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Advanced LIGO Lateral/Tilt Mechanical Coupling Study

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Advanced LIGOLateral/Tilt Mechanical Coupling Study

Ken Smith

January 8, 2004

Ref: 20008299-A

- 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

- 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

- 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

- 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

- 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

- 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

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

- 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