Question

Item 13 This question is about the homogeneous spring-mass model mj + y+ky 0. (a) Let m 10, u37, and k= 30. Which case is this, and how do you know? (By "which case is this," I mean, "Is the system undamped, underdamped, critically damped, or overdamped?") (b) Suppose m 10, 0, and k30. Which case is this, and how do you know? (c) Suppose now that m 10, 20, and k30. Which case is this, and how do you know? Item 14 Solve the homogeneous spring-mass model with m 10, 20, and k 30 Item 15 Apply the initial conditions y(0) 0 and y (0) =2 to your solution to Item 14.

Answer 1

My + mu y + ky = 0 m = 0 mu = 37 k = 30 as mu = 37 notequalto 0 Rightarrow It is not undammed Now mu^2 - 4mk = 37^2 - 4(10)(30) = 1369 - 1200 = 169 > 0 Rightarrow It is over damped system m = 10, mu = 0 k = 30 since mu = 0 Rightarrow It is undammed system m = 10, mu = 20 k = 30 Rightarrow mu^2 - 4mk = 400 - 1200 = -800 < 0 Rightarrow it is under damped system my + mu y + ky = 0 Let m = 10, mu = 20 k = 30 Rightarrow 10y + 20 y + m 30y = 0 The auxiliary equation is 10r^2 + 20r + 30 = 0 Rightarrow r^2 + 2r + 3 = 0 r = -2 plusminus 2 squareroot 2 I/2 Rightarrow r = -1 plusminus 2i