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lec11_handout

# lec11_handout - Dynamic System Response • Input/Output...

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Unformatted text preview: Dynamic System Response • Input/Output Model of Linear Dynamic Systems • Time Response of Dynamic Systems – Solutions to Differential Equations • Transient and Steady State Response • Frequency Response of Dynamic Systems – Review of Complex Variables – Frequency Response Function – Gain and Phase Characteristics • System Integration ME365 Dynamic System Response 1 From Last Time • Time response: 0 • Frequency response: y ⋯ ⋯ ∠ ME365 Dynamic System Response 2 Frequency Response • 1st Order System • 2nd Order System Differential Equation Differential Equation 1 2 Frequency Response Function Frequency Response Function – Magnitude – Magnitude 4 1 1 – Phase ∠ 2 1 1 – Phase tan ∠ 2 tan 1 ME365 Dynamic System Response 3 Frequency Response Ex: A thermocouple can be modeled by a 1st order ODE 0.1 0.003 Find: (1) The frequency response function (2) The steady state response of the thermocouple to an input temp. of 4 cos 30 (3) The steady state response of the thermocouple to a input temp. of 25 ME365 Dynamic System Response 4 cos 30 4 Frequency Response Ex: A thermocouple has a frequency response: 0.003 0.1 1 Find the differential equation that describes the thermocouple. ME365 Dynamic System Response 5 Frequency Response Ex: An accelerometer can be characterized by the following ODE: 4 400 800 where V is the output voltage and A is the acceleration. Find the following: (1) The frequency response function of the accelerometer. 800 4 400 (2) The steady state response due to each of the following inputs: In p u t ( A ) O u tp u t ( V ) In p u t ( A ) 8 .1 0 5 c o s (2 0 π t) 0 .9 0 1 c o s (3 × 2 0 π t) 0 .1 0 0 c o s(9 × 2 0 π t) 0 .3 2 4 c o s (5 × 2 0 π t) O u tp u t ( V ) 0 .1 6 5 c o s(7 × 2 0 π t) 0 .0 6 7 c o s(1 1 × 2 0 π t) ME365 Dynamic System Response 6 Frequency Response Ex: (Continued) Sum all six input signals Summation of Input Signals 10 100 Total Response 80 8 6 4 2 60 0 -2 0 -4 -6 0.05 0.1 0.15 0.2 Output (V) Acceleration Sum all six output responses 40 20 0 -20 0 0.05 0.1 0.15 0.2 -40 -60 -8 -10 -80 -100 Time (sec) Time (sec) Q: What went wrong? ME365 Dynamic System Response 7 Frequency Response • 1st Order System Differential Equation Frequency Response Function 1 – Magnitude (Gain) 1 – Phase ∠ ME365 Dynamic System Response tan 8 Frequency Response • 2nd Order System Differential Equation 1 2 Frequency Response Function 2 1 – Magnitude (Gain) 4 2 1 ∠ tan 1 – Phase ME365 Dynamic System Response 9 Frequency Response - Bode Plot • 1st Order System Frequency Response Function 1 – Magnitude (Gain) 1 – Phase ∠ tan Bode Plot • Plot dB Magnitude vs Log Frequency () ⎜ ⎟ ⇒ dB M agnitude = 20 log 10 ⎛ G j ω ⎞ ⎝ ⎠ • Plot Phase Angle vs Log Frequency ME365 Dynamic System Response 10 Gain and Phase Characteristics dB Magnitude • 1st Order System • →0 20logK 20logK - 20 20logK - 40 20logK - 60 • 0.1 →∞ 1.0 10 1.0 10 100 1000 Phase (deg) -20 -40 -60 -80 0.1 100 1000 Frequency ωτ ME365 Dynamic System Response 11 Frequency Response - Bode Plot • 2nd Order System Frequency Response Function 2 1 – Magnitude (Gain) 4 1 – Phase ∠ 2 tan 1 ME365 Dynamic System Response 12 Gain and Phase Characteristics • 2nd Order System • →0 dB Magnitude • 20logK + 20 20logK 20logK - 20 20logK - 40 20logK - 60 0.001 →∞ 0.01 0.1 1.0 10 100 10 100 Phase (deg) 0 -50 -100 -150 0.001 0.01 0.1 1.0 Frequency ω/ωn ME365 Dynamic System Response 13 Gain and Phase Characteristics ωres is the resonant frequency. 20logK+20 dB Magnitude • 2nd Order System (cont.) – Peak in Magnitude 20logK 0.001 – Critically Damped ∠ ME365 Dynamic System Response Phase (deg) – At ω = ωn : 0.01 0.001 0.01 0.1 1.0 10 100 0.1 1.0 10 100 0 -50 -100 -150 Frequency ω/ωn 14 Ex: Find the frequency response function of the following 1st order system and sketch its Bode plot. 0.05 10 dB Magnitude Frequency Response 0 10 2 10 4 10 Phase (deg) 0 -50 -100 -150 0 10 2 10 4 10 Frequency (rad/sec) ME365 Dynamic System Response 15 Ex: Find the frequency response function of the following 2nd order system and sketch its Bode plot. 400 10000 80000 dB Magnitude Frequency Response 0 10 2 10 4 10 Phase (deg) 0 -50 -100 -150 0 10 2 10 4 10 Frequency (rad/sec) ME365 Dynamic System Response 16 System Integration • Single Device x(t) G(jω) y(t) ME365 Dynamic System Response 17 System Integration • Multiple Devices (assumes no loading effect) x(t) G1(jω) ME365 Dynamic System Response y1(t) x(t) G2(jω) y2(t) x(t) G3(jω) y3(t) 18 System Integration x(t) G1(jω) y1(t) x(t) ME365 Dynamic System Response G2(jω) GT (jω) y2(t) G3(jω) y(t) y(t) 19 ...
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