1Second Order SystemsSecond Order Equations ( )G s22 + 2 +1Kτ ζτ = ss Standard Form2τd2 ydt2 + 2ζτ dy dt + y = Kf (t)Corresponding Differential EquationK = Gainτ = Natural Period of Oscillationζ = Damping Factor (zeta)Note: this has to be 1.0!!!2Origins of Second OrderEquations1. Multiple Capacity Systems in SeriesK1τ1s + 1K2τ2s +1becomeor … Continue reading “Corresponding Differential Equation | My Assignment Tutor”
1Second Order SystemsSecond Order Equations ( )G s22 + 2 +1Kτ ζτ = ss Standard Form2τd2 ydt2 + 2ζτ dy dt + y = Kf (t)Corresponding Differential EquationK = Gainτ = Natural Period of Oscillationζ = Damping Factor (zeta)Note: this has to be 1.0!!!2Origins of Second OrderEquations1. Multiple Capacity Systems in SeriesK1τ1s + 1K2τ2s +1becomeor K1K2τ( ) 1s +1 ( ) τ 2s + 1 K2τs2+ 2ζτs + 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 CharacteristicEquation2τs2+ 2ζτs + 1 = 0 Two complexconjugate roots0 1Overdamped 3Response of 2nd Order Systemto Step Inputs UnderdampedEq. 5-51Fast, oscillations occurFaster than overdamped, nooscillationCritically dampedEq. 5-50OverdampedEq. 5-48 or 5-49Sluggish, no oscillations Ways to describe underdamped responses:• Rise time • Time to first peak• Settling time • Overshoot• Decay ratio • Period of oscillationResponse of 2nd Order Systemsto Step Input ( 0 1(5-48, 5-49)Example 5.5• Heated tank + controller = 2nd order system(a) When feed rate changes from 0.4 to 0.5kg/s (step function), Ttank changes from 100to 102°C.Find gain (K) of transfer function:8Example 5.5• Heated tank + controller = 2nd order system(b) Response is slightly oscillatory, with firsttwo maxima of 102.5 and 102.0°C at 1000and 3600 S.What is the complete process transferfunction?Example 5.5• Heated tank + controller = 2nd order system(c) Predict tr:9Example 5.6• Thermowell + CSTR = 2nd order system(a)Find τ, ζ:( )( )( ) 3 1 10 11( ) + +=′′T s s sT sreactormeas CSTR ThermocoupleRoad Map for 2nd Order EquationsStandard FormStepResponseSinusoidalResponse(long-time only)(5-63)Other InputFunctions-Use partialfractions Underdamped0 1(5-48, 5-49) 10Example 5.6• Thermowell + CSTR = 2nd order system(a)Find τ, ζ:( )( )( ) 3 1 10 11( ) + +=′′T s s sT sreactormeasExample 5.6• Thermowell + CSTR = 2nd order system(b) Temperature cycles between 180 and 183°C, withperiod of 30 s.Find ω, : Aˆ11Example 5.6• Thermowell + CSTR = 2nd order system(c) Find A (actual amplitude of reactor sine wave):
