Consider the double mass/double spring system shown below.- click to expand.Both springs have spring constants k, and both masses have mass m; each spring is subject to a damping force of F frictionWe can write the resulting system of second-order DEs as a first-order system, w'(t) = Aw(t), with w = (x₁, x₁, x₂, x₂) TFor values of k = 4, m = 1 and c=21,2 = 0.5±3.2i, V₁23,4= - 0.5 ± 1.13i, V31, the resulting eigenvalues and eigenvectors of A are-0.039 0.248i0.8130.024 +0.153i-0.502-0.134-0.302i'0.409-0.216 - 0.489i0.661and(a) Find a set of initial displacements x₁(0), x₂(0) that will lead to the fast mode of oscillation for this sytem. Assume that the initial velocities wil be zero.(x₁(0), x₂(0))Enter your answer using angle braces, (and ).cx (friction proportional to velocity).(b) At what frequency will the masses be oscillating in this mode?Frequency =rad/s

Given:To find:Initial displacementsFrequency

Consider the double mass/double spring system shown below. - click to expand. Both springs have spring constants k, and both masses have mass m; each spring is subject to a damping force of F friction We can write the resulting system of second-order DEs as a first-order system, w ´(1) = Aw(1), with w = (x₁, x₁, x₂, x₂) ™ For values of k = 4, m = 1 and c = 1, the resulting eigenvalues and eigenvectors of A are 21.2 -0.5±3.2i, V₁ = -0.039-0.248i 0.813 0.024 +0.153i -0.502 -0.134-0.302i 0.409 23.4 = -0.5 ± 1.13i, V3 = -0.216 - 0.489i 0.661 and - cx (friction proportional to velocity). (a) Find a set of initial displacements x₁ (0), x₂(0) that will lead to the fast mode of oscillation for this sytem. Assume that the initial velocities wil be zero. (0), x₂(0)) = Enter your answer using angle braces, (and). (b) At what frequency will the masses be oscillating in this mode? Frequency = rad/s

Consider the double mass/double spring system shown below. - click to expand. Both springs have spring constants k, and both masses have mass m; each spring is subject to a damping force of F friction We can write the resulting system of second-order DEs as a first-order system, w ´(1) = Aw(1), with w = (x₁, x₁, x₂, x₂) ™ For values of k = 4, m = 1 and c = 1, the resulting eigenvalues and eigenvectors of A are 21.2 -0.5±3.2i, V₁ = -0.039-0.248i 0.813 0.024 +0.153i -0.502 -0.134-0.302i 0.409 23.4 = -0.5 ± 1.13i, V3 = -0.216 - 0.489i 0.661 and - cx (friction proportional to velocity). (a) Find a set of initial displacements x₁ (0), x₂(0) that will lead to the fast mode of oscillation for this sytem. Assume that the initial velocities wil be zero. (0), x₂(0)) = Enter your answer using angle braces, (and). (b) At what frequency will the masses be oscillating in this mode? Frequency = rad/s

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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Concept explainers

Free Damped Vibrations

Free vibration is a type of vibration in which the external force is removed after giving the initial displacement to a body/system. In other words, in this type of vibration, force is applied at the initial displacement of the system, and after the initial displacement, force is removed, and the system vibrates at the natural frequency.

Question

Consider the double mass/double spring system shown below.
- click to expand.
Both springs have spring constants k, and both masses have mass m; each spring is subject to a damping force of F friction
We can write the resulting system of second-order DEs as a first-order system, w'(t) = Aw(t), with w = (x₁, x₁, x₂, x₂) T
For values of k = 4, m = 1 and c
=
21,2 = 0.5±3.2i, V₁
23,4= - 0.5 ± 1.13i, V3
1, the resulting eigenvalues and eigenvectors of A are
-0.039 0.248i
0.813
0.024 +0.153i
-0.502
-0.134-0.302i'
0.409
-0.216 - 0.489i
0.661
and
(a) Find a set of initial displacements x₁(0), x₂(0) that will lead to the fast mode of oscillation for this sytem. Assume that the initial velocities wil be zero.
(x₁(0), x₂(0))
Enter your answer using angle braces, (and ).
cx (friction proportional to velocity).
(b) At what frequency will the masses be oscillating in this mode?
Frequency =
rad/s

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