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p09fl16 - Lecture 16 Oscillator chains(9 Oct 09 Hour exam...

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Lecture 16: Oscillator chains (9 Oct 09) Hour exam: take home, hand out 16 Oct, due 19 Oct. at 5 PM. A. Review: coupled oscillator problem 1. Coupled pendulums: the eigenvalue problem gives the normal mode frequencies. Use the eigenvectors to expand the general solution and satisfy initial conditions using amplitudes and phases of the normal modes. 2. Planar equilateral triangle normal modes: made the kinetic energy di- agonal using Jacobi coordinates to separate cm and internal degrees of freedom r , s . Can then get to “standard” form by scaling one of the internal vectors ( s ) so that the nominal mass parameter is the same for both. B. linear chain 1. FW Sec.24 2. Longitudinal oscillations of 1D oscillator chain (without specifying the ends..) L = T - V = m 2 X j ˙ η 2 j - K 2 X j ( η j +1 - η j ) 2 m ¨ η j = - K [2 η j - η j +1 - η j - 1 ] 3. The coupled equations are the same for mass points connected by a string of tension τ and small amplitude y j of transverse vibration.
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