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Advanced LIGO Lateral/Tilt Mechanical Coupling Study

Advanced LIGO Lateral/Tilt Mechanical Coupling Study. Ken Smith January 8, 2004 Ref: 20008299-A. Overview.

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Advanced LIGO Lateral/Tilt Mechanical Coupling Study

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  1. Advanced LIGOLateral/Tilt Mechanical Coupling Study Ken Smith January 8, 2004 Ref: 20008299-A

  2. Overview • ASI’s understanding is that many of the LIGO design requirements are related to a desire to fully decouple the in-plane (lateral and torsion) and out-of-plane (vertical and tilt) stiffness of the mechanical system • The issues relate to tilt/horizontal confusion of the lateral seismometers, and the effect on low frequency control; thus the primary concern is with stiffness purity, and less so with coincidence of stage cg’s • Related requirements: • Coincidence of the actuator plane and the lower LZMP plane to 1 mm • Flatness of the spring blades • Other alignment requirements • Joe Giaime is in process of restating these requirements in terms of off-diagonal terms of the system stiffness matrix (or flexibility matrix) • These requirements have a direct effect on the design of the springs and flexures, so ASI has been investigating system performance in this regard

  3. Overview (2) • During our investigation, the following were observed: • Even the “ideal” isolation system, with perfect alignment of actuators and LZMP, has some coupling between lateral translation and tilt rotation due to gravity-related terms in the stiffness matrix • This coupling, though small, is not avoidable in the current design paradigm • The amount of coupling due to gravity is approximately 40 times larger than the coupling caused by 1 mm error in aligning the actuator plane with the LZMP plane • Misalignment error of 1 mm results in a virtual pivot point ~800m from the LZMP plane, in the absence of the gravity-induced coupling • The gravity-induced coupling results in a virtual pivot point ~20m from the LZMP plane • These observations prompted ASI to elevate the issue to the LIGO project

  4. Analysis Approaches • Three complementary analysis methods independently give evidence that the gravity-induced coupling effect is real • Free-body diagrams • Comprehensive closed-form solution of the flexure rod stiffness • NASTRAN finite element analysis with preload stiffening included • Each approach is described in more detail on the following pages

  5. Approach 1: Free-Body Diagrams • Assume the flexure rods are rigid pin-ended links between the UZMP and LZMP; apply a lateral force V to stage 2 at the LZMP, and an opposite force to stage 1 at the LZMP (as the actuators will do) • Stage 2: applied forces are V and P2 (gravity load); reaction provided by stage 1 at UZMP. Note that moment balance implies P2d = Vh. • Stage 1: applied forces are V and P2 (from stage 2) and -V from actuator; reaction provided by stage 0. Moment balance implies thatstage 0 must provide a moment reaction Vh (= P2d) • The moment from stage 0 implies that the system will tilt P2 P2 Stage 2 Stage 1 UZMP Plane UZMP Plane V V h h Actions shown in red Reactions shown in green Vh d LZMP Plane LZMP Plane V V P2 P2

  6. Approach 2: Closed-Form Flexure Rod Solution • See ASI technical note 20007235-B, “Analysis of LIGO Flexure Rods” • Lateral/tilt stiffness matrix of a single flexure rod: This term shows that a moment is reacted to the supporting stage from a shear through the LZMP (the first column of the stiffness matrix gives the forces/moments to induce a unit translation with zero rotation) • Lateral displacements v and tilt • rotations q at LZMP • subscript 3 = suspended stage • subscript 5 = supporting stage • Shear forces V and moments M at LZMP • subscript 3 = suspended stage • subscript 5 = supporting stage

  7. Approach 3: NASTRAN Finite Element Model • Developed a simplified model of the idealized system • Stage 0 is ground • Stages 1 and 2 are rigid bodies, with mass properties similar to ETF • Leaf springs idealized as vertical-only spring elements, rigid in other directions • Flexure rods modeled as bar elements, with preload-induced stiffening • Stage cg’s and actuation planes perfectly located at LZMP of flexure rods

  8. NASTRAN Model Stage 0: GROUND Stage 0/1 springs and flexures r = 25 in Stage 1: 1914 lb RTOR = 22.3 in, RTILT = 16.4 in Stage 1/2 springs and flexures r = 25 in Stage 2: 3470 lb RTOR = 19.5 in, RTILT = 18.7 in Model shown “stretched”; stages are actually coincident under 1g

  9. Model Results • Stage displacements from 1 lb force applied to stage 2 and reacted at stage 1 (in plane of LZMP) in the X direction at the stage centers • Ratio of translational displacement to tilt rotation is approximately 675” • Effect of 0.04” misalignment is approximately 1/40th as severe Case 1: Perfect alignment of forces with LZMP Case 2: Forces applied 0.04” above the LZMP Case 3: Forces applied 0.04” below the LZMP Stage 1 tilt unaffected by misalignment Stage 2 tilt slightly affected by misalignment, but misalignment effects are much smaller than the bias seen in the “perfect” case 1

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