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1 4-4, SAS IP Ettore Majorana 9 Parametric Excitation of an Inverted Pendulum Incorporated to the Seismic Attenuation System of the Gravitational Wave Interferometer. Hideharu Ishizaki, Kazuhiro Agatsuma, Ettore Majorana, and Ryutaro Takahashi Abstract Seismic Attenuation System is used for an anti-vibration mechanism of laser interferometer type gravitational wave detectors. Inverted Pendulum, which is a part of the SAS, play a role of low frequency horizontal vibration isolation for the ground vibration. Since the IP is located at the first stage of a chain of filters, it is exposed to the vertical component of the ground vibration without damp. Therefore, we considered behavior when the IP received longitudinal vibration. We confirmed that the parametric excitation may occur on the IP. The parametric excitation is a vertical-horizontal coupled vibration to occur at the frequency of around two times of the natural frequency. It is thought that the microseisms may trigger the parametric excitation on this occasion. SAS SAS IP IP IP IP SAS Inverted Pendulum TAMA3 IP buckling of the long Seismic Attenuation Ssystem column SAS 5 mhz 7 * Instituto Nazionale di Fisica Nucleare Italy

2 IP SAS IP IP parametric excitation microseisms TAMA3 8 IP IP l m m kg O x = A cos ωt A m ω s t s O x = A cos ωt θ l m m g ẍ A simple pendulum The pivot O is vibrated vertically. Stability chart of a pendulum m g m s g ẍ = g ω A cos ωt. = d /dt θ ml θ m g ω A cos ωt sin θ = ωt = τ θ = ω 4 d θ dτ a = ω ω, q = 4A l ω = g/l sin θ θ d θ dτ a q cos τ θ = Mathieu s equation 9 3 a q stability chart 9 5 a, q q = A ω a. l a q ω ω

3 SAS IP SAS IP a =, 4, 9, ω/ω = 4/a ω ω,, 3,,, 4, 5 a = ω ω 9 ω x coupling 5 parametric excitation 9 5 IP foundation IP 7 flex joint leg top table flex joint O x IP y x L J Top table G θ Mg l O Leg Flex joint X = b cos Ωt y IP vertical vibration model. Leg top table G Mg N J kg m M kg flex joint g m s IP x X m X = b cos Ωt b m Ω s t s Flex joint l m ρ kg m 3 A m I m 4 E Pa Flex joint x x ux, t m y vx, t m Flex joint T J l u T = A ρ t dx } v dadx dt t [ u M t dx }] v. dt t x=l Ũ J flex joint 6 Ũ = l A E l u v ρg x u X dadx A } y v dadx [ Mg x u X ] x=l. W J 7 l W = λ u } v dx. flex joint inextensibility condition u v = λ N Lagrangian multiplier Lagrangian L = T Ũ W Hamilton s principle t δ Ldt = t 3

4 δ variational t t t t t [ u EA t t [ l [ u ρa t d X dt EA u v [ u M t d X dt } ] l v δu g } λ u g δu λ ] x=l ] δudx [ u ] ] l δu dt v [ l ρa v t EA u } v } EI 4 v 4 λ v δvdx } v v EI 3 v 3 λ v [ ] [ M v t δv EI v δ v ] l ] dt =. [ u EA x=l x u x l : ρa t d X dt g ] l δv N x λ u =, 3 x = : u =, 4 u x = l : M t d X dt g N x λ u =. 5 y x l : ρa v t v N x EI 4 v 4 λ v =, 6 x = : v =, v =, 7 x = l : M v t N v x EI 3 v 3 λ v =, N x = EA EI v = 8 u } v flex joint x 3 8 λ 3,5 ρa, M u t d X dt, g λ λ u N x u d ρa X dt N x M d X dt g =, 3 g =. 5 3 x N x λ u EA u } v d X dt ρa c g x = c. flex joint x = 4 7 u/ = c = x v λu EA u dx} d X x ρa dt g = c. 4 c = x = l u = u l l v λu l EA u l dx} ρal d X dt g =. 3 5 N x M λ = N x M d X dt u d X dt g g. 5 [ u/] max u l /l 3, 5 N x = EA u l l ρal M u l v dx} d X dt g. 4

5 SAS IP SAS IP σ Pa ε Hooke s law σ = Eε N x A = E u l l, u l = N x l EA. N x = Mg flex joint N x N x = EA EA l N x l ρal EA M N x v dx} d X dt g λ 9 N x x N x ,9 ρa v t M g d X v dt EI 4 v 4 =, 6 v =, M v t M v g d X dt = at x =, 7 v EI 3 v 3 =, EI v = at x = l. 8 δv t t [ l ρa v t M g d X v [ M v [ t M M g d X dt dt g d X dt EI 4 v 4 EI 3 v 3 v EI 3 v 3 ] [ EI v δ v ] l } δvdx } ] δv } δv ] x= x=l dt =. vx, t t x 6 8 modified Galerkin method 7, 6, 8 ξ = x l, η = v r, t = ω t, ω = Ω ω, β = b l, µ = M ρal, n x = N x EI/l, g = ρal3 EI g. r m radius of gyration of area ω s I r = A, ω = EI l ρa [ t ξ= η t [ µ η t ξ= t µ g βω η cos ωt ξ 4 η ξ 4 η µ g βω cos ωt [ µ η g βω cos ωt ξ 3 η ξ 3 ] [ η ξ δ η ξ ] ξ= ξ= } δηdξ ξ 3 η ξ 3 } ] δη } ] δη ξ= dt =. j ζ j ξ σ j t η j η ξ, t = j ζ j ξσ j t, j =,,, δη σ j δη = i ζ i δσ i. i =,,, 3,3 [ t ξ= δσ i ζ j σ j t i j ξ= µ g βω cos ωt ζ j σ j ζ 4 j σ j }ζ i dξ [ µζ j σ j µ g βω cos ωt ζ j σ j j } ] ζ j σ j ζ i [ σ j µ ] g βω cos ωt ζ j ζ }ζ i j j ξ= ξ= [ ] ] ξ= ζ iζ j σ j dt =. j ξ= ξ= 5

6 ζ j σ j = d σ j dt, ζ i = dζ i dξ, ζ j = dζ j dξ, = d ζ j dξ, ζ j = d3 ζ j dξ 3, ζ4 j = d4 ζ j dξ 4, i, j =,,, δσ i t t [ ] δσ i dt = t i [ ] = M ij σ j K ij j j } µ g βω cos ωt N ij σ j =. 4 K ij = M ij = ζ i ζ j dξ µζ i ζ j, ζ i ζ j dξ, N ij = ζ iζ jdξ. N x cantilever j ζ j = c j cos j πξ, j =,,, 5 c j j ,5 i, j = c = 4 m M = k K = n N = ζ dξ µζ = 3π 8 π ζ dξ = π4 3, ζ dξ = π 8 µ, σ = σ 4 m σ k µ g βω } cos ωt n σ =, σ k µg σ µβω n cos ωt σ =. 6 m k/n m 6 β = µ = σ k/mσ = k/m Ω s π Ω = 4 /6 3 8/π ω EI l ρa Ω 3.56 EI l ρa 4.% 7, 9,,, 5, 6 Ω ω k m 6 β =, µ = µg M ρal 3 k/n = ρal EI g π = Mg /4 π. EI 4 l P cr = π 4 EI l P cr N buckling 7,, 6 τ = ωt, d σ dt = ω d σ 4 dτ 6 d σ dτ a q cos τ σ =. 7 a = 4 k ω µg = m k/n q = k µβ = m k/n Ω Mg, Ω P cr Ω Mb ω ρal π /4 = bmω P cr. 7, 6

7 SAS IP SAS IP Stability chart of a IP Mg/P cr b M Ω q = Ω a = Ω Ω IP flex joint Ω E = 86 GPa ρ = 8. 3 kg m 3 l = 47 3 m d = 8. 3 m A = πd /4 m I = πd 4 /64 m 4 Ω s.5 khz a Mg/P cr a = Ω = Ω Mg P cr 7 IP Mg Ω s Ω = Ω Mg 8 P cr P = Mg Ω Ω P P =, Ω = Ω P cr 6, 7, Ω P cr TAMA3 Mg Ω 3 mhz 8 Large-scale Cryogenic Gravitational wave Telescope 9, SAS IP R&D LCGT IP 8 IP SAS IP IP IP 8 IP 7 4 P cr Ω Ω 5 mhz Mg/P cr Ω, P cr Experimental set up in Kamioka 8 7

8 / IP Ω 7 mhz IP Geophone Accelerometer 6Hz 48 4 Hz. m/ Hz 3 9 IP coherence function Γ Γ = W XY W Y Y W XX Displacement spectra of IP, and the ground s vibration H, V 8 X IP Y cross spectrum W XY W XX, W Y Y power spectrum Ω IP. ±. Hz IP gain IP.3 Hz microseisms 3 IP Coherence.5.. Frequency [Hz] 6 ω n = Ω ω q = µβω n ω nm = µg k/n = Ω ω Mg/Pcr Mg/P cr Mg, P cr Ω g b Coherence function of H, V 8 The domain surrounded with an oval has good sensor sensitivity. d σ dt ω n q cos ωt σ = 9 8

9 SAS IP SAS IP 9 4, 5 ω n n = ω = Ω /ω 8 IP σ = C cos ωt S sin ωt 9 cos ωt sin ωt cos ωt, sin ωt ωt ω n ωt q ω = ω n ± q 5 7 Ω Ω 9 7. Hz 9 5 d σ dt γω n dσ dt ω n q cos ωt σ εω n σ 3 =. γ damping ratio Flex joint F N F = kx ± k x 3 x m k N m k N m 3 m kg F m = k m x ± k k m k x3 = ωnx ± ωnεx 3 ω n = k/m, ε = k /k F O kx k x 3 kx kx k x 3 x Model of restoring forece. IP 7 hardening spring softening spring 7 7 IP 9 ω n ω σ = C / cos ωt S / sin ωt [ ωn q 4 } ω C / γω n ωs / C/ 3 C /S/ ] cos ωt [ ωn 3 4 εω n q 4 ω } S / γω n ωc / 3 4 εω n S/ 3 C / / ] S sin ωt 4 εω nc/ εω nc / S/ ω nq C / cos 3 ωt 3 4 εω nc/ S / 4 εω ns/ 3 ω nq S / sin 3 ωt =. ω/ = = harmonic balance method 5 q } ω ω C / γ S / 4 ω n ω n 3 C 4 ε / 3 C /S/ =, q } ω ω S / γ C / 4 ω n ω n 3 S 4 ε / 3 C / S / =. C /, S / σ T σ rms = σ T dt = C / S / T T = 4π/ω 9

10 σ rms [ σ = ω εq ω 6 ωn 4 εq ωn 4 ω ε q } ] / / q 4 q 4γ ω ω n ω n } / ε q 4 q ω, ω n [ σ = ω εq ω 6 ωn 4 εq ωn 4 ω ε q ω n } ] / / q 4 q 4γ ω ε q 4 q ω ω n } / ω n 4 σ, σ ω Hz σ ω/ω n ω σ ω n = 7 mhz γ =.88 ε =. q q =.77. ω/ω n q β β = β ω = ω n q µ n ωn m ωn ω ω n } ωn β β µ n ωn } m ωn β ω ωn β β β β ω n σ, σ σ σ ω n = 4 mhz.3 Hz. Hz q cos ωt IP Frequency Response Gain Diagram of IP by Parametric Excitation. ω ω ω = ω n a b c ω ω Γ 9 σ = C cos ωt S sin ωt 9 cos ωt ω ω ω ω n IP a b jump phenomenon c hysteresis phenomenon,, 5, 6 Ω Ω 8

11 SAS IP SAS IP q a c q = a a ω c σ = c b ω a c 5 β β x = σ, t = t d x dt ω n q cos ωt x =. 9 q = x = c cos ψ, dx dt = cω n sin ψ, ψ = ω n t φ. 3 c, φ 9 q ẋ = y, ẏ = ω q cos ωt x ẋ = dx/dt, ẏ = dy/dt c, φ t 3 ċ cos ψ c φ sin ψ =, ċ sin ψ c φ cos ψ = ω n q c cos ωt cos ψ. ċ = ω nq c cos ωt sin ψ, φ = ω nq cos ωt cos ψ. ω n ω/ ω x = c cos ψ = c cos t ϑ 4 ψ ω n t φ = ω/t ϑ φ = ϑ ω n ω t. ċ, φ ψ T = π averaging method ċ = T T ċdψ = π π = cq ωn sin ϑ, ω φ = ϑ ω n ω = T T = π φ ω π dt = ω q n ω ϑ = ωn ω q ω n ω ċ ω dt φdψ cos ϑ, cos ϑ. ċ, ϑ du dt = U = c cos ϑ, q ωn dv dt = V = c sin ϑ ω ω n ω } V, q ωn ω ω n ω } U. du/dt, dv/dt ν q ωn ω ω n ω ν q ωn ω ω n ω = ν q ωn ω n ω = ω q ωn ν = ± ω n ω 5 ω U = C e νt C e νt, q ω n ω V = C ω n ω e νt 6 ν C q ω n ω ω n ω ν e νt.

12 C, C 9 4 5,6 4 c = U V, ϑ = tan V U ν c ν c IP 9 d x dt ω nx = q ω nx cos ωt dx/dt d dt dx dt ωn x } = q ωnx dx cos ωt. 8 dt x t x = C e νt cosω n t φ = q ω 3 nc e νt 4 φ } = q ω 3 nc e νt 4 [ cos ω ω n cos ω ω n ω n ω ωt ω n ω ωt } ] ωt T φ ωt [ cos ω ωn t φ ω ω n cos ω ω n t φ ω ω n ] tt 4 ω ω n ω < ω n ω ω n E > t = ω = ω n E = e νt e νt T E > IP t C, φ x, dx/dt T t t t T e νt e νt T const. x C e νt cosω n t φ, dx dt C ω n e νt ω n t φ. 8 T = π E t T E = q ωncos ω n tx dx t dt dt t T = q ωnc e νt ω n cos ωt t } cosω n t φ sinω n t φ dt = q ωnc e νt ω ωtπ n cosωt ω ωt ωn cos ω ωt φ ωn } sin ω ωt φ ωdt 8 Seismic Attenuation System SAS Inverted Pendulum IP buckling 5 mhz SAS IP IP IP LCGT IP

13 SAS IP SAS IP IP parametric excitation microseisms IP IP IP R. Takahashi, K. Arai, D. Tatsumi, M. Fukushima, T. Yamazaki, M.-K. Fujimoto, K. Agatsuma, Y. Arase, N. Nakagawa, A. Takamori, K. Tsubono, R. DeSalvo, A. Bertolini, S. Márka, and V. Sannibale: Operational status of TAMA3 with the seismic attenuation system SAS, Classical and Quantum Gravity., 5, A. Takamori: Low Frequency Seismic Isolation for Gravitational Wave Detectors, PhD thesis, Department of Physics School of Science University of Tokyo. 3 S. Márka, A. Takamori, M. Ando, A. Bertolini, G. Cella, R. DeSalvo, M. Fukushima, Y. Iida, F. Jacquier, S. Kawamura, Y. Nishi, K. Numata, V. Sannibale, K. Somiya, R. Takahashi, H. Tariq, K. Tsubono, J. Ugas, N. Viboud, C. Wang, H. Yamamoto, and T. Yoda: Anatomy of the TAMA SAS seismic attenuation system, Classical and Quantum Gravity, 9, G. Losurdo, D. Passuello, P. Ruggi: The control of the Virgo Susperattenuator revised I. Inertial Damping: present and future, VIR-NOT-FIR-39-38, 6. 5 A. Takamori, P. Raffai, S. Márka, R. De- Salvo, V. Sannibale, H. Tariq, A. Bertolini, G. Cella, N. Viboud, K. Numata, R. Takahashi, and M. Fukushima: Inverted pendulum as low-frequency pre-isolation for advanced gravitational wave detectors, Nuclear Instruments and Methods in Physics Research, A 58, TAMA3 SAS 7. 7 TAMA3 SAS IP R. Takahashi, A. Takamori, E. Majorana: Behavior of an inverted pendulum in the Kamioka mine, LCGT ff meeging, June. 9 :,,, pp.3-, pp pp.75-76, pp.- 4. S. P. Timoshenko and J. M. Gere: Theory of Elastic Stability. Second edition, McGraw- Hill Book Co., pp p.79, pp :,, pp

14 4 N. N. Bogolyubov, YU. A. Mitropolskiï, p.7, pp JSME pp.63-67, pp pp H. Yabuno, Y. Ide, and N. Aoshima: Nonlinear Analysis of a Parametrically Excited Cantilever Beam, Effect of the Tip Mass on Stationary Response, JSME International Journal Series C, Vol. 4, pp K. Kuroda: Status of LCGT, Classical and Quantum Gravity, 7, 844. LCGT Collaboration LCGT 3 9. K. Yamamoto, S. Kamagasako, T. Uchiyama, S. Miyoki, M. Ohashi, K. Kuroda, T. Tomaru, R. Takahashi, D. Tatsumi, S. Telada, A. Araya, and A. Takamori : Measurement of seismic motion at Large-scale Cryogenic Gravitational wave Telescope project site, JGW Document JGW-T8., pp , 5 pp WOLFRAMRESEARCH: M athematica

(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0

(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0 1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45

天文台報 表紙

天文台報表紙