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Unformatted text preview: ME 530.343: Design and Analysis of Dynamic Systems Spring 2009 Lecture 2 Linearization Friday Januray 30, 2009 1 Todays Objectives Why linearize? Pendulum example Taylor series expansion Reading: Palm 1.51.7 2 Why Linearize? Classic massspringdamper is linear. Classic pendulum is nonlinear. Most real mechanical systems are governed by nonlinear differential equationsExact analytical (closedform) solution difficult to find Often, computers are used to numerically integrate nonlinear ODEs: Matlab, Maple, Mathematica, Mathcad, Working Model, Autolev, etc. However, in this course we wish to approximate a nonlinear differential equation with a linear one. Why? serve as a starting point for analytical solution without aid of computer essential for Laplace Transforms can quickly determine stability in the neighborhood of a solution determine the effect of a parameter in system behavior physical insight essential for some control design tools 1 3 Classic Pendulum Equation of motion: + g L sin( ) = 0 nonlinear, homogeneous, constantcoefficient, 2 nd order, ODE Possible solution methods: 1. Numerical integration (Eular, PredictorCorrector, RungaKutta, ...) 2. Analytical solution w/ Jacobian elliptical functions. 3. Analytical solution w/ small angle approx. (sin( ) ) 3.1 Small angle approximation Taylor series sin( ) because sin( ) =  3 3! + 5 5! 7 7! + cos( ) 1 because cos( ) = 1 2 2! + 4 4! 6 6! + Note that sin( ) is a better approximation than cos( ) 1 matches more derivatives 2 In light of the smallangle approximation, revisit pendulum problem: sin( ) + g L = 0 n = r g L Underdamped system, given initial values (0) , (0). Note that, by convention, (0) = d dt ( t ) t =0 , and NOT d dt [ (0)] the latter is always zero (derivative of a constant). Solution: ( t ) = (0) n sin( n t ) + (0) cos( n t ) n = 2 n = 2 q L g However, if you determine it experimentally (e.g., using string, tape measure, stop watch), you will get a slightly different response....
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This note was uploaded on 03/02/2010 for the course MECH 530.343 taught by Professor Sun during the Spring '08 term at Johns Hopkins.
 Spring '08
 Sun
 Damper

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