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Dynamic Stability Analysis of Linear Time-varying Systems via an Extended Modal Identification Approach

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Abstract

The problem of linear time-varying(LTV) system modal analysis is considered based on time-dependent state space representations, as classical modal analysis of linear time-invariant systems and current LTV system modal analysis under the “frozen-time” assumption are not able to determine the dynamic stability of LTV systems. Time-dependent state space representations of LTV systems are first introduced, and the corresponding modal analysis theories are subsequently presented via a stability-preserving state transformation. The time-varying modes of LTV systems are extended in terms of uniqueness, and are further interpreted to determine the system’s stability. An extended modal identification is proposed to estimate the time-varying modes, consisting of the estimation of the state transition matrix via a subspace-based method and the extraction of the time-varying modes by the QR decomposition. The proposed approach is numerically validated by three numerical cases, and is experimentally validated by a coupled moving-mass simply supported beam experimental case. The proposed approach is capable of accurately estimating the time-varying modes, and provides a new way to determine the dynamic stability of LTV systems by using the estimated time-varying modes.

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References

  1. HEYLEN W, LAMMENS S, SAS P. Modal analysis theory and testing[M]. Leuven: Katholieke Universiteit Leuven, 2007.

  2. WALSH P L, LAMANCUSA J S. A variable stiffness vibration absorber for minimization of transient vibrations[J]. Journal of Sound and Vibration, 1992, 158(2): 195–211.

  3. AU F T K, JIANG R J, CHEUNG Y K. Parameter identification of vehicles moving on continuous bridges[J]. Journal of Sound and Vibration, 2004, 269(1–2): 91–111.

  4. JOSHI A. Free vibration characteristics of variable mass rockets having large axial thrust/acceleration[J]. Journal of Sound and Vibration, 1995, 187(4): 727–736.

  5. VERBOVEN P, CAUBERGHE B, GUILLAUME P, et al. Modal parameter estimation and monitoring for on-line flight flutter analysis[J]. Mechanical Systems and Signal Processing, 2004, 18(3): 587–610.

  6. CHU M, ZHANG Y, CHEN G, et al. Effects of joint controller on analytical modal analysis of rotational flexible manipulator[J]. Chinese Journal of Mechanical Engineering, 2015, 28(3): 460–469.

  7. KRISHNAMURTHY K, CHAO M C. Active vibration control during deployment of space structures[J]. Journal of Sound and Vibration, 1992, 152(2): 205–218.

  8. AVENDANO-VALENCIA L D, FASSOIS S D. Stationary and non-stationary random vibration modelling and analysis for an operating wind turbine[J]. Mechanical Systems and Signal Processing, 2014, 47(1–2): 263–285.

  9. GARIBALDI L, FASSOIS S. MSSP special issue on the identification of time varying structures and systems[J]. Mechanical Systems and Signal Processing, 2014, 47(1–2): 1–2.

  10. WU M-Y. On stability of linear time-varying systems[J]. International Journal of Systems Science, 1984, 15(2): 137–150.

  11. ZADEH L A. Frequency analysis of variable networks[J]. Proceedings of the Institute of Radio Engineers, 1950, 38(3): 291–299.

  12. RAMNATH R V. Multiple scales theory and aerospace applications[M]. Reston: American Institute of Aeronautics and Astronautics, 2010.

  13. WU M-Y. A new concept of eigenvalues and eigenvectors and its applications[J]. IEEE Transactions on Automatic Control, 1980, 25(4): 824–826.

  14. KAMEN E W. The poles and zeros of a linear time-varying system[J]. Linear Algebra and Its Applications, 1988, 98: 263–289.

  15. O’BRIEN R T, JR., IGLESIAS P A. Poles and zeros for time-varying systems[C]//The 16th American Control Conference, Evanston, IL, USA, June 4–6, 1997: 2 672–2 676.

  16. O’BRIEN R T, JR., IGLESIAS P A. On the poles and zeros of linear, time-varying systems[J]. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 2001, 48(5): 565–577.

  17. ZENGER K, YLINEN R. Poles and zeros of multivariable linear time-varying systems[C]//The 15th IFAC World Congress, Barcelona, Spain, July 21–26, 2002: 261–266.

  18. PETSOUNIS K A, FASSOIS S D. Parametric time-domain methods for the identification of vibrating structures-a critical comparison and assessment[J]. Mechanical Systems and Signal Processing, 2001, 15(6): 1 031–1 060.

  19. POULIMENOS A G, FASSOIS S D. Parametric time-domain methods for non-stationary random vibration modelling and analysis—a critical survey and comparison[J]. Mechanical Systems and Signal Processing, 2006, 20(4): 763–816.

  20. SPIRIDONAKOS M D, FASSOIS S D. Non-stationary random vibration modelling and analysis via functional series time-dependent ARMA(FS-TARMA) models – a critical survey[J]. Mechanical Systems and Signal Processing, 2014, 47(1–2): 175– 224.

  21. VERHAEGEN M, YU X. A class of subspace model identification algorithms to identify periodically and arbitrarily time-varying systems[J]. Automatica, 1995, 31(2): 201–216.

  22. LIU K. Identification of linear time-varying systems[J]. Journal of Sound and Vibration, 1997, 206(4): 487–505.

  23. LIU K. Extension of modal analysis to linear time-varying systems[J]. Journal of Sound and Vibration, 1999, 226(1): 149–167.

  24. LIU K, DENG L. Identification of pseudo-natural frequencies of an axially moving cantilever beam using a subspace-based algorithm[J]. Mechanical Systems and Signal Processing, 2006, 20(1): 94–113.

  25. SHOKOOHI S, SILVERMAN L M. Identification and model reduction of time-varying discrete-time systems[J]. Automatica, 1987, 23(4): 509–521.

  26. MAJJI M, JUANG J-N, JUNKINS J L. Time-varying eigensystem realization algorithm[J]. Journal of Guidance, Control, and Dynamics, 2010, 33(1): 13–28.

  27. MAJJI M, JUANG J-N, JUNKINS J L. Observer/Kalman-filter time-varying system identification[J]. Journal of Guidance, Control, and Dynamics, 2010, 33(3): 887–900.

  28. BELLINO A, FASANA A, GANDINO E, et al. A time-varying inertia pendulum: analytical modelling and experimental identification[J]. Mechanical Systems and Signal Processing, 2014, 47(1–2): 120–138.

  29. JHINAOUI A, MEVEL L, MORLIER J. A new SSI algorithm for LPTV systems: Application to a hinged-bladed helicopter[J]. Mechanical Systems and Signal Processing, 2014, 42(1–2): 152–166.

  30. SHMALIY Y S. Continuous-time systems[M]. Dordrecht, Netherlands: Springer, 2007.

  31. WU M-Y, HOROWITZ I M, DENNISON J C. On solution, stability and transformation of linear time-varying systems[J]. International Journal of Control, 1975, 22(2): 169–180.

  32. WU M-Y. Solvability and representation of linear time-varying systems[J]. International Journal of Control, 1980, 31(5): 937–945.

  33. D’ANGELO H. Linear time-varying systems: analysis and synthesis[M]. Boston: Allyn & Bacon, 1970.

  34. KAILATH T. Linear systems[M]. Englewood Cliffs, NJ: Prentice Hall, 1980.

  35. WILKINSON J H. The algebraic eigenvalue problem[M]. Oxford: Clarendon Press, 1965.

  36. MARKUS L, YAMABE H. Global stability criteria for differential systems[J]. Osaka Mathematical Journal, 1960, 12(2): 305–317.

  37. MA Z-S, LIU L, ZHOU S-D, et al. Modal parameter estimation of the coupled moving-mass and beam time-varying system[C]//The ISMA2014 International Conference on Noise and Vibration Engineering, Leuven, Belgium, September 15–17, 2014: 587–596.

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Authors and Affiliations

  1. School of Aerospace Engineering, Beijing Institute of Technology, 100081, Beijing, China

    Zhisai MA, Li LIU & Sida ZHOU

  2. Department of Mechanical Engineering, KU Leuven, 3001, Leuven, Belgium

    Zhisai MA, Frank NAETS, Ward HEYLEN & Wim DESMET

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  1. Zhisai MA
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  2. Li LIU
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  3. Sida ZHOU
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  4. Frank NAETS
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  5. Ward HEYLEN
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  6. Wim DESMET
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Correspondence to Zhisai MA.

Additional information

Supported by the China Scholarship Council, National Natural Science Foundation of China(Grant No. 11402022), the Interuniversity Attraction Poles Programme of the Belgian Science Policy Office(DYSCO), the Fund for Scientific Research – Flanders(FWO), and the Research Fund KU Leuven.

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MA, Z., LIU, L., ZHOU, S. et al. Dynamic Stability Analysis of Linear Time-varying Systems via an Extended Modal Identification Approach. Chin. J. Mech. Eng. 30, 459–471 (2017). https://doi.org/10.1007/s10033-017-0075-7

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  • DOI: https://doi.org/10.1007/s10033-017-0075-7

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