overline{a}}_i	Amplitude of the vibration response
Cdf	Dynamic friction coefficient
CD	Damping coefficient
Csf	Static friction coefficient
DP	Penetration depth
eF	Force exponent
E	Elastic modulus of gears
f(t)	Excitation force
f(t1,t2,t3)	Cosine similarity between virtual and experimental signals
fn1, fn2	Rotation frequencies of input and output gears, respectively
fz	Mesh frequency
g	gth iteration
h(t)	Transfer function of the gearbox
k	Number of sensitive model parameters
K	Contact stiffness
L	Number of sample points
M	Number of mode order
n	Number of parameters
p	Model parameters in the dynamic model
pL	Lower bound of the model parameters
ps	Sensitive model parameters
psL	Lower bound of the sensitive model parameters
psU	Upper bound of the sensitive model parameters
pU	Upper bound of the model parameters
r1, r2	Equivalent radius
rij	Distance between fireflies i and j
R2	Coefficient of determination
R(p)	Difference between the generated and physical signals
t1	Normalized contact stiffness
t2	Normalized contact damping coefficient
t3	Normalized force exponent
Vdt	Dynamic transonic speed
Vst	Static transonic speed
X	Variable matrix of sampling points
Xe	Physical signal in the time domain
Xg	Generated signal in the time domain
\overline{y}	Mean value of the simulation response
yi	Result of sample point i based on the complex simulation model
{\tilde{y}}_i	Result of sample point i based on the constructed surrogate model
y(t)	Vibration response
Y	Response vector of sampling points
z1	Tooth number of the driving gear
z2	Tooth number of the driven gear
α	Random parameter that controls movement randomization
β	Unknown coefficient vector
β0	Attractiveness factor
ε	Error vector
γ	Brightness absorption coefficient
ν1, ν2	Poisson’s ratios of the gears
θ1	Phase of the exciting force
{\overline{\theta }}_i	Phase of the vibration response
ρ	Density of gears
\mathbf{\Im }	Random vector (k-dimensional)

Acknowledgements

This work was supported in part by the National Key R&D Program of China (Grant No. 2018YFB1702400), the National Natural Science Foundation of China (Grant Nos. 52275111, 52205100, and 52205101), and the Guangdong Basic and Applied Basic Research Foundation, China (Grant Nos. 2021A1515110708 and 2023A1515012856).

Conflict of Interest

The authors declare that they have no conflict of interest.

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