530_343lecture24

530_343lecture24 - ME 530.343: Design and Analysis of...

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ME 530.343: Design and Analysis of Dynamic Systems Spring 2009 Lecture 24 – Solutions for Coupled ODEs Week of April 27, 2009 Today’s Objectives Complete two-rotor example Modes Reading: Palm 12.4 might provide some insights. Recall Two-rotor system eigenvalues and corresponding eigenvectors: λ 1 = 0 u 1 = c 1 ± 1 1 ² λ 2 = k J A + k J B u 2 = c 2 ± - J B J A 1 ² Modal frequencies: ω 1 = λ 1 = 0 ω 2 = λ 2 = q k J A + k J B Final solution: x ( t ) = c 1 ± 1 1 ² + c 2 ± 1 1 ² t + ± - J B J A 1 ² ( a 3 cos( ω 2 t ) + a 4 sin( ω 2 t )) Continue Example Step 12: Assign values to the inertias and torsional stiffness: J A = J B = 2 kgm 2 k = 4 Nm/rad ω 2 = q 4 2 + 4 2 = 2 rad/s ± θ A ( t ) θ B ( t ) ² = c 1 ± 1 1 ² + c 2 ± 1 1 ² t + ± - 1 1 ² ( a 3 cos( ω 2 t ) + a 4 sin( ω 2 t )) Step 13: Assign initial conditions and solve for constants: Given θ A (0), θ B (0), ˙ θ A (0), ˙ θ B (0) Set up equation to solve for c 1 and a 3 : ± θ A (0) θ B (0) ² = c 1 ± 1 1 ² + ± - 1 1 ² a 3 = ± 1 - 1 1 1 ²± c 1 a 3 ² Take derivative of x : 1
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± ˙ θ A ˙ θ B ² = c 2 ± 1 1 ² + ± - 1 1 ² ( - 2 a 3 sin(2 t ) + 2 a 4 cos(2 t ) Set up equation to solve for c 2 and a 4 : ± ˙ θ A (0) ˙ θ B (0) ² = c 2 ± 1 1 ² + ± - 1 1 ² 2 a 4 = ± 1 - 2 1 2
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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.

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530_343lecture24 - ME 530.343: Design and Analysis of...

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