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 ID: 27185 Download Pdf

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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 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):

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