In this paper, a discrete system model and its equation of motion for beams with arbitrary supports at two ends are established. These supports include elastic, rigid and free supports in translation and rotation directions. Based on theory of oscillatory matrices, a series of qualitative properties of frequencies and modes of this system are derived. The basic properties include: non-zero frequencies are distinct; the ith displacement mode has i - 1 nodes; nodes of ith mode and (i + 1)th mode interlace.
Some additional important qualitative properties owned by rotation modes and strain modes are given as well.
