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 Ffrictionca' (friction proportional to velocity). We can write the resulting system of second-order DEs as a first-order system, t'(t) = Au(t), with = (₁, ₁, ₂, For values of k= 4, m= 1 and c= 1, the resulting eigenvalues and eigenvectors of A are -0.039-0.248 0.813 A₁,2 -0.53.21, ₁- 0.024 +0.153 -0.502 -0.134-0.3021 0.409 -0.2160.4891 0.661 (a) Find a set of initial displacements (0), 2(0) that will lead to the fast mode of oscillation for this sytem. Assume that the initial velocities will be zero. ((0), ₂(0)) Enter your answer using angle braces, (and). A3,4 -0.5 1.134. = 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 Ffrictionca' (friction proportional to velocity). We can write the resulting system of second-order DEs as a first-order system, t'(t) = Au(t), with = (₁, ₁, ₂, For values of k= 4, m= 1 and c= 1, the resulting eigenvalues and eigenvectors of A are -0.039-0.248 0.813 A₁,2 -0.53.21, ₁- 0.024 +0.153 -0.502 -0.134-0.3021 0.409 -0.2160.4891 0.661 (a) Find a set of initial displacements (0), 2(0) that will lead to the fast mode of oscillation for this sytem. Assume that the initial velocities will be zero. ((0), ₂(0)) Enter your answer using angle braces, (and). A3,4 -0.5 1.134. = 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 Ffriction -cz' (friction proportional to velocity).
We can write the resulting system of second-order DEs as a first-order system, t' (t) = Au(t), with = (₁, 21, 22, 2₂) I
For values of k = 4, m = 1 and c = 1, the resulting eigenvalues and eigenvectors of A are
-0.039-0.248i
0.813
A₁2=-0.5±3.2i, v₁ =
0.024 +0.153i
-0.502
-0.134-0.302i
0.409
-0.2160.489
0.661
(a) Find a set of initial displacements (0), 2(0) that will lead to the fast mode of oscillation for this sytem. Assume that the initial velocities wil be zero.
A3,4 -0.5± 1.13i, z =
and
(2₁ (0), ₂(0)) =
Enter your answer using angle braces, (and).
(b) At what frequency will the masses be oscillating in this mode?
Frequency
rad/s

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