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Math · Advanced Math
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Consider a mass-spring-dashpot system mar (t) car (t) kr (t) Focos(wt), (4) where m 100 kg, c 0 N s/m, k 16 N/m, and F 300 N. (In this case we can assume that the dashpot has been disconnected.) Assume the following initial conditions for (t): z(0) 0 (5) i.e., the spring starts at rest, subject to a steady, periodic forcing Focos(wt). (a) What is the natural angular frequency wo of this system (4),(5)? (Hint: Consider the associated homogeneous equation.) (b) Assume that w #wo. Solve the initial value problem (4), (5) by first computing the homogeneous and particular solutions (zH (t) and ap(t), respectively). (c) Using your answer from part (b), write down the special case when w J 1/2 What is the period of the solution? (Hint Consider the identity 2 sin (A B)/2 sin((A B)/2) cos B cos A.) Plot your solution with OS t S 125, and attach the plot. What phenomenon does this solution show? (d) Now let w wo. Again solve the the initial value problem (4),(5). Plot your solution with 0 S t S 125, and attach the plot. What phenomenon does this solution show?

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