EP3740820B1

EP3740820B1 - Horological oscillator

Info

Publication number: EP3740820B1
Application number: EP19700538.2A
Authority: EP
Inventors: Mohammad Hussein Kahrobaiyan; Ilan Vardi; Simon Henein; Billy NUSSBAUMER; Etienne THALMANN
Original assignee: Ecole Polytechnique Federale de Lausanne EPFL
Current assignee: Ecole Polytechnique Federale de Lausanne EPFL
Priority date: 2018-01-18
Filing date: 2019-01-17
Publication date: 2021-12-22
Anticipated expiration: 2039-01-17
Also published as: WO2019141789A1; EP3740820A1

Description

Technical Field

The present invention relates to the technical field of horology. More particularly, it relates to horological oscillators comprising at least two coupled inertial masses.

State of the art

Coupled horological oscillators are known in the documents EP2491463 , CH700747 and FR322419 . In each of these documents, the oscillator comprises two balance wheels are coupled to each other by means of teeth. These teeth are either provided on the periphery of the balances, or on gear wheels concentric with each balance, such that the balances turn synchronously in opposite directions. A restoring force is provided by a hairspring associated with each balance, and an escapement is arranged to interact with one or both balances.
The aim of these arrangements is to improve the isochronism of the oscillator, and to minimize the influence of gravity and shocks thereupon.
However, the use of teeth to couple the balances in rotation results in nonnegligible friction losses. Plus, the use of a conventional escapement in these arrangements is associated with a significant loss of energy in the gear train of the movement in which the oscillator is integrated. In essence, the stop-and-go discretization of time introduces energy losses such as audible ticking and significant accelerations of the movement's gear wheels. In order to overcome this issue, moves have been made to develop oscillators that operate without a conventional escapement, by exploiting multiple degree-of-freedom oscillators. Examples of such are described in EP2894521 , EP3095010 . These documents contain a definitive discussion of the conditions required for isochronism to be present in an oscillator, which thus need not be repeated at length here.
However, these oscillators are sensitive to the direction of the gravity vector, i.e. to their orientation in space. Plus, they are also sensitive to angular and linear shocks, and as a result are unsuitable for use in a wristwatch or other portable timepiece.
Another attempt to overcome these limitations is disclosed in figures 10-13 of document US 2017/269551 . In this realization, a pair of inertial bodies, each supported by a corresponding parallelogram flexure pivot system, are arranged to translate along parallel axes in opposite, substantially rectilinear directions. The parallelogram pivot systems prohibit rotation of the inertial bodies, with the inertial bodies maintaining the same orientation in space with respect to the frame. The two bodies are joined by an S-shape coupler link which is joined to the midpoint of each body on an outer side thereof, and passes between the bodies. This coupler link itself rotates about an anchor point fixed near the center of the system in order to couple the translations of the oscillating inertial bodies such that when one translates to the right, the other translates to the left and vice versa. Such an arrangement is clearly susceptible to rotational shocks and accelerations in the plane of the oscillator, since such accelerations will augment or diminish the amplitude of oscillation depending on the direction of the rotational shock in relation to the direction of translation of the inertial bodies at the moment of the application of the shock, and hence cannot give satisfactory isochronism.
US 9 465 363 describes an oscillator system in which four inertial bodies, supported by flexure pivots, are connected in a square in a rotationally-symmetric manner by flexible blade flexures. These flexures are all joined at one end to a central ring arranged to be driven in a circular or oval pathway by a crank, and the other ends of the flexures are attached to respective levers extending away from the inertial bodies. The inertial bodies are hence obliged to oscillate in opposite directions by opposed pairs, each pair acting at 90° to the other. This arrangement results in undesired bending of the flexures during oscillation, which causes variations in the distances between the ends of the flexures where they join rigid levers attached to the inertial bodies. This hence results in the oscillations of each opposed pair of inertial bodies significantly influencing the other, lending component to the oscillation of each pair which is determined by the state of the other pair. This is again clearly unsatisfactory from an isochronism perspective.
US 1 595 169 describes an oscillator comprising a single inertial body arranged to translate in two directions in its plane, according to a substantially circular pathway. This arrangement is clearly influenced by both translational and rotational shocks and is hence unsatisfactory. FR 1 539 670 describes an oscillator comprising two inertial bodies, each arranged to rotate about an axis, and a coupler link, coupling the two inertial bodies.
Finally, figure 57 of WO 2015/104692 describes an oscillator based on an orbiting inertial body describing a bidirectional translation in a plane, following a circular or oval pathway. Compensating masses attached to levers serve to eliminate undesired forces. Again, this system is susceptible to shocks and is hence unsuitable for use in portable applications such as wristwatches.
An object of the present invention is thus to propose horological oscillators which are exempt from the above-mentioned drawbacks, and thereby to create oscillators which substantially satisfy Newton's model for isochronism and are insensitive to gravity, linear shocks and angular shocks. Such oscillators are hence suitable for integration in a wristwatch or similar.

Disclosure of the invention

More specifically, the invention relates to a horological oscillator comprising a first inertial body arranged to rotate with respect to a first axis, and a second inertial body arranged to rotate about a second axis parallel to said first axis. These inertial bodies can, for instance, be balance wheels of any convenient form. At least one elastic element is provided, this elastic element being arranged to apply a restoring torque to at least one of said inertial bodies so as to urge said inertial body towards a neutral position. For instance, this elastic element may be arranged between an inertial body and a supporting framework which may be fixed or mobile.
According to the invention, the oscillator further comprises a substantially rigid coupler link comprising at least one bar or rod attached directly or indirectly at a first pivot point to said first inertial body and likewise directly or indirectly at a second pivot point to said second inertial body such that said inertial bodies are able to rotate synchronously about their respective axes in the same or opposite directions of rotation, depending on the geometry of the oscillator. These pivot points can be constituted either by pin pivots, or flexure pivots such that the inertial bodies can pivot with respect to the coupler link at these points. The angular velocities of these inertial bodies may be of the same magnitude or, in the case that the moment of inertia and/or the radius at which the coupler link is attached to each inertial body is different, may be of different magnitudes. Said pivot points are distinct from, and hence remote from, the axes of rotation of the inertial bodies, and the coupler link is in consequence arranged to couple the inertial bodies by transmitting force between them according to a vector joining said first and said second pivot points.
As a result, isochronism and sensitivity to gravity and shocks are improved with respect to a conventional single balance wheel, and the frictional losses due to the use of teeth in the prior art are significantly reduced. The same advantages are present with respect to the other more complex coupled oscillators mentioned above, since the claimed coupling in rotation tends to cancel out the effect of gravity and both linear and rotational shocks, and likewise improves the isochronism. Furthermore, the claimed arrangement is particularly suitable for being driven by a crank mechanism rather than a conventional escapement, which further permits reducing energy losses in a timepiece movement.
Advantageously, said first inertial body is mounted pivotally on a supporting element such as a fixed frame or the coupler link of another oscillator by means of a first flexure pivot and said second inertial body is mounted pivotally on said supporting element by means of a second flexure pivot, said flexure pivots not only serving as pivots, but also as said elastic elements. This arrangement simplifies manufacture, since it reduces the number of parts that need to be individually manufactured and assembled. Monobloc constructions are hence possible.
Advantageously, the coupler link is subject to a restoring torque with respect to each of said inertial bodies, said restoring torque being provided at at least one, preferably at each, of said first and second pivot points, for instance by means of an appropriate elastic element such as a coupling spring or by the inherent elasticity of a flexure pivot. Further advantageously, said coupler link is attached said first inertial body by means of a further flexure pivot, and is attached to said second inertial body by means of a yet further flexure pivot, said flexure pivots constituting said elastic elements. In this case again, the flexure pivots serve also to provide the restoring torque between the coupler link and the inertial bodies. Simplicity and efficient manufacture are thus assured.
Advantageously, a first line joining said first axis to said first pivot point is parallel to a second line joining said second axis to said second pivot point. Said first and second lines are preferably perpendicular to a third line joining said first pivot point to said second pivot point when said oscillator is in a neutral position, i.e. when the net force generated by the various elastic elements are not tending to urge the coupler link in any particular direction. This geometry results in an oscillator that is insensitive to gravity, linear shocks and angular shocks, and which is also dynamically balanced for longer stroke than in any other case.
Advantageously, the oscillator comprises at least two masses mounted movably, e.g. slidably, on said coupler link, one on either side of the center of mass of said coupler link. These masses, which can be displaced against friction or otherwise be provided with suitable blocking means to ensure that they remain in position during oscillation, permit modification of isochronism defect without affecting the nominal frequency of oscillation. This latter can be modified in the normal way, for instance by means of adjustable masses such as screws provided on at least one of the inertial bodies.
Advantageously, a first and second horological oscillator according to the invention can be combined together into an oscillator system, these oscillators being coupled in series or in parallel. This provides an oscillator system with two degrees of freedom that can easily be driven by a crank arrangement. The isochronism can thus be improved, and the influence of gravity and shocks can be diminished.
In order to mount two oscillators in series, the inertial bodies of the first horological oscillator can each be mounted pivotally on the coupler link of the second horological oscillator such that the two oscillators preferably act perpendicularly to one another. The coupler link of the second oscillator can be shaped accordingly, e.g. by being provided with a hollow frame-shaped part. Such an oscillator system can be driven simply by a pin-and-slot crank arrangement, with the pin being situated on the coupler link of the first oscillator. This pin can then describe unidirectional two-dimensional trajectories which are ideally circular but may also be elliptical or oval. Such an oscillator system, if properly dynamically balanced, is insensitive to gravity, linear shocks and angular shocks.
Alternatively, the oscillator system can be arranged in parallel, i.e. with both oscillators being joined at a common point and each oscillating with respect to a frame. In such a case, the first horological oscillator and the second horological oscillator are preferably arranged to act at substantially 90° to each other, and are coupled by means of a rigid body from which extend respective linkages each attached to a respective coupler link. This rigid body can, e.g. support a pin which can be driven as part of a pin-and-slot crank arrangement as mentioned above. In the context of the invention, "parallel" is to be interpreted functionally, in a similar manner to in electronics, rather than geometrically, since the two oscillators are geometrically orthogonal but operate in parallel as they are driven from the same point and each oscillate independently.
Advantageously, at least one of the linkages may comprise one of:

a single-bar linkage comprising pinned pivots or flexure pivots;
a double-bar linkage comprising pinned pivots or flexure pivots;
a single or double blade flexure;
at least one rigid bar provided with a single blade flexure at each end;
a Robert's four-bar linkage comprising pinned pivots or flexure pivots;
at least one four-bar Watt linkage comprising pinned pivots or flexure pivots;
a compound parallelogram linkage comprising pinned pivots or flexure pivots.

Advantageously, the first horological oscillator and the second horological oscillator are arranged such that they are related by a 90° rotation in the plane of the oscillator system. Alternatively, they can be arranged such that one is a mirror reflection of the other about a plane equidistant from the coupler link of each horological oscillator, said plane being perpendicular to the plane of the oscillator system. In either case, the influence of gravity, linear and angular shocks can be all be minimized, and the mirror reflection arrangement is particularly compact.
Advantageously, the first and second horological oscillators are coupled by means of a two-layer coupling system. To this end, the first horological oscillator comprises a first rigid body arranged in a first layer and attached to the coupler link of said first horological oscillator, and the second horological oscillator comprises a second rigid body arranged in a second layer and attached to the coupler link of said second horological oscillator. The system is arranged such that the rigid bodies are constrained to translate together and are free to pivot one with respect to the other by means of a rotary joint, flexures or similar. Such an arrangement permits great flexibility in design, and can utilize particularly advantageous arrangements of flexures or conventional pivots so as to achieve the desired guidance in translation of the rigid bodies. More than two layers can be provided if desired.
The invention also relates to a timepiece movement comprising an oscillator or an oscillator system as defined above.
Advantageously, the timepiece movement comprises a source of energy kinematically connected with said oscillator or said oscillator system by means of a crank attached to at least one coupler link. The oscillator or oscillator system can thus regulate the running of the timepiece movement without any energy being lost to ticking.

Brief description of the drawings

Further details of the invention will appear more clearly upon reading the description below, in connection with the following schematic figures which illustrate:

Fig. 1: an embodiment of an oscillator according to the invention in plan view;
Fig. 2-15: variations on the oscillator of figure 1;
Fig. 16 and 17a, b: examples of flexure-based pivots which can be used with oscillators according to the invention;
Fig. 18: a 2-DOF example of an oscillator with two 1-DOF oscillators in series;
Fig. 19-20: an oscillator system according to the invention with two oscillators in series;
Fig. 21, 22, and 24 - 26a-c: variations of pivots suitable for use with parallel oscillator systems according to the invention;
Fig. 23: an embodiment of an oscillator system according to the invention comprising two individual oscillators arranged in parallel;
Fig. 27a-b - 29: embodiments of oscillators according to the invention based around flexure pivots;
Fig. 30: an embodiment of an oscillator system according to the invention in a series arrangement and based around flexure pivots;
Fig. 31a-b: a flexure-based two-layer pivot suitable for use with an oscillator system according to the invention;
Fig. 32: an embodiment of a flexure-based oscillator system according to the invention;
Fig. 33a-b and 34: diagrams illustrative of oscillators according to the invention provided with certain measurements;
Fig. 35a-b: examples of oscillators according to the invention provided with means for adjusting isochronism;
Fig. 36a-e: an oscillator system according to the invention, respectively in: front view (Fig. 36a), rear view (Fig. 36b), front isometric view (Fig. 36c), exploded view (Fig. 36d), front isometric view with certain components removed (Fig. 36e); and
Fig. 37a-b: simplified views of crank arrangements suitable for driving oscillators and oscillator systems according to the invention.

Embodiments of the invention

In the present specification, all units are SI base units or SI derived units (including radians for angles) unless otherwise explicitly stated. Furthermore, terms such as "rigid" and "flexible" are to be understood according to usual usage in the art. For instance, a "rigid" element will not undergo elastic deformation which substantially affects its shape and/or function under the typical axial and bending forces imposed during normal operation. On the other hand, a "flexible" element is intended to deform elastically (e.g. in bending) when subjected to the typical bending forces imposed during normal operation, this elastic bending being of a degree and with a spring constant appropriate for its function. An element can be described as being "rigid" in one or more directions and "flexible" in others, however if an element is simply described as "rigid", it is "rigid" in every direction. An element simply described as "flexible" in one direction can be presumed "rigid" in the others. Also, tolerances as are usual in the art are to be assumed for all relations, ratios, measurements, directions etc., even when the term "substantially" is not used.
Furthermore, and according to the normal understanding of the term in the art, a "rotation" means that that the angular orientation of an element changes with respect to the framework element or elements upon which it is mounted. In other words, the element in question pivots with respect to its support. This is distinct from a two-dimensional translation as in some of the prior art discussed above, in which the angular orientation of the element in question remains substantially unchanged with respect to said framework, the element undergoing a translation in an orbital motion in a plane according to a circular or oval pathway. To parametrize this distinction for the avoidance of doubt, a "rotation" in the sense of the present invention will always have an axis of rotation contained within or passing through the element in question (or an element rotationally-integrated therewith) at a predetermined, substantially unchanging point, whether this axis is defined by a conventional pinned pivot or by a flexure pivot. In the latter case, the axis will typically be defined by the intersection of the extension of respective longitudinal axes of a pair of flexures, considered in their unstressed state. The central point of an orbital translation, on the other hand, will intersect the element in question at an ever-changing point, and the element in question will not change its angular orientation with respect to the framework.
Figure 1 illustrates a first embodiment of an oscillator according to the invention.
This oscillator comprises a first inertial body (141) and a second inertial body (143), each anchored to a fixed frame (140) by a respective pivot (A), (D). These pivots (A), (D) are arranged such that the inertial bodies (141), (143) can rotate (i.e. pivot) about respective axes which are mutually parallel, these axes being defined by the pivots (A) and (D).
Inertial bodies (141) and (143) are illustrated as having the form of conventional balance wheels connected to each other by a coupler link (142) which can be a substantially rigid rod or bar of any convenient cross-section, this rigidity being present in every direction, as is equally the case in every embodiment. Other forms of inertial bodies (141), (143) are also possible, and this point also applies to all the embodiments below and need not be repeated each time. In any case, the inertial bodies contribute at least 70%, preferably at least 75% or even at least 85% of the inertia which, together with the restoring torque (see below), determines the frequency of oscillation of the oscillator. This applies equally to all embodiments and need not be repeated below.
Coupler link (142) is attached pivotably to first inertial body (141) at a first pivot point by a pivot (B), and is likewise attached pivotably to second inertial body (143) at a second pivot point by a further pivot (C). All pivots in this embodiment are typical pinned pivots, and are remote from the axes of rotation (A), (D) of the inertial bodies (141), (143). This latter point is equally the case in all of the embodiments discussed below, since no rotational coupling could occur between the inertial bodies (141), (143) if the pivot points (B), (C) each corresponded to a respective axis of rotation (A), (D). In essence, in such a case no moment could be applied to either of the inertial bodies (141), (143) by the coupler link (142). Furthermore, coupler link (142) is of unitary construction and is substantially rigid over at least a certain portion of its length, which is to say that it is unarticulated and cannot undergo any deformation in any direction that may influence its function or its interaction with the inertial bodies (141), (143) under the influence of the forces to which it is subject during operation of the oscillator. This applies to all of the embodiments, and in the case of flexure pivots (see below), the coupler link (142) is at least 10 times, preferably at least 20 times, at least 50 times or at least 100 times stiffer than said flexure pivots. The rigid part of the coupler link extends at least 50%, preferably at least 80% or at least 90% of the distance between the pivot points (B), (C). This applies to all embodiments. In the case of pinned pivots at the pivot points (B), (C), the rigid part of the coupler link extends at least between said pivot points, and would typically extend beyond them: in other words, in such a case the entire coupler link (142) is substantially rigid. It should also be noted that the rigidity of the coupler link (142) assures that it substantially cannot bend or articulate perpendicular to an axis intersecting the pivot points (B), (C) and is thus constrained to move only as permitted by the rotational displacements of the inertial bodies (141), (143). Also, the coupler link (142) substantially cannot be displaced in a direction perpendicular to an axis intersecting the pivot points (B), (C) at any given moment under the forces applied during operation. Furthermore, the trajectory of the coupler link (142) is defined exclusively by the relative positions of the axes of rotation (A), (D) and of the pivot points (B), (C). Again, all of this applies to all embodiments.
In the present example, the coupler link (142) is substantially massless and its contribution to the oscillations of the oscillator can be ignored. As a parametrization, the moment of inertia of the coupler link (142) around pivot

(A) is at least one order of magnitude, preferably at least two orders of magnitude, less than that of first inertial body (141), and likewise about pivot
(B) with respect to the second inertial body (143). It is clear from the geometry of the system that the coupler link (142) transmits force between the two inertial bodies (141), (143) according to a force vector substantially aligned along an axis λ2 joining the first and second pivot points (B), (C), which is also the case in all other embodiments and is independent of the geometry of the coupler link (142), and indeed more complex coupler link shapes are described below in the context of other embodiments. The force vector substantially obeys this condition at all times. A moment about each axis of rotation (A), (D) is thus applied by the coupler link (142) to each of the inertial bodies (141), (143). Furthermore, in all embodiments the coupler link (142) is not attached to any framework element, and is supported exclusively on the inertial bodies (141), (143).

A first elastic element (144) such as a hairspring or any other convenient spring arranged to apply a restoring torque is attached between first inertial body (141) and the fixed frame (140) so as to urge the first inertial body (141) towards a neutral position. Likewise, a second elastic element (147) is similarly arranged between rigid body (143) and the fixed frame (140) to apply a similar restoring torque. It should be noted that one of the elastic elements (144) or (147) can be omitted if desired.
A first coupling spring (145) is attached between the first inertial body (141) and the coupler link (142) so as to also apply a restoring torque between these two elements and to tend to bring them into a predetermined mutual angular orientation, and a second coupling spring (146) is likewise attached between the second inertial body (143) and the coupler link (142).
The oscillator of figure 1 is dynamically balanced when:

1- It is statically balanced, meaning that the center of mass of the oscillator remains fixed. This yields that the center of mass of first inertial body (141) is at pivot (A), and the center of mass of second inertial body (143) is at pivot (D); note that the mass of the coupler link (142) is considered as being negligible.
2- The angular momentum of the oscillator about an arbitrary fixed point remains constant. This yields that pivots (A) and (D) are in the different sides of the line λ ₂ connecting pivots (B) and (C), and
3- $J_{141} {\dot{ω}}_{141} = J_{143} {\dot{ω}}_{143}$

J

₁₄₁

J

₁₄₃

ω

₁₄₁

ω

₁₄₃

The oscillator is insensitive to gravity, linear shocks and angular shocks, for s certain stroke, and can be driven via a pin and crank arrangement (not illustrated on figure 1; see figure 37b), with the pin being situated at a convenient location on coupler link (142), similarly to that described below in the context of other embodiments. Alternatively, this oscillator can cooperate with a conventional escapement in the known manner, which also applies to the other 1-DOF embodiments mentioned below.
Figure 1 represents the general case of this type of oscillator according to the invention. Several special cases will now be discussed below, without there being a need to re-describe them in detail.
In figure 2, λ₁ and λ₃ intersect at a point E when the oscillator is in its resting state. As a result, this oscillator is dynamically balanced when:

1- It is statically balanced yielding that the center of mass of first inertial body (141) is at pivot (A), the center of mass of second inertial body (143) is at pivot (D), and again the mass of the coupler link (142) is negligible.
2- Pivots (A) and (D) are on different sides of the line λ ₂ connecting pivots (B) and (C).
3- $\overline{BE} \overline{CD} J_{141} = \overline{CE} \overline{AB} J_{143}$

J

₁₄₁

J

₁₄₃

AB

CD

CE

BE

λ

₁

λ

₃

λ

₁

λ

₃

The oscillator in this configuration is insensitive to gravity, linear shocks and angular shocks. It should be noted that this oscillator is dynamically balanced for short strokes.
In order to drive the oscillator, a pin (not illustrated) or similar attachment means can be provided on the coupler link (142), or alternatively principle on one of the inertial bodies (141), (143). Optimally, this pin can be provided at the mid-point (O) of coupler link (142). The pin can then be driven by any convenient crank mechanism (see for instance figure 37b), the axis of the crank being subject to a driving torque produced by a motor (e.g. a mainspring). This driving torque causes the oscillator to oscillate, which serves to regulate the rate at which the crank turns. In the embodiment of figure 1, an eccentric crank and connecting rod attached to the coupler link (142) is suitable (see figure 37b), although other types of crank are also possible. Since such cranks are well-known in mechanics, it is not necessary to illustrate them in detail. This applies equally to all other embodiments where no specific driving arrangement is mentioned.
In figure 3, λ₁ and λ₃ are parallel in the resting state of the oscillator, which implies a relatively longer or shorter connecting link (142) than that illustrated in figure 1, or moving one of the pivot points (A) or (B) so as to bring these axes parallel. In this case, the oscillator is dynamically balanced when:

1- It is statically balanced yielding that the center of mass of first inertial body (141) is at pivot (A), the center of mass of second inertial body (143) is at pivot (D), and again that the mass of the coupler link (142) is negligible.
2- Pivots (A) and (D) are on different sides of the line λ ₂ connecting pivots (B) and (C).
3- $\overline{CD} J_{141} = \overline{AB} J_{143}$

where J ₁₄₁ is the moment of inertia of first inertial body (141) about its pivot point (A), J ₁₄₃ is the moment of inertia of second inertial body (143) about its pivot point (D), AB is the distance between pivots (A) and (B) and CD is the distance between pivots (C) and (D).
This oscillator is insensitive to gravity, linear shocks and angular shocks. It should be noted that this oscillator is dynamically balanced for a longer stroke compared to the case where λ ₁ and λ₃ are intersecting. The trajectory of the point (O), the mid-point of the line segment BC, is a good approximation of a straight line. This characteristic can be exploited to mount two oscillators in parallel arrangement to make a 2-DOF oscillator, as will become clear below. When AB = CD, the oscillator is dynamically balanced for a longer stroke and the trajectory of the point (O) is a better approximation of a straight line.
In figure 4, λ₁ and λ₃ are parallel to each other and perpendicular to line λ₂ when the oscillator is in a neutral position (see figure 4), it is dynamically balanced when:

1- It is statically balanced so that the center of mass of first inertial body (141) is at pivot (A), the center of mass of second inertial body (143) is at pivot (D), and again the mass of the coupler link (142) is negligible.
2- Pivots (A) and (D) are in the different sides of the line λ ₂ connecting pivots (B) and (C).
3- $\overline{CD} J_{141} = \overline{AB} J_{143}$

J

₁₄₁

J

₁₄₃

AB

CD

This oscillator is insensitive to gravity, linear shocks and angular shocks. This oscillator is dynamically balanced for longer strokes compared to the case where λ ₂ is not perpendicular to the lines λ ₁ and λ ₃. In addition, the trajectory of point (O) of this oscillator is a better approximation for a straight line compared the case where λ ₂ is not perpendicular to the lines λ ₁ and λ ₃. It should be noted that for this oscillator with AB = CD is dynamically balanced for longer strokes and the point (O) has is a better approximation of a straight line compared to previously-mentioned cases.
Figure 5 illustrates the general case of an oscillator according to the invention, in which the coupler link (152) is not presumed to be essentially massless. In horological applications, even an almost-negligible coupler link mass can introduce errors which decrease chronometric precision over the millions of cycles that such oscillators carry out. In this embodiment, the mass of the coupler link (152) is compensated by having inertial bodies (151), (153) whose centers of mass do not lie on their axes of rotation. As defined by pivots (A) and (D).
The oscillator illustrated in figure 5 comprises a first inertial body (151) and a second inertial body (153), each anchored to a fixed frame (150) by a respective pivot (A), (B). Inertial bodies (151) and (153) are connected to each other by a coupler link (152) which is substantially rigid, with first inertial body (151) being connected to the coupler link (152) at a first point by pivot (B) and second inertial body (153) being connected to the coupler link (152) at a second point by pivot (C).
Similar to the variants of figures 1-4, four elastic elements (154), (155), (156) and (157) are arranged similarly to the elastic elements (144), (145), (146) and (147) respectively of figures 1-4, and carry out the same corresponding function as coupling springs. As a result, these do not need to be re-described here.
The center of mass of inertial bodies (151) and (153) and coupler link (152) are denoted by (G ₁), (G₃ ), and (G ₂) respectively. δ ₁ is the line connecting (A) to (G ₁), δ ₂ is the line connecting (B) to (G ₂), δ' ₂ is the line connecting (C) to (G ₂) and δ ₃ is the line connecting (D) to (G ₂). Again, λ ₁ is the line connecting (A) and (B), λ ₂ is the line connecting (B) and (C), and λ ₃ is the line connecting (C) and (D). The angle between λ ₁ and δ ₁ is ϕ ₁, the angle between λ ₂ and δ ₂ is ϕ ₂ , the angle between λ ₂ and δ' ₂ is ϕ' ₂ , and the angle between λ ₃ and δ ₃ is ϕ ₃.
The oscillator of this embodiment is dynamically balanced when:

1- It is statically balanced meaning that the center of mass of the oscillator remains constant. To that end, the following equations should be satisfied: $m_{1} \overline{{AG}_{1}} \overline{BC} = m_{2} \overline{{CG}_{2}} \overline{AB}$
$m$ $_{3} \overline{{DG}_{3}} \overline{BC} = m_{2} \overline{{BG}_{2}} \overline{CD},$
$φ_{1} = φ_{2}^{'},$
$φ_{3} = π + φ_{2},$
where m ₁, m ₂ and m ₃ are the mass of first inertial body (151), coupler link (152) and second inertial body (153), respectively. AG ₁ is the distance between (A) and (G ₁), BC is the distance between (B) and (C), CG ₂ is the distance between (C) and (G ₂), AB is the distance between (A) and (B), DG ₃ is the distance between (D) and (G₃ ), BG ₂ is the distance between (B) and (G ₂), and CD is the distance between (C) and (D).
2- The total angular momentum of the oscillator about an arbitrary fixed point is substantially constant: $\frac{d H_{q}}{dt} = 0$
where H _q is the vector of the total angular momentum of the oscillator about an arbitrary fixed point (q).

The oscillator of this embodiment is insensitive to gravity, linear shocks and angular shocks, for short stroke.
As before, several special cases will now be described.
Figure 6 illustrates a special case of the oscillator of figure 5, in which the lines λ₁ and λ ₃ intersect at a point (E) when the oscillator is in a neutral position.
This oscillator is dynamically balanced when:

1- It is statically balanced so that the following conditions are satisfied: $m_{1} \overline{{AG}_{1}} \overline{BC} = m_{2} \overline{{CG}_{2}} \overline{AB}$
$m$ $_{3} \overline{{DG}_{3}} \overline{BC} = m_{2} \overline{{BG}_{2}} \overline{CD},$
$φ_{1} = φ_{2}^{'},$
$φ_{3} = π + φ_{2},$
where m ₁, m ₂ and m ₃ are the mass of first inertial body (151), connecting link (152) and second inertial body (153), respectively. AG ₁ is the distance between (A) and (G ₁), BC is the distance between (B) and (C), CG ₂ is the distance between (C) and (G ₂), AB is the distance between (A) and (B), DG ₃ is the distance between (D) and (G₃ ), BG ₂ is the distance between (B) and (G ₂), and CD is the distance between (C) and (D).
2- The total angular momentum of the oscillator about an arbitrary fixed point is substantially constant so that $J_{151} ω_{151}^{\cdot} = J_{152} ω_{152}^{\cdot} + J_{153} ω_{153}^{\cdot},$
where J ₁₅₁ is the moment of inertia of body (151) about the point (A), J ₁₅₂ is the moment of inertia of body (152) about the point (E), J ₁₅₃ is the moment of inertia of body (153) about the point (D), ω ₁₅₁ is the angular velocity of first inertial body (151), ω ₁₅₂ is the angular velocity of coupler link (152), and ω ₁₅₃ is the angular velocity of second inertial body (153). Equation (6) can be rewritten as follows, $J_{151} \overline{BE} \overline{CD} = J_{152} \overline{AB} \overline{CD} + J_{153} \overline{AB} \overline{CE},$

BE

CE

The oscillator of this embodiment is insensitive to gravity, linear shocks and angular shocks, for short strokes.
Figure 7 illustrates a special case of the oscillator of figure 5, where the lines λ ₁ and λ ₃ are parallel.
This oscillator is dynamically balanced when:

1- It is statically balanced so that the following equations are satisfied: $m_{1} \overline{{AG}_{1}} \overline{BC} = m_{2} \overline{{CG}_{2}} \overline{AB}$
$m$ $_{3} \overline{{DG}_{3}} \overline{BC} = m_{2} \overline{{BG}_{2}} \overline{CD},$
$φ_{1} = φ_{2}^{'},$
$φ_{3} = π + φ_{2},$
where m ₁, m ₂ and m ₃ are the mass of first inertial body (151), connecting link (152) and second inertial body (153), respectively. AG ₁ is the distance between (A) and (G ₁), BC is the distance between (B) and (C), CG ₂ is the distance between (C) and (G ₂), AB is the distance between (A) and (B), DG ₃ is the distance between (D) and (G₃ ), BG ₂ is the distance between (B) and (G ₂), and CD is the distance between (C) and (D).
2- Pivots (A) and (D) are on different sides of the line λ ₂ connecting pivots (B) and (C).
3- The total angular momentum of the oscillator about an arbitrary fixed point is substantially constant: $J_{151} ω_{151}^{\cdot} = J_{153} ω_{153}^{\cdot},$
where J ₁₅₁ is the moment of inertia of body (151) about point (A), J ₁₅₃ is the moment of inertia of body (153) about point (D), ω ₁₅₁ is the angular velocity of body (151), and ω ₁₅₃ is the angular velocity of body (173). Equation (8) can be rewritten as follows, $J_{151} \overline{BE} \overline{CD} = J_{153} \overline{AB} \overline{CE},$

BE

CE

This oscillator is insensitive to gravity, linear shocks and angular shocks, for longer strokes compared to the previous case.
Figure 8 illustrates a special case of the oscillator of figure 5, wherein the center of mass of the coupler link (G ₂) lies on the line λ ₂ connecting pivots (B) and (C).
This oscillator is dynamically balanced when:

1- It is statically balanced so that the center of mass (G ₁) of first inertial body (151) lies on the line λ ₁, the center of mass (G ₃) of second inertial body (153) lies on the line λ ₃ and the following equations are satisfied: $\begin{array}{l} m_{1} \overline{{AG}_{1}} \overline{BC} = m_{2} \overline{{CG}_{2}} \overline{AB} \\ m \\ _{3} {\overline{DG}}_{3} \overline{BC} = m_{2} \overline{{BG}_{2}} \overline{CD,} \end{array}$
where m ₁, m ₂ and m ₃ are the mass of first inertial boy (151), coupler link (152) and second inertial body (153), respectively. AG ₁ is the distance between (A) and (G₁), BC is the distance between (B) and (C), CG ₂ is the distance between (C) and (G ₂), AB is the distance between (A) and (B), DG ₃ is the distance between (D) and (G₃ ), BG ₂ is the distance between (B) and (G ₂), and CD is the distance between (C) and (D).
2- The total angular momentum of the oscillator about an arbitrary fixed point is substantially constant.

This oscillator is insensitive to gravity, linear shocks and angular shocks, for short strokes.
Figure 9 illustrates a special case of the oscillator of figure 5, wherein the lines λ ₁ and λ ₃ are parallel.
This oscillator is dynamically balanced when:

1- It is statically balanced so that the center of mass (G ₁) of first inertial body (151) lies on the line λ ₁, the center of mass (G ₃) of second inertial body (153) lies on the line λ ₃ and the following equations are satisfied: $\begin{array}{l} m_{1} \overline{{AG}_{1}} \overline{BC} = m_{2} \overline{{CG}_{2}} \overline{AB} \\ m \\ _{3} {\overline{DG}}_{3} \overline{BC} = m_{2} \overline{{BG}_{2}} \overline{CD,} \end{array}$
where m ₁, m ₂ and m ₃ are the mass of first inertial body (151), coupler link (152) and second inertial body (153), respectively. AG ₁ is the distance between (A) and (G ₁), BC is the distance between (B) and (C), CG ₂ is the distance between (C) and (G ₂), AB is the distance between (A) and (B), DG ₃ is the distance between (D) and (G₃ ), BG ₂ is the distance between (B) and (G ₂), and CD is the distance between (C) and (D).
2- Pivots (A) and (D) are on different sides of the line λ ₂ connecting pivots (B) and (C).
3- $\overline{CD} J_{151} = \overline{AB} J_{153}$

where J ₁₅₁ is the moment of inertia of first inertial body (151) about pivot point (A) and J ₁₅₃ is the moment of inertia of second inertial body (153) about pivot point (D).

This oscillator is insensitive to gravity, linear shocks and angular shocks, for longer strokes as compared to the previous case.
It should be noted that when λ ₂ is perpendicular to λ ₁ and λ ₃, and/or AB = CD, and/or (G ₂) lies on point (O) (the mid-point of the line segment BC), the oscillator is dynamically balanced for relatively longer strokes.
Figure 10 illustrates a special case of the oscillator of figure 9, where the center of the coupler link (G ₂) is at point (O), the mid-point of the line segment BC, and also the line λ ₂ is perpendicular to the lines λ ₁ and λ ₃ when the oscillator is in a neutral position, i.e. all the torques provided by the elastic elements are balanced such that there is no net force or torque tending to try to displace the coupler link.
This oscillator is dynamically balanced when:

1- It is statically balanced so that the center of mass (G ₁) of first inertial body (151) lies on the line λ ₁, the center of mass (G ₃) of second inertial body (153) lies on the line λ ₃ and the following equations are satisfied: $\begin{array}{l} m_{1} \overline{{AG}_{1}} = \frac{1}{2} m_{2} \overline{AB} \\ m \\ _{3} \overline{{DG}_{3}} = \frac{1}{2} m_{2} \overline{CD}, \end{array}$
where m ₁, m ₂ and m ₃ are the mass of first inertial body (151), coupler link (152) and second inertial body (153), respectively. AG ₁ is the distance between (A) and (G ₁), BC is the distance between (B) and (C), AB is the distance between (A) and (B), DG ₃ is the distance between (D) and (G₃ ), and CD is the distance between (C) and (D).
2- Pivots (A) and (D) are on the different sides of the line λ ₂ connecting pivots (B) and (C).
3- $\overline{CD} J_{151} = \overline{AB} J_{153}$
where J ₁₅₁ is the moment of inertia of first inertial body (151) about pivot point (A) and J ₁₅₃ is the moment of inertia of second inertial body (153) about pivot point (D).

This oscillator is insensitive to gravity, linear shocks and angular shocks, for relatively long strokes. It should be noted that the inertial bodies (151) and (153) can be identical, meaning that J ₁₅₁ = J ₁₅₃, m ₁ = m ₃ , CD = AB, and $\overline{{AG}_{1}} = \overline{{DG}_{3}} = \frac{m_{2}}{2 m_{1}} \overline{AB}$
. This enables the oscillator to be insensitive to gravity in case of flexure pivots being used as pivots (A) and (D) for supporting the inertial bodies (151), (153).
Figure 11 illustrates a special case of the oscillator of figure 8 where AB = AD, and BC = CD. In this embodiment, the coupler link (152) extends beyond pivot (C) and its center of mass G₂ does not lie between pivots (B) and (C), but beyond pivot (C).
This oscillator is dynamically balanced when:

1- It is statically balanced so that the center of mass (G ₁) of first inertial body (151) lies on the line λ ₁, the center of mass (G ₃) of second inertial body (153) lies on the line λ ₃ and the following equations are satisfied: $\begin{array}{l} m_{1} \overline{{AG}_{1}} \overline{BC} = m_{2} \overline{{CG}_{2}} \overline{AB} \\ m \\ _{3} {\overline{DG}}_{3} \overline{BC} = m_{2} \overline{{BG}_{2}} \overline{CD,} \end{array}$
where m ₁, m ₂ and m ₃ are the mass of first inertial body (151), coupler link (152) and second inertial body (153), respectively. AG ₁ is the distance between (A) and (G ₁), BC is the distance between (B) and (C), CG ₂ is the distance between (C) and (G ₂), AB is the distance between (A) and (B), DG ₃ is the distance between (D) and (G₃ ), BG ₂ is the distance between (B) and (G ₂), and CD is the distance between (C) and (D).
2- Pivots (A) and (D) are on the same side of the line λ ₂ connecting pivots (B) and (C).
3- $\begin{array}{l} J_{152}^{B} = m_{2} \overline{{BG}_{2}} \overline{BC} - J_{151}^{A} + m_{1} \overline{{AG}_{1}} \overline{AB}, \\ J_{153}^{D} = - J_{151}^{A} + m_{1} \overline{{AG}_{1}} \overline{AB} - m_{3} \overline{{DG}_{3}} \overline{CD}, \end{array}$

J_{151}^{A}

J_{152}^{B}

J_{153}^{D}

The oscillator is insensitive to gravity, linear shocks and angular shocks, for long strokes.
Figure 12 illustrates a variant of the embodiment of figure 1, in which the simple pin pivots (A) and (D) have been replaced by coupled back to back crossed pivots.
This embodiment comprises a first inertial body (1306) attached to a fixed frame (1308) by two crossing rods (1301) and (1302)which cross one another. The crossing rods (1301) and (1302) are each attached to the fixed frame (1308) on one of their extremities by respective pivots (1310) and (1315) and to the rigid body (1306) on their other extremities by respective pivots (1309) and (1316). A first elastic element (1319) such as a hairspring, leaf spring or any other convenient arrangement is attached between the frame (1308) and the rod (1301), and a second elastic element (1320) is likewise provided linking the frame (1308) to the rod (1302). These elastic elements (1319), (1320) each provide a restoring force so as to urge the first inertial body (1306) towards a neutral angular position.
A second inertial body (1305) is provided, which is again attached to the fixed frame (1308) by two crossing rods (1303) and (1304), each associated with a respective elastic element (1321), (1322). The crossing rods (1303) and (1304) are each attached to the fixed frame (1308) on one extremity by respective pivots (1311) and (1314) and to the second inertial body (1305) on the other extremity by respective pivots (1312) and (1313). First inertial body (1306) and second inertial body (1305) are connected to each other by a coupler link (1307) as before, this coupler link (1307) being is attached to first inertial body (1306) at a first point by pivot (1318) and to second inertial body (1305) at a second point by pivot (1317). Geometrically, the ensemble formed by first inertial body (1306), its rods (1301), (1302) and the corresponding elastic elements (1319), (1320) is geometrically related to the corresponding ensemble formed by second inertial body (1305), its rods (1303), (1304) and the corresponding elastic elements (1321), (1322) by a 180-degree rotation in the x-y plane. Note that no coupling springs are provided at pivot points (1317) and (1318) which join the coupler link (1307) to each inertial body (1305), (1306).
Instead of using pins for pivots (1310) to (1318), it is also possible to use elastic flexible blades or beams provided with cols as substitutes for the rods (1301), (1302), (1303) and (1304). These possibilities are particularly advantageous since the elastic elements (1319), (1320), (1321), (1322) can be formed integrally e.g. by using the inherent flexibility of the blades or by means of cols, see for instance the book, F. Cosandier, S. Henein, M. Richard, L. Rubbert, The Art of Flexure Mechanism Design, EPFL Press 2017, namely the pivot systems named "separated cross-spring pivot" and "separated cross-spring pivot with four necked-down flexure hinged". More generally, any other convenient known flexible pivot can be used.
Figures 13 a-d illustrate further embodiments of oscillators according to the invention. These variants incorporate coupled back-to-back remote center compliance (RCC) pivots rather than the crossed rods of figure 12, and contains similar elements to the coupled pair of crossed pivots described above. The numbering of the elements starting at (1401) correspond to the respective elements numbered starting at (1301), and hence need not be repeated at length here. The difference with the embodiment of figure 12 is that the remote center compliance (RCC) pivots each comprise pairs of beams (1401), (1402) and (1403), (1404) whose longitudinal axes intersect at a point in space beyond the extremities of the beams of each respective pair, defining respective centers of rotation R ₁ and R ₂ which act as respective pivot points. As before, flexible pivots can be used instead of conventional pin-based pivots, and as such can comprise two flexure blades (cf. pivot RCC à deux lames from Henein op cit) as illustrated in figure 13d, or two rigid beams with necked-down flexures (e.g. cols) at their extremities (cf. pivot RCC à quatre co/sfrom Henein op cit). More generally, one can replace the pivots (1409)-(1416) by any convenient flexible pivot from the literature.
Figure 13c illustrates a variant of the embodiment of figure 13a in which the coupler link 1407 has been placed in a particular manner by overhanging terminal portions of each inertial body (1405), (1406) such that they overlap, the coupler link (1407) being arranged between these overhanging portions. The significance of this arrangement is discussed further below in the context of figure 14.
In the implementation depicted in figure 13d, the inertial bodies (1505) and (1506) are connected to the fixed frame (1508) respectively by flexure blades (1503)-(1504) and (1501)-(1502). These blades are placed in such way that they form pairs at a 180-degree rotational symmetry with respect to the center of the oscillator (cf. pairs (1501)-(1504) and (1502)-(1503)). This way, their axial loads (due to gravity) are opposite and hence have the same magnitude but opposite signs. As a result, when one beam is subjected to tensile axial load, the other one is subjected to compressive axial load with equal magnitude. This improves the gravity insensitivity of the pivot. Coupler link (1507) is in this case a rigid beam with flexure blades at each end serving to connecting the first inertial body (1505) to the second inertial body (1506) in a similar fashion to the other embodiments described above.
For the oscillator of figure 13a (and also that of figure 13d, mutatis mutandis), the fixed frame (1408) and inertial bodies (1405), (1406) can be exchanged such that the inertial bodies (1405), (1406) become the fixed frame (1408) and the fixed frame (1408) is separated into two inertial bodies (1405), (1406), which are then connected by the coupler link (1407) as illustrated in figure 13b.
In order to have gravity insensitivity, the position of the center of mass of the oscillator should not change during its rotation. This can be obtained by having the center of mass of each pivot of the pair at its respective center of rotation. More generally, in the case of a coupled pair of ideal pivots, the combination of the following properties guarantee that the overall center of mass does not move:

the centers of rotations of the pivots are placed substantially symmetrically with respect to the center of the oscillator;
the centers of mass of the pivot are placed substantially symmetrically with respect to the center of the oscillator;
the pivots rotate substantially the same angle in the same direction; and
the masses of the pivots are substantially the same.

Any two points symmetrical with respect to the origin (i.e. to the mid-point of the line joining the centers of rotation R₁ and R₂) which rotate by the same angle in the same direction around respective centers of rotation placed symmetrically with respect to the origin stay symmetrical with respect to the origin. This is the case when two identical pivots are placed symmetrically with respect to the center of the oscillator and the connecting rod is placed such that they both rotate by the same angle. In practice, there are parasitic shifts of the centers of rotation of the flexible pivots which cause the angular position of the two pivots to be slightly different and causes a parasitic shift of the overall center of mass.
In respect of the kinematics of the embodiments of figures 12 and 13 a-d, each pair of flexible pivots in each of the oscillators of these figures has one degree of freedom, namely in rotation in the x-y plane. Each pivot of the pair has its own rotation axis (R ₁ and R₂) but the rotations are coupled. In the case of a crossed pivot such as in figure 12, the corresponding rotation axis is perpendicular to the x-y plane, and intersects the crossing point of the beams in question. In the case of an RCC pivot such as in figures 13 a-d, the corresponding rotation axis is perpendicular to x-y plane, intersecting the crossing point of the extension of the longitudinal axis of the beams in question.
In each of these embodiments, the inertial bodies can be coupled so as to turn in the same direction as illustrated in figure 12, or in opposite directions as illustrated in figure 14. This latter embodiment couples the two inertial bodies such that when one turns clockwise, the other turns anticlockwise. This same principle can be applied to the flexure-based oscillator of figure 13d, and it is not necessary to describe this further here. Having the inertial bodies turn in the same direction has the advantage of having the intrinsic parasitic shifts of each pivot compensate each other. This reduces the sensitivity to linear accelerations such as gravity. However, such an oscillator is sensitive to angular accelerations. Having one inertial body turn in the opposite direction to the other cancels this sensitivity but does not benefit from parasitic shift compensation.
The position where the connecting link (1407), (1507) is attached to the inertial bodies (1405), (1406), (1505), (1506) defines the coupling between the sets of pivots. In order that both sets of pivots rotate the same angle, they need to rotate in the same direction (like in figures 12 and 13 a-d), and the connecting rod needs to have the same length L as the distance a between the rotation axes R₁, R₂ of the pivots. The rod must be connected in a way that the vectors r from a connecting point on a pivot to its respective rotation center forms a parallelogram with the other sides being the connecting rod and the segment between the rotation axes of the pivots (see figure 15).
This same principle is also applied in the embodiment of figure 13c mentioned above.
A variant of the oscillator of figure 12 can also be implemented using a pivot with four necked-down flexures shown highly schematically in figure 16. This variant applies the pivot presented on p.129 of Helmer and Clavel, Conception systématique de structures cinématiques orthogonales pour la microrobodique, These EPFL, No. 3365 (2006). Rather than having beams which cross like the pivot croisé a quatre cols from Henein op cit, one of the beams (1504) connecting the fixed part (1508) to the inertial body (1505) passes around the structure by means of a frame. This planar architecture is advantageous for easy manufacturing, since no beams are actually required to cross in separate planes parallel to the xy plane.
Figures 17a and 17b illustrate this principle applied to the oscillator of figure 13a. In this variant, a first pivot ensemble (1501, 1502) is attached to the second inertial body (1506) on a first side thereof at the point illustrated with a corresponding chain line, and likewise a second pivot ensemble (1503, 1504) is attached to the first inertial body (1505) on a second side thereof as illustrated by the corresponding chain line. As a result, the inertial bodies (1505), (1506) are sandwiched between the pivot ensembles (1501, 1502), (1503, 1504).
Figure 18 illustrates a two degree of freedom (2-DOF) oscillator comprising two inertial bodies (301), (302) connected together in series and taking the form of balance wheels. This oscillator does not form part of the invention.
First inertial body (301) is connected to another inertial body (302) by a pivot (303) situated on an extension of a second inertial body (302) which serves as a coupler link (307) and is fixedly attached to said second inertial body (302) at a second point (307). The second inertial body (302) is itself anchored to a fixed frame (300) by a pivot (304), the part of the second inertial body (302) acting as coupler link (307) extending between the two pivots (304), (303). A torsional spring (305) is arranged between the two inertial bodies (301) and (302) so as to apply a restoring force therebetween, and a further torsional spring (306) is likewise arranged between second inertial body (302) and the fixed frame (300). A driving pin (P) is provided on the first inertial body so as to be driven by a crank (see figure 37a), which will drive the oscillators synchronously.
The center of mass of first inertial body (301) lies on the rotation axis of pivot (303), where first inertial body (301) is mounted pivotally on second inertial body (302). The center of mass of the combination of second inertial body (302) and first inertial body (301) lies on the rotation axis of pivot (304) where second inertial body (302) is mounted pivotally on fixed support 300.
In order to have isotropy, the following equation should be satisfied, $\frac{J_{301}}{k_{305}} = \frac{J_{302}}{k_{306}}$
where J ₃₀₁ is the moment of inertia of the first inertial body (301) about its pivot (303), J ₃₀₂ is the moment of inertia of second inertial body (302) together with first inertial body (301) about pivot (304), k ₃₀₅ is the stiffness of torsional spring (305) and k ₃₀₆ is the stiffness of torsional spring (306)
This oscillator is insensitive to gravity and linear shocks; however it is sensitive to angular shocks.
For practical reasons, the line δ_x connecting pivot (303) and the driving pivot (P) is best arranged perpendicular to the line δ_y connecting pivot (303) and pivot (304). In addition, the distance between pivots (303) and (P) is chosen to equal the distance between pivots (303) and (304).
As a result, when driving pin (P) is driven by a simple crank with a pin-and-slot or other convenient arrangement (see figure 37a), it describes an oval or substantially circular trajectory, depending on how well balanced and regulated the oscillator is.
Figure 19 illustrates an oscillator system comprising two oscillators (O₁) and (O₂) according to the invention arranged in series. Each of these oscillators (O₁), (O₂) comprises an oscillator according to one of the embodiments of figures 1-11, first oscillator (O₁) being mounted on the coupler link (315) of second oscillator (O₂).
First oscillator (O₁) comprises two inertial bodies (311) and (313) each fixed to the coupler link (315) of oscillator (O ₂) at appropriate pivot points. To this end, coupler link (315) comprises a hollow frame-shaped portion, upon which first oscillator (O₁) is mounted.
Concerning the first oscillator (O ₁), its first inertial body (311) is connected to the coupler link (315) of second oscillator (O ₂) by a pivot (A ₁). This coupler link (315) thus serves as a support frame for the first oscillator (O ₁). Its second inertial body (313) is connected to the same coupler link (315) by a further pivot (D ₁). The inertial bodies (311) and (313) of oscillator (O₁) are connected to each other by a respective coupler link (312). This coupler link (312) is attached to first inertial body (311) by a pivot (B ₁) and is attached to second inertial body (313) by a pivot (C ₁). The second oscillator (O ₂) comprises two respective inertial bodies (314) and (316) coupled together by a respective coupler link (315). Inertial body (314) is anchored to a fixed frame (310) by a pivot (A ₂), and inertial body (316) is anchored thereto by a pivot (D ₂). The coupler link (315) of the second oscillator (O ₂) is attached to inertial body (314) by pivot (B ₂) and attached to inertial body (316) by pivot (C ₂). A torsional spring (317) is arranged between inertial body (311) and the coupler link (315) of the second oscillator (O ₂), and another torsional spring (318) is arranged between inertial body (311) and the coupler link (312) of the first oscillator (O ₁). Furthermore, a torsional spring (319) is arranged between inertial body (313) and the coupler link (312) of the first oscillator (O ₁), another torsional spring (320) between inertial body (313) and the coupler link (315) of the second oscillator (O ₂),
Moving to the second oscillator (O ₂), a torsional spring (321) is arranged between inertial body (314) and the fixed frame (310), another torsional spring (322) is arranged between inertial body (314) and the coupler link (315), another torsional spring (323) is arranged between inertial body (316) and the coupler link (315), and a torsional spring (324) is arranged between inertial body (316) and the fixed frame (310). The driving pin (P), which is driven by a crank, is attached to the coupler link (312) of oscillator (O ₁) at a convenient location, for instance at the mid-point thereof.
Oscillators (O ₁) and (O ₂) are dynamically balanced as described above in the context of figures 1-11. Note that the total mass and moment of inertia of the coupler link of oscillator (O ₂) comprise the mass and moment of inertia of the coupler link (315) of the second oscillator (O ₂) as well as the mass and moment of inertia of oscillator (O ₁).
The two oscillators (O ₁) and (O ₂) are arranged such that they act perpendicularly to each other. A movement of the pin exclusively in the y direction will cause the first oscillator (O₁) to be actioned without disturbing the second oscillator (O ₂), whereas a movement of the pin exclusively in the x direction will cause the second oscillator (O ₂) to be actioned without disturbing the first oscillator (O ₁). Moving the pin (P) in a circular or oval trajectory will cause both oscillators (O ₁), (O ₂) to be actioned synchronously. This applies to all the oscillator systems described below, and need not be repeated each time.
This oscillator is insensitive to gravity, linear shocks and angular shocks.
Figure 20 illustrates a variant of the oscillator system of figure 19, in which each oscillator (O ₁), (O ₂) is similar to that of figure 10 in that in a neutral position the axes joining adjacent pivot points intersect at right angles. Furthermore, the line λ_x connecting pivots (B ₂) and (C ₂) is perpendicular to the line connecting pivots (A ₂) and (B ₂) and also to the line connecting pivots (C ₂) and (D ₂). The line λ_y connecting pivots (B ₁) and (C ₁) is perpendicular to the line connecting pivots (A ₁) and (B ₁) and also to the line connecting pivots (C ₁) and (D₁ )
This oscillator is insensitive to gravity, linear shocks and angular shocks for long strokes.
Figures 21 and 22 illustrate solutions for coupling 1-DOF oscillators in parallel so as to form a 2-DOF oscillator system.
Figure 21 illustrates a three-bar coupling element, in which a rigid body (400), which supports a driving pin (P), is connected to a first oscillator (O₁) by two parallel rigid bars (401) and (407) and connected to a second oscillator (O ₂) by a further single rigid bar (402). Rigid bar (401) is connected to rigid body (400) by pivot (403) and is connected to second oscillator (O₁) by pivot (405). Rigid bar (407) is connected to rigid body (400) by pivot (408) and is connected to first oscillator (O₁) by pivot (409). Rigid bar (402) is connected to rigid body (400) by pivot (404) and is connected to second oscillator (O ₂) by pivot (406). Oscillators (O₁) and (O ₂) can be of any convenient above-mentioned type.
This coupling arrangement is statically determinate.
In the case in which rigid bar (402) is perpendicular to rigid bars (401) and (407) when the coupling arrangement is at rest, the trajectories of pivots (405) and (409) are approximately straight lines parallel to the y-axis and the trajectory of pivot (406) is approximately a straight line parallel to the x-axis.
The coupling arrangement of figure 22 differs from that of figure 21 in that only one rigid bar (401) links rigid body (400) to the first oscillator (O ₁). The longitudinal axis of bar (401) extends such that it intersects driving pin (P) when the system is in a neutral position with no resultant force tending to try to displace the rigid body 400.
This two-bar arrangement is under-constrained, which results in an extra degree of freedom compared to the arrangement of figure 21, namely rotation of the body (400) about the pin (P). In principle, this degree of freedom remains unexcited during oscillations, so should not pose any practical problems.
Figure 23 illustrates an oscillator comprising two oscillators (O₁) and (O ₂) each according to the oscillator of figure 10, coupled functionally in parallel by means of the coupling arrangement of figure 22 such that the axis of action of one oscillator is substantially orthogonal to that of the other, i.e. the directions of displacement of the coupler links of each oscillator (O ₁), (O ₂) during oscillation is substantially orthogonal. This is the case in all of the parallel-coupled oscillators described below.
The optimal connecting point at which the coupling element is connected to each of the oscillators is the mid-point of each coupler link (417), (418) respectively. This attachment is effected by means of a respective pivot (415), (416), the former lying on the mid-point of the coupler link (417) of first oscillator (O₁), and the latter lying on the mid-point of the coupler link (418) of second oscillator (O ₂). Displacement of the driving pin (P) in the y direction thus excites first oscillator (O ₁), and displacement of the driving pin (P) in the x direction excites second oscillator (O ₂), Circular displacements of the driving pin (P) excites both oscillators (O ₁), (O ₂) synchronously.
Since the coupling arrangements of figure 21 and 22 can undergo parasitic shifts, that is to say undesired translations of the various pivots (403), (404), (407) on the body (400) during rotary movement of the drive pin (P), it can be advantageous to use coupling systems which compensate for this.
Figure 24 illustrates a variant of a coupling arrangement in which the above-mentioned parasitic shift is compensated by means of double parallelogram pivots. To this end, rigid body (420) supporting the driving pin (P) is connected indirectly to the oscillators (O₁) and (O ₂). Rigid body (420) is connected by two rigid bars (421) and (422) to an intermediate stage (423). Intermediate stage (423) is connected to oscillator (O₁) by two further rigid bars (424) and (425). Rigid bars (421), (422), (424) and (425) are parallel. Rigid bar (421) is connected to rigid body (420) by pivot (430) and is connected to intermediate stage (423) by pivot (433). Rigid bar (422) is connected to rigid body (420) by pivot (431) and connected to intermediate stage (423) by pivot (432). Rigid bar (424) is connected to oscillator (O₁) by pivot (437) and connected to intermediate stage (423) by pivot (436). Rigid bar (425) is connected to first oscillator (O₁) by pivot (435) and connected to intermediate stage (423) by pivot (434). Rigid body (420) is connected by a rigid bar (426) to intermediate stage (423). Intermediate stage (423) is connected to second oscillator (O ₂) by two rigid bars (428) and (429). Rigid bars (426), (428), and (429) are parallel. Rigid bar (426) is connected to rigid body (420) by pivot (438) and connected to intermediate stage (423) by pivot (439). Rigid bar (428) is connected to second oscillator (O ₂) by pivot (443) and connected to intermediate stage (423) by pivot (442). Rigid bar (429) is connected to second oscillator (O ₂) by pivot (440) and connected to intermediate stage (423) by pivot (441).
In the optimal case, rigid bars (421), (422), (424), and (425) are perpendicular to rigid bars (426), (428) and (429) when the system is in a neutral position. Furthermore, the trajectories of pivots (440) and (443) are substantially parallel to the x-axis and the trajectories of pivots (435) and (437) are substantially parallel to the y-axis. As a result, the trajectories of pivots (430) and (431) with respect to first oscillator (O₁) are approximately straight lines parallel to the x-axis. The trajectory of pivot (438) with respect to oscillator (O ₂) is approximately a straight line parallel to the y-axis, and any parasitic shift is substantially eliminated.
Figure 25 illustrates a further coupling arrangement based on a pair of Robert's four-bar linkages, which convert rotational motions to approximately straight-line motions.
In this variant, the rigid body (470) supporting the driving pin (P) is connected to each of oscillators (O₁) and (O ₂) by a respective Robert's four-bar linkage. Rigid body (470) is connected to rigid body (471) by pivot (478). Rigid body (471) is connected to first oscillator (O₁) by two rigid bars (472) and (473). Rigid bar (472) is connected to rigid body (471) by pivot (475) and connected to first oscillator (O₁) by pivot (474). Rigid bar (473) is connected to rigid body (471) by pivot (477) and connected to first oscillator (O₁) by pivot (476). Rigid body (470) is connected to rigid body (479) by pivot (486). Rigid body (479) is connected to second oscillator (O ₂) by two rigid bars (480) and (481). Rigid bar (480) is connected to rigid body (479) by pivot (483) and connected to second oscillator (O ₂) by pivot (482). Rigid bar (481) is connected to rigid body (479) by pivot (485) and connected to second oscillator (O ₂) by pivot (484).
Pivot (478) lies on the straight line connecting pivots (474) and (476). The line connecting pivots (475) and (477) is parallel to the line connecting pivots (474) and (476). The distances between pivots (474) and (475); (475) and (478); (478) and (477); and (477) and (476) are equal. The distance between pivots (474) and (476) is two times the distance between pivots (475) and (477).
Pivot (486) lies on the straight line connecting pivots (484) and (482). The line connecting pivots (485) and (483) is parallel to the line connecting pivots (484) and (482). The distances between pivots (482) and (483); (483) and (486); (486) and (485); and (485) and (484) are equal. The distance between pivots (484) and (482) is two times the distance between pivots (485) and (483).
The trajectories of pivot (482) and (484) are approximately two straight lines parallel to the x-axis. The trajectories of pivot (474) and (476) are approximately two straight lines parallel to the y-axis. The trajectory of pivot (478) with respect to first oscillator (O₁) is approximately a straight line parallel to the x-axis. The trajectory of pivot (486) with respect to second oscillator (O ₂) is approximately a straight line parallel to the y-axis. A similar arrangement can be formed using flexure pivots rather than pinned pivots.
Figures 26a-c illustrate another variant of a coupling element, in this case distributed over two layers (L ₁) and (L ₂). These figures are partial and schematic, and do not illustrate all the details of the entire oscillator system. First oscillator (O₁) is in or adjacent to layer (L ₁) and second oscillator (O ₂) is in or adjacent to layer (L ₂). These oscillators can be of any type disclosed in the present specification, each hence comprising two inertial bodies, for a total of four inertial bodies for the oscillator system. In layer (L ₁), rigid body (450) is connected to first oscillator (O₁) by two rigid bars (451) and (452), these bars (451) and (452) notably being connected to the coupler link of said first oscillator (O ₁). Rigid bar (451) is connected to rigid body (450) by pivot (453) and is connected to first oscillator (O₁) by pivot (454). Rigid bar (452) is connected to rigid body (450) by pivot (455) and connected to first oscillator (O₁) by pivot (456). Rigid bars (451) and (452) are related to each other by a 180-degree in-plane rotation about the mid-point of the line segment (453)-(455). The center of mass of rigid body (450) lies on the mid-point of the line segment (453)-(455). The trajectories of pivots (454) and (456) are approximately straight lines parallel to the y-axis. With respect to first oscillator (O₁), the trajectory of the mid-point of the line segment (453)-(455) is approximately a straight line parallel to the x-axis.
In layer (L ₂), rigid body (460) is connected to second oscillator (O ₂) by two rigid bars (461) and (462), these bars (461) and (462) likewise being connected to the coupler link of said second oscillator (O ₂). Rigid bar (461) is connected to rigid body (460) by pivot (463) and connected to second oscillator (O ₂) by pivot (464). Rigid bar (462) is connected to rigid body (460) by pivot (465) and connected to second oscillator (O ₂) by pivot (466). Rigid bars (461) and (462) are related to each other by a 180-degree in-plane rotation about the mid-point of the line segment (463)-(465). The center of mass of rigid body (460) lies on the mid-point of the line segment (463)-(465). The trajectories of pivots (464) and (466) are approximately straight lines parallel to the x-axis. With respect to second oscillator (O ₂), the trajectory of the mid-point of the line segment (463)-(465) is approximately a straight line parallel to the y-axis.
Rigid body (450) of oscillator (O₁) is connected to the rigid body (460) of the second oscillator (O ₂) by pivot (457), which may be of any convenient type ensuring that the rigid bodies (450), (460) are unified such that they are constrained to translate together but can rotate with respect to each other. The driving pin (P) can be attached to either rigid body (450) or rigid body (460).
Other arrangements of two-layer coupling elements which constrain the first and second oscillators (O₁), (O ₂) to oscillate at 90° to each other, and in which the rigid bodies (450), (460) are constrained to translate in translation but can rotate with respect to each other are also possible.
Since flexure pivots are more adapted to series fabrication than conventional fixed pivots and torsional springs, the following figures present embodiments of oscillators according to the invention which are based around flexures.
To this end, figure 27a illustrates a flexure-based dynamically balanced 1-DOF oscillator based on that of figure 10. The first inertial body (41) is anchored to the frame (40) by a flexure pivot comprising two blades (43) and (44), the extension of whose longitudinal axes cross at point (A) so as to form a remote center compliance (RCC) pivot. Point (A) is the center of rotation (and hence the axis of rotation) of the pivot in question. The second inertial body (42) is similarly anchored to the fixed frame (40) by two further blades (45) and (46) the extension of whose axes cross at point (D), which likewise forms the center of rotation and hence axis of rotation of the pivot in question. Inertial bodies (41) and (42) are connected to each other by a coupler link comprising a rigid beam (47) with a flexible blade at each extremity thereof.
The center of mass of first inertial body (41) lies on point (A) and the center of mass of second inertial body (42) lies on point (D). Inertial bodies (41) and (42) are substantially dynamically identical, i.e. they each have equal masses and moments of inertia about their respective centers of mass. Each point on the midline of coupler link (47) is substantially equidistant from point (A) and point (D). Flexible blades (43) and (45) are substantially elastically identical and related to each other by a 180-degree in-plane rotation about the mid-point (O) of coupler link (47). Flexible blades (44) and (46) are substantially elastically identical and related to each other by a 180-degree in-plane rotation about the mid-point (O) of coupler link (47).
Figure 27b illustrates a special case of the oscillator of figure 27a in which the line passing through (A) and (D) is the perpendicular bisector of the bar (57) serving as the coupler link and vice-versa when the oscillator is at rest. In this embodiment, the reference signs (50)-(57) correspond respectively to the reference signs (40)-(47) of figure 27a, and need not be re-described here.
As before, a pin can be provided as appropriate on one of the inertial bodies or on the coupler link, arranged to be driven by a crank as discussed above in the context of figure 1. Again, this applies to each of the following embodiments where appropriate.
Figure 28 illustrates a further variant of an oscillator according to the invention, wherein the coupler link (508) is a substantially rigid beam with negligible mass. This embodiment is analogous to that of figure 1
This oscillator comprises first and second inertial bodies (507) and (509) connected by a coupler link (508) with negligible mass. First inertial body (507) is anchored to a fixed frame (500) by an RCC (remote center compliance) pivot comprising two flexible blades (501) and (502) whose extended axes cross at point (A), which is on the midline on the inertial body (507). Second inertial body (509) is likewise anchored to the fixed frame (500) by a further RCC (remote center compliance) pivot comprising two flexible blades (504) and (505) whose extended axes cross at point (D), which is again on the midline of corresponding inertial body (509). The coupler link (508) is connected to first inertial body (507) by a single-blade flexure pivot (503), whose center of rotation lies at the mid-point (B) of the flexible blade (503). The coupler link (508) is furthermore connected to second inertial body (509) by a further single-blade flexure pivot (506) having its center of rotation of the pivot lies at the mid-point (C) of the flexible blade (506).
Inertial bodies (507) and (509) are substantially dynamically identical, i.e. they have the same mass and the same moment of inertia about their respective pivot points. They are related to each other by a 180-degree in-plane rotation about point (O), which is the mid-point of line segment BC. The length of line segment AB is equal to the length of line segment CD. The flexible blades of each of the pairs (501) and (504); (502) and (505); (503) and (506) are kinematically substantially identical and are related to each other by a 180-degree in-plane rotation about point (O) to cancel the effect of gravity on oscillator stiffness. The center of mass of coupler link (508) lies on point (O). In the ideal case, line BC is perpendicular to lines AB and CD.
Figure 29 illustrates an embodiment of an oscillator according to the invention wherein the coupler link (530) has significant mass and moment of inertia, and is analogous to that of figure 10.
First inertial body (529) is anchored to a fixed frame (520) by an RCC (remote center compliance) pivot comprising two flexible blades (521) and (522) crossing at point (A), which is at the center of rotation of the pivot. Second inertial body (531) is anchored to the fixed frame (520) by a further RCC (remote center compliance) pivot likewise comprising two flexible blades (525) and (526) crossing at point (D). The coupler link (530) is connected to first inertial body (529) by an RCC flexure pivot comprising two yet further flexible blades (523) and (524) the extensions of whose longitudinal axes cross at a point (B), which is the center of rotation of the pivot. The coupler link (530) is likewise connected to second inertial body (531) by a yet further RCC flexure pivot comprising another two flexible blades (527) and (528), the extension of whose longitudinal axes crosses at point (C), which is again the center of rotation of the pivot.
Inertial bodies (529) and (531) are dynamically substantially identical, i.e. they have the same mass and the same moment of inertia about their respective pivot points, and are related to each other by a 180-degree in-plane rotation about point (O), the mid-point of line segment BC. The length of line segment AB is equal to the length of line segment CD. The flexible blades of each of the pairs: (521) and (525); (522) and (526); (523) and (527); (524) and (528) are kinematically identical and are related to each other by a 180-degree in-plane rotation about point (O) to cancel the effect of gravity on oscillator stiffness. The center of mass (G ₂) of the coupler link (530) lies on point (O). The center of mass (G ₁) of the first inertial body (529) lies on line AB. The center of mass (G ₂) of second inertial body (531) lies on line CD. In the ideal case, line BC is perpendicular to lines AB and CD.
Figure 30 illustrates a flexure implementation of a 2-DOF dynamically balanced oscillator system comprising two serially connected double inertial bodies. This oscillator system is flexure-based version of that of figure 19, where the rigid pivots and torsional springs are replaced by flexure pivots.
First inertial body (541) of second oscillator (O ₂) is anchored to a fixed frame (540) by an RCC pivot comprising two flexible blades (547) and (548) whose longitudinal axes cross at point (A ₂). Second inertial body (543) of the second oscillator (O ₂) is anchored to the fixed frame (540) by an RCC pivot comprising two flexible blades (557) and (558) whose axes cross at point (D ₂). Inertial bodies (541) and (543) are connected to each other by a coupler link (542). The coupler link (542) is connected to first inertial body (541) by an RCC pivot comprising two flexible blades (549) and (550) whose longitudinal axes cross at point (B ₂), and is connected to inertial body (543) by an RCC pivot comprising two flexible blades (559) and (560) whose longitudinal axes cross at point (C ₂).
Moving to the first oscillator (O ₁), its first inertial body (544) is connected to the coupler link (542) of the second oscillator (O ₂) by an RCC pivot comprising two flexible blades (551) and (552) whose longitudinal axes cross at point (A ₁). Corresponding second inertial body (546) is connected to coupler link (542) by an RCC pivot comprising two flexible blades (554) and (555) whose longitudinal axes cross at point (D ₁). Inertial bodies (544) and (546) are connected to each other by corresponding coupler link (545). Coupler link (545) is connected to inertial body (544) by a single-blade flexure pivot (553) and is connected to inertial body (546) by a single-blade flexure pivot (556). The mid-points of flexible blades (553) and (556) lie substantially on points (B ₁) and (C ₁), respectively.
The driving pin (P) lines substantially on the intersection of the lines λ_x and λ_y when the system is at rest, where λ_x is a straight line connecting (B ₂) and (C ₂), and λ_y is a straight line connecting (B ₁) and (C ₁). It should be noted that the lines λ_x and λ_y are substantially perpendicular. The driving pin (P) lies on the mid-point of line segment B ₁ C ₁ and also on the mid-point of line segment B ₂ C ₂ . The flexible blades of the following pairs are related to each other by a 180-degree in-plane rotation about (P) to cancel the effect of gravity on stiffness of the oscillator; (547) and (557); (548) and (558); (549) and (559); (550) and (560); (551) and (554); (552) and (555); (553) and (556).
When the oscillator system of figure 30 is integrated in a timepiece movement and is attached to a crank, the pin (P) is advantageously be biased away from the intersection of lines λ_x and λ_y when the crank is not rotating and the movement is at rest. This permits the crank to apply sufficient force when subject to a driving torque to cause the oscillator system to self-start without having to be displaced manually or subjected to a shock.
In the alternate case of an oscillator system comprising oscillators of figures 27-29 coupled in parallel, rather than using the pinned coupling elements of figures 21-26, the pinned pivots be exchanged for flexure pivots. This is achieved by simply replacing the pinned pivots with flexible blades of the type used for connecting link (47) of figure 27a, or single-blade flexures either side of a substantially rigid bar, of the type used for connecting link (508) of figure 28 and also used in figures 31a-b below. Since these adaptations are well within the ability of the skilled person, they do not need to be illustrated and described at length here.
Figures 31a and 31b, however, illustrate a particular variant of the double-layer coupling element of figure 26 specially adapted for a flexure-based construction, in which two layers (L ₁), (L ₂), are superposed one with respect to the other. The same considerations as described above in the context of figure 26 equally apply here, and need not be repeated.
In first layer (L ₁), a rigid body (701) is connected to first oscillator (O₁) by two rigid bars (700) and (702), these bars (700) and (702) notably being connected to the coupler link of the first oscillator (O ₁). Rigid bar (700) is connected to first rigid body (701) by a single-blade flexure pivot (704) and is connected to first oscillator (O ₁) by another single-blade flexure pivot (705). Rigid bar (702) is likewise connected to second rigid body (701) by single-blade flexure pivot (706) and is connected to first oscillator (O₁) by single-blade flexure pivot (707). First rigid body (701) has a plurality of mounting holes (709), (710), (711) and (712), which can be of any number or can be substituted by other mounting means.
The mid-points of flexible blades (704) and (706) lie on a straight line which is perpendicular to rigid bars (700) and (702). Rigid body (700), flexible blade (704) and flexible blade (705) are respectively related to rigid body (702), flexible blade (706) and flexible blade (707) by a 180-degree in-plane rotation.
In layer (L ₂), second rigid body (721) is connected to second oscillator (O ₂) by two rigid bars (720) and (722), these bars (720) and (722) likewise being connected to the coupler link of the second oscillator (O ₂). Rigid bar (720) is connected to rigid body (721) by single-blade flexure pivot (724) and connected to second oscillator (O ₂) by another single-blade flexure pivot (725). Rigid bar (722) is connected to rigid body (721) by single-blade flexure pivot (726) and connected to second oscillator (O ₂) by single-blade flexure pivot (727). Rigid body (723) also has mounting holes (729), (730), (731) and (732) corresponding to those of the first layer (L ₁), and is connected to rigid body (721) by a flexure pivot comprising two flexible blades (728) and (733) situated in different planes parallel to the xy plane. A single blade flexure in this role is also sufficient. The mid-points of flexible blades (724), (726) and the crossing point of blades (728) and (733) lie on a straight line which is perpendicular to rigid bars (720) and (722). Rigid body (720), flexible blade (724) and flexible blade (725) are respectively related to rigid body (722), flexible blade (726) and flexible blade (727) by a 180-degree in-plane rotation about the crossing point of blades (728) and (733).
First oscillator (O₁) has a DOF along the y-axis and second* oscillator (O ₂) has a DOF along the x-axis. Rigid bodies (701) and (723) are connected to each other by four screws passing through the corresponding mounting holes, or by other convenient means such as welding, soldering, glue, pins, or similar.
In a further variant of the system of figure 31, the bar and blade flexures, (720), (724), (725); (722), (726), (727); (700), (704), (705) and (702), (706), (707) in one or (ideally) both layers can be replaced by simple blade flexures.
Figure 32 illustrates a flexure implementation of the oscillator system of figure 23. This system comprises two independent double inertial body 1-DOF oscillators connected in parallel to form a 2-DOF oscillator system.
Rigid body (740) supporting the driving pin (P) is connected to first oscillator (O₁) by a rigid bar (741) which is connected to rigid body (740) by a single-blade flexure pivot (747) and is connected to first oscillator (O₁) by a single-blade flexure pivot (743). Rigid body (740) is likewise connected to second oscillator (O ₂) by a further rigid bar (742) which is connected to rigid body (740) by another single-blade flexure pivot (748) and is connected to second oscillator (O ₂) by a further single-blade flexure pivot (744). Rigid bars (741) and (742) extend in superposition to their respective oscillators (O ₁), (O ₂) without impinging on them.
The coupler link (745) of first oscillator (O₁) is substantially oscillates parallel to the y-axis and the coupler link (746) of second oscillator (O ₂) is substantially oscillates parallel to the x-axis. The mid-points of flexible blade (743) and coupler link (745) are coincident, as are the mid-points of flexible blade (744) and coupler link (746).
Figures 33a and 33b illustrate the variables used below in the following discussion of tuning the isochronism of a 1-DOF oscillator according to figure 10.
As a reminder, this oscillator comprises two substantially identical inertial bodies each with moment of inertia J ₁ about their respective pivot points (A) and (D). The inertial bodies are connected to each other by a coupler link of mass m ₂ and moment of inertia J ₂ about its center of mass which lies on the mid-point (O) of the line segment BC. The distance between (A) and (B) is equal to the distance between (C) and (D) and is denoted by H. The distance between (B) and (C) is denoted by d.
The rotation angles of the first and second inertial bodies are θ ₁ and θ ₃ respectively, and the rotation angle of the coupler link is θ ₂. The following equations give θ ₂ and θ ₃ as functions of θ ₁, $\begin{matrix} θ_{2} = \frac{H}{d} θ_{1}^{2} + O (θ) (_{1}^{3}), & θ_{3} = θ_{1} + O (θ) (_{1}^{4}) . \end{matrix}$
Given the kinematic parameters, the kinetic energy K of the oscillator as a function of θ ₁ can be derived follows, $K = \frac{1}{2} J_{0} [1 + (4 \frac{J_{2}}{J_{0}} {(\frac{H}{d})}^{2} - \frac{m_{2} H^{2}}{J_{0}}) θ_{1}^{2}] {\dot{θ}}_{1}^{2},$
where J ₀ = 2J ₁ + m ₂ H ² .
Considering a general restoring torque M(θ ₁) while noting that M is an odd function of θ ₁ due to the symmetries of the oscillator, the Taylor's series expansion of the restoring torque around the nominal position of the oscillator (θ ₁ = 0) is as follows, $M = k_{0} θ_{1} + k_{2} θ_{1}^{3} + O (θ) (_{1}^{5}),$
where k ₀ and k ₂ are constant. Given the restoring torque, the potential energy V of the oscillator is derived as follows, $V = \frac{1}{2} k_{0} θ_{1}^{2} + \frac{1}{4} k_{2} θ_{1}^{4} + O (θ) (_{1}^{6}) .$
Given the kinetic and potential energies, the isochronism defect of the oscillator is $1 - \frac{T}{T} = \frac{1}{2} (\frac{k_{2}}{k_{0}} - 4 \frac{J_{2}}{J_{0}} {(\frac{H}{d})}^{2} + \frac{m_{2} H^{2}}{J_{0}}) θ_{1}^{2} + O (θ) (_{1}^{4}),$
where T is the period of oscillations and $T_{0} = 2 π \sqrt{\frac{J_{0}}{k_{0}}}$
is the nominal period.
Isochronism can be achieved up to second order by setting, $\frac{J_{2}}{J_{0}} = \frac{1}{4} (\frac{m_{2} d^{2}}{J_{0}} + {(\frac{d}{H})}^{2} \frac{k_{2}}{k_{0}}) .$
Figure 34, which corresponds to the oscillator of figure 28, illustrates the effect of parasitic shift of the pivots on isochronism.
As a reminder, this oscillator comprises two substantially identical inertial bodies of mass m ₁ and moment of inertia J ₁, about their respective centers of rotation (A) and (D). The inertial bodies are connected to each other by a coupler link of mass m ₂ and moment of inertia J ₂ about its center of mass which lies on the mid-point (O) of the line segment BC. Each of the flexure pivots anchoring the inertial bodies to the fixed frame comprises two identical flexible blades of length L and angle 2α with respect to each other. The distance between the points where the flexible blades are attached to the inertial bodies and the crossing point of the extensions of the axes of each pair of flexible blades is r. The distance between A and B is H which is equal to the distance between C and D. The distance between B and C is d
Denoting the rotation angles of the inertial bodies by θ ₁, the rotation angle of the coupler link θ ₂ is, $θ_{2} = (\frac{H + 2 λ}{d}) θ_{1}^{2} + O (θ) (_{1}^{3}),$
where $λ = \frac{L}{15 \cos α} [9 {(\frac{r}{L})}^{2} + 9 (\frac{r}{L}) + 1] .$
It should be noted that this takes into account the parasitic shift of the rotation center of the flexure pivots, where the parasitic shift is ${λθ}_{1}^{2} + O (θ) (_{1}^{3})$
.
The kinetic energy K of the oscillator is $K = \frac{1}{2} J_{0} [1 + (4 \frac{J_{2}}{J_{0}} {(\frac{H + 2 λ}{d})}^{2} + \frac{8 m_{1} λ^{2}}{J_{0}} - \frac{m_{2} H^{2}}{J_{0}}) θ_{1}^{2}] {\dot{θ}}_{1}^{2},$
where J ₀ = 2J ₁ + m ₂ H ² .
Considering the following Taylor's series expansion for a general restoring torque, $M = k_{0} θ_{1} + k_{2} θ_{1}^{3} + O (θ) (_{1}^{5}),$
where k ₀ and k ₂ are constant, the isochronism defect of the oscillator is, $1 - \frac{T}{T} = \frac{1}{2} (\frac{k_{2}}{k_{0}} - 4 \frac{J_{2}}{J_{0}} {(\frac{H + 2 λ}{d})}^{2} + \frac{m_{2} H^{2}}{J_{0}} - \frac{8 m_{1} λ^{2}}{J_{0}}) θ_{1}^{2} + O (θ) (_{1}^{4}),$
where T is the period of oscillations and $T_{0} = 2 π \sqrt{\frac{J_{0}}{k_{0}}}$
is the nominal period.
We are able to achieve isochronism up to second order by setting, $\frac{J_{2}}{J_{0}} = \frac{d^{2}}{4 {(H + 2 λ)}^{2}} (\frac{m_{2} H^{2}}{j_{0}} + \frac{k_{2}}{k_{0}} - \frac{8 m_{1} λ^{2}}{J_{0}}) .$
Figures 35a and 35b illustrate the application of the above-mentioned teaching on isochronism to practical oscillators which permit the adjustment of isochronism independently of the frequency of oscillation. The oscillator of figure 35a corresponds to that of figure 10, and that of figure 25b corresponds to that of figure 28.
The coupler link (800), (810) of each oscillator carries a pair of slidable masses (801) and (802) in figure 35a, and (811) and (812) in figure 35b. Displacement of these masses (801), (802), (811), (812) along the respective coupler link (800), (810) tunes the isochronism of the oscillator since such displacement changes the moment of inertia of the coupler link J ₂ without changing mass of the coupler m ₂. Sufficient friction, or other suitable blocking means, are provided so as to ensure that the slidable masses (801), (802) remain in position. Hence, isochronism can be tuned without changing the nominal frequency. It should be noted that to avoid changing the center of mass of the coupler links (800), (810), the respective masses (801), (802); (811), (812) should be moved symmetrically with respect to the center of mass of the coupler link (800), (810). However, an asymmetric movement may be advantageous if it is required to move the center of mass of the coupler link (800), (810) in order to balance the oscillator.
In respect of the nominal frequency of the oscillators, this can be tuned in the conventional manner known in horology, namely by providing adjustable inertia-blocks such as radially-adjustable screws at convenient locations (typically in the outer edge) on the inertial masses.
Figures 36a to 36e illustrate an embodiment of an oscillator system according to the invention, which incorporates the above teachings. In figure 36e several elements have been removed so as to better show the underlying structures. This oscillator system comprises two independent dynamically balanced 1-DOF oscillators (O ₁), (O ₂) mounted in parallel in a manner similar to that shown in figure 32.
This oscillator comprises a driving pin (P) connected to each 1-DOF oscillator by a two-bar coupling element similar to that of figure 32, each connection incorporating a substantially rigid bar and a pair of single-blade blade flexible pivots. This oscillator also incorporates mechanisms for adjusting isochronism and frequency.
The first dynamically balanced 1-DOF oscillator (O₁) comprises two substantially identical inertial bodies, the first of which comprises two relatively heavy masses (905) and (906) rigidly attached to a rigid body (901). A pair of inertia-blocks in the form of radially-adjustable screws (913) and (914) are mounted in appropriate holes located in the periphery of masses (905) and (906) so as to permit the frequency of oscillations to be tuned by modifying the inertia of the inertial body as is generally known. The second inertial body of the first oscillator (O₁) comprises again two relatively heavy masses (907) and (908) rigidly attached to a rigid body (902). Again, radially-adjustable screws (915) and (916) are provided in the periphery of masses (907) and (908) as before.
Rigid body (901) is rigidly attached to rigid body (950) and rigid body (902) is rigidly attached to rigid body (949). Rigid body (949) is anchored to the fixed frame (900) by a flexure pivot comprising two flexible blades (930) and (931), the extension of whose longitudinal axes intersect at the theoretical pivot point that they define. Rigid body (950) is similarly anchored to the fixed frame (900) by a similar flexure pivot comprising blades (932) and (933). Rigid bodies (949) and (950) are connected to each other by a coupler link (951), which is connected to rigid body (949) by a single-blade flexure pivot (934), and to rigid body (950) by another single-blade flexure pivot (935). Sliding masses (938) and (939) are mounted such that they can be translated along coupler link (951) so as to permit tuning of the isochronism by changing the moment of inertia of the coupler link (951) around its center of gravity.
The second dynamically balanced 1-DOF oscillator (O ₂) is constructed similarly to the first oscillator (O ₁). Its first inertial body comprises two relatively heavy masses (909) and (910) rigidly attached to a rigid body (903). Again, two screws (917) and (918) are provided on the masses (909) and (910) to tune the frequency of oscillations by changing the inertia of the inertial body. The second inertial body comprises two relatively heavy masses (911) and (912) rigidly attached to a rigid body (904). Again, two further screws (919) and (920) are provided on masses (911) and (912) to tune the frequency of oscillations by changing the inertia of the inertial body. Rigid body (903) is rigidly attached to rigid body (923) and rigid body (904) is rigidly attached to rigid body (921). Rigid body (921) is anchored to the fixed frame (900) by a flexure pivot comprising two flexible blades (924) and (925), the extension of whose longitudinal axes cross. Rigid body (923) is anchored to the fixed frame (900) by a similar flexure pivot comprising two flexible blades (926) and (927). Rigid bodies (921) and (923) are connected to each other by a coupler link (922). This latter is connected to rigid body (921) by a single-blade flexure pivot (928), and connected to rigid body (923) by a further single-blade flexure pivot (929). Sliding masses (936) and (937) mounted slidably on coupler link (922) are again arranged to tune isochronism by changing the moment of inertia of the coupler link (922) around its center of gravity.
Driving pin (P) is connected to each of the 1-DOF oscillators by a flexure-based two-bar coupling element which comprises a rigid body (940) supporting the driving pin (P). This rigid body (940) is connected to the coupler links of the oscillators (O ₁), (O ₂). Rigid body (940) is connected to coupler link (951) by a rigid bar (941) and is connected to coupler link (922) by a rigid bar (942), these rigid bars (941), (942) being substantially perpendicular to each other. Rigid bar (941) is connected to rigid body (940) by a single-blade flexure pivot (946) and is connected to rigid body (943) by another single-blade flexure pivot (945). Rigid bar (942) is also connected to rigid body (940) by a further single-blade flexure pivot (948) and is connected to rigid body (944) by a yet further single-blade flexure pivot (947). Rigid body (943) is rigidly attached to coupler link (951) and rigid body (944) is rigidly attached to coupler link (922)
Mid-points of flexible blades (945) and (947) lie on the mid-points of coupler links (951) and (922) respectively. Flexible blades (924), (925), (930) and (931) are substantially elastically identical to flexible blades (926), (927), (932) and (933), respectively. Flexible blades (924) and (925) are related to flexible blades (926) and (927) by a 180-degree in-plane rotation about the mid-point of coupler link (922). Flexible blades (930) and (931) are related to flexible blades (932) and (933) by a 180-degree in-plane rotation about the mid-point of coupler link (951). Rigid bar (941) is perpendicular to rigid bar (942). Coupler link (922) is perpendicular to coupler link (951).
Four counter-masses (952), (953), (954) and (955) are provided on each respective inertial body (901), (902), (903), (904) for static balancing of the oscillators. These masses can be moved toward or away from the rotation centers of the inertial bodies (901), (902), (903), (904).
This oscillator system is insensitive to gravity, insensitive to all linear and angular shocks. It does not transmit torque to the support, and hence has extremely high Q-factor oscillations. The architecture is particularly compact and suitable for integration in a wristwatch. The isochronism and the frequency can be tuned independently, and each individual oscillator (O ₁), (O ₂) is independently tunable. Finally, the driving pin (P) moves with a relatively long stroke.
Experiments with a prototype of this oscillator system have shown that, when driving pin (P) is driven by a simple pin-and-slot crank arrangement (see figure 37a) by a conventional timepiece movement, it self-starts when P is biased away from its illustrated location when the crank is stopped, and carries out substantially circular displacements.
Figures 37a and 37b illustrate schematically simple crank arrangements suitable for driving a 2-DOF oscillator system and a 1-DOF oscillator respectively when these are integrated in a timepiece movement (7). These diagrams are highly schematic and are not to scale.
In figure 37a, a source of energy (M) such as a driving spring or similar, drives a wheel (1), to which is rigidly attached a bar (2) comprising a slot (3). Bar (2) is typically arranged along a radius of the wheel (1) and extends beyond the outer periphery of the wheel (1). Other arrangements are of course possible. Driving pin (P) as described above is located slidingly in the slot (3), and as the source of energy (M) causes the wheel (1) to rotate about its own axis (1a) and the crank to cause the driving pin (P) to translate in orbital motion about the axis (1a) of the wheel (1). This driving system applies to the 2-DOF oscillator of figure 18, and also to all the 2-DOF oscillator systems described above.
In figure 37b, a source of energy (M) such as a driving spring or similar, drives a wheel (1), to which is pivotally mounted a connecting rod (4) by means of an eccentric (5). This connecting rod (4) is pivotally connected to coupler link (6) of a 1-DOF oscillator of any type mentioned above, the rest of the oscillator not being illustrated. Complete rotations of the wheel (1) thus cause the coupler link (6) to carry out substantially linear oscillations since it is constrained by the geometry of the pivots of the oscillator. The length of stroke of the coupler link (6) (and hence of the oscillator of which it is a part) is determined by the geometry of the wheel (1), eccentric (5) and connecting rod (4).
Although the invention has been described by reference to specific embodiments, variations thereto are possible without departing from the scope of the invention as defined in the claims. It is particularly noted that the shapes of the various elements of the oscillators of the invention are by no means limiting. On this point, each inertial mass can be shaped as desired, and the shapes of the coupler links can also be freely chosen within large bounds, provided that the various pivot points are correctly located.

Claims

Horological oscillator comprising:
- a first inertial body (141; 151; 1305; 1405; 1505; 301; 311; 314; 41; 51; 507; 529; 544; 541; 901; 903) arranged to rotate about a first axis (A; A₁; A₂);

- a second inertial body (143; 153; 1306; 1406; 1506; 302; 313; 316; 42; 52; 509; 531; 546; 543; 902; 904) arranged to rotate about a second axis (D; D₁; D₂) substantially parallel to said first axis (A; A₁; A₂);

- at least one elastic element (144, 155, 156, 157; 154, 155, 156, 157; 1319, 1320, 1321, 1322; 1419, 1420, 1421, 1422; 1501, 1502, 1503, 1504, 1507; 305, 306; 317, 318, 319, 320; 321, 322, 323, 324; 43, 44, 45, 46; 53, 54, 55, 56; 501, 502, 503, 504, 505, 506; 521, 522, 523, 524, 525, 526, 527, 528; 551, 552, 553, 554, 555, 556, 547, 548, 549, 550, 557, 558, 559, 560; 924, 925, 926, 927, 928, 929, 930, 931, 932, 933, 934, 935) arranged to apply a restoring torque to at least one of said inertial bodies so as to urge said inertial body towards a neutral position;

- a coupler link (142; 152; 1307; 1407; 1507; 307; 312; 315; 47; 57; 508; 530; 545; 542; 745; 746; 800; 810; 922; 951) attached at a first pivot point thereof to said first inertial body (141; 151; 1305; 1405; 1505; 301; 311; 314; 41; 51; 507; 529; 544; 541; 901; 903) and at a second pivot point thereof to said second inertial body such that said inertial bodies are arranged to rotate synchronously about their respective axes (A; A₁; A₂; D; D₁; D₂), said first and second pivot points being distinct from said first and second axes (A; A₁; A₂; D; D₁; D₂), characterised by the fact that said coupler link (142; 152; 1307; 1407; 1507; 307; 312; 315; 47; 57; 508; 530; 545; 542; 745; 746; 800; 810; 922; 951) is substantially rigid, comprises a bar or rod and is arranged to transmit force between said first inertial body (141; 151; 1305; 1405; 1505; 301; 311; 314; 41; 51; 507; 529; 544; 541; 901; 903) and said second inertial body (143; 153; 1306; 1406; 1506; 302; 313; 316; 42; 52; 509; 531; 546; 543; 902; 904) according to a vector substantially aligned along an axis (λ2) joining the first and second pivot points.
Horological oscillator according to claim 1, wherein said first inertial body (1505; 41; 51; 507; 529; 544; 541; 901; 903) is mounted pivotally on a supporting element (40; 50; 500; 520; 542; 540; 900) by means of a first flexure pivot (1501, 1502; 43, 44; 53, 54; 501, 502; 521, 522; 551, 552; 547, 548; 926, 927; 932, 933) and said second inertial body (1505; 42; 52; 509; 531; 546; 543; 902; 904) is mounted pivotally on said supporting element (40; 50; 500; 520; 542; 540; 900) by means of a second flexure pivot (1503, 1504; 45, 46; 55, 56; 504, 505; 525, 526; 554, 555; 557, 558; 924, 925; 930, 931), said flexure pivots constituting said elastic elements.
Horological oscillator according to claim 1 or 2, wherein said coupler link (142; 152; 1307; 1407; 1507; 307; 312; 315; 47; 57; 508; 530; 545; 542; 922; 951) is subject to a restoring torque with respect to each of said inertial bodies, said restoring torque being provided at at least one of said first and second points.
Horological oscillator according claim 3, wherein said coupler link is a substantially rigid bar attached to said first inertial body (507; 529; 544; 541; 901; 903) by means of a further flexure pivot (503; 523, 524; 553; 549, 550; 929; 935) and attached to said second inertial body (509; 531; 543; 546; 902; 904) by means of a yet further flexure pivot (506; 527, 528; 556; 559, 560; 928; 934), said flexure pivots constituting said elastic elements.
Horological oscillator according to one of the preceding claims, wherein a first line (λ₁) joining said first axis to said first point is parallel to a second line (λ₃) joining said second axis to said second point, said first and second lines preferably being perpendicular to a third line (λ₂) joining said first point to said second point when said oscillator is in a neutral position.
Horological oscillator according to one of the preceding claims, further comprising at least two masses (801, 802; 811, 812; 936, 937; 938, 939) mounted movably on said coupler link (800; 810; 522; 951), at least one of said masses being situated on each side of the center of mass of said coupler link.
Oscillator system comprising:
- a first horological oscillator (O₁) according to any preceding claim; and

- a second horological oscillator (O₂) according to any preceding claim coupled to said first horological oscillator (O₁).
Oscillator system according to claim 7, wherein the inertial bodies (311, 313; 544, 546) of said first horological oscillator (O₁) are each mounted pivotally on the coupler link (315; 542) of said second horological oscillator (O₂) such that said first oscillator (O₁) acts in a direction which is preferably substantially perpendicular to the direction of action of the second oscillator (O₂).
Oscillator system according to claim 7, wherein said first horological oscillator (O₁) and said second horological oscillator (O₂) are mutually coupled by means of a rigid body (740; 940) from which extend respective linkages (741, 742; 941, 942) each attached to a respective coupler link (745, 746; 922, 951), said first horological oscillator (O₁) and said second horological oscillator (O₂) being preferably arranged to act substantially perpendicular to each other.
Oscillator system according to claim 9, wherein at least one of said linkages comprises at least one of:
- a single-bar linkage comprising pinned pivots or flexure pivots;

- a double-bar linkage comprising pinned pivots or flexure pivots;

- a single or double blade flexure;

- at least one rigid bar provided with a single blade flexure at each end;

- a Robert's four-bar linkage comprising pinned pivots or flexure pivots;

- at least one four-bar Watt linkage comprising pinned pivots or flexure pivots;

- a compound parallelogram linkage comprising pinned pivots or flexure pivots.
Oscillator system according to one of claims 9-10, wherein said first horological oscillator (O₁) and said second horological oscillator (O₂) are arranged such that they are related by a substantially 90° rotation in the plane of the oscillator system about an axis orthogonal to a predetermined point in said plane.
Oscillator system according to one of claims 9-10, wherein said first horological oscillator (O₁) and said second horological oscillator (O₂) are arranged such that one is a mirror reflection of the other about a plane equidistant from the coupler link of each horological oscillator, said plane being perpendicular to the plane of the oscillator system.
Oscillator system according to claim 7, wherein:
- said first horological oscillator (O₁) comprises a first rigid body (450; 701) arranged in a first layer (L₁) and attached to said coupler link of said first horological oscillator (O₁));

- said second horological oscillator (O₂) comprises a second rigid body (460; 721) arranged in a second layer (L₂) and attached to said coupler link of said second horological oscillator (O₂); and

- said rigid bodies (450, 460; 701, 721) are constrained to translate together and are free to pivot one with respect to the other.
Timepiece movement comprising an oscillator according to one of claims 1-6 or an oscillator system according to one of claims 7-13.
Timepiece movement according to claim 14 comprising a source of energy kinematically connected with said oscillator or said oscillator system by means of a crank attached to at least one of said coupler links.