Problem 3 Unilateral Laplace transform analysis of transfer function zeros in a flywheel system. Return to the system of three flywheels: 221 23 b3 +0₂ byl As in Homework 6, J₁

= J2 = J3 = 1, b₁ = b2 = b3 = b = b5 = 1. Answer the following: 1. The three coupled first-order ODEs governing this system where derived to be U ₁ = 29₁ +₂+u 2₂=₁-302 +03 03-0₂-203 Apply the unilateral Laplace transform to each ODE and solve for ₁, ₂, and f, in terms of {1(0), 2(0), 3(0)} and û. This involves a bit of algebra but yields the complete IVP solu- tion for each dependent variable, albeit in the Laplace domain. Split up the final expressions for 21, 22, and 3 in terms of the zero-input response (the part with the initial conditions) plus the zero-state response (the part with the transfer function times û). 2. The transfer function associated with ₁ (denoted H₁ in prior homework) is H₁ = s²+58 +5 (8+1)(8+2)(8+4) This transfer function has two zeros. Label them z₁ and 22 such that 22 <₁ <0. Let u(t) = e²¹¹µ(t), t≥0. Find initial conditions {₁(0), ₂(0¯), N₂(0¯)} such that ₁ = 0, t≥0. Hint: ₁ = 0 if and only if f₁ = 0. 3. Consider the ICs and input from the previous part, i.e. u(t) = e²¹tu(t), t≥ 0. Show that N₂(t) = N₂(0-)eit Na(t) = N₂(0)eit t20™ Hint: the easiest way is to show (2) satisfies the ODEs for ₂ and 3, 0₂=91 35₂ +93 123 = 22-203 (2) 4. Now consider the input u(t) = etu(t), t≥ 0. Find new initial conditions (2₁(0¯), N₂(0¯), N3(0)} such that ₁ = 0, t≥ 0. Also show N₂(t) = N₂(0¯)et and N3(t) = N3(0¯)e²²ª, t≥ 0¯. 4