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


This chapter gives an introduction to the qualitative theory in Structural Mechanics, providing a brief history of its development, describing the content and methods of the study, and explaining the significance of the theory in research and application. In addition, this chapter presents main results of qualitative theory in Structural Mechanics covered in the book, which makes it easier for readers to select portions of the book for further study based on their personal need and interest.


  1. 1.
    Courant R, Hilbert D (1962) Methods of mathematical physics, vol I, 1953, vol II. Interscience Publishers, New YorkGoogle Scholar
  2. 2.
    Faber G (1923) Beweis, dass unter allen homogenen membranen van gleicher fläche und gleicher spannung die kreisfömige den tiefsten grundton gibt. S B Math-Nat KI Bayer Akad Wiss, 169–172Google Scholar
  3. 3.
    Gantmacher FP, Krein MG (1961) Oscillation matrices and kernels and small vibrations of mechanical systems. U. S. Atomic Energy Commission, WashingtonGoogle Scholar
  4. 4.
    Gladwell GML (2004) Inverse Problems in Vibration. 2nd edn. Springer, Dordrecht (1986, 1st edn, Martinus Nijhoff Publishers, Dordrecht)Google Scholar
  5. 5.
    Hu HC, Liu ZS, Wang DJ (1996) Influence of small changes of support location on vibration mode shape. Acta Mech Sin 28(1):23–32 (in Chinese)Google Scholar
  6. 6.
    Krahn E (1924) Uber eine von Raylegh formulierte minimaleigenschaft des kreises. Math Ann 1924(94):97–100Google Scholar
  7. 7.
    Leung AYT, Wang DJ, Wang Q (2004) On concentrated mass and stiffness in structural theories. Int J Struct Stab Dyn 4:171–179CrossRefGoogle Scholar
  8. 8.
    Liu ZS, Hu HC, Wang DJ (1994) Effect of small variation of support location on natural frequencies. In: Proceedings of international conference on vibration engineering, ICVE’94, Beijing. International Academic Publishers, Singapore, pp 9–12Google Scholar
  9. 9.
    Liu ZS, Hu HC, Wang DJ (1996) New method for deriving eigenvalue rate with respect to support location. AIAA J 34(4):864–865CrossRefGoogle Scholar
  10. 10.
    Pleijel A (1956) Remarks on Courant’s nodal line theorem. Comm Pure Appl Math 1956(4):543–550CrossRefGoogle Scholar
  11. 11.
    Polterovich I (2009) Pleijel’s nodal domain theorem for free membranes. Proc Am Math Soc 137(3):1021–1024CrossRefGoogle Scholar
  12. 12.
    Wang DJ, Hu HC (1982) A unified proof for the positive definiteness and compactness of two kinds of operators in the theory of elastic structure. Acta Mech Sin 14(2):111–121 (in Chinese)Google Scholar
  13. 13.
    Wang DJ, Hu HC (1982) A unified proof of the general properties of the linear vibrations in the theory of elastic structures. J Vibr Shock, 1(1):6–16 (in Chinese)Google Scholar
  14. 14.
    Wang DJ, Hu HC (1983) A unified proof for the positive-definiteness and compactness of two kinds of operators in the theories of elastic structures. In: Proceedings of the China-France symposium on finite element methods. Science Press China, New York, pp 6–16Google Scholar
  15. 15.
    Wang DJ, Hu HC (1985) Positive definiteness and compactness of two kinds of operators in theory of elastic structures. Sci Sin Ser A 28(7):727–739Google Scholar
  16. 16.
    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
  17. 17.
    Wang QS, Wang DJ (1987) Construction of the discrete system for the rod by partial natural modes and frequencies data. J Vibr Eng 1987(1):83–87 (in Chinese)Google Scholar
  18. 18.
    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
  19. 19.
    Wang QS, Wang DJ, He BC (1992) Qualitative properties of frequencies and modes of discrete system of continuous second-order systems. J Vibration and Shock, 11(3):7–12 (in Chinese)Google Scholar
  20. 20.
    Wang QS, Wang DJ (1994) An inverse mode problem for continuous second-order systems. In: Proceedings of international conference on vibration engineering, ICVE’94, Beijing. International Academic Publishers, Singapore, pp 167–170Google Scholar
  21. 21.
    Wang QS, Wang DJ (1997) Qualitative properties of frequency spectrum and modes of arbitrary supported beams in vibration. Acta Mech Sin 29(5):540–547 (in Chinese)Google Scholar
  22. 22.
    Wang QS, Huang PC (2008) The influence for the frequency of beams by adding internal support. Mod Vibr Noise Technol 6:104–107Google Scholar
  23. 23.
    Zheng ZJ, Chen P, Wang DJ (2013) Oscillation property of the vibrations for finite element models of Euler beam. Qtr J Mech & Appl Math, 66(4):587–608Google Scholar
  24. 24.
    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

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© 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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