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