M3 C3 k2 C2 Figure Q1
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Transcribed Image Text:y3 F;(1)
13
k3
C3
m2
| C2
V |F:(0)
F;(t)
k
Figure Q1
![Q1 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 y(t) = Y sin wt .
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, k; = (300
x 4) N/m, k2= (200 × 4) N/m and k3= (100 × 4) N/m while damping, c= (15 × 8)
Ns/m, c2 = (10 × 8) Ns/m and c3 = (5 × 8) Ns/m, respectively, as simplified in Figure
Q1. Neglect the effect of gravitational force.
(a) Sketch a free body diagram for each steel grating that includes the mass's action
and reaction forces.
(b) Determine the equation of motion in a matrix form using Newton's second law,
[m]ÿ + [c]ý + [k]y = F.
(c) By omitting the damping and external force parameter, deduce and express the
general solution in the form of ([k] – w²[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 non-
trivial solution and F1,2,3(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 wi, w2 > wi and w3 > w2 > wi.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F65d0b228-441a-4371-8d66-b2d3fccad1d7%2F97437e4b-efb2-4ce6-b3ce-d646cf9c7045%2Fm5zod8_processed.png&w=3840&q=75)
Transcribed Image Text:Q1 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 y(t) = Y sin wt .
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, k; = (300
x 4) N/m, k2= (200 × 4) N/m and k3= (100 × 4) N/m while damping, c= (15 × 8)
Ns/m, c2 = (10 × 8) Ns/m and c3 = (5 × 8) Ns/m, respectively, as simplified in Figure
Q1. Neglect the effect of gravitational force.
(a) Sketch a free body diagram for each steel grating that includes the mass's action
and reaction forces.
(b) Determine the equation of motion in a matrix form using Newton's second law,
[m]ÿ + [c]ý + [k]y = F.
(c) By omitting the damping and external force parameter, deduce and express the
general solution in the form of ([k] – w²[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 non-
trivial solution and F1,2,3(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 wi, w2 > wi and w3 > w2 > wi.
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