Problem 1 - A Single-Degree-of-Freedom (SDOF) system with a mass of M is supported by a stiffness K and a dashpot of C and subject to an input motion of u^··_g(t). Its equation of motion is written as:

Mu^··(t) + Cu^·(t) + Ku(t) = -Mu^··_g(t) (1)

where u(t) is the relative displacement of the system to the ground. The natural frequency and the damping ratio of the system is defined as ω₀ = √(K/M) and ξ = C/2Mω₀ respectively.

(a) The Fourier transform of u(t) is defined as u(ω) = FFT{u(t)} = _-∞∫^∞ u(t)e^-iωt dt while the inverse Fourier transform of u(ω) is defined as u(t) = IFFT{u(ω)} = 1/2π _-∞∫^∞ u(ω)e^iωt dω. Apply the Fourier transform to the equation of motion in (1) so that to obtain the displacement in frequency domain u(ω) as:

u(ω) = H(ω)u^··_g(ω) (2)

where H(ω) is the transfer function while the u^··_g(ω) is the Fourier transform of u^··_g(t). Derive the form of H(ω) in terms of the damping ratio ξ and the frequency ratio β = ω/ω₀. Plot the real part, the imaginary part and the modulus of H(ω) separately as function of β(β ∈ [0, 3]) for three values of damping ratio ξ = 2, 5, and 10% and ω₀ = 1 rad/s.

(b) When the SDOF system is subject to an impulse loading, the impulse response function h(t) is defined as:

h(t) = -(1/ω_D)e^-ξω_0t sin(ω_Dt) (3)

where ω_D = ω₀√(1-ξ²) is the damped natural frequency. Prove that h(t) is the Fourier pair of the transfer function H(ω), i.e. h(t) = 1/2π _-∞∫^∞H(ω)e^iωt dω or H(ω) = _-∞∫^∞h(t)e^-iωt dt.

Problem 2 - A stiff superstructure that can be approximated with a rigid mass, M = 2500Mg, is seismically isolated by ten high-damping elastomeric bearings. The stiffness and damping provided by ten bearings are k = 26.5M/V/m and c = 2.45 MN·s/m respectively. Assume that the superstructure-isolation system can be approximated with a SDOF system, whose equation of motion is shown in Eq. (1). It is subject to the fault normal component of the Newhall station in 1994 Northridge earthquake at the base.

a. Compute the natural frequency and damping ratio of the system.

b. Construct a Matlab code to compute the response u(t) of the isolated mass under the given ground motion by the Duhamel's integral (time domain method):

u(t) = 1/ω_{D 0}∫^t-u^··_g(τ)e^{-ξω_0(t- τ)}sin[ω_D(t- τ)]dτ (4)

c. Construct a Matlab code to compute the response u(t) of the isolated mass under the given ground motion by the Fourier integral theorem (frequency domain method):

u(t) = 1/2π _-∞∫^∞H(ω)u^··_g(ω)e^iωt dω (5)

d. Construct a Matlab code to compute the response u(t) of the isolated mass under the given ground motion by the Newmark integration (time domain method).

e. Plot the computed response time histories (i.e. displacement u(t) and the total acceleration a(t)) from all three methods on the same plot and compare with the response of a fixed-base mass. Is the seismic isolation beneficial?

f. Construct and compare the total acceleration and the relative displacement spectra of the fault-normal Newhall record for damping ratios of ξ = 2, 5, 10% and structural periods from 0 to 5 second using both frequency domain and time domain (Newmark) method. If there is any discrepancy between two methods, please explain the reason and remedy measure to remove the discrepancy.

Attachment:- Assignment Files.rar