Numerical experiment on influence of resonance on seismic response of buildings with low damping

摘要

A numerical experiment is performed on an eight-story shear building model subjected to a horizontal triangular pulse of ground. For dynamic analysis, the step-by-step algorithm of constant average acceleration of Newmark is used. With the help of simple approximations of the two extreme eigenfrequencies, we can give to the model any value of Rayleigh damping which does not vary largely from one mode to another. First, a low relative damping ζ = 0.02 is tried. The ground pulse, in about 0.2sec, is transmitted up to the top. After about 2.0sec, it seems that the structure becomes quiet. However, small vibrations remain with amplitudes less than 1.0mm. These waves propagate over the height of cantilever beam, reflect at its two ends and then interfere to each other. Gradually and monotonically the vibration amplitudes increase, which is an autoparametric resonance phenomenon. The distribution of vibration amplitudes, over the height of cantilever beam, exhibit larger values near the middle of height, which decay as they proceed to the two ends. This distribution is similar to that of Schroedinger equation of quantum mechanics. Then, a slightly higher relative damping ζ = 0.03 is tried. After rapid transmission of ground pulse up to the top, all vibrations are practically eliminated in about 2.0sec and no more appear. Consequently, for a low initial damping, an instability phenomenon is observed: A small variation of damping produces dramatic change to dynamic behavior of structure. In order to assure that no resonance will happen, a quite large damping is needed with some safety margin e.g. ζ ≈ 0.05. This can be realized either by additional dampers, widely used last years, or by non-structural elements incorporated in the structure.