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Understanding the physical meaning of phase shift in underdamped systems

  1. Nov 14, 2014 #1
    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
     

    Attached Files:

  2. jcsd
  3. Nov 14, 2014 #2
    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.
     
  4. Nov 15, 2014 #3
    Thank you very much for help!
     
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