1. LSC Meeting Baton Rouge, LA, 8-16-2006 V.Boschi for the HAM-SAS team Ben Abbott, Valerio Boschi, Dennis Coyne, Michael Forte, Jay Heefner, Yu-mei Huang, David Ottaway, Riccardo de Salvo, & Virginio Sannibale HAM-SAS Mechanics Status of modeling DCC G060445-00-R

2. LSC Meeting Baton Rouge, LA, 8-16-2006 Introduction HAM-SAS Attenuation Stages HAM-SAS is a seismic attenuation system expressly designed to fit in the tight space of the LIGO HAM vacuum chamber. Rigid Bodies - 4 Inverted Pendula Legs (IPs) - 4 MGAS Springs: Spring Box (SB) - Optical Table (OT) - Payload (mode cleaner suspensions, etc.)

3. LSC Meeting Baton Rouge, LA, 8-16-2006 Introduction Modeling Approach A state-space model of HAM-SAS mechanical structure have been developed using an Analytical approach. Let’s summarize the approximations used in the model: Lumped system, i.e. rigid body approximation Elastic elements are approximated using quadratic potentials, i.e. small oscillation regime Dissipation mechanisms are accounted using viscous damping which approximate structural/hysteretic damping in the small oscillation regime The system is considered symmetric enough to separate horizontal displacements x, y, and yaw from pitch, roll and vertical displacement z Internal modes of the mechanical structures are not accounted

4. LSC Meeting Baton Rouge, LA, 8-16-2006 Introduction Modeling Approach GAS - Blade stiffness modeled with simple Springs - Hysteretic/structural damping approximated with viscous damping. - Transmissibility saturation modeled using the "magic wand" Inverted Pendulum - Flexural Joint with Ideal pivot point about the attachment point. -Leg, a rigid body - Hysteretic/structural damping approximated with viscous damping.

5. LSC Meeting Baton Rouge, LA, 8-16-2006 IP Table Asymmetric leg length: Horizontal Transm. 30 mHz IP frequency 1.2 Hz Horizontal GAS frequency 105 mHz Little pendulum D.o.f. Contamination

6. LSC Meeting Baton Rouge, LA, 8-16-2006 IP Table Asymmetric leg length: Angular Transm.

7. LSC Meeting Baton Rouge, LA, 8-16-2006 IP Table Courtesy of I.Taurasi (Univ. of Benevento, Italy) ANSYS modeling of Rigid Leg Resonances Resonance frequency with counterweight Resonance frequency without counterweight ~110.6 Hz ~122 Hz Diameter of small flex joint: 1.5 mm Mass of counter weight: 1.212 Kg Ansys shows that counter weight doesn’t reduce significantly the resonances. They can be damped

8. LSC Meeting Baton Rouge, LA, 8-16-2006 Eddy current dampers Before installation t = 4.3 s After installation t = 35 ms Measured and succesfully damped in a prototype without counterweight IP Table Damping of Rigid Leg Resonances Courtesy of I.Taurasi (Univ. of Benevento, Italy)

9. LSC Meeting Baton Rouge, LA, 8-16-2006 IP Table Leg Counterweight tuning

10. LSC Meeting Baton Rouge, LA, 8-16-2006 MGAS Table Asymmetric spring elastic constant k D.o.f. contamination

11. LSC Meeting Baton Rouge, LA, 8-16-2006 MGAS Table Asymmetric spring elastic constant k (expected Quality Factors) Very small effect

12. LSC Meeting Baton Rouge, LA, 8-16-2006 Triple Pendulum Model

13. LSC Meeting Baton Rouge, LA, 8-16-2006 Comparison Triple Pendulum Model We can compare our results with a previous (2003) Mathematica model made by M. Burton Test massIntermediate massUpper mass

14. LSC Meeting Baton Rouge, LA, 8-16-2006 Triple Pendulum + Horizontal Stage Model 30 mHz IP frequency Suspension Resonances 0.67-1.5Hz Little Pendula

15. LSC Meeting Baton Rouge, LA, 8-16-2006 Final considerations We are confident that we can meet the HAM optical table seismic attenuation requirements.