Question

Problem 2: For the following general form of a second order measurement system (Eq. 1), classify the system as underdamped, critically damped or over damped for each of the sets of coefficients given in parts (a) - (d). Also determine the natural frequency (n), the damping ratio (0 and if possible, the ringing frequency (od). (b) k 25, c 44, m- 3 (c) k 125, c 235, m 1100 (d) k 18, c 24, m 8

Answer 1

Given that my^ + cy^ + ky = f (t) taking Laplace transform of above ms^2 y (s) + c sy (s) + ky (s) = F (s) (ms^2 + cs + k) y (s) = f (s) Transfer function = F (s)/y (s) = 1/ms^2 + cs + k F (s)/y (s) = 1/ms^2 + cs + k This is 2^nd order system and compare with standard 2^nd order control system TF = w_n^ /s^2 + 2 w_n s + w_n^2 Then F (s)/y (s) = 1/m/s^2 + c/m s + k/m equation of F (s)/y (s) = s^2 + c/m s + k/m = 0 standard control system equation = s^2 + 2 w_n s + w_n^2 = 0 \r\n compare eq (i) and (ii) then we get, w_n = squareroot k/m rightarrow Natural frequency in radian/ and 2 w_n = c/m z_ = c/2m w_n z_ = c/2m squareroot m/k = c/2 squareroot 1/km z_ = c/2 squareroot km rightarrow ration