^{1}, R. Toncelli

^{2}, M. L. Chiofalo

^{3}, D. Bramanti

^{4}and A. M. Nobili

^{5}

### Abstract

“Galileo Galilei on the ground” (GGG) is a fast rotating differential accelerometer designed to test the equivalence principle (EP). Its sensitivity to differential effects, such as the effect of an EP violation, depends crucially on the capability of the accelerometer to reject all effects acting in common mode. By applying the theoretical and simulation methods reported in Part I of this work, and tested therein against experimental data, we predict the occurrence of an enhanced common mode rejection of the GGG accelerometer. We demonstrate that the best rejection of common mode disturbances can be tuned in a controlled way by varying the spin frequency of the GGG rotor.

Thanks are due to INFN for funding the GGG experiment in its laboratory of San Piero a Grado in Pisa.

I. INTRODUCTION

II. THE NUMERICAL METHOD

A. Dynamical equations: External forces and transfer function

III. RESULTS

A. The common mode rejection factor

B. Nonspinning rotor: Analytical solution and scaling parameter

C. Region of low and high spin frequencies

1. The differential period

2. Spectra of the test mass differential displacements

3. Rejection of dc forces versus the governing parameters

4. Validation of the scaling parameter

D. Region of intermediate spin frequencies

IV. ENHANCED REJECTION BEHAVIOR OF THE GGG ROTOR

V. CONCLUDING REMARKS AND PERSPECTIVES

### Key Topics

- Normal modes
- 9.0
- Suspensions
- 9.0
- Anisotropy
- 6.0
- Elasticity
- 4.0
- Numerical modeling
- 4.0

## Figures

Differential period as a function of the balancing parameter . The various curves refer to different values of the other parameters of the system, as given in Table I (all simulations were performed with the rotor spinning at ).

Differential period as a function of the balancing parameter . The various curves refer to different values of the other parameters of the system, as given in Table I (all simulations were performed with the rotor spinning at ).

Common mode (top panel) and differential mode (bottom panel) relative displacements, divided by the intensity of the acceleration applied, in common mode or differential mode, respectively, as functions of the frequency of the applied force. The rotor is spinning at . The other parameters of the system are typical of the present instrument: , , , , , and .

Common mode (top panel) and differential mode (bottom panel) relative displacements, divided by the intensity of the acceleration applied, in common mode or differential mode, respectively, as functions of the frequency of the applied force. The rotor is spinning at . The other parameters of the system are typical of the present instrument: , , , , , and .

Inverse rejection function vs frequency in the (top) and (bottom) directions for the rotor spinning at . The other system parameters are the same as in Fig. 2.

Inverse rejection function vs frequency in the (top) and (bottom) directions for the rotor spinning at . The other system parameters are the same as in Fig. 2.

Inverse rejection factor of dc forces, , as a function of various system parameters. From top to bottom, the varying parameters are , , , , and the anisotropy . Solid line: nonspinning rotor. Points: rotor spinning at . The parameters are changed one at a time from the values reported in the caption of Fig. 2.

Inverse rejection factor of dc forces, , as a function of various system parameters. From top to bottom, the varying parameters are , , , , and the anisotropy . Solid line: nonspinning rotor. Points: rotor spinning at . The parameters are changed one at a time from the values reported in the caption of Fig. 2.

Results from numerical simulations of the inverse rejection factor of dc forces, , as a function of the scaling parameter . The solid line refers to the zero spin case with isotropic suspensions , and also to the isotropic rotor in the low and high spin frequency regions. Once anisotropy of the suspensions is taken into account (e.g., with ), the rotor spinning at low frequencies gives the results shown as filled circles, while the one spinning at high frequencies gives the results shown as filled triangles. The dashed line has no physical meaning; it simply shows that the filled triangles still lie on a line, though at lower inclination. The system parameters reported in Fig. 2 correspond to .

Results from numerical simulations of the inverse rejection factor of dc forces, , as a function of the scaling parameter . The solid line refers to the zero spin case with isotropic suspensions , and also to the isotropic rotor in the low and high spin frequency regions. Once anisotropy of the suspensions is taken into account (e.g., with ), the rotor spinning at low frequencies gives the results shown as filled circles, while the one spinning at high frequencies gives the results shown as filled triangles. The dashed line has no physical meaning; it simply shows that the filled triangles still lie on a line, though at lower inclination. The system parameters reported in Fig. 2 correspond to .

Top panel: the inverse rejection factor of common mode dc forces, , as a function of the spin frequency . The system parameters are the same as in Fig. 2. The numbered arrows indicate crossing points and minima (see text) and correspond to those shown in the bottom panel. Bottom panel: absolute values of the zeros (dashed lines) and of the poles (solid lines) of the transfer function vs . The horizontal branches correspond to the differential frequencies , and split because of the anisotropy. For within the shaded areas, the response is dominated by the zeros of the transfer function , and therefore the relative displacement in response to common mode dc forces, , is strongly suppressed.

Top panel: the inverse rejection factor of common mode dc forces, , as a function of the spin frequency . The system parameters are the same as in Fig. 2. The numbered arrows indicate crossing points and minima (see text) and correspond to those shown in the bottom panel. Bottom panel: absolute values of the zeros (dashed lines) and of the poles (solid lines) of the transfer function vs . The horizontal branches correspond to the differential frequencies , and split because of the anisotropy. For within the shaded areas, the response is dominated by the zeros of the transfer function , and therefore the relative displacement in response to common mode dc forces, , is strongly suppressed.

Inverse static rejection as a function of the spin frequency. Curves of increasing thickness refer to increasing values of the scaling parameter , 745, and 2070, while keeping fixed. We note that because of the anisotropic central suspension. Note that different values of leave the position of the peaks unaffected.

Inverse static rejection as a function of the spin frequency. Curves of increasing thickness refer to increasing values of the scaling parameter , 745, and 2070, while keeping fixed. We note that because of the anisotropic central suspension. Note that different values of leave the position of the peaks unaffected.

Inverse rejection factor of common mode dc forces, , as a function of the spin frequency at different values of [with the scaling parameter fixed at ]. From bottom to top: , , and . Note the increasing separation in frequency between the peaks from bottom to top, leading to enhanced rejection at higher spin frequencies. For the maximum separation case (top panel) enhanced rejection takes place at .

Inverse rejection factor of common mode dc forces, , as a function of the spin frequency at different values of [with the scaling parameter fixed at ]. From bottom to top: , , and . Note the increasing separation in frequency between the peaks from bottom to top, leading to enhanced rejection at higher spin frequencies. For the maximum separation case (top panel) enhanced rejection takes place at .

## Tables

Legend corresponding to Fig. 1.

Legend corresponding to Fig. 1.

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