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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

Coupled Oscillations

Eric Prebys, FNAL

coupled harmonic oscillators
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


Weak Coupling

Degenerate Case:

Resonance splitting

Lecture 12 - Coupled Resonances


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.

application to accelerators
Application to Accelerators

Planes coupled

x and y motion not


Introduce skew-quadrupole term

General Transfer Matrix

Normal Quad

Lecture 12 - Coupled Resonances


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

coupling in floquet coordinates
Coupling in Floquet Coordinates

In Floquet Coordinate (lecture 10)

Motion given by

Lecture 12 - Coupled Resonances

motion in floquet plane
Motion in Floquet Plane

Phase Changes

Lecture 12 - Coupled Resonances

evolution of perturbation
Evolution of Perturbation

Assume one perturbation per turn.

Evolution of amplitude

Evolution of phase

Lecture 12 - Coupled Resonances

angular gymnastics
Angular Gymnastics


Lecture 12 - Coupled Resonances

difference resonances
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

transformation of variable
Transformation of Variable

Transform into a rotating frame

In one (unperturbed) rotation

Equations Become

Lecture 12 - Coupled Resonances

trial solutions
Trial solutions


Hey, this (finally!) looks like simple coupled harmonic motion

Lecture 12 - Coupled Resonances



Apply usual coupled oscillation formalism. Define normal modes

Lecture 12 - Coupled Resonances



Solve for eigenvalues of M


Lecture 12 - Coupled Resonances

coupled tunes
Coupled Tunes

If there’s no coupling, then

If there’s coupling, then there will always be a tune split

Lecture 12 - Coupled Resonances


The normal modes are given by (homework)

Restrict ourselves to u±0 real

Emmittance oscillates between planes. Period = turns

Lecture 12 - Coupled Resonances

sum resonances
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


We won’t re-do the entire analysis for the sum resonances, but we find that the eigenmodes are


“Stop band width”. The stronger the coupling, the further away you have to keep the tune from a sum resonance.

Lecture 12 - Coupled Resonances

general case
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