Problem 4

Unilateral Laplace transform analysis of another IVP. Consider the system from Problem 3,

Homework 5:

torsional →

spring

k = 1

U

disks rub

with friction

O

c=24

J = 2

J = 1

The dependent variables are the left flywheel angle, 01, and right flywheel angular velocity, 2.

1. Apply the unilateral Laplace transform to the second order ODE for 0₁ and the first order

ODE for ₂. The natural initial conditions are {0₁(0), 6(0), 2₂(0)}. Determine 0₁ and

2₂ in terms of the ICs and û. There is no need to find the corresponding time-domain signals

-the point is to show how the unilateral Laplace transform yields both the zero-input response

and zero-state response parts of the IVP solution.

2. The transfer function associated with ₁ has a zero at s= -1. Therefore, let u = e-¹μ(t),

+20, and find initial conditions (01(0), 0(0), 2(0)}, such that 01 (t) = 0, + > 0. Hint:

use the expression for 8₁ to find ICs for which ₁ = 0.

3. The transfer function associated with 2 has a zero a s = 0. Therefore, let u= u(t), t≥ 0¯,

and find initial conditions {0₁(0), 6(0), ₂(0)} such that ₂(t) = 0, t≥ 0. Hint: find

ICs so ₂ = 0.