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The first 100 (t) mass is denoted as m1, and the total additional 200 (t) mass as m2. Also, the natural frequencies of the building alone, the building with the first 100 (t) mass, and the building with the additional 100 (t) mass are denoted as ωn, ω1, and ω2, respectively. 2 Suppose a base isolated building can be modeled as an underdamped SDOF system with k = 18,935 (kN/m) and m = 2,000 (t). ) With the isolation bearings, the damping ratio is measured as 13%. However, according to the design, a 30% damping ratio is needed.

Usually, the frequency at the peak resonance is used as the natural frequency. 96. 97) where ωD is the displacement resonant frequency. 95, it is seen that only if 1 − 2ξ 2 > 0 or then r > 0. 98 holds, can the system have a peak or resonant value. Otherwise, the magnification factor will be always smaller than unity, except at r = 0. 82, it is seen that when r = 1, the displacement can still be very large, although it has passed the peak value. 101) Note that in the above development, the damping ratio is always smaller than 1.

48, the symbol ∠(⋅) stands for the angle of the complex variable (⋅). 43, respectively. 6 shows the relationship between the real and the imaginary parts of the forcing functions and the responses. The forcing function, f(t), can be represented by a rotating vector with amplitude f0 and angular speed ωf, whose projection on the real axis, marked “Re,” is f0 cos ωft, and on the imaginary axis, marked “Im,” is mf0 sin ωft. 6. Furthermore, the restoring force kx(t) can also be represented by a vector, with amplitude kx0 and phase angle ϕ compared to force f(t).

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