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Consider a vibrating system described by the initial value problem. (A computer algebra system is recommended.) u” + 1 4 u’ + 2u = 2 cos ωt
Question
Consider a vibrating system described by the initial value problem. (A computer algebra system is recommended.) u” + 1 4 u’ + 2u = 2 cos ωt, u(0) = 0, u'(0) = 2 (a) Determine the steady state part of the solution of this problem.
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2021-08-21T07:30:00+00:00
2021-08-21T07:30:00+00:00 1 Answers
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Answer:
Therefore the required solution is
[tex]U(t)=\frac{2(2-\omega^2)^2}{(2-\omega^2)^2+\frac{1}{16}\omega} cos\omega t +\frac{\frac{1}{2}\omega}{(2-\omega^2)^2+\frac{1}{16}\omega} sin \omega t[/tex]
Explanation:
Given vibrating system is
[tex]u”+\frac{1}{4}u’+2u= 2cos \omega t[/tex]
Consider U(t) = A cosωt + B sinωt
Differentiating with respect to t
U'(t)= – A ω sinωt +B ω cos ωt
Again differentiating with respect to t
U”(t) = – A ω² cosωt -B ω² sin ωt
Putting this in given equation
[tex]-A\omega^2cos\omega t-B\omega^2sin \omega t+ \frac{1}{4}(-A\omega sin \omega t+B\omega cos \omega t)+2Acos\omega t+2Bsin\omega t = 2cos\omega t[/tex]
[tex]\Rightarrow (-A\omega^2+\frac{1}{4}B\omega +2A)cos \omega t+(-B\omega^2-\frac{1}{4}A\omega+2B)sin \omega t= 2cos \omega t[/tex]
Equating the coefficient of sinωt and cos ωt
[tex]\Rightarrow (-A\omega^2+\frac{1}{4}B\omega +2A)= 2[/tex]
[tex]\Rightarrow (2-\omega^2)A+\frac{1}{4}B\omega -2=0[/tex]………(1)
and
[tex]\Rightarrow -B\omega^2-\frac{1}{4}A\omega+2B= 0[/tex]
[tex]\Rightarrow -\frac{1}{4}A\omega+(2-\omega^2)B= 0[/tex]……..(2)
Solving equation (1) and (2) by cross multiplication method
[tex]\frac{A}{\frac{1}{4}\omega.0 -(-2)(2-\omega^2)}=\frac{B}{-\frac{1}{4}\omega.(-2)-0.(2-\omega^2)}=\frac{1}{(2-\omega^2)^2-(-\frac{1}{4}\omega)(\frac{1}{4}\omega)}[/tex]
[tex]\Rightarrow \frac{A}{2(2-\omega^2)}=\frac{B}{\frac{1}{2}\omega}=\frac{1}{(2-\omega^2)^2+\frac{1}{16}\omega}[/tex]
[tex]\therefore A=\frac{2(2-\omega^2)^2}{(2-\omega^2)^2+\frac{1}{16}\omega}[/tex] and [tex]B=\frac{\frac{1}{2}\omega}{(2-\omega^2)^2+\frac{1}{16}\omega}[/tex]
Therefore the required solution is
[tex]U(t)=\frac{2(2-\omega^2)^2}{(2-\omega^2)^2+\frac{1}{16}\omega} cos\omega t +\frac{\frac{1}{2}\omega}{(2-\omega^2)^2+\frac{1}{16}\omega} sin \omega t[/tex]