The qualitative properties of the frequency spectrum and modes were studied extensively for the nonhomogeneous single span beams, in which the discrete and continuous systems were included. However, the qualitative properties about the multispan beams were studied and only shown in the book "Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems" published by Gantmakher, F. P. and Krein, M. G. The discrete model given in this book was formed by adding several masses to massless continuous system of the multispan beam, and the adopted method was based on the influence function of the beam, related to the positive systems. Nevertheless, the conditions about the positive systems were not discussed in the above-mentioned book. In this paper the difference discrete models of the multispan .beam with arbitrary supports at two ends are established. According-figthe above book, the flexibility coefficient of the above models corresponding to the positive single span beam systems is given. The oscillating property of the flexibility matrix of the multispan beam is determined based on that of the positive single span beam systems by introducing an auxiliary system. A series of qualitative properties of the frequency spectrum and modes of the multispan beams are obtained.%采用差分法建立了两端任意支承多跨梁的差分离散模型,对只具有两端支承的单跨梁是正系统的情形,导出了对应的多跨梁差分离散系统的柔度系数;从单跨梁正系统的刚度矩阵的符号振荡性出发,通过引入辅助系统,讨论了多跨梁的柔度矩阵的振荡性,确定了对应于单跨梁正系统的多跨梁差分离散系统的频谱和模态的一系列定性性质.
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