Transfer matrix analysis of the elastostatics of one-dimensional repetitive structures
Transfer matrix analysis of the elastostatics of one-dimensional repetitive structures
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DOI：
10.1098/rspa.2006.1669
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发表时间：
2006-08
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影响因子：
--
通讯作者：
N. Stephen
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文献类型：
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作者：
N. Stephen
传递矩阵被广泛地用于工程结构的动力分析，尤其是在静力分析中，并且在处理重复结构时特别有用，对于重复结构，通常可以通过对单个重复单元的分析来确定整个结构的性能，如果结构不是无限大的，则还可以通过边界条件来确定整个结构的性能。对于弹性静力分析，重复单元传递矩阵的非单位特征值是自平衡载荷的衰减率，正如圣维南原理所预期的那样。多个单位特征值与荷载的传递有关，例如拉力或弯矩，并且等效(均化)的连续体属性，如横截面面积、面积二阶矩和泊松比，可以从相关的特征向量和主向量中确定。介绍了各种不同的结果，大多数是新的，其他的来自不同的来源。这些问题包括利用Moore-Penrose逆计算主向量、双正交性和辛正交性及其与互等定理的关系、对复单位本征值的限制、单胞从左到右对称性对刚度和传递矩阵的影响、在没有平移对称性的情况下特征值的偏转以及对可能的Jordan标准形的限制。结果表明，只有重复的单位本征值才能得到非平凡的Jordan块形式，因此不可能存在简并衰变模。目前的弹性静力学分析补充了兰利(兰利，1996)的分析。R.Soc.波动能量学的传递矩阵分析。
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.