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The horizontal displacement y(t) of the left solid is imposed. Then we only study the horizontal displacement (t) of the right solid which has a mass m = 1 kg. The stiffnesses of the springs represented in figure Q1 are k₁= k₂ = 1 N/m. It is assumed that y(t) is harmonic: y(t) = 2 cos(w₁ t) + 4 sin(w₂ t), (1.1)

The horizontal displacement y(t) of the left solid is imposed. Then we only study the horizontal displacement (t) of the right solid which has a mass m = 1 kg. The stiffnesses of the springs represented in figure Q1 are k₁= k₂ = 1 N/m. It is assumed that y(t) is harmonic: y(t) = 2 cos(w₁ t) + 4 sin(w₂ t), (1.1)

Elements Of Electromagnetics
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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Question 1 An engineer is interested in the vibration of the system represented in figure Q1. x(t) y(t) tr m Figure Q1-Dynamic system The horizontal displacement y(t) of the left solid is imposed. Then we only study the horizontal displacement (t) of the right solid which has a mass m = 1 kg. The stiffnesses of the springs represented in figure Q1 are k₁ = k₂ = 1 N/m. It is assumed that y(t) is harmonic: y(t) = 2 cos(w₁ t) + 4 sin(w₂ t), (1.1) with w₁ = 1 rad/s, w2= 2 rad/s. a) Show that the displacement of the right mass is solution of the following equation: mä(t) + (k₁ + k₂)x(t) = k₁ y(t) (1.2) b) Calculate the natural frequency, the damping ratio and the input associated with equation (1.2). c) Calculate the harmonic forced response (t) associated with equation (1.2). d) Assuming that the stiffnesses are both equal to the same value denoted by k (then k₁ = k₂ = k), calculate the values of k that yield a large response amplitude.
Transcribed Image Text:Question 1 An engineer is interested in the vibration of the system represented in figure Q1. x(t) y(t) tr m Figure Q1-Dynamic system The horizontal displacement y(t) of the left solid is imposed. Then we only study the horizontal displacement (t) of the right solid which has a mass m = 1 kg. The stiffnesses of the springs represented in figure Q1 are k₁ = k₂ = 1 N/m. It is assumed that y(t) is harmonic: y(t) = 2 cos(w₁ t) + 4 sin(w₂ t), (1.1) with w₁ = 1 rad/s, w2= 2 rad/s. a) Show that the displacement of the right mass is solution of the following equation: mä(t) + (k₁ + k₂)x(t) = k₁ y(t) (1.2) b) Calculate the natural frequency, the damping ratio and the input associated with equation (1.2). c) Calculate the harmonic forced response (t) associated with equation (1.2). d) Assuming that the stiffnesses are both equal to the same value denoted by k (then k₁ = k₂ = k), calculate the values of k that yield a large response amplitude.
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