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1 COD u d2 u + ku =0 (1) dt2 u = a exp(pt) (2) p = ± k (3) k>0k = ω 2 exp(±iωt) (4) k<0k = γ 2 exp(±γt) (5)

2 ω: γ: 2 1 u(t) =A exp(iωt)+b exp( iωt) (6) u(t) =A cos(ωt)+b sin(ωt) (7) u(t) =A cos(ωt + φ) (8) 1.2 B Q S x B Q S t 4 4 d 2 u + K(s)u = 0 (9) ds2 1 B y K(s) = ρ(s) x (s) (10) s t s u x y s

3 : (9) B y (s) (11) x 4 x y4 4 y N S x S N 1: s K(s) (9) (9)

4 1 u s K(s) K(s) k>0 k <0 k β(s) φ(s) u = A β(s) cos(φ(s)+φ 0 ) = A(β(s)) 1 2 cos(φ(s)+φ0 ), (12) ds dφ φ(s) =, β(s) ds = 1 β = β 1 (13) du ds ( 1 = A β 2) 1 2 β cos(φ + φ 0 ) Aβ 1 2 sin(φ + φ0 )β 1 ( 1 = A β 2) 1 2 β cos(φ + φ 0 ) Aβ 1 2 sin(φ + φ0 ) [( ] = Aβ β 2) cos(φ + φ 0 ) sin(φ + φ 0 ) (14) s β d 2 ( u = A 1 ) [( ] β 3 ds 2 2 β 1 β 2 2) cos(φ + φ 0 ) sin(φ + φ 0 ) + {( Aβ 1 1 [β 2 2) cos(φ + φ 0 ) β sin(φ + φ 0 )β 1] } cos(φ + φ 0 )β 1 = Aβ 3 2 [( 1 ) 4 (β ) 2 cos(φ + φ 0 )+ ( 1 2) ββ cos(φ + φ 0 ) ( 1 2) β sin(φ + φ 0 )+ ( 1 2) β sin(φ + φ 0 ) cos(φ + φ 0 ) ]

5 [( = Aβ 1 2 cos(φ + φ0 )β 2 1 ) (β ) [( = uβ 2 1 ) ( ] 1 (β ) 2 + ββ 4 2) 1 ( ] 1 ββ 2) 1 (15) d 2 u + K(s)u = 0 (16) ds2 d 2 [( u + K(s)u = uβ 2 1 ) ( ] 1 (β ) 2 + ββ ds2 4 2) 1 + K(s)u [( = uβ 2 1 ) ( ] 1 (β ) 2 + ββ 4 2) 1+K(s)β 2 }{{} =0 β0 (17) ( 1 ) ( 1 (β ) 2 + ββ 4 2) 1+K(s)β 2 =0 (18) β(s) (18) β φ 12πν 1 β 1 L 2πν = L 0 ds β = φ(l) φ(0) (19) η u θ φ β ν = 1 ν ds β (20) d dθ ds dθ = = ( ds dθ 1 ( dθ ds ) d ds (21) ) = ( 1 ) 1 = νβ, (22) νβ

6 d = (νβ) d (23) dθ ds η = u = A cos(φ + φ 0 ) (24) β dη ds = A sin(φ + φ 0 )φ = A sin(φ + φ 0 )β 1 (25) dη dθ = (νβ)[ 1A sin(φ + φ 0 )β 1 ]= νasin(φ + φ 0 ) (26) d 2 η = (νβ) d dη dθ 2 ds dθ =(νβ) d ds [ νasin(φ + φ 0)] (27) = (νβ)[ νacos(φ + φ 0 )φ ]= (ν) 2 A cos(φ + φ 0 )= ν 2 η (28) d 2 η dθ 2 + ν2 η = 0 (29) : η = u/ β(s) θ(s) =φ(s)/ν η(θ) = A exp(±iνθ) =A exp(±iφ) (30) η(θ) = A cos(νθ)+bsin(νθ) (31) = A cos(φ)+bsin(φ) (32) η(θ) = A cos(νθ + φ 0 )=Acos(φ + φ 0 ) (33) 1 φ 1 2πν 1 ν ν s η(s) u(s) s β(s) η A u β β β β K(s) K(s + L) =K(s) β β(s + L) =β(s) (18) β β

7 u(s) =x(x) y(x) u(s) s=s0,u (s) s=s0 u(s + L) =u(s) Closed Orbit u(s) = C 1 ( ) ( /2436)[A] = [A] 0.1A 10 1 / = ma 10 1 / = y =0.75 sin(2π(3+1/3)x) /3 3

8 2: 3 1 1/31-1/3=2/3 4 3, n f = f rev (n ± ν frac ) (34) f : f rev :

9 ν frac : 9 d 2 u + K(s)u =0 ds2 K(s) s u 9 u 2 0.5

10 : y =0.75 sin(2π(3+1/3)x) : y =0.75 sin(2π(1/3)x), y=0.75 sin(2π(2/3)x π)

11 1.4 SPring = Spectrum Analyzer TGout RF TGout f RF + ν frac f rev (= f rev ( ν frac )) RF TGout ν frac f rev ( f rev ) ( f rev )f RF +ν frac f rev 30 m RF RF 0 / /2 1/2 1 2

12 frf ~ frf x frev Spectrum Analyzer RFin TGout 0~0.5 x frev MIX from RF Est 5: 6:

13 7: 0 /180

14 8:

15 2 s x(s) 1 8 GeV 16 GeV 0 GeV Spring-8 40 cm 0.1% (= 0.001) 40 cm = 0.4 mm d 2 x ds (K(s) 1 2 ρ(s) )x = 1 δp 2 (35) ρ(s) p 0 ρ SPring-8 ρ K(s) ρ 36 d 2 x ds K(s)x = 1 δp 2 (36) ρ(s) p 0 x x = x β + x p d 2 (x β + x p ) + K(s)(x ds 2 β + x p )= 1 δp (37) ρ(s) p 0 2 d 2 x β ds + K(s)x 2 β =0 (38) d 2 x p ds + K(s)x 2 p = 1 δp ρ(s) p 0 (39) (40)

16 x p x p (s) =η(s) δp p 0 dx p (s) ds d 2 x p (s) ds 2 d 2 x p ds = η(s) δp p 0 (41) δp = η(s) (42) p 0 K(s)x p = 1 δp ρ δp η(s) + K(s)η(s) δp = 1 p 0 p 0 ρ η(s) + K(s)η(s) = 1 ρ p 0 (43) δp p 0 (44) (45) η(s + L) =η(s) 2.1 RF δl L = αδp (46) p 0 1 δl = δt = δfrev L T f rev RF RF RF h f RF = hf rev

17 RF COD 3 δν ν = ξ δp p 0 (47) d 2 u + K(s)u = 0 (48) ds2 1 B y K(s) = ρ(s) x (s) (49) K(s) ρ K(s) K(s) 4 4 SPring

18 3.1 RF 4 COD COD Closed Orbit Distortion d 2 u + K(s)u = 0 (50) ds2 u(s), u(s) u(s+l) =u(s) u(s) =0 u(s) =0 COD 4.1 COD s u(s) BPM: Beam Position Monitor; COD SPring BPM COD 30 m BPM BPM BPM BPM 4

19 f RF 9: BPM COD COD BPM

20 10: 12 11AM 4 GUI

21 11: COD

22 4.3 COD 0 d 2 u + K(s)u = F (s) (51) ds2 F (s) β(s) η, θ ( 1 ) ( 1 (β ) 2 + ββ 4 2) 1+K(s)β 2 = 0 (52) η = β 1 1 ds 2 u, θ = ν β, dθ ds = 1 νβ, ds = νβ (53) dθ d dθ = ds dθ d ds = 1 dθ ds d ds = νβ d ds (54) θ d 2 = νβ d dθ 2 ds { = ν 2 β [ (νβ) d ds ] β d ds + β d2 ds 2 = ν 2 β d ds } [ β d ds ] (55) d 2 { η = ν 2 β β d } dθ 2 ds + β d2 η ds 2 { = ν 2 β β d } ds + β d2 (β 1 ds 2 2 u) (56) 56 s d 1 1 ds (β 2 u) = d 2 = β d ds 2 (β 2 u) = ds 2 β 3 2 β u + β 1 2 u ( u 1 ) 2 β 1 β u { ( β 1 2 u 1 )} 2 β 1 β u (57)

23 = 1 ( 3 2 β 2 β u 1 ) 2 β 1 β u { + β 1 2 u 1 [ β 2 (β ) 2 u + β 1 β u + β 1 β u ]} 2 = β 1 2 {u + u [ 1 2 β 1 β 1 ] 2 β 1 β [( 1 + u 4 2) + 1 β 2 (β ) 2 1 ]} 2 β 1 β { [ = β u β 1 β u + u 4 β 2 (β ) 2 1 ]} 2 β 1 β (58) θ d 2 [ ( η = ν 2 β {β β 1 dθ 2 2 u 1 )] 2 β 1 β u [ ( ( + β β u β 1 β u + u 4 β 2 β 2 1 ))]} 2 β 1 β { = ν 2 β 3 2 u + u [ β 1 β β 1 β ] + u [ 1 2 β 2 (β ) β 2 (β ) 2 1 ]} 2 β 1 β { [ = ν 2 β u + u 4 β 2 (β ) 2 1 ]} 2 β 1 β [ = ν 2 β u + uβ 2 4 (β ) 2 1 ] 2 ββ }{{} =K(s)β 2 1 = ν 2 β 3 2 u + K(s)u β 2 u }{{} =F (s) = ν 2 β 3 2 F (s) ν 2 β 1 2 u = ν 2 β 3 2 F (s) ν 2 η (59) d 2 η dθ + 2 ν2 η = ν 2 β 3 2 F (s) (60) d 2 η dθ + 2 ν2 η = ν 2 f(θ) (61)

24 η(θ) = u(s) = φ(s) = ν θ+2π f(σ) cos(σ π θ)dσ (62) 2 sin(πν) θ β 1 2 (s) s+l { } 1 β 2 (ξ) cos(πν φ(ξ)+φ(s) F (ξ)dξ (63) 2 sin(πν) s ds (64) β s = s 1 u(s) = β 1 2 (s)β 1 2 (s 1 ) cos(πν + φ(s) φ(s 1 )) 2 sin(πν) (65) =F (s 1 )l (66) s1-l/2 l s1+l/2 F(s1) φ(s) φ(s 1 ) sin(πν) 0COD 63

No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2

No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2 No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j

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