Second Order Systems Second Order Equations W Standard Form dt W dy dt Kf Corresponding Differential Equation K  Gain  Natural Period of Oscillation  Damping Factor zeta Note this has to be
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Second Order Systems Second Order Equations W Standard Form dt W dy dt Kf Corresponding Differential Equation K Gain Natural Period of Oscillation Damping Factor zeta Note this has to be

0 brPage 2br Origins of Second Order Equations 1 Multiple Capacity Systems in Series become or W 2 Controlled Systems to be discussed later 3 Inherently Second Order Systems Mechanical systems and some sensors Not that common in chemical process co

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Second Order Systems Second Order Equations W Standard Form dt W dy dt Kf Corresponding Differential Equation K Gain Natural Period of Oscillation Damping Factor zeta Note this has to be




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Presentation on theme: "Second Order Systems Second Order Equations W Standard Form dt W dy dt Kf Corresponding Differential Equation K Gain Natural Period of Oscillation Damping Factor zeta Note this has to be"— Presentation transcript:


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Second Order Systems Second Order Equations ]W Standard Form dt ]W dy dt Kf Corresponding Differential Equation K = Gain = Natural Period of Oscillation = Damping Factor (zeta) Note: this has to be 1.0!!!
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Origins of Second Order Equations 1. Multiple Capacity Systems in Series become or ]W 2. Controlled Systems (to be discussed later) 3. Inherently Second Order Systems Mechanical systems and some sensors Not that common in chemical process control Examination of the Characteristic Equation Two complex conjugate roots Underdamped 0 < < 1 Two equal real

roots Critically Damped = 1 Two distinct real roots Overdamped > 1
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Response of 2 nd Order System to Step Inputs Fast, oscillations occur Underdamped Eq. 5-51 Faster than overdamped, no oscillation Critically damped Eq. 5-50 Sluggish, no oscillations Overdamped Eq. 5-48 or 5-49 Ways to describe underdamped responses: Rise time Time to first peak Settling time Overshoot Decay ratio Period of oscillation Response of 2 nd Order Systems to Step Input ( 0 < < 1) 1. Rise Time: is the time the process output takes to first reach the new steady-state value. 2. Time to First

Peak: is the time required for the output to reach its first maximum value. 3. Settling Time : is defined as the time required for the process output to reach and remain inside a band whose width is equal to 5% of the total change in . The term 95% response time sometimes is used to refer to this case. Also, values of 1% sometimes are used. 4. Overshoot: OS = (% overshoot is 100 ). 5. Decay Ratio : DR = (where is the height of the second peak). 6. Period of Oscillation : is the time between two successive peaks or two successive valleys of the response. KM sin cos Eq. 5-51
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Response of 2 nd Order Systems to Step Input 0 < < 1 ]t Note that < 0 gives an unstable solution as ] , and OS (5-52) (5-53 exp S] OS >@ ln ln OS OS Above (5-56) (5-54) exp S] OS DR (5-55) Above (5-57) (5-60) cos Relationships between OS, DR, P and , for step input to 2 nd order system, underdamped solution ]W KM
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Response of 2 nd Order System to Sinusoidal Input Output is also oscillatory Output has a different amplitude than the input Amplitude ratio is a function of , W (see Eq. 5-63) Output is phase shifted from the input Frequency must be in

radians/time!!! (2 radians = 1 cycle) P = time/cycle = 1/( ), 2 SQ = , so P = 2 (where = frequency in cycles/time) Sinusoidal Input, 2 nd Order System (Section 5.4.2) Input = A sin t, so A is the amplitude of the input function is the frequency in radians/time At long times (so e xponential dies out), is the output amplitude >@ ]ZW ZW KA (5-63) Bottom line: We can calculate how the output ampl itude changes due to a sinusoidal input Note: There is also an equation for the maximum amplitude ratio (5-66) Note log scale
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Road Map for 2 nd Order Equations Standard Form Step

Response Sinusoidal Response (long-time only) (5-63) Other Input Functions -Use partial fractions Underdamped 0 < < 1 (5-51) Critically damped = 1 (5-50) Overdamped > 1 (5-48, 5-49) Relationship between OS, P, t and , (pp. 119-120) Example 5.5 Heated tank + controller = 2 nd order system (a) When feed rate chan ges from 0.4 to 0.5 kg/s (step function), T tank changes from 100 to 102 C. Find gain ( ) of transfer function:
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Road Map for 2 nd Order Equations Standard Form Step Response Sinusoidal Response (long-time only) (5-63) Other Input Functions -Use partial fractions

Underdamped 0 < < 1 (5-51) Critically damped = 1 (5-50) Overdamped > 1 (5-48, 5-49) Relationship between OS, P, t and , (pp. 119-120) Example 5.5 Heated tank + controller = 2 nd order system (a) When feed rate chan ges from 0.4 to 0.5 kg/s (step function), T tank changes from 100 to 102 C. Find gain ( ) of transfer function:
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Example 5.5 Heated tank + controller = 2 nd order system (b) Response is slightly oscillatory, with first two maxima of 102.5 and 102.0 C at 1000 and 3600 S. What is the complete process transfer function? Example 5.5 Heated tank + controller = 2 nd

order system (c) Predict
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Example 5.6 Thermowell + CSTR = 2 nd order system (a) Find , 10 reactor meas CSTR Thermocouple Road Map for 2 nd Order Equations Standard Form Step Response Sinusoidal Response (long-time only) (5-63) Other Input Functions -Use partial fractions Underdamped 0 < < 1 (5-51) Critically damped = 1 (5-50) Overdamped > 1 (5-48, 5-49) Relationship between OS, P, t and , (pp. 119-120)
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10 Example 5.6 Thermowell + CSTR = 2 nd order system (a) Find , 10 reactor meas Example 5.6 Thermowell + CSTR = 2 nd order system (b) Temperature cycles

between 180 and 183 C, with period of 30 s. Find , :
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11 Example 5.6 Thermowell + CSTR = 2 nd order system (c) Find A (actual amplitude of reactor sine wave):