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skyscraper project

skyscraper project - Ryan Devine SID 17197266 ME 133 Spring...

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Ryan Devine SID: 17197266 ME 133 Spring 2007 Matlab Assignment The building’s behavior will correspond exactly to a three-spring, three-mass, three-damper system with all the elements connected in series. Having been given the mass, damping, and free-response characteristics of the building, it is up to us to figure out its inherent stiffness and apply that to our model of the building to find its response. For this, we use the Rayleigh’s Quotient technique. From some basic analysis of the model system, we find that the vector of static displacements due to gravity is X = [x1 x2 x3] = [1 1.666 2]. By newton’s method, we find that the mass, stiffness, and damping matrices of the system are: [m 0 0] [2 -1 0] [2 -1 0] M = [0 m 0] K = k[-1 2 -1] C = c[-1 2 -1] [0 0 m] [0 -1 1] [0 -1 1] where m = 18,144 kg, k is unknown, and c = 4570 Ns/m. So when we set up W^2 = (XKX*)/(XMX*) (where X is given as a row vector and X* denotes the transpose of X, which is a column vector, and W is 2*pi*f = 4*pi), we find that k = 14.3e6. When we check this conclusion via eig(K,M), we find that the first natural frequency squared is 156. Since (4*pi)^2 is about 157, this is quite close and our k is accurate. From here on, it’s MATLAB. The response of the three floors of the building over the first 7 seconds of the earthquake is shown in the figure below: Figure 1: Response of the three floors of the building to the 8cm earthquake. Data 1 = x1, Data 2 = x2, Data 3 = x3. From this figure, it is clear that the building will suffer serious structural damage from the earthquake.
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When we incorporate the base-isolator system proposed, and set its stiffness to 0.5e6, we
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