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Qualitative Properties of Vibration and Static Deformation Associated with Discrete Systems of Beams

  • Dajun WangEmail author
  • Qishen Wang
  • Beichang (Bert) He
Chapter

Abstract

The focus of the first six sections of this chapter is on qualitative properties of the finite difference model or the related physical model of a beam. We will set up the governing equations of motion and boundary conditions associated with the finite difference system; derive various modal qualitative properties of the discrete model under different boundary constraints, by applying the theory of oscillatory matrices and the concept of conjugate beams; and establish qualitative properties in static deformation of the finite difference system of a well-constrained beam.

References

  1. 1.
    Gantmacher FP, Krein MG (1961) Oscillation matrices and kernels and small vibrations of mechanical systems. U. S. Atomic Energy Commission, WashingtonGoogle Scholar
  2. 2.
    Gladwell GML (1985) Qualitative properties of vibrating systems. Proc Roy Soc Lond A 401:299–315CrossRefGoogle Scholar
  3. 3.
    Gladwell GML (2004) Inverse problems in vibration, 2nd edn. Springer, Dordrecht (1986, 1st edn, Martinus Nijhoff Publishers, Dordrecht)Google Scholar
  4. 4.
    Gladwell GML, England AH, Wang DJ (1987) Examples of reconstruction of an Euler-Bernoulli beam from spectral data. J Sound Vibr 19(1):81–94CrossRefGoogle Scholar
  5. 5.
    Gladwell GML (1991) Qualitative properties of finite-element models II: the Euler-Bernoulli beam. Q J Mech Appl Math 44(2):267–284CrossRefGoogle Scholar
  6. 6.
    He BC, Wang DJ, Low KH (1989) Inverse problem for a vibrating beam using a finite difference model. In: Proceedings international conference on noise & vibration, Singapore, B, pp 92–104Google Scholar
  7. 7.
    He BC, Wang DJ, Wang QS (1989) Inverse problem for the finite difference model of Euler beam in vibration. J Vibr Eng 2(2):1–9 (in Chinese)Google Scholar
  8. 8.
    He BC, Wang DJ, Wang QS (1991) The single mode inverse problem of Euler-Bernoulli beam. Acta Mech Solida Sin 12(1):85–89 (in Chinese)Google Scholar
  9. 9.
    Wang DJ, He BC, et al (1990) Inverse problems of the finite difference model of a vibrating Euler beam. In: Proceedings international conference on vibration problems in engineering, June 1990, China, vol 1(1), pp 21–26. International Academic Publishers, SingaporeGoogle Scholar
  10. 10.
    Wang DJ, He BC, Wang QS (1990) On the construction of the Euler-Bernoulli beam via two sets of modes and the corresponding frequencies. Acta Mech Sin 22(4):479–483 (in Chinese)Google Scholar
  11. 11.
    Wang QS, He BC, Wang DJ (1990) Some qualitative properties of frequencies and modes of Euler beams. J Vibr Eng 3(4):58–66 (in Chinese)Google Scholar
  12. 12.
    Wang QS, Wang DJ, He BC (1991) Construction of the difference discrete system for the simple-support beam from the spectra data. Eng Mech 8(4):10–19 (in Chinese)CrossRefGoogle Scholar
  13. 13.
    Wang QS, Wang DJ (2006) Difference discrete system of the Euler beam with arbitrary supports and sign oscillatory property of stiffness matrices. Appl Math & Mech 27(3):393–398CrossRefGoogle Scholar
  14. 14.
    Wang QS, Wu L, Wang DJ (2009) Some qualitative properties of frequency spectrum and modes of difference discrete system of multi-bearing beam. Chin J Theor Appl Mech 41(6):61–68 (in Chinese)Google Scholar
  15. 15.
    Wang QS, Wang DJ, Wu L et al (2009) Sign-oscillating property of stiffness matrix of difference discrete system and qualitative property of beam with overhang. J Vibr Shock 28(6):113–117 (in Chinese)Google Scholar
  16. 16.
    Wang QS, Wang DJ, Wu L et al (2011) Qualitative properties of frequencies and modes of vibrating multi-span beam. Q J Mech Appl Math 64(1):75–86CrossRefGoogle Scholar
  17. 17.
    Wang QS, Zhang LH, Wang DJ (2012) Some qualitative properties of modes of discrete system of beam with overhang. Chin J Theor Appl Mech 44(6):1071–1074 (in Chinese)Google Scholar
  18. 18.
    Wang QS, Wang DJ (2014) Supplementary proof of some oscillation property for discrete systems of rod and beam having rigid modes. Chin Q Mech 35(2):262–269 (in Chinese)Google Scholar
  19. 19.
    Zheng ZJ, Chen P, Wang DJ (2012) Oscillation property of modes for FE models of bars and beams. J Vibr Shock 31(20):79–83 (in Chinese)Google Scholar
  20. 20.
    Zheng ZJ, Chen P, Wang DJ (2013) Oscillation property of the vibrations for finite element models of Euler beam. Q J Mech Appl Math 66(4):587–608CrossRefGoogle Scholar
  21. 21.
    Zheng ZJ, Chen P, Wang DJ (2014) A unified proof to oscillation property of discrete beam models. Appl Math Mech 35(5):621–636CrossRefGoogle Scholar
  22. 22.
    Zheng ZJ (2014) The qualitative vibrational property and modal inverse problems of rods and Euler beams [D]. Department of Mechanics and Engineering Science, College of Engineering, Peking University (in Chinese)Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. and Peking University Press 2019

Authors and Affiliations

  1. 1.Department of Mechanics and Engineering SciencePeking UniversityBeijingChina
  2. 2.School of Physics and Electrical EngineeringAnqing Normal UniversityAnqingChina
  3. 3.HBC ConsultingSeattleUSA
  4. 4.State Key Laboratory for Turbulence and Complex SystemsPeking UniversityBeijingChina

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