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.