PHYS 422 Lecture 19 Coupled Oscillations and Normal Modes II

PHYS 422 Lecture 19 Coupled Oscillations and Normal Modes...

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Unformatted text preview: l d O ill ti d Coupled Oscillations and ormal Modes II Normal Modes II Coupled Pendulums 2 d x k ( ) 2 2 2 2 A A A B B m m x k x x dt d x m x k x x ω + + − = − − = x ( ) 2 B A B m m x k x x dt ω + = x B A – we seek solutions of the form cos cos A B x C t x C t ω ω ′ = = ( ) 2 2 cos cos cos m C t m C t k C C t ω ω ω ω ω ′ − + + − = – this can written in the matrix form ( ) 2 2 cos cos cos m C t m C t k C C t ω ω ω ω ω ′ ′ ′ − + − − = 2 2 2 2 2 2 2 c c c c C C C C ω ω ω ω ω ω ω ⎡ ⎤ + − ⎡ ⎤ ⎡ ⎤ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ′ ′ − + ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ – this is an eigenvalue equation with the values of ω 2 giving non-zero solutions the eigenvalues of the matrix Coupled Pendulums 2 2 2 C ⎤ − ⎤ ⎡ ⎤ 2 2 2 2 c c c c C C C C ω ω ω ω ω ω ω ⎡ ⎤ + ⎡ ⎤ ⎡ ⎤ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ′ ′ − + ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ e can write this as x – we can write this as 2 2 2 2 2 2 2 2 c c C ω ω ω ω ⎡ ⎤ + − − ⎡ ⎤ = ⎢ ⎥ ⎢ ⎥ + ⎦ x B A – in this form, non-zero solutions exist nly if the determinant of the matrix c c C ω ω ω ω − + − ⎣ ⎦ ⎣ ⎦ only if the determinant of the matrix vanishes ( ) 2 2 2 2 4 c c ω ω ω ω + − − = 2 2 2 2 2 2 2 2 2 1 2 or 2 c c c ω ω ω ω ω...
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This note was uploaded on 03/23/2011 for the course PHYS 422 taught by Professor Staff during the Spring '08 term at Purdue.

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PHYS 422 Lecture 19 Coupled Oscillations and Normal Modes...

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