Oscillatory Matrices and Kernels as Well as Properties of Eigenpairs

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


The theory of oscillatory matrices and kernels forms the mathematical foundation for the study of qualitative properties of natural frequencies and mode shapes of bars and beams. This chapter provides an introduction to the theory. The content is drawn largely from the monograph, Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems, written by creators of the theory, Gantmacher and Krein; but Sect. 2.11 and most of Sect. 2.10 are the original work by authors of this book as well as their collaborators Zijun Zheng and Pu Chen.


  1. 1.
    Courant R, Hilbert D (1962) Methods of mathematical physics, vol I, 1953, vol II. InterScience Publishers, New YorkGoogle Scholar
  2. 2.
    Davis C, Kahan WM (1970) The rotation of eigenvectors by a perturbation. III. SIAM J Num Anal 7(1):1–46CrossRefGoogle 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.
    Shigley JE, Mischke CR, Budynas RG (2004) Mechanical engineering design. McGraw-Hill ProfessionalGoogle Scholar
  6. 6.
    Wang QS, Wang DJ (1997) United proof for qualitative properties of discrete and continuous systems of vibrating rod and beam. Acta Mech Sin 29(1):99–102 (in Chinese)Google Scholar
  7. 7.
    Zheng ZJ, Chen P, Wang DJ (2013) Oscillation property of the vibrations for finite element models of Euler beam. Q J Mech Appl Mech 66(4):587–608CrossRefGoogle Scholar
  8. 8.
    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

Personalised recommendations