Copyright 2004 Alliance Spacesystems, Inc. All rights reserved. Confidential and Proprietary Material of ASI. Reproduction without prior written permission of ASI is strictly prohibited by law. PRIVILEGED & CONFIDENTIAL. LIGO BSC Stage 0 Preliminary Transfer Function Assessment Ken Smith April 23, 2004 Ref: A BSC Stage 0 Transfer Function February 25, 2016 ASI ProprietaryCopyright 2004 Alliance Spacesystems, Inc. All rights reserved. Confidential and Proprietary Material. Reproduction without prior written permission of ASI is strictly prohibited by law. PRIVILEGED & CONFIDENTIAL. Page 2 Doc. No A Requirement C.9 from LIGO E The structure must be suitably stiff to allow proper functioning of the servo control systems. The criteria stated below for the various structural elements or stages apply to the structure complete with mounted displacement sensors, dummy seismometers, actuators, and a mass equal to the structure payload, distributed over the optical table. Finite Element models of the complete design must be constructed to quantitatively predict and document compliance with these requirements. In addition to the structure being designed, the finite element model shall also include (on Stage 0) (see External Support Parts for the BSC): the support tubes, mounting caps and bases, crossbeams, crossbeam attachment plates and crossbeam feet (pointed at 45 o, toward the chamber center). Assume that Stage 0 is a free body analysis purposes. The Finite Element model shall also include a dummy payload whose CG is clamped at an X, Y of 9.1, 9.1 (23.1 cm, 23.1 cm); see H.2. Assume a sinusoidal excitation, that applies a force, p(t) = p 0 cos(2*pi*f*t), where f = frequency and t = time, along the actuator's nominal force axis, exerted between the stage holding the seismometer or displacement sensor and the stage to which the other side of the actuator is mounted. On the seismometer/ displacement sensor's stage, this excitation will cause a sinusoidal displacement at the location of the actuator's center of gravity (CG), x a (t) and one at the CG of the seismometer or displacement sensor, x s (t). These displacements shall be measured along the seismometers, the displacement sensors and the actuator's nominal axes. The frequency-domain Fourier coefficient X(f) is defined by x(t) = Re{X(f)*cos(2*pi*f*t)}. X(f) is complex, i.e., X(f) = |X(f)| exp{i phase(X(f))}. BSC Stage 0 Transfer Function February 25, 2016 ASI ProprietaryCopyright 2004 Alliance Spacesystems, Inc. All rights reserved. Confidential and Proprietary Material. Reproduction without prior written permission of ASI is strictly prohibited by law. PRIVILEGED & CONFIDENTIAL. Page 3 Doc. No A Requirement C.9 from LIGO E (Cont.) For common-corner seismometer/actuator and displacement sensor/actuator pairs that share an axis direction on stages 1 and 2: the phase of the transfer function X s /X a shall be greater than -90 degrees for all frequencies below 500 Hz. For all other pairs of seismometer and actuator, and displacement sensor and actuator on stages 1 and 2: the phase of X s /X a shall be greater than -90 degrees for all frequencies below 150 Hz. For all pairs of displacement sensor and actuator on stage 0: the phase of X s /X a shall be greater than -90 degrees for all frequencies below 100 Hz. If a mode that causes a phase excursion has only minimal amplitude in the | X s /X a | of all of the above pairs, and the phases return above -90 degrees above the resonance frequencies, and with Caltech's consensus, the modal frequency may not count against this requirement. (This sort of behavior may result from modes that primarily involve motion of the external structure.) BSC Stage 0 Transfer Function February 25, 2016 ASI ProprietaryCopyright 2004 Alliance Spacesystems, Inc. All rights reserved. Confidential and Proprietary Material. Reproduction without prior written permission of ASI is strictly prohibited by law. PRIVILEGED & CONFIDENTIAL. Page 4 Doc. No A External Structure FEM Completed Based on Caltech-provided drawings Special attention to tube/crossbeam joint flexibility Detailed solid model of local joint Combination of bars and springs tuned to give matching translational and bending stiffness through joint Boot masses at corners, oriented per discussions with Larry Jones Boundary conditions: unconstrained 96 bar elements 4 point masses for corner boots Spring elements for tube/crossbeam flexibility Total mass: 4998 lb BSC Stage 0 Transfer Function February 25, 2016 ASI ProprietaryCopyright 2004 Alliance Spacesystems, Inc. All rights reserved. Confidential and Proprietary Material. Reproduction without prior written permission of ASI is strictly prohibited by law. PRIVILEGED & CONFIDENTIAL. Page 5 Doc. No A Free-Free Modes of Bare External Structure Seven or eight modes below 100 Hz Note frequencies have dropped compared to preliminary model (first mode was 36 Hz, now 21 Hz) Reason for lower frequencies are joint flexibility, and more accurate mass properties, including corner masses Mode shape plots on following pages BSC Stage 0 Transfer Function February 25, 2016 ASI ProprietaryCopyright 2004 Alliance Spacesystems, Inc. All rights reserved. Confidential and Proprietary Material. Reproduction without prior written permission of ASI is strictly prohibited by law. PRIVILEGED & CONFIDENTIAL. Page 6 Doc. No A Free-Free Modes of Bare External Structure (Cont.) Mode 1, Hz Flexural Bending Mode 2, Hz Shear Strain energy distribution: support tubes: 71% crossbeams: 8% tube/beam joint: 21% Strain energy distribution: support tubes: 70% crossbeams: 14% tube/beam joint: 18% BSC Stage 0 Transfer Function February 25, 2016 ASI ProprietaryCopyright 2004 Alliance Spacesystems, Inc. All rights reserved. Confidential and Proprietary Material. Reproduction without prior written permission of ASI is strictly prohibited by law. PRIVILEGED & CONFIDENTIAL. Page 7 Doc. No A Free-Free Modes of Bare External Structure (Cont.) Mode 3, Hz Support Tube Vertical Bending Mode 4, Hz Support Tube Lateral Bending Strain energy distribution: support tubes: 76% crossbeams: 6% tube/beam joint: 18% Strain energy distribution: support tubes: 47% crossbeams: 49% tube/beam joint: 4% BSC Stage 0 Transfer Function February 25, 2016 ASI ProprietaryCopyright 2004 Alliance Spacesystems, Inc. All rights reserved. Confidential and Proprietary Material. Reproduction without prior written permission of ASI is strictly prohibited by law. PRIVILEGED & CONFIDENTIAL. Page 8 Doc. No A Free-Free Modes of Bare External Structure (Cont.) Mode 5, Hz Support Tube Vertical Antisymmetric Mode 6, Hz Crossbeam Twist Strain energy distribution: support tubes: 55% crossbeams: 37% tube/beam joint: 8% Strain energy distribution: support tubes: 38% crossbeams: 32% tube/beam joint: 30% BSC Stage 0 Transfer Function February 25, 2016 ASI ProprietaryCopyright 2004 Alliance Spacesystems, Inc. All rights reserved. Confidential and Proprietary Material. Reproduction without prior written permission of ASI is strictly prohibited by law. PRIVILEGED & CONFIDENTIAL. Page 9 Doc. No A Free-Free Modes of External Structure (Cont.) Mode 7, HzMode 8, Hz BSC Stage 0 Transfer Function February 25, 2016 ASI ProprietaryCopyright 2004 Alliance Spacesystems, Inc. All rights reserved. Confidential and Proprietary Material. Reproduction without prior written permission of ASI is strictly prohibited by law. PRIVILEGED & CONFIDENTIAL. Page 10 Doc. No A SEI Stage 0 on External Structure Bars and plates for hexagonal base structure Solid elements for actuator support piers Stages 1 and 2 treated as rigid masses Idealized springs elements for springs/flexures vertical actuation (3x) tangential actuation (3x) BSC Stage 0 Transfer Function February 25, 2016 ASI ProprietaryCopyright 2004 Alliance Spacesystems, Inc. All rights reserved. Confidential and Proprietary Material. Reproduction without prior written permission of ASI is strictly prohibited by law. PRIVILEGED & CONFIDENTIAL. Page 11 Doc. No A Free-Free Modes of Coupled System 12 rigid body modes between 1.8 and 9.8 Hz 11 elastic modes between 10 Hz and 100 Hz Mode shape plots of first 6 elastic modes on following pages BSC Stage 0 Transfer Function February 25, 2016 ASI ProprietaryCopyright 2004 Alliance Spacesystems, Inc. All rights reserved. Confidential and Proprietary Material. Reproduction without prior written permission of ASI is strictly prohibited by law. PRIVILEGED & CONFIDENTIAL. Page 12 Doc. No A Free-Free Modes of Coupled System (Cont.) Mode 13, Hz Flexural Bending Mode 14, Hz Shear BSC Stage 0 Transfer Function February 25, 2016 ASI ProprietaryCopyright 2004 Alliance Spacesystems, Inc. All rights reserved. Confidential and Proprietary Material. Reproduction without prior written permission of ASI is strictly prohibited by law. PRIVILEGED & CONFIDENTIAL. Page 13 Doc. No A Free-Free Modes of Coupled System (Cont.) Mode 15, Hz Support Tube VerticalBending Mode 16, Hz X Bending of Hex and Support Tubes BSC Stage 0 Transfer Function February 25, 2016 ASI ProprietaryCopyright 2004 Alliance Spacesystems, Inc. All rights reserved. Confidential and Proprietary Material. Reproduction without prior written permission of ASI is strictly prohibited by law. PRIVILEGED & CONFIDENTIAL. Page 14 Doc. No A Free-Free Modes of Coupled System (Cont.) Mode 17, Hz Z Bounce Mode 18, Hz Y Rocking BSC Stage 0 Transfer Function February 25, 2016 ASI ProprietaryCopyright 2004 Alliance Spacesystems, Inc. All rights reserved. Confidential and Proprietary Material. Reproduction without prior written permission of ASI is strictly prohibited by law. PRIVILEGED & CONFIDENTIAL. Page 15 Doc. No A Transfer Function Analysis Coupled system excited by action on stage 1, reaction on stage 0 Three vertical load points, three tangential load points Assumed 1% modal damping Frequency response calculated for the following displacements: Action point on stage 1 in actuation direction Relative displacement across each actuator (this is a substitute for the displacement sensor, since the displacement sensors have not yet been located on the structure) Transfer function computed as the ratio of the relative displacements to the action absolute displacement at the actuated location Total of 36 (6x6) transfer functions computed BSC Stage 0 Transfer Function February 25, 2016 ASI ProprietaryCopyright 2004 Alliance Spacesystems, Inc. All rights reserved. Confidential and Proprietary Material. Reproduction without prior written permission of ASI is strictly prohibited by law. PRIVILEGED & CONFIDENTIAL. Page 16 Doc. No A Typical Collocated Vertical Transfer Function phase dips in suspension modes phase transition at 36 Hz antiresonance antiresonance at 36 Hz, which is the natural frequency of stage 0 and stage 1 with actuators rigidly connected in actuation direction BSC Stage 0 Transfer Function February 25, 2016 ASI ProprietaryCopyright 2004 Alliance Spacesystems, Inc. All rights reserved. Confidential and Proprietary Material. Reproduction without prior written permission of ASI is strictly prohibited by law. PRIVILEGED & CONFIDENTIAL. Page 17 Doc. No A Typical Collocated Tangential Transfer Function phase dips in suspension modes phase transition at 35 Hz antiresonance BSC Stage 0 Transfer Function February 25, 2016 ASI ProprietaryCopyright 2004 Alliance Spacesystems, Inc. All rights reserved. Confidential and Proprietary Material. Reproduction without prior written permission of ASI is strictly prohibited by law. PRIVILEGED & CONFIDENTIAL. Page 18 Doc. No A All 36 Transfer Functions Available for Review Available on project site, in directoryTwo files: bsc_stage0_transfer_funcs.pdf Plots of transfer functions bsc_stage0_transfer_funcs.xls Transfer function data in Excel BSC Stage 0 Transfer Function February 25, 2016 ASI ProprietaryCopyright 2004 Alliance Spacesystems, Inc. All rights reserved. Confidential and Proprietary Material. Reproduction without prior written permission of ASI is strictly prohibited by law. PRIVILEGED & CONFIDENTIAL. Page 19 Doc. No A Discussion LIGO requirement C.9 is not currently satisfied for BSC stage 0, even in the looser sense described in the last paragraph of the requirement The reason for the phase shift is an antiresonance near 35 Hz, though there are also external structure modes that contribute I believe the frequency of the antiresonance corresponds to the frequency of stage 1 supported through rigid actuators, and is therefore influenced by the stiffness of the actuator supports It is unlikely that the antiresonance can be raised above 100 Hz; it would certainly require a substantial reworking of the stage 0-1 actuator supports, and possibly of the entire stage 0 concept