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Dynamical response of the Galileo Galilei on the ground rotor to test the equivalence principle: Theory, simulation, and experiment. I. The normal modes
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10.1063/1.2173075
/content/aip/journal/rsi/77/3/10.1063/1.2173075
http://aip.metastore.ingenta.com/content/aip/journal/rsi/77/3/10.1063/1.2173075

Figures

Image of FIG. 1.
FIG. 1.

(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 .

Image of FIG. 2.
FIG. 2.

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.

Image of FIG. 3.
FIG. 3.

(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.

Image of FIG. 4.
FIG. 4.

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).

Image of FIG. 5.
FIG. 5.

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).

Image of FIG. 6.
FIG. 6.

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

Image of FIG. 7.
FIG. 7.

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

Image of FIG. 8.
FIG. 8.

(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.

Image of FIG. 9.
FIG. 9.

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

Generic image for table
Table I.

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

Generic image for table
Table II.

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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/content/aip/journal/rsi/77/3/10.1063/1.2173075
2006-03-23
2014-04-20
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Dynamical response of the Galileo Galilei on the ground rotor to test the equivalence principle: Theory, simulation, and experiment. I. The normal modes
http://aip.metastore.ingenta.com/content/aip/journal/rsi/77/3/10.1063/1.2173075
10.1063/1.2173075
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