Skip to main content
SpringerLink
Log in

Advanced method to estimate reliability-based sensitivity of mechanical components with strongly nonlinear performance function

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

Based on the random perturbation technique for reliability sensitivity design, some realistic reliability-based sensitivity issues are discussed, some of which have a structure of high nonlinear performance functions. Combining the related theories of the moment method of the reliability analysis, the matrix differential, and the Kronecker algebra, the reliability-based sensitivity method based on the perturbation method is modified if the first four moments of random variables are given. Meanwhile, a reliability-based sensitivity computation method is proposed. Some examples are used to show that using this method can effectively improve the accuracy of the reliability-based sensitivity computation and offer a reliable theoretic basis in engineering.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

X :

a vector of random variables

n :

the dimensions of the vector X

X d :

a certain part of the random parameters

X p :

a random part of the random parameters

ɛ :

a small parameter

g(X):

the performance function

g d (X):

a certain part of the performance function

g p (X):

a random part of the performance function

E(X):

the mean value of random variables

var(X):

the variance of random variables

C3(X):

the third moment of random variables

C4(X):

the fourth moment of random variables

µg :

the mean value of the performance function

σ 2 g :

the variance of the performance function g(X)

θ g :

the third central moment of the performance function g(X)

η g :

the fourth central moment of the performance function g(X)

β :

the reliability index

PDF:

the probability density function

Φ(·):

the standard normal distribution function

φ(·):

the standard normal probability density function

H i (y):

the Hermite polynomial

R(β):

the reliability of the system

\( \frac{{dR\left( \beta \right)}} {{d\bar X^T }} \) :

sensitivity of reliability with respect to the mean value of random variables

\( \frac{{dR\left( \beta \right)}} {{d\operatorname{var} \left( X \right)}} \) :

sensitivity of reliability with respect to the variance of random variables

I n :

the n × n unit matrix

U n×n :

the permutation matrix

Q :

the load effect on the tension bar

d 0 :

the inside diameter of the tubular section

d 1 :

the outside diameter of the tubular section

r :

material strength of a tension bar

l :

the length of the shaft where the dynamic stress is maximum

E :

the elastic modulus of the shaft

ω :

the rotational speed of the rotor system

u :

the unbalance value of the rotor system

ρ :

the correlation coefficient

PRSM:

the reliability sensitivity computation method based on the perturbation method

PRSMM:

the modified reliability sensitivity method based on the perturbation method

CDM:

the central difference method

MCS:

the Monte-Carlo simulation method.

References

  1. Ben-Haim, Y. Robust reliability of structures. Advances in Applied Mechanics 31, 1–41 (1997)

    Article  Google Scholar 

  2. Liaw, L. D. and DeVries, R. I. Reliability-based optimization for robust design. International Journal of Vehicle Design 25(1–2), 64–77 (2001)

    Article  Google Scholar 

  3. Zhang, Y. M., He, X. D., Liu, Q. L., and Wen, B. C. Reliability-based optimization and robust design of coil tube-spring with non-normal distribution parameters. Proceedings of the Institution of Mechanical Engineers, Part C—Journal of Mechanical Engineering Science 219(6), 567–576 (2005)

    Article  Google Scholar 

  4. De Lataillade, A., Blanco, S., Clergent, Y., Dufresne, J. L., El Hafi, M., and Fournier, R. Monte-Carlo method and sensitivity estimations. Journal of Quantitative Spectroscopy and Radiative Transfer 75(5), 529–538 (2002)

    Article  Google Scholar 

  5. Melchers, R. E. and Ahammed, M. A fast approximate method for parameter sensitivity estimation in Monte-Carlo structural reliability. Computers and Structures 82(1), 55–61 (2004)

    Article  Google Scholar 

  6. Shiraishi, F., Tomita, T., Iwata, M., Berrada, A. A., and Hirayama, H. A reliable Taylor seriesbased computational method for the calculation of dynamic sensitivities in large-scale metabolic reaction systems: algorithm and software evaluation. Mathematical Biosciences 222(2), 73–85 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Karamchandani, A. and Cornell, C. A. Sensitivity estimation within first and second order reliability methods. Structural Safety 11(2), 95–107 (1991)

    Article  Google Scholar 

  8. Zhao, Y. G. and Ono, T. Moment methods for structural reliability. Structural Safety 23(1), 47–55 (2001)

    Article  Google Scholar 

  9. Wu, Y. T., Shah, C. R., and Baruah, A. K. D. Progressive advanced-mean-value method for CDF and reliability analysis. International Journal of Materials and Product Technology 17(5–6), 303–318 (2002)

    Article  MATH  Google Scholar 

  10. Zhang, Y. M., He, X. D., Liu, Q. L., Wen, B. C., and Zheng, J. X. Reliability sensitivity of automobile components with arbitrary distribution parameters. Proceedings of the Institution of Mechanical Engineers, Part D-Journal of Automobile Engineering 219(D2), 165–182 (2005)

    Article  Google Scholar 

  11. Zhang, Y. M., Wen, B. C., and Chen, S. H. PFEM formalism in Kronecker notation. Mathematics and Mechanics of Solids 1(4), 445–461 (1996)

    Article  MATH  Google Scholar 

  12. Zhang, Y. M., Wen, B. C., and Liu, Q. L. Reliability sensitivity for rotor-stator systems with rubbing. Journal of Sound and Vibration 259(5), 1095–1107 (2003)

    Article  Google Scholar 

  13. Zhang, Y. M., Wen, B. C., and Liu, Q. L. First passage of uncertain single degree-of-freedom nonlinear oscillators. Computer Methods in Applied Mechanics and Engineering 165(1–4), 223–231 (1998)

    Article  MATH  Google Scholar 

  14. Zhang, Y. M., Liu, Q. L., and Wen, B. C. Quasi-failure analysis on resonant demolition of random structural systems. AIAA Journal 40(3), 585–586 (2002)

    Article  Google Scholar 

  15. Wu, Y. T. and Mohanty, S. Variable screening and ranking using sampling-based sensitivity measures. Reliability Engineering and System Safety 91(6), 634–647 (2006)

    Article  Google Scholar 

  16. Kreinovich, V., Beck, J., Ferregut, C., Sanchez, A., Keller, G. R., Averill, M., and Starks, S. A. Monte-Carlo-type techniques for processing interval uncertainty, and their potential engineering applications. Reliable Computing 13(1), 25–69 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  17. Wu, Y. Adaptive importance sampling (AIS)-based system reliability sensitivity analysis method. Proceedings of the IUTAM Symposium, Springer-Verlag, New York, 550 (1993)

    Google Scholar 

  18. Kim, C., Wang, S., and Choi, K. K. Efficient response surface modeling by using moving leastsquares method and sensitivity. AIAA Journal 43(11), 2404–2411 (2005)

    Article  Google Scholar 

  19. Youn, B. D. and Choi, K. K. A new response surface methodology for reliability-based design optimization. Computers and Structures 82(2–3), 241–256 (2004)

    Article  Google Scholar 

  20. Lee, S. H. and Kwak, B. M. Response surface augmented moment method for efficient reliability analysis. Structural Safety 28(3), 261–272 (2006)

    Article  Google Scholar 

  21. Brewer, W. J. Kronecker products and matrix calculus in system theory. IEEE Transactions on Circuits and Systems 25(9), 772–781 (1978)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mechanical Engineering and Automation, Northeastern University, Shenyang, 110819, P. R. China

    Yi-min Zhang  (张义民), Li-sha Zhu  (朱丽莎) & Xin-gang Wang  (王新刚)

Authors
  1. Yi-min Zhang  (张义民)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Li-sha Zhu  (朱丽莎)
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xin-gang Wang  (王新刚)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yi-min Zhang  (张义民).

Additional information

Communicated by Li-qun CHEN

Project supported by the Key National Science and Technology Special Project on “Hign-Grade CNC Machine Tools and Basic Manufacturing Equipments” (No. 2010ZX04014-014), the National Natural Science Foundation of China (No. 50875039), and the Program for Changjiang Scholars and Innovative Research Team in University

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, Ym., Zhu, Ls. & Wang, Xg. Advanced method to estimate reliability-based sensitivity of mechanical components with strongly nonlinear performance function. Appl. Math. Mech.-Engl. Ed. 31, 1325–1336 (2010). https://doi.org/10.1007/s10483-010-1365-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-010-1365-x

Key words

Chinese Library Classification

2000 Mathematics Subject Classification

Search

Navigation