Unformatted text preview: Summary – Forced Response of an Underdamped 2nd
order System to a Step Input The forced response of a second
order system to a unit step input: 2
2 + 2ζω n y + ω n y = Kω n h ( t ) y has been seen to take on the following form: ⎧
⎛
⎞⎫
ζω
⎪
⎪
y ( t ) = K ⎨1 − e−ζω n t ⎜ cosω d t + n sinω d t ⎟ ⎬ ωd
⎝
⎠⎪
⎪
⎩
⎭ where ω d = ω n 1 − ζ 2 . This response is dictated by the location of the poles of the transfer function for this system: p1, 2 = −ζω n ± jω d We have seen that there are three important loci of poles in the complex plane: • Constant damping ratio: radial lines from the origin at an angle of φ . Moving the system poles along these radial lines represents constant %OS . • Constant settling time: straight line parallel to the imaginary axis. Moving the system poles along these lines represents constant 2% settling time, t s = 4 / ζω n • Constant undamped natural frequency: circular arc centered at the origin and with a radius of ω n . Moving the system poles around these circular arcs represents constant undamped natural frequency ω n imag{ p}
ωn constant damping ratio, ζ ωd φ φ = cos −1 (ζ )
real { p} constant ω n constant settling time, t s −ω n ...
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 Fall '10
 Meckle
 φ, 2%, the00

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