It is important for the LIGO project to estimate the tabletop motion (translational and rotatory) because there is no satisfactory data acquisition channel that describes the motion of a table. A simple way to estimate this would be to calculate the motion of a mirror on the required table from the data acquisition channels and multiply the mirror motion with the mirrortabletop transfer function. The result is the tabletop motion. This experiment verified the accuracy of this process by using the calculated tabletop motion to predict the motion of a different mirror on the same table. The results show that calculating the tabletop motion from the motion of mirror placed on it is feasible.
INTRODUCTION
The Laser Interferometer Gravitational Wave Observatory (LIGO) is a project undertaken to detect gravitational waves. The Interferometer uses several mirrors suspended from a frame. The frame sits on a table that is isolated from ground noise. A simulation package called E2E is being developed to simulate the working of the interferometer. It is necessary to understand how the tabletop moves in response to the ground noise to create an effective simulation.
HAM1 tabletop
MMT3
MMT1
Isolation stack
Fig 1 – Schematic diagram of the HAM1 table with MMT1 and MMT3 on it
This experiment is aimed at calculating the HAM1 tabletop motion from the MMT1 mirror placed on top of it. The accuracy of the calculation was tested by checking how closely this calculated tabletop motion predicted the motion of the MMT3 mirror that is also placed on the same table.
METHODS
The following steps were taken:

The readings of the coil magnets behind the MMT1 and MMT3 were obtained as a function of time for a certain period of time.

The translational and the yaw motion of MMT1 and MMT3 were calculated as a function of time.

The Fast Fourier Transform (FFT) of the translational and Yaw motions of MMT1 were taken. This yielded the various frequency components of the motions, or the translational and yaw motions as functions of frequency.

These frequency functions were multiplied with the Small Optic Suspension (SOS) transfer functions for translational and yaw motion to obtain the HAM1 translational and yaw motions as a function of frequency.

This HAM1 motion was multiplied with the Large Optic Suspension (LOS) to obtain the translational and yaw motions of MMT3 as functions of frequency.

The Inverse FFT of the calculated MMT3 motions were taken to yield the predicted translational and yaw motions of MMT3 as a function of time.

The correlation between the calculated MMT3 motions and acquired MMT3 motions (step 1 & 2) were calculated.
The result of this process is two correlation coefficients, one for the translational motion and the other for the yaw motion.
RESULTS
The correlation coefficients for the MMT3 motions are (the Matlab code for these calculations is here):

Corr.
Coeff
( r )

error

Translational

0.7714

0.0077

Yaw

0.7242

0.0037

DISCUSSION
The HAM1 tabletop motion is believable if there are peaks at the resonance frequencies of the table.
The correlation of the recorded and calculated motions of MMT3 is a direct indicator of the effectiveness of the process to calculate HAM1 motion from MMT1. The closer to 1 the absolute values of these coefficients are, the more accurate the calculated HAM1 motion.
These coefficients can certainly be improved if the location and orientation of MMT1 and MMT3 on HAM1 are taken into account.
Regardless, the results testify that estimating tabletop motions from the optics placed upon them is a feasible process statistically.
REFERENCES
Findley. T, and K. Rogillio. 2004. Matlab code for calculating MMT3 motion.