Question

1 point) Consider the initial value problem my "tou' + 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(t), 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 the applied force in Newtons is 50 if0 st/2, F(t) = if t > π/2. a Solve the initial value problem, using that the displacement y(t) and velocity y'(t) remain continuous when the applied force is discontinuous. For 0 < t < π/2, y(t)- E help (formulas) For t > π/2, y(t) 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 approximates y(t) for very large positive values of For very large positive values of t, y(t) E help (formulas)

Answer 1

my'' + cy' + ky = F(t) y (0) = 0 y'(0) = 0 m = 2kg c = 8 kg/s k = 80 n/m f(t) = {50 0 lessthanorequalto t lessthanorequalto pi/2 0 pi/2 < t implies 2y'' + 8y' + 80y = F(t) y'' + 4y' + 40y = 1/2 F (t) let 0 lessthanorequalto t lessthanorequalto pi/2 implies F(t) = 50 y'' + 4y' + 40y = 25 corresponding homogeneous equation is given by y'' + 4y' + 40 y = 0 implies (D^2 + 4D + 40) y = 0 D = d/dt D^2 = dt^2/dt^2 Auxillary equation m^2 + 4m + 40 = 0 m = -4 plusorminus squareroot 16 - 160/2 = - 4 plusorminus squareroot - 144/2 = -4 plusorminus squareroot 12i/2 m = - 2 + 6i , - 2 - 6i hence y_cf(t) = e^- 2t (c_1 cos 6t + c_2 sin 6t) To find p.I f (D) = (D^2 + 4D + 40) P.I = 1/f (D) q(x) = 1/(D^2 + 4D + 40) . 25e^0t = 25/40 hence y (t) = e^- 2t (c_1 cos 6t + c_2 sin 6t) + 25/40 now y (0) =