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Coupled Oscillations

Coupled Oscillations. Eric Prebys , FNAL. Coupled Harmonic Oscillators. Equations of motion. Define uncoupled frequencies:. Try a solution of he form:. Multiply the top by the bottom:. . Weak Coupling. Degenerate Case:. Resonance splitting. Formalism. General coupled equation.

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Coupled Oscillations

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  1. Coupled Oscillations Eric Prebys, FNAL

  2. Coupled Harmonic Oscillators Equations of motion Define uncoupled frequencies: Try a solution of he form: Multiply the top by the bottom: Lecture 12 - Coupled Resonances

  3. Weak Coupling Degenerate Case: Resonance splitting Lecture 12 - Coupled Resonances

  4. Formalism General coupled equation General solution  Lecture 12 - Coupled Resonances i.e. ω2 are the eigenvalues of M and (a,b) are the linear combinations of x and y which undergo simple harmonic motion.

  5. Application to Accelerators Planes coupled x and y motion not independent Introduce skew-quadrupole term General Transfer Matrix Normal Quad Lecture 12 - Coupled Resonances

  6. Skew quad So the transfer matrix for a skew quad would be: For a normal quad rotated by ϕ it would be (homework) Lecture 12 - Coupled Resonances

  7. Coupling in Floquet Coordinates In Floquet Coordinate (lecture 10) Motion given by Lecture 12 - Coupled Resonances

  8. Motion in Floquet Plane Phase Changes Lecture 12 - Coupled Resonances

  9. Evolution of Perturbation Assume one perturbation per turn. Evolution of amplitude Evolution of phase Lecture 12 - Coupled Resonances

  10. Angular Gymnastics Recall Lecture 12 - Coupled Resonances

  11. Difference Resonances Focus on case when The sum terms will oscillate quickly, so we focus in the difference terms Note that Sum of emittances in transverse planes stays constant! Lecture 12 - Coupled Resonances

  12. Transformation of Variable Transform into a rotating frame In one (unperturbed) rotation Equations Become Lecture 12 - Coupled Resonances

  13. Trial solutions Try Hey, this (finally!) looks like simple coupled harmonic motion Lecture 12 - Coupled Resonances

  14. Rearrange Apply usual coupled oscillation formalism. Define normal modes Lecture 12 - Coupled Resonances

  15. Solutions Solve for eigenvalues of M Recall Lecture 12 - Coupled Resonances

  16. Coupled Tunes If there’s no coupling, then If there’s coupling, then there will always be a tune split Lecture 12 - Coupled Resonances

  17. The normal modes are given by (homework) Restrict ourselves to u±0 real Emmittance oscillates between planes. Period = turns Lecture 12 - Coupled Resonances

  18. Sum Resonances Returning to the original equations, but examining the sum terms Same sign! Following the same math as before, we get In other words, emittances can grow in both planes simultaneously. Lecture 12 - Coupled Resonances

  19. We won’t re-do the entire analysis for the sum resonances, but we find that the eigenmodes are integer “Stop band width”. The stronger the coupling, the further away you have to keep the tune from a sum resonance. Lecture 12 - Coupled Resonances

  20. General Case Although we won’t derive it in detail, it’s clear that if motion is coupled, we can analyze the system in terms of the normal coordinates, and repeat the analysis in the last chapter. In this case, the normal tunes will be linear combinations of the tunes in the two planes, and so the general condition for resonance becomes. This appears as a set of crossing lines in the nx,ny “tune space”. The width of individual lines depends on the details of the machine, and one tries to pick a “working point” to avoid the strongest resonances. Lecture 12 - Coupled Resonances

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