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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 Today’s 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 Newton’s Law: τ k = k ( θ A- θ B ) τ b = b ( ˙ θ A- ˙ θ B ) J A ¨ θ A =- τ k- τ b J B ¨ θ B = τ k + τ b 1 J A ¨ θ A + b ( ˙ θ A- ˙ θ B ) + k ( θ A- θ B ) = 0 J B ¨ θ B + b ( ˙ θ B- ˙ θ A ) + k ( θ B- θ A ) = 0 Matrix Form We will now put the two equations into a single matrix equation: Let x = θ A θ B ˙ x = ˙ θ A ˙ θ B ¨ x = ¨ θ A ¨ θ B We want to write equations of motion in the form: M ¨ x + B ˙ x + K x = 0 M...
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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