^{1}, A. Sirr

^{1}, L. Ju

^{1,a)}and D. G. Blair

^{1}

### Abstract

LaCoste linkage vibration isolators have shown excellent performance for ultra-low frequency vertical vibration isolation. However, such isolators depend on the use of conventional pre-stressed coil springs, which suffer from creep. Here, we show that compressional Euler springs can be configured to create a stable tension unit for use in a LaCoste structure. In a proof of concept experiment, we demonstrate a vertical resonance frequency of 0.15 Hz in an Euler-LaCoste configuration with 200 mm height. The system enables the use of very low creep maraging steel as spring elements to eliminate the creep while minimising spring mass and reducing the effect of parasitic resonances. Larger scale systems with optimized Euler spring boundary conditions should achieve performance suitable for applications on third generation gravitational wave detectors such as the proposed Einstein telescope.

The authors thank John Winterflood for valuable discussions. This project was supported by the Australian Research Council and the Western Australian Government Centre of Excellence Scheme. It is part of the research program of the Australian Consortium for Gravitational Astronomy.

I. INTRODUCTION

II. THEORY OF LACOSTE LINKAGES AND EULER SPRINGS

A. LaCoste linkage

B. Euler springs

III. EULER-LACOSTE LINKAGE

IV. EXPERIMENTAL RESULTS

A. Force-displacement characteristics Euler spring module

B. Tuning of Euler-LaCoste linkage

1. Frequency tuning by x-offset tuning

2. Height tuning

V. DISCUSSIONS

### Key Topics

- Creep
- 13.0
- Elasticity
- 11.0
- Boundary value problems
- 9.0
- Buckling
- 5.0
- Gravitational wave detectors
- 5.0

##### F16D

##### F16F

##### F16F1/00

##### F16F15/00

##### G02B23/00

## Figures

Schematic concept of the LaCoste linkage consisting of a zero length spring and flexure pivoted arm.

Schematic concept of the LaCoste linkage consisting of a zero length spring and flexure pivoted arm.

Force-displacement diagram of different types of spring with initial physical length *L* _{0}. The dots represent the force applied to the springs. The slopes of the lines are the spring rate of these different springs, respectively. Note that the stretching of the spring Δ*l* is different from the physical length change of the spring Δ*L* = *L* − *L* _{0}.

Force-displacement diagram of different types of spring with initial physical length *L* _{0}. The dots represent the force applied to the springs. The slopes of the lines are the spring rate of these different springs, respectively. Note that the stretching of the spring Δ*l* is different from the physical length change of the spring Δ*L* = *L* − *L* _{0}.

Schematic of tunable LaCoste system.

Schematic of tunable LaCoste system.

Force-angular displacement characteristic of a LaCosta Linkage with non-zero length spring or non-zero *x* offset.

Force-angular displacement characteristic of a LaCosta Linkage with non-zero length spring or non-zero *x* offset.

Effect of *x* offset tuning on the slope of the force-angular displacement curve. It can be seen that at small angle the force-angular displacement is quite linear and by tuning the *x* offset, it is possible to achieve a low spring rate and thus low resonant frequency. Here, *x* < 0 means the top mount is at the left side relative to the pivot.

Effect of *x* offset tuning on the slope of the force-angular displacement curve. It can be seen that at small angle the force-angular displacement is quite linear and by tuning the *x* offset, it is possible to achieve a low spring rate and thus low resonant frequency. Here, *x* < 0 means the top mount is at the left side relative to the pivot.

Euler buckling spring in different boundary conditions: (a) hinged-hinged elastic column in buckled mode; (b) clamped-clamped elastic column in buckled mode.

Euler buckling spring in different boundary conditions: (a) hinged-hinged elastic column in buckled mode; (b) clamped-clamped elastic column in buckled mode.

Force-displacement characteristic of typical Euler springs.

Force-displacement characteristic of typical Euler springs.

Principle of the tensile Euler springs arrangement which uses tension wires pulling from opposite ends. This structure is normally unstable.

Principle of the tensile Euler springs arrangement which uses tension wires pulling from opposite ends. This structure is normally unstable.

Stable tensile Euler spring unit able to work as a “zero length spring” in a LaCoste frame. (a) Four pairs of blades are clamped and attached to two rigid rings with non-conflicting wire structures for tension. (b) A monolithic Euler spring unit with three pairs of blades constructed by electric discharge machining. Similar sets of non-conflicting extension wires as the ring design are attached to the two triangular frames of the spring module to form a tension spring module.

Stable tensile Euler spring unit able to work as a “zero length spring” in a LaCoste frame. (a) Four pairs of blades are clamped and attached to two rigid rings with non-conflicting wire structures for tension. (b) A monolithic Euler spring unit with three pairs of blades constructed by electric discharge machining. Similar sets of non-conflicting extension wires as the ring design are attached to the two triangular frames of the spring module to form a tension spring module.

Prototype Euler-LaCoste linkage as a very low frequency oscillator. (a) Schematic diagram; (b) Photo of the apparatus. The Euler spring blades are 2 cm wide, 0.5 mm thick, and 260 mm long between the clamps. The distance between the top and bottom mount of the Euler spring unit *L* ∼ 300 mm.

Prototype Euler-LaCoste linkage as a very low frequency oscillator. (a) Schematic diagram; (b) Photo of the apparatus. The Euler spring blades are 2 cm wide, 0.5 mm thick, and 260 mm long between the clamps. The distance between the top and bottom mount of the Euler spring unit *L* ∼ 300 mm.

Force-displacement characteristic of the Euler spring unit. The unit under test is arranged as shown in Fig. 9(a) so that it operates under tension. The original spring unit length is 260 mm. The force was applied by directly adding weight under the unit, which buckled under the weight about 21 kg.

Force-displacement characteristic of the Euler spring unit. The unit under test is arranged as shown in Fig. 9(a) so that it operates under tension. The original spring unit length is 260 mm. The force was applied by directly adding weight under the unit, which buckled under the weight about 21 kg.

Tuning of the Euler-LaCoste linkage with *x* offset. Here, Δ*x* is the change of the offset value.

Tuning of the Euler-LaCoste linkage with *x* offset. Here, Δ*x* is the change of the offset value.

Euler-LaCoste linkage period of oscillation with height adjustment. Each curve represents different final length *L* indicated in the legend.

Euler-LaCoste linkage period of oscillation with height adjustment. Each curve represents different final length *L* indicated in the legend.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content