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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Notes
- 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 \( s \ne b \).
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Wang, D., Wang, Q., He, B. (2019). Oscillatory Matrices and Kernels as Well as Properties of Eigenpairs. In: Qualitative Theory in Structural Mechanics. Springer, Singapore. https://doi.org/10.1007/978-981-13-1376-9_2
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DOI: https://doi.org/10.1007/978-981-13-1376-9_2
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