1. In this paper, we would like to deal with the vibration of buildings standing on the surface of a semi-infinite homogeneous elastic solid. As the following investigation shows, a remarkable fact is involved in the present problem, namely, the fact that the motion of buildings is affected by the existence of elastic base and the elastic base by that of buildings.
To the analysis of this phenomenon we applied the theory developed by I. Toriumi. Consequently the assumption he used is also adopted in the present paper. The authors' hearty thanks are due to him to which we owe very much.
2. Vertical motion. When the incident wave is a purely vertical harmonic oscillation traveling in the direction perpendicular to the free surface, the problem may be solved by the process explained in Fig. 1. The amplitude and phase-lag of the building oscillation are given by the formula (3.8) and Fig. 2 and Fig. 3, where
S0=(ρ
0/ρ)·(
l0/
r0) is used as a parameter. Since the ratio of densities of the building and the ground does not differ much from 1/4,
S0 may be approximately determined by the geometrical configuration of a building only. We find it is nearly equal to
l0/4
r0=1/4β. The graph of the amplitude is similar to that of the magnification of a seismometer. Fig. 4 shows that the value of the maximum amplitude is considerably smaller than in the case of rocking or horizontal motions.
3. Rocking motion. When a wave of purely horizontal oscillation is incident upon the surface where a rigid cylindrical building is standing, a rocking motion is induced. Fig. 5 and Fig. 6 explain the process by which the analysis is perfomed. The results are shown in Fig. 7-Fig. 15. Fig, 7 and Fig. 8 give the amplitude and the phase-lag of the horizontal displacement of the center of gravity. If the displacement of the building is same with that of the ground,
X becomes unity. In Fig. 9 and Fig. 10, the resonance amplitude and the corresponding frequency are given. The amplitude and the phase-lag of the center of gravity for the angular motion can be seen in Fig. 11 and Fig. 12, and its resonance amplitude is shown in Fig. 13. If the center of gravity remains stil,
Y′=-1. Parameter in the figures are the same with those adopted in the former case.
Since the waves are incident upon the surface, it is natural for the base of the building to move accordingly. However, the motion of the ground is also affected by the existence of the building. The results of the calculation of this effect are given in Fig. 14 and Fig. 15, which show the amplitude and the phase-lag of the additional displacement of the center of the base. We must notice the fact that seismometers installed at the basement do not record the same motion with those installed where the building is absent.
4. Horizontal motion. Next, we shall consider the horizontal vibration of a building assuming a model of one-mass system. (See Fig. 16.) The process of the calculation is illustrated in Fig. 17, and the analytical expression of the motion of the center of gravity is expressed by Eq. (5.13). The results are given in Fig. 18 and Fig. 19. Fig. 20 gives the horizontal displacement of the center of the base. In this figure, the interesting fact is that the curves all meet at a point on the
a0-axis. This point (
a0=1) gives th condition of resonance, which is shown in Eq. (5.19). Fig. 21 shows the maximum amplitude. In this figure we used
S0 as a parameter, the values of which are assumed to be 0.3, 0.4 and 0.5, which represent fairly well buildings with different heights. (We assumed that the building has no damping and
v=1.)
Fig. 22 gives the apparent damping coefficient when the building itself is without damping. On the analogy of the magnification of a seismograph we u
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