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530_343lecture02 - ME 530.343 Design and Analysis of...

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ME 530.343: Design and Analysis of Dynamic Systems Spring 2009 Lecture 2 – Linearization Friday Januray 30, 2009 1 Today’s Objectives Why linearize? Pendulum example Taylor series expansion Reading: Palm 1.5–1.7 2 Why Linearize? Classic mass-spring-damper is linear. Classic pendulum is nonlinear. Most real mechanical systems are governed by nonlinear differential equations -Exact analytical (closed-form) 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
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3 Classic Pendulum Equation of motion: ¨ θ + g L sin( θ ) = 0 nonlinear, homogeneous, constant-coefficient, 2 nd order, ODE Possible solution methods: 1. Numerical integration (Eular, Predictor-Corrector, Runga-Kutta, ...) 2. Analytical solution w/ Jacobian elliptical functions. 3. Analytical solution w/ small angle approx. (sin( θ ) θ ) 3.1 Small angle approximation Taylor series sin( θ ) θ because sin( θ ) = θ - θ 3 3! + θ 5 5! - θ 7 7! + · · · cos( θ ) 1 because cos( θ ) = 1 - θ 2 2! + θ 4 4! - θ 6 6! + · · · Note that sin( θ ) θ is a better approximation than cos( θ ) 1 matches more derivatives 2
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In light of the small-angle approximation, revisit pendulum problem: sin( θ ) θ ¨ θ + g L θ = 0 ω n = g L Underdamped system, given initial values θ (0) , ˙ θ (0). Note that, by convention, ˙ θ (0) = d dt θ ( t ) t =0 , and NOT d dt [ θ (0)] – the latter is always zero (derivative of a constant). Solution: θ ( t ) = ˙ θ (0) ω n sin( ω n t ) + θ (0) cos( ω n t ) τ n = 2 π ω n = 2 π 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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