Suspension Dynamics Modeling for (TAMA and) LIGO and KAGRA. Mark Barton July GW Seminar 7/24/15

Transcription

1 Suspension Dynamics Modeling for (TAMA and) LIGO and KAGRA Mark Barton July GW Seminar 7/24/15

2 Prehistory Started modeling the X- pendulum I developed for TAMA300 at ICRR Originally MathemaNca v2.2 (now v10.1)!

3 Glasgow Matlab Models Calum Torrie modeled the GEO triple as part of his PhD (LIGO- P v1). Lots of hand- coded Matlab hard to check and hard to modify. Also lots of simplificanons to the physics. Ken Strain extended Calum's code to model first aligo QUAD prototype. These were being used to design aligo suspensions. Needed something to compare these models against and allow for future improvements.

4 The MathemaNca Toolkit, PendUNl.nb The X-pendulum model was developed into a general toolkit. Implemented as a Mathematica package, PendUtil.nb, for specifying different configurations (e.g., quad, triple etc) in a (relatively) user-friendly way Supported features: 6-DOF rigid bodies for masses (no internal modes) Springs described by an elasticity tensor and a vector of pre-load forces Massless wires (i.e., no violin modes) but detailed elasticity model from beam equation Arbitrary frequency-dependent damping on all sources of elasticity Symbolic up to the point of minimizing the potential to find the equilibrium position Calculates elasticity and mass matrices semi-numerically (symbolic partial derivatives of functions with mostly numeric coefficients) Eigenfrequencies and eigenmodes calculated numerically Arbitrary frequency dependent damping on each different elastic element Transfer functions Thermal noise plots Export of state-space matrices to Matlab and E2E

5 Based on standard method of mass/snffness matrices Express the potennal energy of the system in terms of the coordinates: E P = E P ( x 1,...x n ) = E P ( x) Express the kinenc energy of the system in terms of the coordinates and velocines: E K = E K ( x 1,...x n,!x 1,...!x n ) Minimize the potennal energy to find the equilibrium values of the coordinates. ( ( ) ) T x eq = x 1( eq),...x n eq Compute the matrix of second derivanves of potennal, a.k.a., the potennal energy matrix or the snffness matrix. E P = E P ( x eq ) x x eq ( ) T K( x x eq ) K : K ij = E P x i x j x=x eq Compute the matrix of second derivanves of kinenc energy, a.k.a., the kinenc energy matrix or the mass matrix: E K = 1 2!xT M!x M : M ij = E K!x i!x j!x=0 x=x eq Write matrix equanons of monon and solve for parncular numerical frequencies using!x = iωx = 2πifx!!x = (2π f ) 2 x M!!x = Kx + Cx external + f external x = ( K ( 2π f ) 2 M) 1 ( Cx external + f external ) Or, assume a sinusoidal solunon with no external forces Ke i = ω i 2 Me i Do a simultaneous diagonalizanon to obtain the eigenfrequencies f i = ω i 2π and eigenmodes e i : x i ( t) = x eq + e i e ω it See Goldstein, "Classical Mechanics", 3 rd Ed.

6 Rigid- body mass coordinates Six coordinates for center of mass (COM): x, y, z, yaw (Y), pitch (P), roll (R) Three extra "body" coordinates for non- COM points such as wire agachments = x space y space z space x COM y COM z COM = + x COM y COM z COM + cosy siny 0 siny cosy cos P 0 sin P sin P 0 cos P The model definer needs to provide a list of the COM coordinates of all the masses cos R sin R 0 sin R cos R cos P cosy cosy sin Psin R cos RsinY cos RcosY sin P + sin RsinY cos PsinY cos RcosY + sin Psin RsinY cos Rsin PsinY cosy sin R sin P cos Psin R cos P cos R x body y body z body x body y body z body

7 KineNc energy There is typically a linear term in terms of mass m plus angular term in terms of moment of inerna tensor I: ( ) E K = 1 2 m(!x 2 +!y 2 +!z 2 ) ω x ω y ω z I xx I xy I zx I xy I yy I yz I xz I yz I zz ω x ω y ω z I is in body coordinates so the angular velocity vector needs to be transformed: ω x ω y ω z = sin P 0 1 cos Psin R cos R 0 cos P cos R sin R 0 Y!!P!R The model definer needs to supply a MathemaNca expression for the kinenc energy for all masses, with terms similar to the above.

8 GravitaNonal energy E G = mgz The model definer has to supply a MathemaNca expression giving the total gravitanonal potennal for all masses.

9 Wire energy Wire energy is broken into four terms: Longitudinal stretch based on straight- line distance Bending near the endpoints Extra longitudinal stretch due to bending TJ Torsion E P(wire_torsional ) = 1 2 A + GJ 2 e Δθ T Model definer needs to supply a list of parameters for each wire, including Young's modulus, length, moment of area, agachment points, etc, etc. E P( wire_bending) = YM 1 2 E P(wire_longitudinal ) = 1 2 k w E P wire_ extra _ stretch ( ) = T L d 2 y dl 2 ( l( t) l 0 ) 2 dy dl L 2 dl 2 dl + YM L d 2 z dl 2 2 dl

10 Flexure correcnon Wire bending terms use some complicated 3D geometry and are slow to calculate. It turns out there's a shortcut: the wire behaves as if agached with a hinge at a distance a from actual break- off point, where a = EI T E=Young's modulus, I = second moment of area, T = tension This trick was not used in the MathemaNca toolkit but has been used in the exported Matlab code.

11 Springs Simple zero- length spring model connecnng a point on one mass to a point on another. 6x6 matrix of elasnc constants plus a 6x1 vector of pre- load forces: E P(spring) = 1 ( 2 Δx Δy Δz ΔY ΔP ΔR ) K xx K xy K xz K xy K xp K xr K xy K yy K yz K yy K yp K yr K xz K yz K zz K zy K zp K zr K xy K yy K zy K YY K YP K YR K xp K yp K zp K YP K PP K PR K xr K yr K zr K YR K PR K RR Model definer has to provide a list of springs with masses, agachment points and elasnc constant and pre- load informanon. Δx Δy Δz ΔY ΔP ΔR + f x f y f z f Y f P f R ( ) Δx Δy Δz ΔY ΔP ΔR

12 As well as the coordinates used in the normal mode analysis ("vars"), two other sorts are important: "params" the support and other objects which are stanonary during normal modes but move for transfer funcnons "floats" things like wire- spring juncnons with no associated mass Extra coordinates

13 SNffness matrix for extra coordinates It is convenient to calculate a master snffness matrix with parnal derivanves between all types of coordinates: "vars" "floats" K C XQ C XS K master = C QX Q C QS C SX C SQ S K, Q and S give the coupling among vars, floats and params within their respecnve groups. C XQ, C QS and C SX give the coupling between groups. Because the "floats" are dependent on the others, they need to be eliminated: "params" "vars" "floats" "params" K effective = K C XQ Q 1 C QX C SX(effective) = C SX C SQ Q 1 C QX S effective = S C SQ Q 1 C QS

14 Damping Lossiness in elasnc components can be represented by a complex elasnc constant: k k 0 ε ω ( ( ) + i ε ( ω )) The real and imaginary parts should sansfy the Krämers Krönig relanonship: ( ) 1 = 2 π PV ε ( x) ε ω dx ε ω x ω If losses are small, the real part can be assumed to be constant: k k 0 ( 1+ iφ ( f )) The model definer can specify a different damping funcnon for each elasnc component. The toolkit keeps track of which damping funcnon applies to which coefficients in the master snffness matrix. ( ) = 2 π PV ε ( x) 1 x ω dx

15 Wire/Fibre damping Wire φ usually has a frequency- independent "structural" term and a frequency dependent "thermoelasnc " term f f HHzL ThermoelasNc damping has a characterisnc peak at the Nmescale of heat flow across the wire. HLTS HSTS

16 Thermal Noise Suspension thermal noise is a potennal liminng factor in GW detectors. Noise is given in terms of damping by FluctuaNon DissipaNon Theorem: x 2 ( ( )) ( ω ) = 4k BT Re Y ω Y is the admigance (v/f) at a test point where the thermal noise is to be calculated. ω 2

17 DissipaNon DiluNon (i) In a simple mass- spring system, the quality factor of the oscillanon depends purely on the spring material: 1 Q = φ material However systems such as pendulums and wires are used in GW detectors because with certain geometries, the damping (dissipanon of energy) is "diluted": Q = D φ material 1 φ material

18 DissipaNon DiluNon (ii) The dissipanon dilunon factor D depends on the fracnon of the energy involved in first- order stress changes of the material. Pulling a pendulum or violin string sideways creates only a second- order or smaller stretch - > dilunon. Quick test: calculate the contribunon to the potennal matrix for each potennal term twice, once with and without all tensions zeroed.

19 CalculaNon Procedure (i) The model calculanon notebook is run. It loads the model defininon notebook, which loads the toolkit, the model defininon, and a default set of numerical values. The calculanon notebook can then selecnvely override some of the numerical values. The wire and spring lists are processed to create a list of potennal terms, each with a damping funcnon. The total potennal is computed, with numerical values for all quannnes except the "var" coordinates. (Usually the wire bending potennal terms are omiged for speed.) The total potennal is minimized to find the equilibrium posinon.

20 CalculaNon Procedure (ii) Numerical values for everything but the coordinates are subsntuted into the potennal terms. The potennal terms are differennated to find the snffness matrix elements. Each term is processed separately for two reasons: for speed (each term depends on only a few coordinates, so most derivanves are zero and don't need to be computed) to keep contribunons with different damping funcnons separate. The process is repeated with the tension switched off, to allow dissipanon dilunon to be calculated.

21 CalculaNon Procedure (iii) Versions of the snffness matrix with and without damping are calculated. The snffness matrix without damping (a totally numerical matrix) is used to calculate the eigenfrequencies and eigenmodes. The snffness matrix with damping (a mostly numerical matrix that is a MathemaNca funcnon of frequency, f) is used for all other purposes. MathemaNca funcnons are provided to allow transfer funcnons from/to selected inputs/outputs, thermal noise plots at selected test points, eigenmode shape plots etc. If the small but Nme- consuming wire bending/torsion potennal terms were omiged at the beginning, they are computed and added in.

22 Models for LIGO Models defined for all the LIGO suspensions (no KAGRA yet): QUAD: single chain of AdvLIGO quad pendulum, with 4 masses, 6 blade springs and 14 wires. BSFM, HSTS, HLTS: 3 masses, 6 blade springs and 10 wires. OFMC, TMTS: 2 masses, 2 or 4 blades, 6 wires HAUX, HTTS, OFIS: 1 mass, assorted blade/wire combinations Many toy models

23 2 blade springs 2 wires top mass 2 blade springs 4 wires upper intermediate mass 2 blade springs 4 wires intermediate mass 4 fibres (or wires) optic (or reaction mass) Example model: LIGO quad pendulum

24 Sample Output (i) Table of Mode Freqs/Shapes Model " TMproducNonTM" The aligo quad suspension main chain, with monolithic final stage (i.e., fused silica fibres suppornng the test mass)

25 Sample Output (ii) Individual Mode Shape

26 Sample Output (iii) Transfer Function

27 Sample Output (iv) Thermal Noise

28 State Space Terminology For simulanons, it is convenient to write the equanons of monon in state- space format. In tradinonal notanon:!x y = A B C D x: vector of state coordinates u: vector of inputs y: vector of outputs A: state matrix (maps state to rate of change) B: input matrix (maps u to rate of change) C: output matrix (maps state to output) D: feedthrough matrix (maps inputs directly to output) x u

29 SS State/Inputs/Outputs for Suspension Models In mechanical simulanons, the EOMs are second- order, so the full state needs to be the coordinates ("vars") plus the velocines: x x!x A good set of inputs is the support coordinates ("params") plus input forces on the main coordinates: u = s!s f x A good set of outputs is all the main coordinates plus the reacnon forces on the support: y = x f s!x!!x = A x!x + Bu

30 Full State Space for Suspension Models!x!!x 2 N x 1 = 0 N x N x I N x N x M 1 K N x N x M 1 E A N x N x 2 N x 2 N x x!x 2 N x N x N s 0 N x N s 0 N x N x M 1 C N x N s M 1 E B N x N s M 1 N x N x s!s f x ( 2 N s +N x ) 1 x f s ( N x +N s ) 1 = I N x N x C N s N x 0 N x N x 0 N s N x ( N x +N s ) 2 N x x!x 2 N x N x N s 0 N x N s 0 N x N x S N s N s E D N x N s 0 N s N x ( N x +N s ) ( N s +N x ) s!s f x ( 2 N s +N x ) 1 K is the snffness matrix; M is the mass matrix C is the coupling snffness matrix; S is the support snffness The E matrices are arrays of velocity damping coefficients. Sizes of matrices and vectors are given as <Nrows x Ncolumns>.

31 Matlab Export All the components of the SS A/B/C/D matrices can easily be exported from MathemaNca as numbers or Matlab code. For symbolic export, it's beger to export M and calculate M - 1 in Matlab.

32 Usage at LIGO MathemaNca models were used to study thermal noise and asymmetrical suspensions. The GEO Matlab models were retained but had the hand- wrigen state- space matrices replaced with improved code generated by the corresponding MathemaNca model. Matlab models were used for analysis of experimental data and for control design. All MathemaNca models are hosted on the SUS SVN (Subversion version control system) at hgps://redoubt.ligo- wa.caltech.edu/websvn/lisnng.php? repname=sus&path=%2ftrunk%2fcommon %2FMathemaNcaModels %2F#path_trunk_Common_MathemaNcaModels_ (needs LIGO.org credennals). Matlab models are in the same SVN but scagered about.

33 References LIGO- T Models of the Advanced LIGO Suspensions in MathemaNca LIGO- T Models of the Advanced LIGO Suspensions in MATLAB hgps://awiki.ligo- wa.caltech.edu/aligo/ Suspensions/OpsManual (needs LIGO credennals or see LIGO- E ). LIGO- T DissipaNon dilunon LIGO- T Flexure correcnons

34 KAGRA Sekiguchi- san has used some of the ideas and wrigen his own package. Much more automated and easy to use. Probably no need (certainly no plan) to redo KAGRA models with LIGO toolkit. Not too hard to do if there is a need for say detailed thermal noise plots.