Question

(1 point) Consider the initial value problem my"+cy'+ky F(t), y(0) = 0, y'(0) = 0 modeling the motion of a spring-mass-dashpot system initially at rest and subjected to an applied force F(), where the unit of force is the Newton (N). Assume that m 2 kilograms, c 8 kilograms per second, k 80 Newtons per meter, and F(t) 60 cos(8t) Newtons. a. Solve the initial value problem. help (formulas) b. Determine the long-term behavior of the system. Is lim y(t)= 0? If it is, enter zero. If not, enter a function that 100 approximates y(r) for very large positive values of t. For very large positive values of t, y(t) 777 help (formulas)

Answer 1

where Given initial value problem is my" Acy' + ky = f(1)_700)=b, y'ro)=0 m = 2 kg c=8kg, K: 80N/m und Elt) = 60 cosst N - Giren equation become L ly" 8y' + So y = 60 cos 87. . Dividing by 2. y" + 4y +407 - 30 C0587. The Auxillary equation is * E t .72 +48 +40=0 8= -42/ 16-160 | 7 = -4 -144 = -4 7 121 2 = -2761 c = ezt [Grosot+G sinat ] Now. PI: 1 30 cosst 12+47 +40 = 30 1 -64 +47 +40 cosst cos st 30. I 48-24

4 3-6 15 1r+6) 18-6)(8+6) cosst - 15 lote) g2-36 oset = 15 2 18+6) cosst -64-3€ = cosst 15 2 (8+6) -100 200 = -15 18 losst+ 6 cessf / -8 sin 54 +6 81] 40 : G.S. - C.F.AP.I. y = using e 2t [ G cos 6tta sinct] - 3 (-8 sinst + Grosst] initial conditions - from o we get 0 - 6 - 9 663 - GES 20 20 -

How differentiate og? 0 word to y rozēt [ geesot +95in£t 1+ ē7-gesinett Ga6 teset) - 3 / - C4 cos 87 - 48110 3+1 using y'lo)=0. 1.: 03-2C, +66 +3 1641 404 (32) = -2/ 9 +66 + 3 20 20 = 78 7 6 Go | 20 6 - 78 2046 : - 39 G using volues of a 46 in coq?. o. we get 20 60 yit)=7?t/ 900567 -39 smot] -2 [-ssiast +64584] This is required solution A for long term behavior of the system Lilin yet: 0-3 lima [-851787 +660581] juta - | -8 singi +6 cosst] for long term behavior of the system. 40 7760 Z