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Fractional polynomial function in stochastic response surface method for reliability analysis

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Abstract

The explicit polynomial function can replace the actual limit state function for failure probability. In this study, the fractional order is introduced to an extended family of polynomial functions to guarantee a smooth relationship between the sample data and polynomial expression. The fractional polynomial method is proposed to establish the function and analyze the system’s reliability accurately and flexibly. Integer, fractional, positive, and negative orders are considered. The scope of orders, constraint conditions, and sampling method are investigated under different conditions of the proposed method. Furthermore, the calculation process for the fractional polynomial method when the approach is applied to a multivariable system is examined, and engineering examples are provided to illustrate the application of the proposed method. The fractional polynomial function can describe system performance and reliability with few terms and high accuracy. The results calculated by the proposed method agree with those from Monte Carlo simulation.

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Abbreviations

g(x):

Fractional polynomial with multivariable x, x = [xi, …]

β l :

Coefficient corresponding to the term involving the jth order

υ 1 :

Lower bound of the scope of order j

τ 1 :

Upper bound of the scope of order j

a :

Lower bound of variable x

b :

Upper bound of variable x

ε :

Minimum value in the computing process

w k :

Sampling points to determine the order, (k = 1, 2, …, s)

α k :

Quantile of the probability distribution function

c l/d l :

Fractional order j = cl/dl, (l = 1, 2, …)

E( ):

Mathematical expectation

V( ):

Standard deviation

H( ):

Optimization target

L :

Degree of function

Δ:

Allowable error value

G :

Goodness of fit

p i h,:

Order of variable xi in the mixed term h, (h = 1, 2, …)

D :

Set of all viable orders

γ :

Coefficient corresponding to the mixed term

X :

Matrix of independent variables xi

A :

Coefficient vector

Y :

Vector of system responses y(ι), (ι = 1, 2, …)

I I :

Important index of the independent term

I M :

Important index of the mixed term

I λ :

Evaluated term in the established function

S λ :

Influence of term λ

u :

Threshold value of the vibration system

ω:

Frequency of external excitation

m 1 m 2 :

Mass

c1, c2 :

Damping

k 1, k 2 :

Stiffness

MSE :

Mean square error

FPM :

Fractional polynomial method

MCS :

Monte Carlo simulation

SRSM :

Quadratic stochastic response surface method

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Acknowledgments

This project is supported by the National Natural Science Foundation of China (Grant Nos. 52005327, 72071127, and U1708254). The authors would like to thank the anonymous referees and the editor for their valuable comments and suggestions.

Author information

Authors and Affiliations

  1. State Key Laboratory of Mechanical System and Vibration, Department of Industrial Engineering, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China

    Di Zhou & Ershun Pan

  2. Equipment Reliability Institute, Shenyang University of Chemical Technology, Shenyang, 110142, China

    Yimin Zhang

Authors
  1. Di Zhou
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  2. Ershun Pan
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  3. Yimin Zhang
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Corresponding author

Correspondence to Ershun Pan.

Additional information

Di Zhou is a Post Doctor in the School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, China. He received his Ph.D. in Mechanical Engineering and Automation from Northeastern University. His research interests include mechanical reliability analysis and mechanical dynamics.

Ershun Pan is currently a Professor at the Department of Industrial Engineering and Management, School of Mechanical Engineering, Shanghai Jiao Tong University. His research interests are engineering reliability and maintenance strategy.

Yimin Zhang, Ph.D., is a Chang Jiang Scholar of Mechanical Design and Theory and a member of the Academic Degrees Committee of the State Council of China for Mechanical Engineering. His interests include mechanical dynamic design, mechanical reliability design, modern design methodology, and rotor dynamics.

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Zhou, D., Pan, E. & Zhang, Y. Fractional polynomial function in stochastic response surface method for reliability analysis. J Mech Sci Technol 35, 121–131 (2021). https://doi.org/10.1007/s12206-020-1211-3

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  • DOI: https://doi.org/10.1007/s12206-020-1211-3

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