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Uncertain vibration analysis based on the conceptions of differential and integral of interval process

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Abstract

Recently, the authors proposed a new mathematical model called as the “interval process model” for quantifying uncertainty of time–varying parameters by making extension of the interval method into the time domain. In the interval process model, the imprecision of a time-varying parameter at arbitrary time point is described using an interval rather than the precise probability distribution, which makes the interval process model having some advantages over the traditional stochastic process in uncertainty quantification. Further, the authors proposed the important conceptions of limit, continuity, differential and integral of interval process, enriching the theory of interval process model. This paper applies the newly developed conceptions of differential and integral of interval process into the vibration analysis of mechanical structures or systems subjected to uncertain external excitations. By means of this application, the formulations of dynamic bounds of the velocity and acceleration responses are derived for the linear/multiple single degree of freedom (SDOF/MDOF) vibration systems subjected to dynamic uncertain excitations, which can provide some important reference information for reliability analysis and safety design of many practical mechanical structures or systems. The effectiveness of the proposed method are validated by investigating a spring-mass-damper system and a vehicle vibration problem.

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References

  • Attoh-Okine, N.O.: Uncertainty analysis in structural number determination in flexible pavement design—a convex model approach. Constr. Build. Mater. 16(2), 67–71 (2002)

    Google Scholar 

  • Ben-Haim, Y.: Convex models of uncertainty in radial pulse buckling of shells. ASME J. Appl. Mech. 60(3), 683–688 (1993)

    MATH  Google Scholar 

  • Ben-Haim, Y.,: A non-probabilistic concept of reliability. Struct. Saf. 14(4), 227–245 (1994) 

    Google Scholar 

  • Ben-Haim, Y.: A non-probabilistic measure of reliability of linear systems based on expansion of convex models. Struct. Saf. 17(2), 91–109 (1995)

    Google Scholar 

  • Ben-Haim, Y., Elishakoff, I.: Convex models of uncertainty in applied mechanics. Elsevier, Amsterdam (1990)

    MATH  Google Scholar 

  • Bi, R.G., Han, X., Jiang, C., Bai, Y.C., Liu, J.: Uncertain buckling and reliability analysis of the piezoelectric functionally graded cylindrical shells based on the nonprobabilistic convex model. Int. J. Comput. Methods 11(06), 1350080 (2014)

    MathSciNet  MATH  Google Scholar 

  • Chen, X.-Y., Fan, J.-P., Bian, X.-Y.: Theoretical analysis of non-probabilistic reliability based on interval model. Acta Mech. Solida Sin. 30(6), 638–646 (2017)

    Google Scholar 

  • Chen, N., Xia, S., Yu, D., Liu, J., Beer, M.: Hybrid interval and random analysis for structural-acoustic systems including periodical composites and multi-scale bounded hybrid uncertain parameters. Mech. Syst. Signal Process. 115, 524–544 (2019)

    Google Scholar 

  • Clough, R.W., Penzien, J.: Dynamics of Structures. McGraw-Hill, New York (1975)

    MATH  Google Scholar 

  • Crandall, S.H., Mark, W.D.: Random Vibration in Mechanical Systems. Academic Press, New York (2014)

    Google Scholar 

  • Deng, Z., Guo, Z., Zhang, X.: Non-probabilistic set-theoretic models for transient heat conduction of thermal protection systems with uncertain parameters. Appl. Therm. Eng. 95, 10–17 (2016)

    Google Scholar 

  • Elishakoff, I., Cai, G.Q., Starnes, J.H.: Non-linear buckling of a column with initial imperfection via stochastic and non-stochastic convex models. Int. J. Non-Linear Mech. 29(1), 71–82 (1994a)

    MATH  Google Scholar 

  • Elishakoff, I., Elisseeff, P., Glegg, S.A.L.: Nonprobabilistic, convex-theoretic modeling of scatter in material properties. AIAA J. 32(4), 843–849 (1994b)

    MATH  Google Scholar 

  • Faes, M., Moens, D.: Multivariate dependent interval finite element analysis via convex hull pair constructions and the extended transformation method. Comput. Methods Appl. Mech. Eng. 347, 85–102 (2019)

    MathSciNet  Google Scholar 

  • Faes, M., Broggi, M., Patelli, E., Govers, Y., Mottershead, J., Beer, M., Moens, D.: A multivariate interval approach for inverse uncertainty quantification with limited experimental data. Mech. Syst. Signal Process. 118, 534–548 (2019)

    Google Scholar 

  • Feng, X., Zhang, Y., Wu, J.: Interval analysis method based on Legendre polynomial approximation for uncertain multibody systems. Adv. Eng. Softw. 121, 223–234 (2018)

    Google Scholar 

  • Gao, W.: Interval finite element analysis using interval factor method. Comput. Mech. 39(6), 709–717 (2007)

    MATH  Google Scholar 

  • Ghanem, R.G., Spanos, P.D.: Stochastic finite elements a spectral approach. Springer, New York (1991)

    MATH  Google Scholar 

  • Guo, S.-S., Wang, D., Liu, Z.: Probabilistic analysis of random structural intensity for structural members under stochastic loadings. Int. J. Comput. Methods 12(03), 1550013 (2015)

    MathSciNet  MATH  Google Scholar 

  • Jensen, H.A., Mayorga, F., Papadimitriou, C.: Reliability sensitivity analysis of stochastic finite element models. Comput. Methods Appl. Mech. Eng. 296, 327–351 (2015)

    MathSciNet  MATH  Google Scholar 

  • Jiang, C., Li, J.W., Ni, B.Y., Fang, T.: Some significant improvements for interval process model and non-random vibration analysis method. Comput. Methods Appl. Mech. Eng. (2019) (accepted)

  • Jiang, C., Han, X., Lu, G.Y., Liu, J., Zhang, Z., Bai, Y.C.: Correlation analysis of non-probabilistic convex model and corresponding structural reliability technique. Comput. Methods Appl. Mech. Eng. 200(33), 2528–2546 (2011)

    MATH  Google Scholar 

  • Jiang, C., Ni, B.Y., Han, X., Tao, Y.R.: Non-probabilistic convex model process: a new method of time-variant uncertainty analysis and its application to structural dynamic reliability problems. Comput. Methods Appl. Mech. Eng. 268, 656–676 (2014)

    MathSciNet  MATH  Google Scholar 

  • Jiang, C., Zheng, J., Ni, B.Y., Han, X.: A probabilistic and interval hybrid reliability analysis method for structures with correlated uncertain parameters. Int. J. Comput. Methods 12(04), 1540006 (2015)

    MathSciNet  MATH  Google Scholar 

  • Jiang, C., Liu, N.Y., Ni, B.Y., Han, X.: Giving dynamic response bounds under incertain excitations—a non-random vibration analysis method. Chin. J. Theor. Appl. Mech. 48(2), 447–463 (2016a)

    Google Scholar 

  • Jiang, C., Ni, B.Y., Liu, N.Y., Han, X., Liu, J.: Interval process model and non-random vibration analysis. J. Sound Vib. 373, 104–131 (2016b)

    Google Scholar 

  • Li, G., Zhao, G., Zhou, C., Ren, M.: Stochastic elastic properties of composite matrix material with random voids based on radial basis function network. Int. J. Comput. Methods 15(01), 1750082 (2018a)

    MathSciNet  MATH  Google Scholar 

  • Li, J.W., Ni, B.Y., Jiang, C., Fang, T.: Dynamic response bound analysis for elastic beams under uncertain excitations. J. Sound Vib. 422, 471–489 (2018b)

    Google Scholar 

  • Lin, Y.K.: Probabilistic Theory of Structural Dynamics. McGraw Hill, New York (1967)

    Google Scholar 

  • Liu, Y., Wang, X., Wang, L.: Interval uncertainty analysis for static response of structures using radial basis functions. Appl. Math. Model. 69, 425–440 (2019)

    MathSciNet  Google Scholar 

  • Long, X., Elishakoff, I., Jiang, C., Han, X., Hashemi, J.: Notes on random vibration of a vehicle model and other discrete systems possessing repeated natural frequencies. Arch. Appl. Mech. 84(8), 1091–1101 (2014)

    Google Scholar 

  • Long, X.H., Xie, Z.Y., Fan, J., Miao, Y.: Convex model-based calculation of robust seismic fragility curves of isolated continuous girder bridge. Bull. Earthq. Eng. 16(1), 155–182 (2018)

    Google Scholar 

  • Martin, G.H.: Kinematics and Dynamics of Machines, 2nd edn. McGraw-Hill, New York (1982)

    Google Scholar 

  • Meng, Z., Zhou, H.: New target performance approach for a super parametric convex model of non-probabilistic reliability-based design optimization. Comput. Methods Appl. Mech. Eng. 339, 644–662 (2018)

    MathSciNet  Google Scholar 

  • Meng, Z., Zhou, H., Li, G., Yang, D.: A decoupled approach for non-probabilistic reliability-based design optimization. Comput. Struct. 175, 65–73 (2016)

    Google Scholar 

  • Ni, B.Y., Jiang, C., Huang, Z.L.: Discussions on non-probabilistic convex modelling for uncertain problems. Appl. Math. Model. 59, 54–85 (2018)

    MathSciNet  MATH  Google Scholar 

  • Pantelides, C.P.: Stability of elastic bars on uncertain foundations using a convex model. Int. J. Solids Struct. 33(9), 1257–1269 (1996)

    MATH  Google Scholar 

  • Protter, M.H., Morrey, C.B.: Intermediate Calculus, 2nd edn. Springer, New York (1985)

    MATH  Google Scholar 

  • Sofi, A., Romeo, E., Barrera, O., Cocks, A.: An interval finite element method for the analysis of structures with spatially varying uncertainties. Adv. Eng. Softw. 128, 1–19 (2019)

    Google Scholar 

  • Su, H., Li, J., Wen, Z., Fu, Z.: Dynamic non-probabilistic reliability evaluation and service life prediction for arch dams considering time-varying effects. Appl. Math. Model. 40(15), 6908–6923 (2016)

    MATH  Google Scholar 

  • Timoshenko, S., Young, D.H., Weaver Jr., W.: Vibration problems in engineering. Wiley, New York (1974)

    Google Scholar 

  • Truong, V.H., Liu, J., Meng, X., Jiang, C., Nguyen, T.T.: Uncertainty analysis on vehicle-bridge system with correlative interval variables based on multidimensional parallelepiped model. Int. J. Comput. Methods 15(05), 1850030 (2018)

    MathSciNet  MATH  Google Scholar 

  • Wang, R., Wang, X., Wang, L., Chen, X.: Efficient computational method for the non-probabilistic reliability of linear structural systems. Acta Mech. Solida Sin. 29(3), 284–299 (2016)

    Google Scholar 

  • Xiong, C., Wang, L., Liu, G., Shi, Q.: An iterative dimension-by-dimension method for structural interval response prediction with multidimensional uncertain variables. Aerosp. Sci. Technol. 86, 572–581 (2019)

    Google Scholar 

  • Xu, M., Du, J., Chen, J., Wang, C., Li, Y.: An iterative dimension-wise approach to the structural analysis with interval uncertainties. Int. J. Comput. Methods 15(06), 1850044 (2018)

    MathSciNet  MATH  Google Scholar 

  • Yu, Z.S.: Automobile Theory, 2nd edn. China Machine Press, Beijing (1989)

    Google Scholar 

  • Zheng, Y., Qiu, Z.: Uncertainty propagation in aerodynamic forces and heating analysis for hypersonic vehicles with uncertain-but-bounded geometric parameters. Aerosp. Sci. Technol. 77, 11–24 (2018)

    Google Scholar 

  • Zhou, C., Tang, C., Liu, F., Wang, W.: A probabilistic representation method for interval uncertainty. Int. J. Comput. Methods 15(05), 1850038 (2018)

    MathSciNet  MATH  Google Scholar 

Download references

Funding

Funding was provided by the Science Challenge Project (Grant No. TZ2018007), the National Science Fund for Distinguished Young Scholars (Grant No. 51725502), the National Key R&D Program of China (Grant No. 2016YFD0701105) and the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (Grant No. 51621004).

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Authors and Affiliations

  1. State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University, Changsha City, 410082, People’s Republic of China

    Jinwu Li, Chao Jiang & Bingyu Ni

  2. Department of Electrical Engineering, Hunan Polytechnic of Water Resources and Electric Power, Changsha City, 410131, People’s Republic of China

    Lina Zhan

Authors
  1. Jinwu Li
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  2. Chao Jiang
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  3. Bingyu Ni
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  4. Lina Zhan
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Correspondence to Chao Jiang.

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Li, J., Jiang, C., Ni, B. et al. Uncertain vibration analysis based on the conceptions of differential and integral of interval process. Int J Mech Mater Des 16, 225–244 (2020). https://doi.org/10.1007/s10999-019-09470-0

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