Advanced ligo lateral tilt mechanical coupling study
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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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Advanced ligo lateral tilt mechanical coupling study

Advanced LIGOLateral/Tilt Mechanical Coupling Study

Ken Smith

January 8, 2004

Ref: 20008299-A


Overview

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


Overview 2

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


Analysis approaches

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


Approach 1 free body diagrams

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


Approach 2 closed form flexure rod solution

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


Approach 3 nastran finite element model

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


Nastran model

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


Model results

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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