Abstract: By using the method of eigen function expansion and separation of variables，under gravitational action，the definite problem of vertical vibration of horizontal uniform rod which one end maintain level but harmonically oscillate perpendicularly and the other end is free is solved strictly，all the natural frequencies of stable harmonic oscillation of level rod are acquired. It is also found that along with the time increasing continuously，when the frequency of vibration which act on the left end approach to the value of natural frequency of level rod，the rod will resonate which the nodes and wave loops are appeared. The resonant frequency and the place distribution of nodes and antinodes distinguishedly agree to the results of level rod while the left end is clamped and the right end is cantilevered，which is independent to the gravity action. It is indicated when the resonance happens，the place of left end of rod where the vibration source at can be regarded as wave node，which is acquiesced by the existing experiment. At last，the fruit of initial value problem in the process of solving is verified by Laplace transformation method.

Key words: vertical vibration of level rod, eigen function expansion method, separation variables method, natural frequency, resonance phenomenon