Set08_10

# Set08_10 - Dynamic System Response Input/Output Model of...

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Dynamic System Response Input/Output Model of Linear Dynamic Systems Time Response of Dynamic Systems – Solutions to Differential Equations ransient and Steady State Response ME365 Dynamic Response Slide 1 Galen King Purdue University • Transient and Steady State Response Frequency Response of Dynamic Systems – Review of Complex Variables – Frequency Response Function – Gain and Phase Characteristics System Integration

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Linear Systems Linear System Input Output x 1 (t) y 1 (t) Mass-Spring-Damper, Thermocouple, Strain Gauge . .. x 2 (t) y 2 (t) A x 1 (t) + B x 2 (t) omplicate imple omplicate ME365 Dynamic Response Slide 2 Galen King Purdue University Satisfies the Superposition Principal. Can be modeled by Linear Ordinary Differential Equations. Input a sinusoidal signal of frequency f 1 , the output will be a sinusoidal signal with the same frequency f 1 . Complicated Input Simple Inputs Output
Input/Output Model 1) System Identification: use input and output to generate system model 2) System Simulation: use system model and input to predict output ) verse Filtering: use system model and output to infer input Linear System Input Output x(t) y(t) ME365 Dynamic Response Slide 3 Galen King Purdue University 3) Inverse Filtering: Input-Output Equation: n th order linear ODE ( n th order system ) Frequency Response Function Transfer Function a d y dt a d y dt a dy dt a y b x b dx dt b d x dt b d x dt n n n n n n m m m m m m + + + + = + + + + - - - - - - 1 1 1 1 0 0 1 1 1 1

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Time Response - ( Solution to ODE) Given an input-output ODE of a system: The time response of the system y(t) due to a known input x(t) is: a d y dt a d y a dy dt a y b x b dx b d x b d x n n n n n n m m m m m m + + + + = + + + + - - - - - - 1 1 1 1 0 0 1 1 1 1 ME365 Dynamic Response Slide 4 Galen King Purdue University • Steps to solve an ODE: (1) Solve for particular solution y P (t) . (2) Solve for homogeneous solution y H (t) . (3) Combine y P (t) and y H (t) to form total solution y(t). (4) Find unknown coefficients by matching the initial conditions. ( 29 ( 29 ( 29 y t y t y t P H = + Particular Solution (Steady State Solution) Homogeneous Solution (Transient Solution) dD±
Time Response - ( Solution to ODE) 1st Order System (e.g. thermocouple ) where τ is the time constant and K is the static sensitivity . Step Response - The output of the system due to a step change in input. dy dt y K x + = ME365 Dynamic Response Slide 5 Galen King Purdue University Ex: A thermocouple can be modeled by a first order ODE with time constant =0.1 sec and sensitivity K = 0.005 V/ o C. The thermocouple is at room temperature T o = 25 o C when the temperature is suddenly changed to T = 80 o C at time t = 0 sec. What will be the response of the thermocouple?

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Time Response - ( Solution to ODE) Particular Solution ( y P (t) ) The input is a constant T = 80 o C Constant steady state solution: = ME365 Dynamic Response Slide 6 Galen King Purdue University y P ( t ) = V ss Substitute into ODE and solve for V ss .
Time Response - ( Solution to ODE) Homogeneous Solution ( y H (t) ) Let input x ( t ) = 0 and solve: τ d dt y y H H + = 0 ME365 Dynamic Response Slide 7 Galen King Purdue University Let ( 29 y t Ae H t = λ

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## This note was uploaded on 12/26/2011 for the course ME 365 taught by Professor Merkle during the Fall '07 term at Purdue University.

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Set08_10 - Dynamic System Response Input/Output Model of...

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