Of motion damping factor h c 2mo c 2 mk 5 putting

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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.

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