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2019 | OriginalPaper | Chapter

2. Oscillatory Matrices and Kernels as Well as Properties of Eigenpairs

Authors : Dajun Wang, Qishen Wang, Beichang (Bert) He

Published in: Qualitative Theory in Structural Mechanics

Publisher: Springer Singapore

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Abstract

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.

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previous chapter Overview
next chapter Qualitative Properties of Vibration and Static Deformation Associated with Discrete Systems of Strings and Bars
Footnotes
1
For a non-square matrix \( (a_{ij} )_{m \times n} \), it is totally nonnegative if all of its p-th-order minors are nonnegative where p is any positive integer satisfying \( p \le \hbox{min} (m,n) \).
 
2
Here, \( (x,s) \ne (a,b) \) means that \( x = a \) and \( s = b \) cannot occur simultaneously, and neither can \( x = b \) and \( s = a \). Thus, as an example, when \( x = a \), we should have \( s \ne b \).
 
Literature
1.
Courant R, Hilbert D (1962) Methods of mathematical physics, vol I, 1953, vol II. InterScience Publishers, New York
2.
Davis C, Kahan WM (1970) The rotation of eigenvectors by a perturbation. III. SIAM J Num Anal 7(1):1–46CrossRef
3.
Gantmacher FP, Krein MG (1961) Oscillation matrices and kernels and small vibrations of mechanical systems. U. S. Atomic Energy Commission, Washington
4.
Gladwell GML (2004) Inverse Problems in Vibration. 2nd edn. Springer, Dordrecht (1986, 1st edn, Martinus Nijhoff Publishers, Dordrecht)
5.
Shigley JE, Mischke CR, Budynas RG (2004) Mechanical engineering design. McGraw-Hill Professional
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)
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):587–608CrossRef
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)
Metadata
Title
Oscillatory Matrices and Kernels as Well as Properties of Eigenpairs
Authors
Dajun Wang
Qishen Wang
Beichang (Bert) He
Copyright Year
2019
Publisher
Springer Singapore
Book
Qualitative Theory in Structural Mechanics

Print ISBN: 978-981-13-1375-2

Electronic ISBN: 978-981-13-1376-9

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-981-13-1376-9_2

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