Question

Question (b)

Ans : root(7/2) , 16/((5)^(1/2))

9. Consider a mass-spring system as shown in the figure with a body of mass m, a spring and a dashpot. Let k, c and r(t) be the spring constant, the damping constant and driving force, respectively Let y(t) be the displacementMass of the body from the equilibrium with downward direction as positive. b) [7pts] Let m=1, c=1, k=4, and r(t) 8cosut. Determine w such that you get the steady-state vibration of maximum possible amplitude, and find this amplitude Spring Dashpot (a) [5pts) Let m-10, c 40, k 40 and r(t)-0. If the initial displacement is 3m, find the initial velocity such that the body passes through y 0 at t= 6 second.

Answer 1

Equation of motion m d^2 y /dt^2 + C dy/dt + k y = 8 cos (wt) (D^2 + D + 4) y (t) = 8 cos (wt) because D^2 = d^2/dt^2, D = d/dt F (D) y (t) = Q Steady state Solution y (t) = 1/F (D) Q = 1/ (D^2 + D + 4) 8 cos (wt) y (t) = 8 Re [1/ D^2 + D + 4 e^iwt] = 8 Re [1/ (iw)^2 + (iw) + 4 e^iwt] therefore 1/ F (D) e^0t = e^0t/ F(a) = 8 Re [e^iwt/ - w^2 + iw + 4] = 8 Re [e^+ iwt/ (4 - w^2)+ iw times (4 - w^2) - iw/ (4 - w^2) - iw] 8 Re [{cos (wt) + i sin (wt)} {(4 - w^2) - iw}/ (4 - w^2) + w^2] y (t) = 8/ (4 - w^2)^2 + w^2 [(4 - w^2) cos (wt) + w sin (wt)] y (t) = 8/ squareroot (4 - w^2)^2 + w^2 [(4 - w^2/ squareroot (4 - w^2) + w^2 cos (wt)+ w/ squareroot (4 - w^2)^2 + w^2)]
i think you did mistake in typing of option b answer.please let me know .