Question

Ih, head inertia flexible shaft k, b I, motor inertia rn b, motor friction 6. Show that the equations of motion for the disk-drive system are given by: where motor inertia θm = motor position Pn-head position b-shaft damping bm-motor bearing and brush friction Ihead inertia k-shaft stiffness t,n(t) = motor torque input Express the equations of motion in state-variable form using [bmuOh,o,n, θ ] as the state vector. What is the order of this system? Could the dynamics of this system be expressed with fewer state variables? hom

Answer 1

I_m (theta_m) + b(theta_m - theta_n) + b_m theta_m + k(theta_m - theta_n) = Tau_m (t) I_n theta_n + b(theta_n - theta_m) + k(theta_n - theta_m) = 0 theta_m = X_1: theta_m = n - X_2 theta_m = X-1 = x_3 rightarrow (1): theta_n = X-2 = X-4 rightarrow (2) I_m (X_3) + b(X_3 - x_4) + b(X-3) + k(x_1 + x_2) = Tau_m (t) I_n (X_4) + b(X_4 - x_3) + k(X-2 - X_1) = 0 X_3 = -k_1 x_1/I_m + kI_2/I_m - x_2 *= (b + b_m)/I_m + x_4 (b/I_m) +tau (t) x_4 = kx_1/I_n - kX-2/I_n + bX-3/I_n - bX-4/I_m rightarrow (4 from equation (1), (2), (3), and (4) [X_1 X-2 X_3 X_4] = [0 0 1 0 0 0 0 1 -k/I_m k/I_m -(b + b_m)/I_m b/I_m k/I_n - k/ I_n b/I_n -b/I_m] [X_1 X_2 X_3 x_4] + [0 0 1 0] Tau_m (t) \r\n order of the system is 4 Since the inertias are two the dynamic s must have 4 state variables