Research Papers

Vibration Analysis in the Presence of Uncertainties Using Universal Grey System Theory

[+] Author and Article Information
X. T. Liu

Department of Mechanical and
Aerospace Engineering,
University of Miami,
Coral Gables, FL 33146
e-mail: xintianster@gmail.com

S. S. Rao

Fellow ASME
Department of Mechanical and
Aerospace Engineering,
University of Miami,
Coral Gables, FL 33146
e-mail: srao@miami.edu

1Present address: School of Automotive Engineering, Shanghai University of Engineering Science, Shanghai 201620, China.

2Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received March 4, 2017; final manuscript received September 26, 2017; published online February 9, 2018. Assoc. Editor: Stefano Gonella.

J. Vib. Acoust 140(3), 031009 (Feb 09, 2018) (11 pages) Paper No: VIB-17-1088; doi: 10.1115/1.4038940 History: Received March 04, 2017; Revised September 26, 2017

The uncertainty present in many vibrating systems has been modeled in the past using several approaches such as probabilistic, fuzzy, interval, evidence, and grey system-based approaches depending on the nature of uncertainty present in the system. In most practical vibration problems, the parameters of the system such as stiffness, damping and mass, initial conditions, and/or external forces acting on the system are specified or known in the form of intervals or ranges. For such cases, the use of interval analysis appears to be most appropriate for predicting the ranges of the response quantities such as natural frequencies, free vibration response, and forced vibration response under specified external forces. However, the accuracy of the results given by the interval analysis suffers from the so-called dependency problem, which causes an undesirable expansion of the intervals of the computed results, which in some case, can make the results unacceptable for practical implementation. Unfortunately, there has not been a simple approach that can improve the accuracy of the basic interval analysis. This work considers the solution of vibration problems using universal grey system (or number) theory for the analysis of vibrating systems whose parameters are described in terms of intervals or ranges. The computational feasibility and improved accuracy of the methodology, compared to interval analysis, are demonstrated by considering one and two degrees-of-freedom (2DOF) systems. The proposed technique can be extended for the uncertainty analysis of any multi-degrees-of-freedom system without much difficulty.

Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.


Elishakoff, I. , 2017, Probabilistic Methods in Theory of Structures, World Scientific, Singapore. [CrossRef]
Rao, S. S. , 1984, “Multiobjective Optimization in Structural Design in the Presence of Uncertain Parameters and Stochastic Process,” AIAA J., 22(11), pp. 1670–1678. [CrossRef]
Cavalini , A. A., Jr ., Dourado, A. S. , Lara-Molina, F. A. , and Steffen, V. V., Jr ., 2016, “Uncertainty Analysis of a Tilting-Pad Journal Bearing Using Fuzzy Logic Techniques,” ASME J. Vib. Acoust., 138(6), p. 061016. [CrossRef]
Kaufmann, A. , and Gupta, M. M. , 1991, Introduction to Fuzzy Arithmetic, Van Nostrand Reinhold, New York.
Moore, R. E. , Kearfott, R. B. , and Cloud, M. J. , 2009, Introduction to Interval Analysis, Society for Industrial and Applied Mathematics, Philadelphia, PA. [CrossRef]
Dempster, A. P. , 1967, “Upper and Lower Probabilities Induced by a Multi-Valued Mapping,” Ann. Math. Stat., 38(2), pp. 325–339. [CrossRef]
Liu, S. , and Lin, Y. , 2006, Grey Information Theory and Practical Applications, Springer-Verlag, London.
Chen, L. , and Rao, S. S. , 1997, “Fuzzy Finite Element Approach for the Vibration Analysis of Imprecisely-Defined Systems,” J. Finite Elem. Anal. Des., 27(1), pp. 69–83. [CrossRef]
Moens, D. , and Vandepitte, D. , 2005, “A Fuzzy Finite Element Procedure for the Calculation of Uncertain Frequency-Response Functions of Damped Structures—Part I: Procedure,” J. Sound Vib., 288(3), pp. 431–462. [CrossRef]
Qui, Y. , and Rao, S. S. , 2005, “A Fuzzy Approach for the Analysis of Unbalanced Nonlinear Rotor Systems,” J. Sound Vib., 284(1–2), pp. 299–323.
Rao, S. S. , and Annamdas, K. K. , 2013, “A Comparative Study of Evidence Theories in the Modeling, Analysis, and Design of Engineering Systems,” ASME J. Mech. Des., 135(6), p. 061006. [CrossRef]
Ayyub, B. , Guran, A. , and Haldar, A. , 1997, Uncertainty Modeling in Vibration, Control and Fuzzy Analysis of Structural Systems, World Scientific, Singapore. [CrossRef]
Dimarogonas, A. D. , 1995, “Interval Analysis of Vibrating Systems,” J. Sound Vib., 183(4), pp. 739–749. [CrossRef]
Rao, S. S. , and Berke, L. , 1997, “Analysis of Uncertain Structural Systems Using Interval Analysis,” AIAA J., 35(4), pp. 727–734. [CrossRef]
Elishakoff, I. , and Thakker, K. , 2014, “Overcoming Overestimation Characteristic to Classical Interval Analysis,” AIAA J., 52(9), pp. 2093–2097. [CrossRef]
Luo, Y. X. , Huang, H. Z. , and Fan, X. F. , 2006, “The Universal Grey Transfer Matrix Method and Its Application in Calculating the Natural Frequencies of Systems,” Strojniski Vestnik—J. Mech. Eng., 52(9), pp. 592–598.
Rao, S. S. , and Jin, H. L. , 2014, “Analysis of Coupled Bending-Torsional Vibration of Beams in the Presence of Uncertainties,” ASME J. Vib. Acoust., 136(5), p. 051004. [CrossRef]
Deng, J. L. , 1982, “The Control Problem of Grey Systems,” Syst. Control Lett., 1(5), pp. 288–294. [CrossRef]
Liu, S. F. , and Forrest, J. Y. L. , 2010, Advances in Grey Systems Research, Springer, Berlin. [CrossRef]
Gong, Z. , and Forrest, J. Y. L. , 2014, “Editorial: Special Issue on Meteorological Disaster Risk Analysis and Assessment: On Basis of Grey Systems Theory,” Nat. Hazards, 71(2), pp. 995–1000. [CrossRef]
Zou, Q. , Zhou, J. Z. , Zhou, C. , and Chen, S. S. , 2012, “Flood Disaster Risk Analysis Based on Principle of Maximum Entropy and Attribute Interval Recognition Theory,” Adv. Water Sci., 23(3), pp. 323–333.
Zavadskas, E. K. , Turskis, Z. , and Tamosaitience, J. , 2010, “Risk Assessment of Construction Projects,” J. Civ. Eng. Manage., 16(1), pp. 33–46. [CrossRef]
Hsu, C. C. , and Chen, C. Y. , 2003, “Applications of Improved Grey Prediction Model for Power Demand Forecasting,” Energy Convers. Manage., 44(14), pp. 2241–2249. [CrossRef]
Neumaier, A. , 1990, Interval Methods for Systems of Equations, Cambridge University Press, Cambridge, UK.


Grahic Jump Location
Fig. 1

A spring–mass–damper system subjected to base excitation

Grahic Jump Location
Fig. 2

(a) Variation of displacement transmissibility with frequency ratio for ζ=0.1 and (b) variation of displacement transmissibility with frequency ratio for ζ=0.5

Grahic Jump Location
Fig. 3

(a) Variations of force transmissibility with frequency ratio for ζ=0.1 and (b) variations of force transmissibility with frequency ratio for ζ=0.5

Grahic Jump Location
Fig. 4

A spring–mass–damper system subjected to harmonic force

Grahic Jump Location
Fig. 5

A damped two degrees-of-freedom system

Grahic Jump Location
Fig. 6

Free vibration response of the system (x1)

Grahic Jump Location
Fig. 7

Free vibration response of the system (x2)



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In