1. All-mode stack damping by helical stage suspension
John Winterflood1, Li Ju1, Chunnong Zhao1, 2, and David Blair1
1 School of Physics, University of Western Australia
2 Beijing Normal University

2. Motivation To build a isolation system for KAGRA output mode cleaner
Based on UWA isolation system using Euler spring/pendulum
Monolithic Euler Spring to improve the Euler stage performance (new)
Motion cross-coupling to achieve all modes self damping (new)

3. Euler spring vs Conventional Spring
In the early 2000s we investigated using Euler springs for vertical suspension. The basic idea is that when a thin blade spring of length L starts to buckle it should be a close approximation to a conventional spring which has been extended by a length 2L.

4. Force-Displacement Characteristic
Euler spring: No deformation until buckle
Conventional spring: Deform under any load P dl
All geometric anti-spring techniques achieve a low spring-rate over a short deflection range and thus store less elastic energy than an equivalent linear spring. But Euler buckling achieves this with a very simple mechanical structure and really minimizes the mass of spring material required to suspend a load.
Zero displacement point

5. Advantage of Euler Spring
No large static deformation
spring element has small mass thus higher internal frequencies so as not to degrade isolation performance over a larger operating frequency range
Practical consideration
Spring must operation in compression

6. use of lever arms to keep the compressed spring stable
Practical Design
use of lever arms to keep the compressed spring stable
We used a simple pivoting lever arm to keep the compressed spring stable, and arranged two of them as a balanced pair in order prevent coupling to horizontal motion. We designed a real-world implementation to look something like this early design.  It worked quite well and we published the isolation function that we measured using this unit.

7. Non Ideal Behaviour Imperfect boundary condition
The main problem that we have encountered is that any imperfection leads to early buckling of the flex strips at less than their critical load as shown in red. Then the force increases asymptotically to the classical buckling characteristic shown in green. What we would like is the classical sudden square corner onset of buckling that is shown by the dotted green line. Since the gradient of the curve sets the resonance frequency, the rounded characteristic results in a much higher resonance frequency and poorer isolation performance. One possible cause is that some creep or set has occurred in the spring material such that they start off slightly curved and so favour bowing in a particular direction. However in some of our designs - such as the one on the right - with blades mounted side by side, simply pressing on the strips while loaded, to reverse their bending direction, should then give the opposite effect - providing a much flatter or even negative spring rate as shown in blue. But typically this is not observed.
Imperfect boundary condition

8. Monolithic blades to reduce clamping slippage
Monolithic Ends
Difficult to achieve with ordinary machining
Cannot use readily available high tensile sheet
We believe the main reason for the defect is that the high stresses at the boundary region where the blades are clamped allows some slight slippage which is enough to give a preferred buckling direction once it has occurred. Once there is a preferred buckling direction then flex strips start to buckle before they reach critical load and the force displacement curve becomes rounded and non-linear instead of square and linear. If the direction of bow is manually reversed then slippage again occurs in the other direction and the problem remains.
An obvious solution to this boundary condition problem would be to engineer monolithic steps at the end of the strips to avoid high clamping stresses. However this is significantly difficult to achieve with ordinary machining and loses the large advantage of readily available high-tensile sheet spring material. So we propose the following solution.
Clamped Ends

9. Monolithic sheet - no end clamps needed
By putting multiple slits into a single wide strip of material and making each thin strip bend in the opposite direction, we keep all the high stresses in the high-tensile material - in the same way that a monolithic step would, but still making use of readily available sheet material.
In addition we found that even fixing multiple strips into clamps was an onerous task and so we propose to taper the ends of the strips so that they self-wedge into matching sockets without needing any fixing. The required tapers should be easy to machine using the EDM (electric discharge machining) wire cutter at our facility.

10. Add anti-spring for lower & adjustable freq
Constant torque spring
Spring = +
Various anti-spring techniques are possible but we chose this version as it applies minimal force to the hinges. Considering the angular spring coefficient of the lever arm about the hinge, we can add some negative spring coefficient to it by adding some inverted pendulum effect. If the two effects are made equal then the angular spring coefficient goes to zero and the structure produces a constant torque which translates into a constant lifting force. We included this effect in our second design and made the height of the inverse pendulum adjustable to allow the suspension frequency to be changed. This system worked quite well but the problem of the non-ideal force-displacement characteristic was still present and shows up more strongly when the spring coefficients are subtracted.
For constant force
Balanced configuration
Real-world implementation

11. Passively Self-damped Stage
3-D stage (spring-mass/pendulum) has several modes
Horizontal (Pendulum) – usually high Q
Vertical – could be low Q with high fraction of anti-spring nulling
Rotation – usually high Q
Tilt – usually coupled with, and similar to horizontal modes
To have all the normal modes damped without introduce excess noise
Couple high Q mode to low Q mode for self-damping
Requirement
Modes to have similar frequency for effective coupling

12. Multi-wire suspension stack
Characteristics:
Twist modes same frequencies as pendulum
Similar sum & difference (translation & rotation) control force actuation
If vertical modes are also made similar frequency, all motions equally isolated
Vertical & rocking modes damped by lossy springs
Disadvantage
Horizontal & twist modes remain undamped
Radius of gyration

13. Helical stage concept Characteristics:
Same advantages as previous multi-wire stack
Twisting motion couples strongly to vertical, rocking to horizontal, & vice-versa
Thus all modes of motion exercise vertical springs and are thereby damped
Possible Disadvantage:
Vertical creep produces strong unbalanced twisting torque to be counteracted by the local control system
One solution proposed is to unbalance the Euler spring vertical suspension lever arms so that it is deliberately assymmetric in a helical manner. By doing this twisting motion will be strongly coupled to vertical bouncing motion. This will mean that twisting motion will become damped by its coupling to the Euler spring losses.
In addition if you think about rocking motion - the helical lever arms means that rocking motion is strongly cross-coupled to pendulum translation motion and this coupling also occurs through the lever-arms and thus also incurs Euler spring losses. So the result of the helical lever arms is that all of the normal modes of the suspension stack are strongly coupled to the lossy Euler spring vertical suspension and are damped accordingly.
If it turns out that still more damping is required than is readily obtained from the Euler spring losses, then pivoting some mass as before and applying our self-damping technique between it and any other degree of freedom will effectively damp all degrees of freedom as a result of this deliberate helical cross-coupling.

14. 45 degree, Euler-sprung Lever Arm, With spring-rate nulling
Counterweight for centre-of- percussion balancing (could be located on other side of pivot)
Suspension wire parallel and close to Euler spring to minimise load on pivot
Replacing the conceptual 45 degree crank-arm of the previous slide with an Euler spring version could make use of something like this design.
Monolithic crossed- flex pivot
45°
Short, square-corner buckling Euler spring
with well nulled spring-rate should provide good structural damping when it exercises

15. First concept practical design
Here is how it might look when four of them have been added to one stage of an isolation stack.

16. Example: Self-damped pendulum
In the early 2000s we implemented another technique which we called self-damping.  In this technique we deliberately cross-couple pendulum translation motion with isolation stage rocking motion and then viscously damp these separate degrees of freedom against each other.  This gave the advantage of providing good damping for the stack horizontal modes but still providing 1/f^2 pendulum isolation per stage in a completely passive system.
Having the horizontal motion self-damped and the vertical motion naturally damped still leaves the twisting motion about a vertical axis.  This motion is typically less problematic than horizontal motion because there is very little disturbance driving it and its softness allows active damping to be quite effective.  However in some ways its softness and low-damping is quite inconvenient and we would prefer it to be more comparable with the horizontal modes.  For instance this would allow control actuation applied in common mode for translation or differential mode for steering to have very similar coefficients and control loop stability conditions.
Couple pendulum motion with very low frequency rocking motion
use eddy current damping between these two motions

17. Summary Low frequency vertical isolation stage design
Euler spring + anti-spring (low Q)
Monolithic without clamping
Compact self-damped stage
Multi-wire suspension
Cross coupling to achieve damping of all DOF resonant modes for easy control