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

1/08/ AKS-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

1/08/ AKS-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

1/08/ AKS-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

1/08/ AKS-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 P 2 (gravity load); reaction provided by stage 1 at UZMP. Note that moment balance implies P 2 d = Vh. Stage 1: applied forces are V and P 2 (from stage 2) and -V from actuator; reaction provided by stage 0. Moment balance implies that stage 0 must provide a moment reaction Vh (= P 2 d) The moment from stage 0 implies that the system will tilt LZMP Plane UZMP Plane V d h P2P2 LZMP Plane UZMP Plane h V P2P2 P2P2 V VhVh Stage 2Stage 1 Actions shown in red Reactions shown in green V P2P2

1/08/ AKS-6 Approach 2: Closed-Form Flexure Rod Solution See ASI technical note B, “Analysis of LIGO Flexure Rods” Lateral/tilt stiffness matrix of a single flexure rod: Shear forces V and moments M at LZMP subscript 3 = suspended stage subscript 5 = supporting stage Lateral displacements v and tilt rotations  at LZMP subscript 3 = suspended stage subscript 5 = supporting stage 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)

1/08/ AKS-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

1/08/ AKS-8 NASTRAN Model Stage 0: GROUND Stage 1: 1914 lb R TOR = 22.3 in, R TILT = 16.4 in Stage 2: 3470 lb R TOR = 19.5 in, R TILT = 18.7 in Stage 0/1 springs and flexures r = 25 in Stage 1/2 springs and flexures r = 25 in Model shown “stretched”; stages are actually coincident under 1g

1/08/ AKS-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