^{1}, M. L. Chiofalo

^{2}, R. Toncelli

^{3}, D. Bramanti

^{3}, E. Polacco

^{4}and A. M. Nobili

^{4}

### Abstract

Recent theoretical work suggests that violation of the equivalence principle might be revealed in a measurement of the fractional differential acceleration between two test bodies—of different compositions, falling in the gravitational field of a source mass—if the measurement is made to the level of or better. This being within the reach of ground based experiments gives them a new impetus. However, while slowly rotating torsion balances in ground laboratories are close to reaching this level, only an experiment performed in a low orbit around the Earth is likely to provide a much better accuracy. We report on the progress made with the “Galileo Galilei on the ground” (GGG) experiment, which aims to compete with torsion balances using an instrument design also capable of being converted into a much higher sensitivity space test. In the present and following articles (Part I and Part II), we demonstrate that the dynamical response of the GGG differential accelerometer set into supercritical rotation—in particular, its normal modes (Part I) and rejection of common mode effects (Part II)—can be predicted by means of a simple but effective model that embodies all the relevant physics. Analytical solutions are obtained under special limits, which provide the theoretical understanding. A simulation environment is set up, obtaining a quantitative agreement with the available experimental data on the frequencies of the normal modes and on the whirling behavior. This is a needed and reliable tool for controlling and separating perturbative effects from the expected signal, as well as for planning the optimization of the apparatus.

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

I. INTRODUCTION

II. THE GGG ROTOR: OVERVIEW OF THE EXPERIMENT

A. Description of the mechanical structure of the apparatus

B. The differential motion detector system

C. Principle of operation

1. Differential character and common mode rejection

2. Signal modulation and whirl motions

III. THE MODEL

A. The Lagrangian

B. Choice of the generalized coordinates

C. Equilibrium positions and second-order expansion

D. Linearized equations of motion

1. Rotating and nonrotating dampings

IV. THE NUMERICAL METHOD

A. General considerations

B. System parameters

V. RESULTS: THE NORMAL MODES

A. Comparison with the experiment

B. Role of damping

C. Low-frequency limit

D. Anisotropy

E. Scissors’s shape

F. Mode splitting

G. Instability regions

VI. CONCLUDING REMARKS

### Key Topics

- Suspensions
- 36.0
- Capacitance
- 13.0
- Normal modes
- 12.0
- Experiment design
- 10.0
- Gravitational fields
- 6.0

## Figures

(Color) Section through the spin axis of the differential accelerometer inside the vacuum chamber. , vacuum chamber; , motor; OD, optical device (see Sec. II B); , ball bearings; ST, suspension tube; , coupling (balance) arm, located inside the suspension tube, with its three laminar cardanic suspensions (in red); and , center of mass of the two cylinder’s system (in blue the outer cylinder, in green the inner one, each). IP are the internal capacitance plates of the differential motion detector (Sec. II B), OP are the outer ones for whirl control (Sec. ???), and PC is the contactless inductive power coupler providing power to the electronics inside the rotor. The relevant distances, and , of the centers of mass of the inner and outer bodies from their suspension points are also sketched, along with the arm length . and , at the top of the rotor, are the tiltmeter and three-PZTs (at 120° from one another—only one shown) for automated control of low frequency terrain tilts. The drawing is to scale and the inner diameter of the vacuum chamber is .

(Color) Section through the spin axis of the differential accelerometer inside the vacuum chamber. , vacuum chamber; , motor; OD, optical device (see Sec. II B); , ball bearings; ST, suspension tube; , coupling (balance) arm, located inside the suspension tube, with its three laminar cardanic suspensions (in red); and , center of mass of the two cylinder’s system (in blue the outer cylinder, in green the inner one, each). IP are the internal capacitance plates of the differential motion detector (Sec. II B), OP are the outer ones for whirl control (Sec. ???), and PC is the contactless inductive power coupler providing power to the electronics inside the rotor. The relevant distances, and , of the centers of mass of the inner and outer bodies from their suspension points are also sketched, along with the arm length . and , at the top of the rotor, are the tiltmeter and three-PZTs (at 120° from one another—only one shown) for automated control of low frequency terrain tilts. The drawing is to scale and the inner diameter of the vacuum chamber is .

Left hand side: the central laminar cardanic suspension of the GGG rotor, located at the midpoint to the coupling arm in order to suspend it from the suspension tube (shaft). Right hand side: the coupling arm inside the suspension tube (shaft) as seen from the top. Two cardanic laminar suspensions are located at its top and bottom ends. They suspend the test cylinders (not shown here) through two metal rings. The dimensions of the rings depend on the dimensions of the concentric cylinders, which have equal mass and therefore different sizes. The top and bottom rings refer to outer and inner test cylinders, respectively.

Left hand side: the central laminar cardanic suspension of the GGG rotor, located at the midpoint to the coupling arm in order to suspend it from the suspension tube (shaft). Right hand side: the coupling arm inside the suspension tube (shaft) as seen from the top. Two cardanic laminar suspensions are located at its top and bottom ends. They suspend the test cylinders (not shown here) through two metal rings. The dimensions of the rings depend on the dimensions of the concentric cylinders, which have equal mass and therefore different sizes. The top and bottom rings refer to outer and inner test cylinders, respectively.

(Color) Minimal model for the real instrument sketched in Fig. 1 (see text for details). On the left hand side the various parts are drawn with the same colors and labels as in Fig. 1. Here the midpoint of the coupling arm is indicated as MP. , , and refer to the dimensions of the coupling arm and the outer mass and inner mass suspension arms, respectively. , , and are the unit vectors of the corresponding beams. The offset vector , due to construction and mounting imperfections, is also indicated. On the right hand side we sketch one of the cylinders in the rotating reference frame , showing its principal axes of inertia , the position vector of its center of mass, and the angles , which are not the usual Euler angles, as discussed in the text. Below this figure, the small one to the right shows a typical deformation of one of the laminar suspensions of length , for instance, the central one. None of these figures is to scale.

(Color) Minimal model for the real instrument sketched in Fig. 1 (see text for details). On the left hand side the various parts are drawn with the same colors and labels as in Fig. 1. Here the midpoint of the coupling arm is indicated as MP. , , and refer to the dimensions of the coupling arm and the outer mass and inner mass suspension arms, respectively. , , and are the unit vectors of the corresponding beams. The offset vector , due to construction and mounting imperfections, is also indicated. On the right hand side we sketch one of the cylinders in the rotating reference frame , showing its principal axes of inertia , the position vector of its center of mass, and the angles , which are not the usual Euler angles, as discussed in the text. Below this figure, the small one to the right shows a typical deformation of one of the laminar suspensions of length , for instance, the central one. None of these figures is to scale.

Normal modes of the GGG rotor: the frequencies of the normal modes are plotted as functions of the spin frequency . The normal modes as predicted theoretically assuming anisotropic suspensions are shown as 12 solid lines in the case of zero rotating damping and as open circles in the case of nonzero damping (see text). The experimental results are plotted as filled circles and clearly agree with the theoretical predictions. The bisecting dot-dashed line separates the supercritical from the subcritical region. Three vertical thick lines are plotted in correspondence of three instability regions, their thickness referring to the width of the regions (see Sec. V G).

Normal modes of the GGG rotor: the frequencies of the normal modes are plotted as functions of the spin frequency . The normal modes as predicted theoretically assuming anisotropic suspensions are shown as 12 solid lines in the case of zero rotating damping and as open circles in the case of nonzero damping (see text). The experimental results are plotted as filled circles and clearly agree with the theoretical predictions. The bisecting dot-dashed line separates the supercritical from the subcritical region. Three vertical thick lines are plotted in correspondence of three instability regions, their thickness referring to the width of the regions (see Sec. V G).

Normal modes of the GGG rotor. On the left hand side we show a zoom from Fig. 4 in the very low spin frequency region, showing, in particular, the three-natural frequencies of the system in the zero spin case. On the right hand side, we plot a zoom from Fig. 4 in the small frequency region of both axes, showing the splitting into two lines of the low frequency mode because of anisotropy of the suspensions (the dashed line is the line as in Fig. 4).

Normal modes of the GGG rotor. On the left hand side we show a zoom from Fig. 4 in the very low spin frequency region, showing, in particular, the three-natural frequencies of the system in the zero spin case. On the right hand side, we plot a zoom from Fig. 4 in the small frequency region of both axes, showing the splitting into two lines of the low frequency mode because of anisotropy of the suspensions (the dashed line is the line as in Fig. 4).

Normal modes of the GGG rotor. A zoom from Fig. 4 showing one particular case of anticrossing of two modes.

Normal modes of the GGG rotor. A zoom from Fig. 4 showing one particular case of anticrossing of two modes.

Normal modes of the GGG rotor. The lowest-frequency instability region is zoomed in from Fig. 4.

Normal modes of the GGG rotor. The lowest-frequency instability region is zoomed in from Fig. 4.

(Color) Self-centering of one cylinder in the presence of nonrotating damping. Simulated plot showing the motion of the center of mass of the cylinder in the rotating reference frame (one cylinder model, ). The center of mass spirals inward from the initial offset value and large initial oscillations to a final value, much closer to the rotation axis.

(Color) Self-centering of one cylinder in the presence of nonrotating damping. Simulated plot showing the motion of the center of mass of the cylinder in the rotating reference frame (one cylinder model, ). The center of mass spirals inward from the initial offset value and large initial oscillations to a final value, much closer to the rotation axis.

Self-centering of one cylinder in the presence of nonrotating damping. The distance of the center of mass of the cylinder from the rotation axis is plotted as function of the spin frequency , in agreement with Eq. (C3). The same distance in the case of a point mass is plotted as a dashed line. See text for comments on their comparison.

Self-centering of one cylinder in the presence of nonrotating damping. The distance of the center of mass of the cylinder from the rotation axis is plotted as function of the spin frequency , in agreement with Eq. (C3). The same distance in the case of a point mass is plotted as a dashed line. See text for comments on their comparison.

## Tables

Input parameters for the numerical calculations: geometrical dimensions of the real bodies. (A mounting error of has also been used.)

Input parameters for the numerical calculations: geometrical dimensions of the real bodies. (A mounting error of has also been used.)

Input parameters for the numerical calculations: laminar suspensions data. In addition, a conservative value has been used taken from previous measurements of whirl growth (Ref. 14, Fig. 7).

Input parameters for the numerical calculations: laminar suspensions data. In addition, a conservative value has been used taken from previous measurements of whirl growth (Ref. 14, Fig. 7).

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