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Unformatted text preview: lar Frequency: ωo = k / m (3) Natural Frequency: fo = ωo /( 2π) (4) Eq. of Motion： Damping Factor : h = c /( 2mωo ) = c /( 2 mk ) (5) Putting x = X eiω t and using Eq.(2) to Eq.(4), Eq.(1) leads. { 1 − (ω / ωo ) 2 + i 2h (ω / ωo )} X = Xs (6) then, X / Xs = 1 / {1 − ( ω / ωo ) 2 } 2 + 4h 2 ( ω / ωo ) 2 (7) X / Xs = 1/ {1 − ( f / fo ) 2 } 2 + 4h 2 ( f / fo ) 2 (8) or, X / Xs i s c a l l e d “ A m p l i f i c a t i o n Fa c t o r”. The figure shows the amplification Factor changing the
damping factor h. The smaller the damping factor, the
larger the amplification factor at the natural frequency fO.
(Resonance Phenomena) 2m 1.4 Effect of Embedding
Let us now consider the SSI when the foundation is
embedded. In order to investigate the effect of embedding,
forced vibration tests were carried out using a test specimen
as shown below. The figure shows the
resonance curve for
the displacement at
the gravity center of
the test specimen
before and after
backfilling the
excavation with sand. The embedding increases the resonance frequency and that
the resonance curve near the peak is not sharp, but rather
rounded.
The damping factor is used in vibration mechanics to
express the degree of sharpness of this resonance curve.
The damping factor of the test specimen is increased from
5.7% to 6.3% by the embedding effect. 1.5 Coupling System of Building and Ground
When the forces are applied to the building by an earthquake
or by means of an exciter etc., the building and the ground
vibrate with influencing each other.
This system is called “ Coupling System of Building and
Ground”.
The SSI as treated foregoing connects the building to the
ground in this coupling system.
Let us now investigate the influence of the SSI onto the
coupling system. This building is represented in simplified form by the spring
K, connecting the mass point and the basement, and the
dashpot C.
When the basement is fixed, the natural frequency of the
building is set on 2 Hz and the damping factor h on 5 % .
The basement is assumed as a rigid body with a unit
volumetric mass of 0.5 t/m3, and the ground is
homogeneous half space. The figure shows the calculation results for the resonance
curves of the horizontal displacement of the building top
when a horizontal excitation is applied at the top of the
building. In the calculation, embedding depth of the
basement is a parameter. As already described above, the natural frequency of the
building itself under the fixed basement condition is 2 Hz. But, the resonance
frequency shown in
this figure is
considerably lower
than 2 Hz. In earthquake engineering, this resonance frequency is
called “ the resonance frequency of the coupling system “,
and thi...
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This note was uploaded on 03/14/2014 for the course CE 5680 taught by Professor Drgrd during the Summer '14 term at Indian Institute of Technology, Chennai.
 Summer '14
 DrGRD

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