coupled oscillation

coupled oscillation - 1 Equation of motion 2 1 1 1 x x kx x...

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Unformatted text preview: 1. Equation of motion : ) ( 2 1 1 1 x x kx x m--- = η , ) ( 1 2 2 2 x x kx x m--- = η 2. Looking for normal mode in which all masses oscillate at the same frequency ϖ- t i e a t x ϖ 1 1 ) ( = , t i e a t x ϖ 2 2 ) ( = 3. After replacement, eqns. of motion reduce to = ---- 2 1 2 2 a a k m k m η ϖ η η η ϖ 4. For a’s to be nontrivial (non-zero), determinant of the 2*2 matrix should be zero. m k = ∴ 2 ϖ (low freq. with 2 1 a a = ; “move-together(MT)” mode), m k η ϖ 2 2 + = (high freq. with 2 1 a a- = ; “mirror symmetry(MS)” mode) 5. Normal coordinates : 2 ) ( ) ( ) ( 2 1 1 t x t x t + = ξ , 2 ) ( ) ( ) ( 2 1 2 t x t x t- = ξ 6. For MT mode, 1 ≠ ξ & 2 = ξ , while 1 = ξ & 2 ≠ ξ for MS. These are brief summary of what we’ve covered during the section today. But let’s try to keep moving a little bit further into the true meaning of normal mode....
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This note was uploaded on 11/02/2008 for the course PHYS 2218 taught by Professor Wittich,p during the Spring '08 term at Cornell.

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coupled oscillation - 1 Equation of motion 2 1 1 1 x x kx x...

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