Abstract: The qualitative properties of the frequency spectrum andmodes were studied extensively for the nonhomogeneous single spanbeams, in which the discrete and continuous systems were included. However,the qualitative properties about the multispan beams were studied and onlyshown in the book ``Oscillation Matrices and Kernels and Small Vibrations ofMechanical Systems'' published by Gantmakher, F. P. and Krein, M. G. Thediscrete model given in this book was formed by adding several masses tomassless continuous system of the multispan beam, and the adopted method wasbased on the influence function of the beam, related to the positivesystems. Nevertheless, the conditions about the positive systems were notdiscussed in the above-mentioned book. In this paper the difference discretemodels of the multispan beam with arbitrary supports at two ends areestablished. According to the above book, the flexibility coefficient of theabove models corresponding to the positive single span beam systems isgiven. The oscillating property of the flexibility matrix of the multispanbeam is determined based on that of the positive single span beam systems byintroducing an auxiliary system. A series of qualitative properties of thefrequency spectrum and modes of the multispan beams are obtained.