lec05_dynre

lec05_dynre - ME365 Dynamic Response Slide 1 Dynamic System...

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Unformatted text preview: ME365 Dynamic Response Slide 1 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 Response Slide 2 Linear Systems • 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 . 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) Complicated Input Simple Inputs Complicated Output ME365 Dynamic Response Slide 3 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 3) Inverse Filtering: use system model and output to infer input • Input-Output Equation: – n th order linear ODE ( n th order system ) • Frequency Response Function • Transfer Function Linear System Input Output x(t) y(t) 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 1 1 1 1 ME365 Dynamic Response Slide 4 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: • 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. 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 1 1 1 1 y t y t y t P H Particular Solution (Steady State Solution) Homogeneous Solution (Transient Solution) ME365 Dynamic Response Slide 5 Time Response - ( Solution to ODE) Ex: A thermocouple can be modeled by a first order ODE with time constant 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? • 1st Order System (e.g. thermocouple ) where is the time constant and K is the static sensitivity . Step Response- The output of the system due to a step change in input....
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lec05_dynre - ME365 Dynamic Response Slide 1 Dynamic System...

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