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Unformatted text preview: ME 530.343: Design and Analysis of Dynamic Systems Spring 2009 Lecture 22 - Solutions for Coupled ODEs Wednesday, April 22, 2009 Todays Objectives Introduction to the solution of coupled homogeneous ODEs Setting up the eigenvalue problem Coupled Systems Example: two-rotor system 2 heavy rotors (disks) connected by a long, thin shaft. The shaft is modeled as a massless, linear torsional spring/damper. The bearing is modeled as frictionless. For our initial analysis, we will assume a homogeneous system Constants and variables: J A = m A r 2 A 2 J B = m B r 2 B 2 torsional stiffness of shaft: k torsional damping of shaft: b Determine equations of motion using free body diagrams and Newtons Law: k = k ( A- B ) b = b ( A- B ) J A A =- k- b J B B = k + b 1 J A A + b ( A- B ) + k ( A- B ) = 0 J B B + b ( B- A ) + k ( B- A ) = 0 Matrix Form We will now put the two equations into a single matrix equation: Let x = A B x = A B x = A B We want to write equations of motion in the form: M x + B x + K x = 0 M...
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- Spring '08