A three-level offshore platform located in the Helang Oilfield area has a 1500 kg floorsteel grating supported at each level. The structure sometimes is subjected to a verticaloscillation movement during rough sea waves given by function of ?(?) = ? sin ??.If the steel grating only moves in the vertical direction and is supported by oneequivalent spring and damper at each steel grating pole level with stiffness, k1 = (300× T) N/m, k2 = (200 × T) N/m and k3 = (100 × T) N/m while damping, c1 = (15 × U)Ns/m, c2 = (10 × U) Ns/m and c3 = (5 × U) Ns/m, respectively, as simplified in FigureQ1. Neglect the effect of gravitational force.T=2 U=7 m32 F(0)m2k 2Figure Q1 b) Determine the equation of motion in a matrix form using Newton's second law, [m]y+[c]y + [k]y = F.c) By omitting the damping and external force parameter, deduce and express the generalsolution in the form of ([k] – w[m]){Y} = 0.d) Analyze the maximum vertical displacement at each floor when the system's naturalfrequency is equal to the external stimulation frequency. Assume nontrivial solutionand F12,3(t) exhibits harmonic oscillation of sin 0.02t.e) Describe an eigenvector using a three-degree-of-freedom system's mode form as anillustration for wl, w2 > w1 and w3 > w2 > wl.

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Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

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A three-level offshore platform located in the Helang Oilfield area has a 1500 kg floor
steel grating supported at each level. The structure sometimes is subjected to a vertical
oscillation movement during rough sea waves given by function of ?(?) = ? sin ??.
If the steel grating only moves in the vertical direction and is supported by one
equivalent spring and damper at each steel grating pole level with stiffness, k1 = (300
× T) N/m, k2 = (200 × T) N/m and k3 = (100 × T) N/m while damping, c1 = (15 × U)
Ns/m, c2 = (10 × U) Ns/m and c3 = (5 × U) Ns/m, respectively, as simplified in Figure
Q1. Neglect the effect of gravitational force.

T=2 U=7

b) Determine the equation of motion in a matrix form using Newton's second law, [m]y+
[c]y + [k]y = F.
c) By omitting the damping and external force parameter, deduce and express the general
solution in the form of ([k] – w[m]){Y} = 0.
d) Analyze the maximum vertical displacement at each floor when the system's natural
frequency is equal to the external stimulation frequency. Assume nontrivial solution
and F12,3(t) exhibits harmonic oscillation of sin 0.02t.
e) Describe an eigenvector using a three-degree-of-freedom system's mode form as an
illustration for wl, w2 > w1 and w3 > w2 > wl.

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