PENDULUM, PARTICULARLY FOR USE IN A CLOCK

Abstract

A pendulum, particularly for use in a clock Disclosed herein is a pendulum including a movable body connected to a stationary supporting structure for oscillation in a fixed vertical plane, and further including a control and stabilization system for controlling and stabilizing the oscillation amplitude of the movable body. The control and stabilization system includes a braking device which includes a permanent magnet and an electrically conductive element, one of which is attached to the movable body and the other of which is connected to the supporting structure such that when the movable body approaches one end of the oscillation field, the generated magnetic flux causes a braking of the movable body's motion.

The quality factor Q of the pendulum is substantially proportional to the inverse of a power, at least equal to the square, of the amplitude of the oscillation of the movable body when said amplitude is close to the desired amplitude of oscillation.

Claims

1. A pendulum, particularly for use in a clock, including a movable body connected to a stationary supporting structure for oscillation in a vertical plane fixed relative to said supporting structure, in an angular field of oscillation extending around an intermediate angular position of rest, and a control and stabilization system for controlling and stabilizing the amplitude of oscillation of the movable body, including a braking device for damping the oscillatory motion of the movable body, comprising a permanent magnet attached to the movable body and at least one stationary electrically conductive element arranged in a fixed position relative to said supporting structure in proximity of a position of reversal of the motion of the movable body and having a horizontal edge which intersects the plane of oscillation of the movable body, so that when the movable body approaches said position of reversal of motion the magnetic flux generated by the magnet starts to induce in the conductive element eddy currents determining a braking of the motion of the movable body; and a maintaining system for maintaining the oscillation of the movable body, including an actuator device designed to impart successive accelerations to the movable body, a sensor device designed to provide an indication of the position and/or velocity of the movable body in the field of oscillation, and driving means arranged to acquire the indications from the sensor device and drive the actuator device in a predetermined manner; wherein the position of the magnet on the movable body and the position of the at least one conductive element relative to the supporting structure are such that in operation the quality factor of the pendulum is substantially proportional to the inverse of a power, equal to or greater than the square, of the amplitude of oscillation of the movable body when said amplitude is close to the desired amplitude of oscillation.

2. A pendulum according to claim 1, wherein the control and stabilization system is arranged so that the motion of the movable body is reversed when the orthogonal projection of the magnet on the horizontal plane containing said edge of the at least one conductive element is substantially tangent to said edge.

3. A pendulum according to claim 2, wherein the braking device further comprises a second conductive element disposed stationary in proximity of the other angular position of reversal of the motion of the movable body and presenting a horizontal edge intersecting the plane of oscillation of the movable body; the control and stabilization system being arranged so that the motion of the movable body reverses at each of said conductive elements when the orthogonal projection of the magnet on the horizontal plane containing said edges is substantially tangent to one of said edges.

4. A pendulum according to claim 3, wherein said conductive elements are coplanar and said edges thereof are essentially parallel to each other.

5. A pendulum according to claim 1, wherein the sensor device comprises the permanent magnet of said braking device, and a magnetic field detector disposed stationary in the supporting structure of the pendulum and capable of detecting, at each oscillation of the movable body, the magnetic field generated by said permanent magnet.

6. A pendulum according to claim 5, wherein the detector device and/or the actuator device comprise at least one planar winding.

7. A pendulum according to claim 1, wherein the movable body is connected to the supporting structure by means of a suspension arrangement comprising a flexible element which is connected to the movable body and which extends between at least two opposing elements or supporting cheeks having respective symmetrical convex bearing surfaces facing each other, so that when the movable body oscillates said flexible element alternately winds(unwinds) on(from) one and then on(from) the other of said bearing surfaces; and wherein said bearing surfaces have respective profiles and mutual distances such that in operation the period of oscillation of the moving body varies, as the amplitude of oscillation varies, according to a curve presenting an absolute or relative minimum value of the period of the oscillation at a non-zero value of the oscillation amplitude.

8. A pendulum according to claim 7, wherein said means of suspension are arranged so that when the movable body oscillates said permanent magnet moves therewith remaining parallel to itself.

9. A pendulum, particularly for use in a clock, including: a movable body connected to a stationary supporting structure for oscillation in a vertical plane fixed relative to said supporting structure, in an angular field of oscillation extending around an intermediate rest position, and a control and stabilization system for controlling and stabilizing the oscillation amplitude of the movable body, comprising: a braking device for damping the oscillatory motion of the movable body, comprising a permanent magnet and an electrically conductive element, one of which is attached to the movable body and the other is connected to the supporting structure, such that when the movable body approaches one end of the oscillation field the magnetic flux generated by the magnet induces in the conductive element eddy currents causing a braking of the motion of the movable body; wherein the magnet and the conductive element, as well as the position of one of them on the movable body and the position of the other one of them relative to the supporting structure, are such that in the operation the quality factor of the pendulum is substantially proportional to the inverse of a power, at least equal to the square, of the amplitude of the oscillation of the movable body when said amplitude is close to the desired amplitude of oscillation.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Further objects, features and advantages of the invention will appear from the detailed description below, provided purely by way of non-limiting examples with reference to the attached drawings, in which:

[0036] FIG. 1 is a schematic representation of a pendulum according to the present invention,

[0037] FIG. 2 is a diagram showing the variation of the period, in ordinates, as a function of the oscillation amplitude, in abscissae, in the case of a simple gravity pendulum or mathematical pendulum, and in a pendulum according to the invention,

[0038] FIG. 3 is a partial representation in elevation of the pendulum according to FIG. 1 in two operating positions/conditions,

[0039] FIG. 4 is a partial representation in top view of the pendulum in the condition according to FIG. 3,

[0040] FIG. 5 is a logarithmic scale diagram illustrating the measured factor of merit Q in a real pendulum according to the invention, as a function of a parameter x univocally related to the amplitude of oscillation ,

[0041] FIGS. 6 and 7 are partial representations of additional pendulums according to the invention,

[0042] FIGS. 8a through 8d are schematic representations of further variants,

[0043] FIG. 9 is a diagram showing the relative circular error, T/T, as a function of parameter x, in a pendulum in which a flexible suspension element extends into the gap between shaped supporting cheeks, for different values of the free width of the gap FGW,

[0044] FIG. 10 is a schematic representation of an electronic oscillation control and maintaining system that can be used with a pendulum according to the invention,

[0045] FIG. 11 is a diagrammatic representation in elevation view of part of a pendulum according to the invention,

[0046] FIG. 12 is a perspective view of part of the pendulum according to FIG. 11, and

[0047] FIG. 13 is a diagrammatic representation of a pendulum according to the prior art transformed into a pendulum according to the invention by retrofitting some devices.

DETAILED DESCRIPTION OF EMBODIMENTS

[0048] In FIG. 1 with 1 is indicated an improved pendulum according to the invention.

[0049] In the embodiment illustrated therein the pendulum 1 comprises an operationally stationary support structure including a rigid body 2, e.g. of aluminum, in which two pairs of parallel, preferably but not necessarily coplanar, supporting elements or cheeks A, B and C, D, made for example of a metallic material, are arranged.

[0050] In the illustrated example the supporting elements A, B and C, D are cylinders with a circular cross section, arranged with their respective longitudinal axes horizontal and parallel.

[0051] Such cylinders can be made for instance of Al 7075 (Ergal).

[0052] Between adjacent cylinders of each pair A, B and C, D are defined respective interspaces or gaps, having a same predetermined width.

[0053] A flexible suspension element, made of an essentially inextensible metallic material, exhibiting a very low coefficient of thermal expansion and a high Young's modulus value, is indicated 4 in FIG. 1. Such a suspension element 4 is, for example, a tantalum tape, having an approximate thickness of 10 m and a width of 10 mm.

[0054] In other embodiments, a thread or multiple ribbons or threads may be used instead of a tape.

[0055] At the top of body 2 the ends of suspension element 4 are connected to a regulating device 40 whose functions will be described later.

[0056] From the regulating device 40 the two end branches of the suspension element 4 deflect downward at respective horizontal cylindrical deflection rollers or idlers 41 and then extend into the interspaces or gaps defined between the pairs of cylinders or cheeks A, B and C, D. Below these cylinders the tape 4 forms a kind of loop wrapped around an oscillatable body or bob 5 having a predetermined mass. Such a body 5 may be, for example, cylindrical or spherical in shape, and in FIG. 1 its horizontal axis, which is orthogonal to the plane of oscillation, is indicated 5a.

[0057] Between body 5 and cylinders A-D the two branches 4a and 4b of tape 4 are essentially parallel to each other, but this condition is not binding, since in alternative embodiments these branches can be slightly converging or diverging upward.

[0058] By L in FIG. 1 is indicated the vertical distance between the axis 5a of body 5 and the horizontal plane containing the axes of cylinders A-D when body 5 is at its lowest position, and H denotes the distance between the plane containing the axes of cylinders A-D and the horizontal plane containing the axes of idlers 41. As an example, the distance L can be 15 cm or greater, and the distance H can be 5 cm or greater.

[0059] The regulating device 40 is arranged to implement a fine adjustment of the H/L ratio as the temperature changes, so that the thermal expansion of the flexible element 4 is compensated by that of the supporting body 2 and the length L of the pendulum is maintained essentially constant.

[0060] The arrangement described above is such that when body 5 oscillates in a vertical plane below cylinders A-D, the two branches 4a and 4b of tape 4 wind/unwind on/from one and then on/from the other of the curved bearing surfaces of cylinders A, B and C, D, respectively, between which the interspaces or gaps mentioned above are defined.

[0061] By virtue of the winding/unwinding of the tape 4 on/by the cylinders A, B and C, D, in the oscillation the body 5 follows a trajectory having a radius of curvature that is not constant, but varies between a maximum value and an absolute minimum value.

[0062] As is shown in FIG. 2, in the case of an ideal or simple mathematical pendulum the amplitude of the oscillation varies as a function of the period of the oscillation T in accordance with a curve W that is basically an arc of a parabola, having an absolute minimum period value T.sub.0 when the amplitude of the oscillation is 0.

[0063] In contrast, in a real pendulum, such as that described with reference to FIG. 1, the relationship of the period T as a function of the oscillation amplitude is, to an acceptable approximation, of the type represented by the curve Y in FIG. 2, which has an absolute minimum value T.sub.0 of the oscillation period T at a nonzero value .sub.0 of the oscillation amplitude . Such a point .sub.0,T.sub.0 represents an isochronous working point.

[0064] Reverting now to FIG. 1, with the pendulum 1 there is associated an oscillation maintaining system, as a whole denoted by 6, arranged to stabilize the angular amplitude of oscillation , so as to keep the period of oscillation T substantially within a small neighborhood of the minimum value T.sub.0.

[0065] The oscillation maintaining and stabilizing system 6 associated with pendulum 1 in FIG. 1 includes a permanent magnet M attached at the lower surface of the oscillating body 5. Such a magnet M is, for example, cylindrical, of the neodymium-iron-boron type, and is magnetized according to a radial axis with respect to the horizontal axis 5a of the body 5, i.e., according to the vertical longitudinal axis of the magnet M in the condition of FIG. 1.

[0066] The oscillation maintaining and stabilizing system 6 also comprises two electrically conductive elements 7 and 8, preferably coplanar, attached to a stationary support plate 9 connected in a non-illustrated manner to the support structure 2 of the pendulum 1. The elements 7 and 8 are, for example, in the form of parallelepiped plates or ingots, made of copper or other material having a good electrical conductivity, and are arranged on the plate 9 in a horizontally spaced relation.

[0067] In the condition shown in FIG. 1, the magnet M is in a position horizontally intermediate between the positions of elements 7 and 8, at a vertical distance of a few millimeters, say 4-5 millimeters, from the horizontal plane in which the upper surfaces or faces of said elements 7, 8 extend.

[0068] To a good approximation, it can be assumed that the oscillating body 5 and the portions of the flexible element 4 extending between said body and the pairs of cylinders A, B and C, D form on the whole a sort of articulated parallelogram, so that during the oscillation the body 5 and the magnet M remain substantially parallel to themselves.

[0069] As can be seen in FIGS. 1, 3 and 4, elements 7 and 8 have respective upper edges 7a and 8a, facing each other, extending in (preferably) parallel directions, intersecting (preferably at 90 degrees) the plane of oscillation of body 5. When body 5 is near a position of reversal of its oscillation motion the portions of elements 7 and 8 located near said edges 7a, 8a are invested by a portion of the magnetic flux generated by permanent magnet M.

[0070] The mutual horizontal distance S between the edges 7a and 8a of elements 7 and 8 is such that in operation no significant magnetic flux generated by magnet M flows through them, except when body 5 reaches an end position (or position of reversal) of the oscillatory motion.

[0071] A magnetic flux flows either through element 7 or element 8 in the phases of approach to and departure from a position of reversal of motion and generates eddy currents in the conductive element concerned, which currents in turn generate a braking force on the magnet M and, by the Joule effect, cause in that conductive element a thermal dissipation of the energy lost by the oscillating body 5. This dissipation of energy causes a reduction in the quality factor Q of the pendulum, to an extent which depends on the amplitude of oscillation according to a relationship that is a function of the field distribution of the magnet M and the velocity of the oscillating body 5.

[0072] The analytical derivation of this relationship is rather complex, but the important fact is that, as the inventor has found, the Q factor is essentially inversely proportional to a power, at least equal to the square, of the oscillation amplitude when the reversal of the motion occurs as soon as the magnetic flux from magnet M begins to invest element 7 or element 8 at its edge 7a or 8a.

[0073] The graph in FIG. 5 shows the results of measurements made with a prototype pendulum according to FIG. 1 with a length L of 16 cm, with elements A-D having a diameter of 13 mm, a magnet M having a diameter of 6 mm, and with two conductive elements 7 and 8 made of silver, half an ounce each, arranged at a vertical resting distance of 4 mm from the bottom face of the magnet M.

[0074] That graph shows on the ordinate the base-10 logarithm of the measured pendulum quality or merit factor Q as a function of the base-10 logarithm of the maximum horizontal displacement or elongation x of the vertical axis of the magnet M with respect to the direction of that axis at rest (x being measured in the horizontal plane containing the upper surfaces or faces of conductive elements 7 and 8), for different values of the horizontal separation distance S (in mm) between the conductive elements 7 and 8. The horizontal displacement x=x() is univocally correlated with the magnitude of the corresponding angular displacement or amplitude of oscillation of body 5, but is measurable more easily than . Some direct values of Q and x have been marked in the graph in FIG. 5 for ease of reference.

[0075] The slope of the various solid-line curves in the graph in FIG. 5 confirms that the value of Q is essentially proportional to .sup.2 in a range of amplitudes near the position of the edge 7a, or 8a, of the brake 7, or 8.

[0076] In the above graph the curve corresponding to S=50 mm for x greater than or equal to 22 exhibits initially a marked reduction in slope, and then a dotted-line extension where the log Q values have not been measured but are an educated guess of what would be measured if gradually increasing amplitudes were imposed on the oscillating body 5 until the magnet M travels beyond a conductive element 7 or 8, similarly to what happens in a Hatot pendulum according to U.S. Pat. No. 1,527,255 cited above when the end of the magnet provided therein extends through and beyond the metal element acting as a brake at an end position of the oscillating body. Said extension of the curve for S=50 mm has a very small slope, corresponding at most to a proportionality of the Q-factor to .sup.1, and thus would not represent per se an improvement over Harrison's solution in which this relationship is produced by air resistance.

[0077] This confirms that in Hatot's pendulum, in which at each reversal of the oscillation the magnet passes through and beyond a brake before the motion is reversed, the prolonged interaction between the magnet field and the brake causes a considerably increased heat dissipation that is not much dependent on the amplitude of the oscillation. Thus, the solution according to U.S. Pat. No. 1,527,255 does not achieve much stabilization of the amplitude of the pendulum oscillation, and in any case no better than in Harrison's pendulums.

[0078] Returning to FIG. 5, it is important to note that the electromagnetic brake stabilization system described above, in addition to offering a steeper reduction in the factor of merit Q than that produced by air resistance in Harrison's clocks, also allows to select at will, by properly positioning the brake(s), the values of the amplitudes at which one wishes an abrupt reduction in Q to occur and thus makes it easier to achieve the desired operating point.

[0079] According to Harrison's approach, the achievement of the desired working point was, by his own admission, extremely complicated because, since air resistance is difficult to change, very careful design and implementation were required, by trial and error, in order to obtain the desired reciprocal positioning between the working oscillation amplitude and that at which the minimum of the circular error curve occurs, or a desired slope of the circular error curve with the amplitude around the working point, such that it could be used for compensating changes in air resistance.

[0080] This process involved difficult experimental convergence, with significant time expenditure, for choosing the radius of curvature of the suspension cheeks and the characteristics of the suspension tape (material and thickness), as well as the mass of the moving body or bob.

[0081] FIGS. 6 and 7 show two variants of the pendulum shown in FIG. 1. In these figures, parts and elements already described have again been given the same reference numbers used previously.

[0082] FIG. 6 shows a variant of the pendulum in which the movable body 5 is in the shape of a cylinder, however not with a circular cross section, but with a horizontally elongated cross section, in order to reduce aerodynamic dissipation.

[0083] In the variant according to FIG. 7 there are only three supporting cylinders or cheeks, indicated A, B and C, and the two branches 4a and 4b of tape 4 extend into respective gaps defined between the end cylinders A and C and the intermediate cylinder B.

[0084] Otherwise, the structure and modus operandi of the pendulum according to FIG. 7 also substantially correspond to those of the pendulum according to FIG. 1.

[0085] Additional implementation variants are diagrammatically illustrated in FIGS. 8a-8b.

[0086] In the variant according to FIG. 8a the permanent magnet M carried by the oscillating body cooperates with only one stationary conductive element, which in the example shown is element 7 (but alternatively could be element 8). In this type of variant for the purpose of oscillation amplitude stabilization the motion of the oscillating body 5 is electromagnetically braked only once per forward and return oscillation.

[0087] FIG. 8b shows a variant comprising two stationary permanent magnets M1 and M2 cooperating with a conductive element 50 attached to the oscillating body 5. The element 50 has two edges 50a and 50b intended to be invested by the flux generated by the magnets M1 and M2, respectively, when the oscillating body 5 is in the vicinity of one and the other motion reversal position, respectively. In this variant one or both magnets can also be used to detect the position and/or velocity of the oscillating body. In a further variant not shown in the drawings, one of the magnets M1, M2 in FIG. 8b can be dispensed with.

[0088] In the variant according to FIG. 8c, a fixed permanent magnet M is provided, cooperating with two conductive elements 7 and 8 attached to the oscillating body 5. The edge or edges 7a and 8a of these elements, which are closer to each other, are invested by the flux generated by the magnet M when the oscillating body is in the vicinity of one and the other position of motion reversal, respectively. In a further variant not shown in the drawings, one of the conductive elements 7 and 8 in FIG. 8c can be dispensed with.

[0089] The variant according to FIG. 8d comprises two permanent magnets M1 and M2 attached to the oscillating body 5 at angularly spaced positions and cooperating with a stationary conductive element 50. The edges or corners 50a and 50b of such an element 50 are invested by the flux generated by magnets M1 and M2 when the oscillating body 5 is in the vicinity of one and the other motion reversal positions, respectively. In a further variant not shown in the drawings, one of the magnets M1, M2 in FIG. 8d can be dispensed with.

[0090] In variants with two permanent magnets (M1, M2; FIGS. 8b and 8d), it is convenient that these magnets present to the associated conductive element 50 polar ends of the same type, e.g. north type as indicated by n in the aforementioned FIGS. 8b and 8d (where s indicates polarities of the opposite type, i.e. south).

[0091] Also for the variants according to FIGS. 8a to 8d the arrangement and positioning of the magnet(s) and the conductive element(s), on the movable body 5 and relative to the supporting structure 2, are such that in the operation the quality factor Q of the pendulum is substantially proportional to the inverse of a power, equal to (or greater than) the square, of the amplitude of oscillation of the movable body 5 when said amplitude is close to the desired amplitude of oscillation .sub.0.

[0092] A further important aspect when making a cylindrical-cheek pendulum concerns the sizing of the diameter of the cheeks and the width of the gaps defined between them, depending on the length of the pendulum and the weight of the oscillating body 5.

[0093] In this regard, it was found that an additional effect may have a major impact on the curve that correlates the circular error T/T or period T with the amplitude of the oscillation: the spring effect generated by the tape 4 to which the oscillating body 5 is suspended, when said tape winds/unwinds on/from the cheeks A-D.

[0094] This spring effect interacts with the force of gravity and becomes, in a relative sense, increasingly important as the amplitude of oscillation decreases.

[0095] For small oscillation amplitudes such an effect can even change the curve that correlates period T with amplitude in such a way that the absolute minimum value T.sub.0 of the period disappears in the graph of FIG. 2 (which does not take into account the spring effect considered here). This phenomenon is illustrated in the graph in FIG. 9, in which solid line curves are plotted showing the circular error T/T (in ordinates) as a function of the previously defined parameter x (in abscissae), which is uniquely related to the values of the oscillation amplitude , for different values of the free width FGW (in m) of the gaps between the A-D cheeks (by free width of the gaps is meant the difference between the separation distance between the cheeks and the thickness of the tape 4). Also shown in the graph in FIG. 9 is a dashed line curve RF representing the effect of the additional pullback force due to the spring effect described above.

[0096] The curves in FIG. 9 are for a pendulum with a length L equal to 16 cm, with cheeks A-D having a diameter of 13 mm, and FGW values ranging from 0 to 30 m. The curve for FGW=0 corresponds to an embodiment in which the cheeks are resiliently loaded toward each other. The graph shows that for decreasing FGW values smaller than 10 m, the curves that correlate the circular error T/T with x, and ultimately with the amplitude of the oscillation, firstly present a relative (but still useful) minimum value, rather than an absolute minimum, and then for even smaller FGW values no longer present a useful minimum value. Furthermore, for curves that exhibit a minimum (absolute or relative) value of T/T, the value and the abscissa of the latter vary as a function of the free amplitude FGW of the gaps between the cheeks. This is determined by the fact that if FGW is greater than zero, when the amplitude of oscillation is so small that the tape 4 never comes to rest on the cheeks the length of the pendulum is greater than L, implying that the period T is significantly greater than that for the working amplitude. As the amplitude is then increased from that situation, the period T passes through a junction zone in which the contribution to the period T by the transition time of the tape 4 across the free gap is gradually reduced and consequently the period T gradually approaches the values expected in the case of a zero free gap, with the result that a minimum period is produced whose position is determined by both the width FGW of the free gap and the radius of the cheeks.

[0097] As is shown in FIG. 10, the pendulum includes a system for controlling and maintaining the oscillation of the movable body 5, including a sensor 10 designed to provide electrical signals indicative of the passage of the movable body 5 by a predetermined angular position (or of the speed of oscillation of body 5).

[0098] In the exemplary embodiment shown, the sensor 10 comprises a permanent magnet 11 attached to the moving body 5, which conveniently may be the same magnet M described above, and an associated detector 12, e.g. a winding or solenoid, attached to the stationary support structure 2 of the pendulum. However, other sensor means, such as optical sensors, can be used.

[0099] Detector 12 is connected to an electronic circuit as a whole indicated 13 in FIG. 10.

[0100] In the illustrated embodiment such an electronic circuit 13 includes an amplifier 14 having an input connected to the output of the detector 12, and an output connected to a variable attenuator 16. The latter comprises, for instance, a variable-ratio resistive voltage divider, including two resistors 17 and 18, the second of which has a variable resistance.

[0101] An additional amplifier 15 has its input connected to the interconnected terminals of resistors 17 and 18 and its output connected to the drive input of a thrust actuator 20 associated with the movable body 5. Such an actuator 20 is of a per se known type, for example of the solenoid type, such as a so-called voice coil, and is able to generate a magnetic field which, by interacting with a permanent magnet attached to the oscillating body 5, conveniently the same magnet M described above, is capable of applying to body 5 a force adequate to maintain the oscillation, compensating the effects of friction and other forces that would tend to dampen the oscillations of said body 5, and thus maintain the amplitude of the oscillation substantially at the value for which the period T of the oscillation remains at the minimum value corresponding to the isochronous working point. This is accomplished efficiently by applying to the body 5 a sine-wave force in phase with the velocity of said body 5.

[0102] The output of amplifier 14 is also connected to an automatic gain controller AGC, which includes an amplitude detector 19 connected to a first input of a differential amplifier 21, a second input of which is connected to a DC voltage source VS via a resistive voltage divider 22. The output of amplifier 21 is connected to the variable resistor 18 of resistive attenuator 16 in such a way that the attenuation introduced by that attenuator is controlled by said amplifier 21.

[0103] The intensity of the force applied to the oscillating body 5 is regulated in a closed-loop by the automatic gain controller AGC.

[0104] The electronic circuit 13 is devised so that its gain G exhibits an essentially zero phase rotation, so that the known null-phase oscillation condition for the ring gain GB (where is the resonator transfer function) is verified at the natural frequency of the pendulum, at which the phase rotation of is null. As is well known, however, the oscillation condition for the modulus of the ring gain is |G|=1 in steady state, and since is proportional to the factor Q, it follows that G must be inversely proportional to Q. So, if the value of Q varies as .sup.2, the gain G must be proportional to the square of the desired oscillation amplitude. This fact makes it possible on the one hand to measure Q without disturbing the oscillations, by means of a gain measurement, and on the other hand to control the oscillation amplitude by varying the gain G.

[0105] The Q-factor measurements shown in FIG. 5 were obtained in that way.

[0106] The electromagnetic brakes 7 and 8, which in a certain range of oscillation amplitudes reduce the value of Q in a measure inversely proportional to the square of the amplitude of the oscillation, thus achieve a passive stabilization of the amplitude of the oscillation, which to a certain extent is possible even without an automatic gain controller because in principle it depends only on the gain G.

[0107] The actuator device 20 can be of any other known type and in particular, if the Q factor is very high, also of an optical type based on the use of radiation pressure.

[0108] FIGS. 11 and 12 schematically show an embodiment of the position or speed sensor 10-12 and the actuator device 20.

[0109] In that embodiment the detector 12 comprises two electrically insulating, parallel, vertical, facing plates 30 and 31 on which, for example by a printed circuit technique, respective planar windings 32 and 33, essentially in the shape of an 8, are provided and connected to each other.

[0110] The plates 30 and 31 are connected to a stationary support structure not shown, and are located below plate 9 which bears conducting elements 7 and 8 (FIG. 11), so that when body 5 oscillates a part of the magnetic flux generated by the magnet M carried by said body 5 periodically passes through said windings 32 and 33.

[0111] An additional insulating plate 34 is arranged horizontally below plate 9, between plates 30 and 31, and bears the winding 12 described earlier with reference to FIG. 10, which is also conveniently planar and provided by a printed circuit technique. The relative position of plate 34 with respect to plates 32 and 33 is such that the mutual inductance between the windings of sensor 12 and actuator 20 is cancelled.

[0112] The arrangement is such that a substantial part of the flux lines from the lower pole of the magnet M close back into the other pole of that magnet without passing again through the above-mentioned windings 12, 32 and 33 after the first time.

[0113] Below plate 34 and between plates 30 and 31 there is arranged an additional vertical insulating plate 35 bearing the circuit components 13, 16 and AGC of FIG. 10.

[0114] FIG. 13 illustrates a further embodiment of the present invention implemented by a retrofitting of a conventional pendulum 1. In the illustrated example, the pendulum 1 comprises an operationally oscillating body 5 carried by a substantially rigid rod 40 and suspended from a supporting structure 2 by means of a so-called knife blade 41. Such a pendulum may be of a purely mechanical type, or of an electromechanical type.

[0115] For the stabilization of the oscillation amplitude, according to the invention the pendulum is equipped with a permanent magnet M attached to the oscillating body 5 and with two conductive elements 7 and 8 made and arranged as more fully described hereinbefore (or equipped with the conductive elements and magnets of the alternative embodiments described above with reference to FIGS. 8a-8d).

[0116] With the retrofitting described above, it is possible to appreciably improve the performance of an ordinary conventional pendulum.

[0117] It is understood that embodiments presented herein are meant to be exemplary. Although the present disclosure has been described in detail with reference to certain preferred configurations thereof both in the specification and in the claims, other versions are possible. Embodiments of the present disclosure can comprise any combination of compatible devices/features described herein and/or shown in the figures, and these embodiments should not be limited to those expressly illustrated and discussed. For instance and not by way of limitation, the appended claims could be modified to be multiple dependent claims so as to combine any combinable combination of elements within a claim set, or from differing claim sets. Claims depending on one independent claim could be modified so as to depend from a different independent claim. Therefore, the spirit and scope of the disclosure should not be limited to the versions described above.

[0118] While the foregoing written description of the disclosure enables one of ordinary skill to make and use what is considered presently to be the best mode thereof, those of ordinary skill will understand and appreciate the existence of variations, combinations, and equivalents of the specific embodiments, methods, systems, and examples herein. The disclosure should therefore not be limited by the above described embodiments, methods, systems, and examples. Furthermore, certain terminology has been used for the purposes of descriptive clarity, and not to limit the present disclosure. It is therefore intended that the following appended claims include all such alterations, modifications and permutations as fall within the true spirit and scope of the present disclosure. No portion of the disclosure is intended, expressly or implicitly, to be dedicated to the public domain if not set forth in the claims.