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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