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Review Problem 8 solution

Review Problem 8 solution - 8.Consider the case where u t =...

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8.Consider the case where ( ) cos4 , 0 u t t t = and ( ) 0 0 x = while ( ) 0 0 x = dotnosp . Obtain the solution for ( ) x t utilizing the phasor method. Repeat the problem but this time utilize the Laplace transform method. Is the answer the same? Should it be? From Problem 1, ( ) ( ) ( ) 0 0 2 2 2 2 5 2 5 U s s x x X s s s s s + + = + + + + + dotnosp . As the initial conditions are both zero ( ) ( ) 2 2 5 U s X s s s = + + Phasor Method The phasor method, as shown in Lecture 35, gives the steady state portion of the particular solution when the input is sinusoidal. For this problem, as the initial conditions are zero, the homogeneous solution is zero for all points in time. Thus the phasor solution gives the steady state portion of the total solution. The abrupt start of the input causes a transient solution to occur which is not captured by this method. First find ( ) ( ) ( ) 2 1 2 5 X S G S U S S S = = + + Recall, for a sinusoidal input, the phasor method gives the general solution, ( ) ( ) ( ) 0 cos p x t u G j t ω ω β φ = + + , where ( ) ( ) Im tan Re G j G j ω φ ω = , for the case where the input is ( ) 0 cos u t ω β
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