# Understanding the physical meaning of phase shift in underdamped systems

1. Nov 14, 2014

### 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

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2. Nov 14, 2014

### zoki85

sin θ = ωdn, 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(ωdt+θ)/sinθ
and
y2(t)=1-sin(ωnt+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.

3. Nov 15, 2014

### mamech

Thank you very much for help!