Measuring phase shift angle in underdamped step response

[mamech]

Hello all

I know that the step response of a second order underdamped system is :
y(t) = 1 – (e^(-ζωn*t) / sin(θ))*sin(ωd*t + θ)

Where sin(θ)= sqrt(1-ζ^2) , ωd = sqrt(ωn^2 – ωn^2 * ζ^2 )

my question is: I can see on the simulation ωd and I can calculate it ,
but I want also to see the phase shift angle θ

I know already that θ= asin(sqrt(1-ζ^2) ) , but this is analytical, can I see and measure it on the graph?

thanks

 

[zoki85]

sin θ = ω_d/ω_n, which is ratio of a damped frequency and undamped frequency.
You may compare graphs of underdamped and undamped functions:
y₁(t)=1-e^-ζt⋅sin(ω_dt+θ)/sinθ
and
y₂(t)=1-sin(ω_nt+π/2)

Observe how difference between intersection points of the curves with line y(t)=1 changes throughout oscillations due to different frequencies.
Phase shift angle has a clear meaning in a time domain only if you compare functions having same frequencies.

 

[mamech]

Thank you very much for help!