Transcription of Second Order Systems
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Second Order Systems
1 Second Order SystemsSecond Order equations ()1222++=ssKsG Standard Form 2d2ydt2+2 dydt+y=Kf(t)Corresponding Differential EquationK = Gain = Natural Period of Oscillation = Damping Factor (zeta)Note: this has to be !!!2 Origins of Second Order Equations1. Multiple Capacity Systems in SeriesK1 1s+1K2 2s+1becomeorK1K2 1s+1() 2s+1()K 2s2+2 s+12. Controlled Systems (to be discussed later)3. Inherently Second Order Systems Mechanical Systems and some sensors Not that common in chemical process controlExamination of the Characteristic Equation 2s2+2 s+1=0 Two complex conjugate rootsUnderdamped0 < < 1 Two equal real rootsCritically Damped = 1 Two distinct real rootsOverdamped > 13 Response of 2ndOrder System to StepInputsFast, oscillations occurUnderdampedEq. 5-51 Faster than overdamped, no oscillationCritically dampedEq. 5-50 Sluggish, no oscillationsOverdampedEq. 5-48 or 5-49 Ways to describe underdamped responses: Rise time Time to first peak Settling time Overshoot Decay ratio Period of oscillationResponse of 2ndOrder Systemsto StepInput ( 0 < < 1)1.
2 Origins of Second Order Equations 1.Multiple Capacity Systems in Series K1 τ1s+1 K2 τ2s +1 become or K1 K2 ()τ1s +1 ()τ2s+1 K τ2s2 +2ζτs+1 2.Controlled Systems (to be discussed later) 3.Inherently Second Order Systems
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