Consider the second-order system +2ζωn+ωn2 with initial conditions y(0) = y0, (0) = 0. Introduce state variables x1=y, x2=. Phase plots for an underdamped (ζ = 0.25), critically damped (ζ = 1), and...

Consider the second-order system
+2ζωn+ω_n
²

with initial conditions y(0) = y₀,
(0) = 0. Introduce state variables x₁=y, x₂=. Phase plots for an underdamped (ζ = 0.25), critically damped (ζ = 1), and overdamped (ζ = 2) case with ω_n
= 1 rad/s and y₀
= 1 are shown in Figure:

a. Plot a histogram for the distance from the initial point x₁(0) = 1, x₂(0) = 0 to the steadystate equilibrium point x₁(∞) = 0, x₂(∞) = 0 along the trajectories in state space if the damping ratio is uniformly distributed between 0 and 2. Note that the distance from the initial point (1,0) to the point [x₁(t), x₂(t)] along the trajectory is given by

(i) Repeat part (a) for the case where ζ = 0.25, and the natural frequency ω_n
is uniformly distributed between 0 and 100 rad/s.

(ii) Repeat part (a) for the case where ζ = 1, and the natural frequency ωn is uniformly distributed between 0 and 12.5 rad/s.

(iii) Repeat part (a) for the case where ζ ~ U(0, 2), ω_n
~ U(0, 100), and y₀
~ U(0, 1).