Seismic Attenuation System Synthesis by Reduced Order Models from Multibody Analysis Valerio Boschi , Riccardo DeSalvo, Virginio Sannibale California Institute of Technology Pierangelo Masarati, Giuseppe Quaranta Politecnico di Milano 1 Multibody Dynamics 2007 Milan, Italy 26-06-07
Outline � LIGO experiment � HAM-SAS attenuation system � Reduced Model Extraction Technique (P. Masarati) � MBDyn HAM-SAS Model � Results and comparison with Experimental data � Conclusions 2 Multibody Dynamics 2007 Milan, Italy 26-06-07
Gravitational Waves General Relativity Gravity is not a force, but a property of space-time "Mass tells space-time how to curve, and space-time tells mass how to move.“ John Archibald Wheeler Einstein’s Equations: When matter moves, or changes its configuration, its gravitational field changes. This change propagates outward as a ripple in the curvature of space-time : a Gravitational Wave 3 Multibody Dynamics 2007 Milan, Italy 26-06-07
Gravitational Waves What do they do to matter ? GW “stretch and compress” every object on their way in perpendicular directions at the frequency of the GW Time Leonardo da Vinci’s Vitruvian man The effect is greatly exaggerated!! If the Vitruvian man were 4.5 light years tall with feet on hearth and head touching the nearest star, he would grow by only a ‘hair width’ To directly measure gravitational waves, we need an instrument able to measure tiny relative changes in length, or strain h= Δ L/L 4 Multibody Dynamics 2007 Milan, Italy 26-06-07
LIGO AdLIGO Optical Layout End mirror Fabry-Perot arm cavity The experimental challenge: 4 km measure differences in length to one part in 10 21 or 10 -18 m !! Mode Power cleaner Pre- Input mirror Recycling Beam splitter Stabilized mirror Laser Nd:YAG Signal 4 km 180 W Recycling “Reflected” mirror photodiode Output Mode cleaner GW signal “Antisymmetric” photodiode 5 Multibody Dynamics 2007 Milan, Italy 26-06-07
LIGO The Observatories LIGO Hanford Observatory (LHO) H1 : 4 km arms H2 : 2 km arms LIGO Livingston Observatory (LLO) L1 : 4 km arms 6 Multibody Dynamics 2007 Milan, Italy 26-06-07
LIGO LIGO Scientific Collaboration (LSC) Caltech Seismic Attenuation System A very complex experiment group requires a huge collaboration Number of colleges, universities and reasearch institutions: 55 There are approximately 560 people in the LSC 7 Multibody Dynamics 2007 Milan, Italy 26-06-07
LIGO Seismic Noise Seismic Noise test mass (mirror) Thermal Residual gas (Brownian) scattering Noise Beam splitter LASER Wavelength & Radiation amplitude photodiode pressure fluctuations "Shot" noise 8 Quantum Noise Multibody Dynamics 2007 Milan, Italy 26-06-07
LIGO BSC and HAM vacuum chambers Hanford Observatory • The Corner Station houses the laser, detector, and all of the optics except the End Test Masses. 2 km photodiode 2 km laser • Each vacuum chamber has an independently supported, seismically HAM chamber isolated table on which the optics are mounted. BSC chamber • The beam tubes are 1.2 m diameter, 4 km photodiode low oxygen stainless steel 4 km laser • BSCs are approximately 5.5 m high and hold the beam splitter and the main interferometer mirrors. • HAMs are smaller chambers used for the Mode Cleaner and the Recycling cavity mirrors. 9 Multibody Dynamics 2007 Milan, Italy 26-06-07
HAM-SAS HAM-SAS Attenuation Stages HAM-SAS is a seismic attenuation system designed to provide 70-80 dB of horizontal and vertical attenuation above 10 Hz and to fit in the tight space of the LIGO HAM vacuum chamber. Rigid Bodies � Optical Table (OT) and Payload � Top Platform � 4 MGAS Springs disposed on a 1.1 x 1 m rectangular configuration. Top + Intermediate Platforms + Springs = Spring Box (SB) � Intermediate Platform � 4 Inverted Pendula Legs (IPs) disposed on a 1.1 x 0.9 m diamond configuration. � Base Platform 10 Multibody Dynamics 2007 Milan, Italy 26-06-07
HAM-SAS The Inverted Pendulum (IP) � IPs are tunable mechanical oscillators widely used for their good horizontal seismic attenuation performance. � Resonant frequencies of tens of millihertz and attenuation factors of 80 dB have been reached experimentally in many systems. � A counterweight placed below the IP pivot point allows the center of percussion (COP) of the system to be tuned, and substantially improves the attenuation at high frequency. TAMA inverted pendulum driven at the shaker resonance IP Legs Actuator 11 Multibody Dynamics 2007 Milan, Italy 26-06-07
HAM-SAS The MGAS � The MGAS filter is a vertical oscillator, developed by the CIT SAS group, which uses a crown of curved blades radially compressed in a horizontal plane for the mechanical vertical compliance. � The blades are clamped on one end to a plate, and connected on the other end to a small disk � Acting on the position of the clamps one can change the compression of the blade, and tune the MGAS resonant frequency down to 100 mHz MGAS 12 Multibody Dynamics 2007 Milan, Italy 26-06-07
HAM-SAS Modeling approach Key required features for HAM-SAS prototyping: •Finite elements for the dynamics of lumped/distributed components •Exact kinematics of the joints •Possible evaluation of friction, freeplay, … •Control sensors and actuators dynamics and control logic • Possibility to extract Reduced Order Models (ROM) for control design All these points may be addressed using the multibody/multidisciplinary approach Currently multibody approach is defined as the merge and blends of various disciplines such as structural dynamics, multi-physics mechanics, control theory and so on, all tackled following classical computational mechanics methodologies. 13 Multibody Dynamics 2007 Milan, Italy 26-06-07
MBDyn POLIMI multibody software MultiBody Dynamics software developed at POLIMI since early '90s Main application field is rotorcraft dynamics; however, it is currently exploited (@ POLIMI and by 3 rd parties) in many projects involving: � Rotorcraft (helicopters & tiltrotors) � Aircraft landing gears � Robotics and mechatronics � Automotive � Wind turbines � Human body dynamics MBDyn is free software, meaning that the source code is available for freedom of use, modification and distribution according to the very broad GPL license conditions http://www.aero.polimi.it/~mbdyn/ 14 Multibody Dynamics 2007 Milan, Italy 26-06-07
MBDyn POLIMI multibody software The following problems may be addressed by MBDyn: • Nonlinear mechanics of rigid and flexible bodies • Exact constraint kinematics by means of Lagrange multipliers • Complex, nonlinear, composite-ready beams • Smart materials (e.g. embedded piezo-electrics) • Hydraulic components • Electric networks • Active control logic • Aerodynamic steady and unsteady forces Special features: • Direct connection with Matlab/Simulink environment • Real-time capabilities under Linux/RTAI extension, for “virtual” hardware-in-the-loop tests 15 Multibody Dynamics 2007 Milan, Italy 26-06-07
Reduced Order Model from Multibody Code The Problem Implicit Differential-Algebraic Initial-Value Problem: = y y t = 0 � F ( , , ) y ( ) t y 0 0 Most integration schemes fall into the scheme ∑ ∑ = + � y a y h b y − − k i k i j k j = = i 1, n j 0, n Its perturbation yields δ = δ � y hb y k 0 k The “correction” phase results in ( ) δ � F + hb F y = -F � /y 0 /y 16 Multibody Dynamics 2007 Milan, Italy 26-06-07
Reduced Order Model from Multibody Code The Problem Eigenanalysis (non-symmetric, structurally non-definite matrices): right & left (transpose) problems: ( ) ( ) λ λ T T F + F Y = 0 , F + F Y = 0 � � /y /y /y /y adj Requirement: “exact” matrices. F , F � /y /y 17 Multibody Dynamics 2007 Milan, Italy 26-06-07
Reduced Order Model from Multibody Code A Solution Issue: MBDyn computes ( ) J = ⋅ F + dCoef F � /y /y rather than the required matrices... Proposed solution: extract eigenvalues as ( ) ( ) ( ) Λ⋅ + − = J dCoef J dCoef Y 0 where λ ⋅ + dCoef 1 Λ = λ ⋅ − dCoef 1 (resembles the inverse of Tustin's transform from continuous to discrete) Possible optimizations for large scale problems discussed in the paper. 18 Multibody Dynamics 2007 Milan, Italy 26-06-07
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