530_343lecture02

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

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 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 3 Classic Pendulum Equation of motion: + g L sin( ) = 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( ) ) 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 small-angle 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....
View Full Document

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.

Page1 / 9

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online