Transfer matrix analysis of the elastostatics of one-dimensional repetitive structures

被引:20
作者
Stephen, N. G. [1 ]
机构
[1] Univ Southampton, Sch Engn Sci, Southampton SO17 1BJ, Hants, England
来源
PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES | 2006年 / 462卷 / 2072期
关键词
transfer; symplectic; matrix; elastostatic; pseudo-inverse; Jordan canonical form;
D O I
10.1098/rspa.2006.1669
中图分类号
O [数理科学和化学]; P [天文学、地球科学]; Q [生物科学]; N [自然科学总论];
学科分类号
07 ; 0710 ; 09 ;
摘要
Transfer matrices are used widely for the dynamic analysis of engineering structures, increasingly so for static analysis, and are particularly useful in the treatment of repetitive structures for which, in general, the behaviour of a complete structure can be determined through the analysis of a single repeating cell, together with boundary conditions if the structure is not of infinite extent. For elastostatic analyses, non-unity eigenvalues of the transfer matrix of a repeating cell are the rates of decay of self-equilibrated loading, as anticipated by Saint-Venant's principle. Multiple unity eigenvalues pertain to the transmission of load, e.g. tension, or bending moment, and equivalent (homogenized) continuum properties, such as cross-sectional area, second moment of area and Poisson's ratio, can be determined from the associated eigen- and principal vectors. Various disparate results, the majority new, others drawn from diverse sources, are presented. These include calculation of principal vectors using the Moore-Penrose inverse, bi- and symplectic orthogonality and relationship with the reciprocal theorem, restrictions on complex unity eigenvalues, effect of cell left-to-right symmetry on both the stiffness and transfer matrices, eigenvalue veering in the absence of translational symmetry and limitations on possible Jordan canonical forms. It is shown that only a repeating unity eigenvalue can lead to a non-trivial Jordan block form, so degenerate decay modes cannot exist. The present elastostatic analysis complements Langley's (Langley 1996 Proc. R. Soc. A 452, 1631-1648) transfer matrix analysis of wave motion energetics.
引用
收藏
页码:2245 / 2270
页数:26
相关论文
共 50 条
[31]   Competing superconducting instabilities in the one-dimensional p-band degenerate cold fermionic system [J].
Bois, V. ;
Capponi, S. ;
Lecheminant, P. ;
Moliner, M. .
PHYSICAL REVIEW B, 2015, 92 (07)
[32]   Determining dynamic characteristics of mechanical systems by the method of constructing one-dimensional spectral portraits of matrices [J].
Kurzin, V. B. .
JOURNAL OF APPLIED MECHANICS AND TECHNICAL PHYSICS, 2008, 49 (01) :84-92
[33]   AN APPROXIMATE ANALYTICAL SOLUTION FOR ONE-DIMENSIONAL IMBIBITION PROBLEM IN LOW-PERMEABILITY POROUS MEDIA [J].
Li, Lu ;
Wang, Min ;
Shi, An-Feng ;
Liu, Zhi-Feng ;
Wang, Xiao-Hong ;
Yu, Jin-Biao ;
Cao, Wei-Dong .
JOURNAL OF POROUS MEDIA, 2020, 23 (07) :683-694
[34]   Determining dynamic characteristics of mechanical systems by the method of constructing one-dimensional spectral portraits of matrices [J].
V. B. Kurzin .
Journal of Applied Mechanics and Technical Physics, 2008, 49 :84-92
[35]   Time-reversal-invariant scaling of light propagation in one-dimensional non-Hermitian systems [J].
Rivero, Jose D. H. ;
Ge, Li .
PHYSICAL REVIEW A, 2019, 100 (02)
[36]   High-sensitivity synchronous image encryption based on improved one-dimensional compound sine map [J].
Wang, Xingyuan ;
Zhang, Maozhen .
IET IMAGE PROCESSING, 2021, 15 (10) :2247-2265
[37]   A novel fast image encryption scheme based on a new one-dimensional compound sine chaotic system [J].
Tang, Jianeng ;
Zhang, Feng ;
Ni, Hui .
VISUAL COMPUTER, 2023, 39 (10) :4955-4983
[38]   Green Element solution of one-dimensional counter-current spontaneous imbibition in water wet porous media [J].
Delijani, Ebrahim Biniaz ;
Pishvaie, Mahmoud Reza .
JOURNAL OF PETROLEUM SCIENCE AND ENGINEERING, 2010, 70 (3-4) :302-307
[39]   Computing the complex wave and dynamic behavior of one-dimensional phononic systems using a state-space formulation [J].
Assis, George F. C. A. ;
Beli, Danilo ;
de Miranda Jr, Edson J. P. ;
Camino, Juan F. ;
Dos Santos, Jose Maria C. ;
Arruda, Jose Roberto F. .
INTERNATIONAL JOURNAL OF MECHANICAL SCIENCES, 2019, 163
[40]   FAST STRUCTURED DIRECT SPECTRAL METHODS FOR DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS, I. THE ONE-DIMENSIONAL CASE [J].
Shen, Jie ;
Wang, Yingwei ;
Xia, Jianlin .
SIAM JOURNAL ON SCIENTIFIC COMPUTING, 2016, 38 (01) :A28-A54