6 - DynamicResponse6 filled

Engineering purdue university 2 slide 10 5 me575

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Unformatted text preview: ⎦ s + ζωn ωd 2 K +A +B s ( s + ζ ω n )2 + ω d 2 ( s + ζ ω n )2 + ω d 2 y ( t ) = K + Ae − ζω n t cos (ω d t ) + Be − ζω n t sin (ω d t ) y (t ) = K − K 1−ζ ME575 Session 6 – Dynamic Response 2 ⎛ 1−ζ e − ζω n t sin (ω d t + ϕ ) , ϕ = tan − 1 ⎜ ⎜ ζ ⎝ School of Mechanical Engineering Purdue University 2 ⎞ ⎟ ⎟ ⎠ Slide 10 5 ME575 Handouts Unit Step Response of 2nd Order System 1.6K yMAX 1.4K OS Unit Step Response 1.2K K 0.8K Td 0.6K 0.4K 0.2K 0 1 tP 2 0 3 4 5 6 7 8 Time [sec] 9 10 11 tS 12 13 14 School of Mechanical Engineering Purdue University ME575 Session 6 – Dynamic Response 15 Slide 11 Step Response of 2nd Order System • Peak Time (tP) tP = π π = ωd ωn 1 − ζ 2 • Percent Overshoot (%OS) −πζ %OS = 100e 1−ζ 2 • Settling Time (tS) Time required for the response to be within δ% of the final (steady-state) value: (steady- ts = − ⎛δ ⎞ ln ⎜ ζωn ⎝ 100 ⎟ ⎠ 1 ME575 Session 6 – Dynamic Response School of Mechanical Engineering Purdue University Slide 12 6 ME575 Handouts Effect of Additional Zeros G (s) = 1 s +1 1.2α 12 4 s + 0.6 s + 1 ⇒ ωn = 2, ζ = 0.6 poles: p1,2 = −1.2 ± j1.6 zeros: z1 = −1.2α Fast zero: almost no effect Slow zero: significant effect ME575 Session 6 – Dynamic Response RHP zero: undershoot LHP zero: overshoot School of Mechanical Engineering Purdue University Slide 13 RHP Zeros and Undershoot Lemma 4.2 Assume a stable LTI system has TF with unity static gain and a RHP RHP (unstable) zero at s = c > 0. Let ts be the δ% band settling time of the system. Then, the unit step response exhibits an undershoot Mu satisfying Mu ≥ Proof: (See textbook) 1− δ ects − 1 Lemma 4.2 shows that, when a system has nonminimum-phase nonminimum(or RHP) zeros, there is a trade-off between having a fast step traderesponse and having small undershoot! ME575 Session 6 – Dynamic Response School of Mechanical Engineering Purdue University Slide 14 7 ME575 Handouts Slow LHP Zeros and Overshoot Lemma 4.3 Assume a stable LTI system has TF with unity static gain and a stable zero at stable s = c < 0. Further, assume that A1: The system has a d...
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This note was uploaded on 12/09/2011 for the course ME 575 taught by Professor Meckl during the Fall '10 term at Purdue University-West Lafayette.

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