Consider the double mass / triple spring system shown belownWWWWWWWWAllspringshave spring constant k=7 and all masses have mass m=1kg.Each spring is subject to a damping force of Ffriction = -3X, Using thestandard conversion to a first-order system, he would obtain thesystem0100-14 -3 7 000010 14-3W²³³ (+) =7W = Aw, where w(t) =The eigenvalues of A are approximately.71,2 -1.5±4.33; and 713,4 -1.5 ± 2.18%The eigenvector for 2₁-1.5 +4.331 is approximatelyeigenvector for 73 -1.5 +2.18; is approximatelyXilt)xỉ (tX2 (t)X₂(+))0.07 +0.21t-1-007-0.21₂0.21 -0.31ż1-0.21 -0.31and the(a). Write general solution wilt) of system of the differential equation above(b). Find an initial condition that will result in the masses displaying simpleoscillations at relatively high frequency I the fart mode) provided the relacityis zero at t-o. Jutify the your answer.[c]. Suppose that there is no friction, i.e., Friction -o. Which of the followingcould be an eigenvalue of the new corresponding matrix A?.तरा, नश + नये, √, - 1.5+4337.

Given the system of DE as, where The eigenvalues and its corresponding eigenvectors are and.

Answered: Consider the double mass / triple…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Consider the double mass / triple spring system shown belown
WWWWWWWW
All
springs
have spring constant k=7 and all masses have mass m=1kg.
Each spring is subject to a damping force of Ffriction = -3X, Using the
standard conversion to a first-order system, he would obtain the
system
0100
-14 -3 7 0
0001
0 14-3
W²³³ (+) =
7
W = Aw, where w(t) =
The eigenvalues of A are approximately.
71,2 -1.5±4.33; and 713,4 -1.5 ± 2.18%
The eigenvector for 2₁-1.5 +4.331 is approximately
eigenvector for 73 -1.5 +2.18; is approximately
Xilt)
xỉ (t
X2 (t)
X₂(+))
0.07 +0.21t
-1
-007-0.21₂
0.21 -0.31ż
1
-0.21 -0.31
and the
(a). Write general solution wilt) of system of the differential equation above
(b). Find an initial condition that will result in the masses displaying simple
oscillations at relatively high frequency I the fart mode) provided the relacity
is zero at t-o. Jutify the your answer.
[c]. Suppose that there is no friction, i.e., Friction -o. Which of the following
could be an eigenvalue of the new corresponding matrix A?.
तरा, नश + नये, √, - 1.5+4337.

Expert Solution

Step 1: Writing down the given information

Given the system of DE as

$w with rightwards arrow on top apostrophe open parentheses t close parentheses equals A w with rightwards arrow on top$ , where $A equals open square brackets table row 0 1 0 0 row cell negative 14 end cell cell negative 3 end cell 7 0 row 0 0 0 1 row 7 0 cell negative 14 end cell cell negative 3 end cell end table close square brackets.$

The eigenvalues and its corresponding eigenvectors are

$lambda subscript 1 comma 2 end subscript equals negative 1.5 plus-or-minus 4.33 i comma space v subscript 1 equals open square brackets table row cell 0.07 plus 0.21 i end cell row cell negative 1 end cell row cell negative 0.07 minus 0.21 i end cell row 1 end table close square brackets comma space v subscript 2 equals open square brackets table row cell 0.07 minus 0.21 i end cell row cell negative 1 end cell row cell negative 0.07 plus 0.21 i end cell row 1 end table close square brackets$ and

$lambda subscript 3 comma 4 end subscript equals negative 1.5 plus-or-minus 2.18 i comma space v subscript 3 equals open square brackets table row cell negative 0.21 minus 0.31 i end cell row 1 row cell negative 0.21 minus 0.31 i end cell row 1 end table close square brackets comma space v subscript 4 equals open square brackets table row cell negative 0.21 plus 0.31 i end cell row 1 row cell negative 0.21 plus 0.31 i end cell row 1 end table close square brackets$ .