Patent Description:
<CIT> discloses a process to drive an oscillating mechanism. A mentioned aim of this document is to replace an intermittent regulation by a continuous regulation but it fails to clearly disclose how the principles exposed apply to a timekeeper such as a watch. In particular, the constructions are not described as isotropic harmonic oscillators and the described architectures do not result in planar motion of the oscillating mass as in the present invention.

<CIT> discloses a rotational resonator for a timekeeper. The disclosed resonator comprises two masses mounted in a cantilevered manner on a central support, each mass oscillating circularly around an axis of symmetry. Each mass is attached to the central support via four springs. The springs of each mass are connected to each other to obtain a dynamic coupling of the masses. To maintain the rotational oscillation of the masses, an electromagnetic device is used that acts on ears of each mass, the ears containing a permanent magnet. One of the springs comprises a pawl for cooperation with a ratchet wheel in order to transform the oscillating motion of the masses into a unidirectional rotational movement. The disclosed system therefore is still based on the transformation of an oscillation, that is an intermittent movement, into a rotation via the pawl which renders the system of this publication equivalent to the escapement system known in the art and cited above.

<CIT> is related to a mechanical rotating resonator for a timekeeper. This patent is mainly directed to the description of springs used in such a resonator as disclosed in <CIT> discussed above. Here again, the principle of the resonator thus uses a mass oscillating around an axis.

<CIT> discloses a torsion oscillator that oscillates around a vertical axis. Again, this is similar to the escapement of the prior art and described above.

An aim of the present invention is thus to improve the known systems and methods.

A further aim of the present invention is to provide a system that avoids the intermittent motion of the escapements known in the art.

A further aim of the present invention is to propose a mechanical planar isotropic harmonic oscillator.

Another aim of the present invention is to provide an oscillator that may be used in different time-related applications, such as: time base for a chronograph, timekeeper (such as a watch), speed governor.

The present invention solves the problem of the escapement by eliminating it completely or, alternatively, by a family of new simplified escapements which do not have the drawbacks of current watch escapements.

The result is a much simplified mechanism with increased efficiency.

In one embodiment, the invention concerns a mechanical planar isotropic harmonic oscillator as defined by the features of appended independent claim <NUM>
Dependent claims <NUM> to <NUM> define particular embodiments of the mechanical isotropic harmonic oscillator of claim <NUM>.

In one embodiment, the invention concerns a timekeeper such as a clock as defined by the features of dependent claim <NUM>.

In one embodiment, the timekeeper is a wristwatch or a chronograph.

In one embodiment, the oscillator defined in the present application is used as a time base for a chronograph measuring fractions of seconds requiring only an extended speed multiplicative gear train, for example to obtain <NUM> frequency so as to measure <NUM>/<NUM>th of a second.

In one embodiment, the oscillator or oscillator system defined in the present application is used as speed regulator for striking either musical clocks and watches, or music boxes, thus eliminating unwanted noise and decreasing energy consumption, and also improving musical or striking rhythm stability.

These embodiments and others will be described in more detail in the following description of the invention.

The present invention will be better understood from the following description and from the drawings which show.

As is well-known, in <NUM> Isaac Newton published Principia Mathematica in which he proved Kepler's laws of planetary motion, in particular, the First Law which states that planets move in ellipses with the Sun at one focus and the Third Law which states that the square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit, see reference [<NUM>].

Less well-known is that in Book I, Proposition X, of the same work, he showed that if the inverse square law of attraction (see <FIG>) was replaced by a linear attractive central force (since called Hooke's Law, see <FIG> and <FIG>) then the planetary motion was replaced by elliptic orbits with the Sun at the center of the ellipse and the orbital period is the same for all elliptical orbits. (The occurrence of ellipses in both laws is now understood to be due to a relatively simple mathematically equivalence, see reference [<NUM>], and it is also well-known that these two cases are the only central force laws leading to closed orbits, see reference [<NUM>].

Newton's result for Hooke's Law is very easily verified: Consider a point mass moving in two dimensions subject to a central force <MAT> centered at the origin, where r is the position of the mass, then for an object of mass m, this has solution <MAT> for constants A<NUM>, A<NUM>, φ<NUM>, φ<NUM> depending on initial conditions and frequency <MAT>.

This not only shows that orbits are elliptical, but that the period of motion depends only on the mass m and the rigidity k of the central force. This model therefore displays isochronism since the period <MAT> is independent of the position and momentum of the point mass (the analogue of Kepler's Third Law proved by Newton).

Isochronism means that this oscillator is a good candidate to be a time base for a timekeeper as a possible embodiment of the present invention.

This has not been previously done or mentioned in the literature and the utilization of this oscillator as a time base is an embodiment of the present invention. This oscillator is also known as a harmonic isotropic oscillator where the term isotropic means "same in all directions.

Despite being known since <NUM> and its theoretical simplicity, it would seem that the isotropic harmonic oscillator, or simply "isotropic oscillator,", has never been previously used as a time base for a watch or clock, and this requires explanation.

It would seem that the main reason is the fixation on constant speed mechanisms such as governors or speed regulators, and a limited view of the conical pendulum as a constant speed mechanism.

For example, in his description of the conical pendulum which has the potential to approximate isochronism, Leopold Defossez states its application to measuring very small intervals of time, much smaller than its period, see reference [<NUM>, p.

Bouasse devotes a chapter of his book to the conical pendulum including its approximate isochronism, see reference [<NUM>, Chapitre VIII]. He devotes a section of this chapter on the utilization of the conical pendulum to measure fractions of seconds (he assumes a period of <NUM> seconds), stating that this method appears perfect. He then qualifies this by noting the difference between average precision and instantaneous precision and admits that the conical pendulum's rotation may not be constant over small intervals due to difficulties in adjusting the mechanism. Therefore, he considers variations within a period as defects of the conical pendulum which implies that he considers that it should, under perfect conditions, operate at constant speed.

Similarly, in his discussion of continuous versus intermittent motion, Rupert Gould overlooks the isotropic oscillator and his only reference to a continuous motion timekeeper is the Villarceau regulator which he states: "seems to have given good results. But it is not probable that was more accurate than an ordinary good-quality driving clock or chronograph," see reference [<NUM>, <NUM>-<NUM>]. Gould's conclusion is validated by the Villarceau regulator data given by Breguet, see reference [<NUM>].

From the theoretical standpoint, there is the very influential paper of James Clerk Maxwell On Governors, which is considered one of the inspirations for modern control theory, see reference [<NUM>]. Moreover, isochronism requires a true oscillator which must preserve all speed variations. The reason is that the wave equation <MAT> preserves all initial conditions by propagating them. Thus, a true oscillator must keep a record of all its speed perturbation. For this reason, the invention described here allows maximum amplitude variation to the oscillator.

This is exactly the opposite of a governor which must attenuate these perturbations. In principle, one could obtain isotropic oscillators by eliminating the damping mechanisms leading to speed regulation.

The conclusion is that the isotropic oscillator has not been used as a time base because there seems to have been a conceptual block assimilating isotropic oscillators with governors, overlooking the simple remark that accurate timekeeping only requires a constant time over a single complete period and not over all smaller intervals.

We maintain that this oscillator is completely different in theory and function from the conical pendulum and governors, see hereunder in the present description.

<FIG> illustrates the principle of the conical pendulum and <FIG> a typical conical pendulum mechanism.

<FIG> illustrates a Villarceau governor made by Antoine Breguet in the <NUM>'s and <FIG> illustrates the propagation of a singularity for a plucked string.

Two types of isotropic harmonic oscillators having unidirectional motion are possible. One is to take a linear spring with body at its extremity, and rotate the spring and body around a fixed center. This is illustrated in <FIG>: Rotating spring. Spring <NUM> with body <NUM> attached to its extremity is fixed to center <NUM> and rotates around this center so that the center of mass of the body <NUM> has orbit <NUM>. The body <NUM> rotates around its center of mass once every full orbit, as can be seen by the rotation of the pointer <NUM>.

This leads to the body rotating around its center of mass with one full turn per revolution around the orbit as illustrated in <FIG>. : Example of rotational orbit. Body <NUM> orbits around point <NUM> and rotates around its axis once for every complete orbit, as can be seen by the rotation of point <NUM>.

This type of spring will be called a rotational isotropic oscillator and will be described in Section <NUM>. In this case, the moment of inertia of the body affects the dynamics, as the body is rotating around itself.

Another possible realization has the mass supported by a central isotropic spring, as described in Section <NUM>. In this case, this leads to the body having no rotation around its center of mass, and we call this orbiting by translation. This is illustrated in <FIG>: Translational orbit. Body <NUM> orbits around center <NUM>, moving along orbit <NUM>, but without rotating around its center of gravity. Its orientation remains unchanged, as seen by the constant direction of pointer <NUM> on the body.

In this case, the moment of inertia of the mass does not affect the dynamics.

Our time base using an isotropic oscillator will regulate a mechanical timekeeper, and this can be implemented by simply replacing the balance wheel and spiral spring oscillator with the isotropic oscillator and the escapement with a crank fixed to the last wheel of the gear train. This is illustrated in <FIG>: On the left is the classical case. Mainspring <NUM> transmits energy via gear train <NUM> to escape wheel <NUM> which transmits energy intermittently to balance wheel <NUM> via anchor <NUM>. On the right is our mechanism. Mainspring <NUM> transmits energy via gear train <NUM> to crank <NUM> which transmits energy continuously to isotropic oscillator <NUM> via the pin <NUM> travelling in a slot on this crank. The isotropic oscillator is attached to fixed frame <NUM>, and its center of restoring force coincides with the center of the crank pinion.

In order to realize an isotropic harmonic oscillator, in accordance with the present invention, there requires a physical construction of the central restoring force. One first notes that the theory of a mass moving with respect to a central restoring force is such that the resulting motion lies in a plane. It follows that for practical reasons, the physical construction should realize planar isotropy. Therefore, the constructions and embodiments described here will mostly be of planar isotropy, but not limited to this embodiment, and there will also be an example of <NUM>-dimensional isotropy.

In order for the physical realization to produce isochronous orbits for a time base, the theoretical model of Section <NUM> above must be adhered to as closely as possible. The spring stiffness k is independent of direction and is a constant, that is, independent of radial displacement (linear spring). In theory, there is a point mass, which therefore has moment of inertia J = <NUM> when not rotating. The reduced mass m is isotropic and also independent of displacement. The resulting mechanism should be insensitive to gravity and to linear and angular shocks. The conditions are therefore.

Planar isotropy may be realized in two ways.

Note that gravity does not affect the spring when it is in the axial direction. However, these realizations have the disadvantage of having the spring and its support both rotating around their own axes, which introduces spurious moment of inertia terms which reduce the theoretical isochronism of the model. Indeed, considering the point mass of mass m and then including a isotropic support of moment of inertia I and constant total angular momentum L, then if friction is ignored, the equations of motion reduce to
<MAT>.

This equation can be solved explicitly in terms of Jacobi elliptic functions and the period expressed in terms of elliptic integrals of the first kind, see reference [<NUM>] for definitions and similar applications to mechanics. A numerical analysis of these solutions shows that the divergence from isochronism is significant unless the moment of inertia I is minimized.

We now list which of the theoretical properties of Section <NUM> hold for these realizations. In particular, for the rotating cantilever spring.

The realizations which appear to be most suitable to preserve the theoretical characteristics of the harmonic oscillator are the ones in which the central force is realized by an isotropic spring, where the term isotropic is again used to mean "same in all directions.

A simple example is given in <FIG> illustrating a simple planar isotropic spring with an orbiting mass <NUM>, a y-coordinate spring <NUM>, an x- coordinate spring <NUM>, a y-spring fixation to ground <NUM>, an x-spring fixation to ground <NUM>, a horizontal ground <NUM>, the y-axis being vertical so parallel to force of gravity. In this figure, the two springs Sx <NUM> and Sy <NUM> of rigidity k are placed such that spring Sx <NUM> acts in the horizontal x-axis and spring Sy <NUM> acts in the vertical y-axis. There is a mass <NUM> attached to both these springs <NUM>, <NUM> and having mass m. The geometry is chosen such that at the point (<NUM>, <NUM>) both springs are in their neutral positions.

One can now show that this mechanism exhibits isotropy to first order, as illustrated in <FIG>. Assuming now a small displacement d r = (dx, dy), then up to first order, there is a restoring force Fx in the x direction of -k dx and a restoring force Fy in y direction of -k dy. This gives a total restoring force <MAT> and the central linear restoring force of Section <NUM> is verified. It follows that this mechanism is, up to first order, a realization of a central linear restoring force, as claimed.

In these realizations, gravity affects the springs <NUM>, <NUM> in all directions as it changes the effective spring constant. However, the springs <NUM>, <NUM> does not rotate around its own axis, minimizing spurious moments of inertia, and the central force is directly realized by the spring itself. We now list which of the theoretical properties of Section <NUM> hold for these realizations (up to first order).

Many planar springs have been proposed and if some may be implicitly isotropic, none has been explicitly declared to be isotropic. In the literature, Simon Henein [see reference <NUM>, p. <NUM>, <NUM>] has proposed two mechanisms which exhibit planar isotropy. But these examples, as well as the one just described above, do not exhibit sufficient isotropy to produce an accurate timebase for a timekeeper, as a possible embodiment of the invention described herein.

An embodiment illustrated in <FIG>, comprises two serial compliant four-bar <NUM> is also called parallel arms linkage, which allows, for small displacements, translations in the X and Y directions. Another embodiment, illustrated in <FIG>, comprises four parallel arms <NUM> linked with eight spherical joints <NUM> and a central bellow <NUM> connecting the mobile platform <NUM> to the ground.

Therefore, more precise isotropic springs have been developed. In particular, the precision has been greatly improved and this is the subject of several embodiments described in the present application.

In these realizations, the spring does not rotate around its own axis, minimizing spurious moments of inertia, and the central force is directly realized by the spring itself. These have been named isotropic springs because their restoring force is the same in all directions.

A basic example of an embodiment of the oscillator made of planar isotropic springs is illustrated in <FIG>. Said figure illustrates a mechanical isotropic harmonic oscillator comprising at least a two degrees of freedom linkage L1/L2 made by appropriate guiding means (for example sliding means, or linkages, springs etc.), supporting an orbiting mass P with respect to a fixed base B with springs S having isotropic and linear restoring force K properties.

In order to place the new oscillator in a portable timekeeper as an exemplary embodiment of the present invention, it is necessary to address forces that could influence the correct functioning of the oscillator. These include gravity and shocks.

The first method to address the force of gravity is to make a planar isotropic spring which when in horizontal position with respect to gravity does not feel its effect.

<FIG> illustrates an example of such a spring arrangement as a <NUM> degree of freedom planar isotropic spring construction. In this design, gravity has negligible effect on the planar motion of the orbiting mass when the plane of mechanism is placed horizontally. This provides single direction minimization of gravitational effect. It comprises a fixed base <NUM>, Intermediate block <NUM>, a frame holding the orbiting mass <NUM>, an orbiting mass <NUM>, an y-axis parallel spring stage <NUM> and an x-axis parallel spring stage <NUM>.

However, this is adequate only for a stationary clock/watch. For a portable timekeeper, compensation is required. This can be achieved by making a copy of the oscillator and connecting both copies through a ball or universal joint as in <FIG>. In the realization of <FIG>, the center of gravity of the entire mechanism remains fixed. Specifically, <FIG> shows a gravity compensation in all directions for planar isotropic spring. Rigid frame <NUM> holds time base comprising two linked non-independent planar isotropic oscillators <NUM> (symbolically represented here). Lever <NUM> is attached to the frame <NUM> by a ball joint <NUM> (or XY universal joint). The two arms of the lever are telescopic thanks to two prismatic joints <NUM>. The opposing ends of the lever <NUM> are attached to the orbiting masses <NUM> by ball joints. The mechanism is symmetric with respect to the point <NUM> at center of joint <NUM>.

Linear shocks are a form of linear acceleration, so include gravity as a special case. Thus, the mechanism of <FIG> also compensates for linear shocks.

Effects due to angular accelerations can be minimized by reducing the distance between the centers of gravity of the two masses as shown in <FIG> by modifying the mechanism of the previous section shown in <FIG>. Precise adjustment of the distance "l" shown in <FIG> separating the two centers of gravity allows for a complete compensation of angular shocks including taking account the moment of inertia of the lever itself. This only takes into account angular accelerations will all possible axes of rotation, except those on the axis of rotation of our oscillators.

Specifically, <FIG> illustrates gravity compensation in all directions for planar isotropic spring with added resistance to angular acceleration. This is achieved by minimizing the distance "I" between the center of gravity of the two orbiting masses. Rigid frame <NUM> holds a time base comprising of two linked non- independent planar isotropic oscillators <NUM> (symbolically represented here). Lever <NUM> is attached to the frame <NUM> by a ball joint <NUM> (or x-y universal joint). The two arms of the lever <NUM> are telescopic thanks to two prismatic joints <NUM>. The opposing ends of the lever <NUM> are attached the orbiting masses <NUM> by ball joints <NUM>. The mechanism is symmetric with respect to the point O at center of joint <NUM>.

<FIG> illustrates another embodiment of a Realization of gravity compensation in all directions for a planar isotropic spring using flexures. In this embodiment, a rigid frame <NUM> holds a time base comprising two linked non-independent planar isotropic oscillators <NUM> (symbolically represented here). Lever <NUM> is attached to a frame <NUM> by x-y a universal joint made of leaf spring <NUM> and flexible rod <NUM>. The two arms of the lever <NUM> are telescopic thanks to two leaf springs <NUM>. The opposing ends of the lever <NUM> are attached the orbiting masses <NUM> by the two leaf springs <NUM> which form two x-y universal joints.

<FIG> illustrates an alternate realization of gravity compensation in all directions for a planar isotropic spring using flexures. In this variant, both ends of lever <NUM> are connected to the orbiting masse <NUM> connected to springs <NUM> in the oscillator by two perpendicular flexible rods <NUM>.

<FIG> illustrates another realization of gravity compensation in all directions for an isotropic spring using flexures. In this embodiment, fixed plate <NUM> holds time base comprising two linked symmetrically placed non-independent orbiting masses <NUM>. Each orbiting mass <NUM> is attached to the fixed base by three parallel bars <NUM>, these bars are either flexible rods or rigid bars with a ball joint <NUM> at each extremity. Lever <NUM> is attached to the fixed base by a membrane flexure joint (not numbered) and vertical flexible rod <NUM> thereby forming a universal joint. The extremities of the lever <NUM> are attached to the orbiting masses <NUM> via two flexible membranes <NUM>. Part <NUM> is attached rigidly to part <NUM>. Part <NUM> and <NUM> are attached rigidly to the lever <NUM>.

Oscillators lose energy due to friction, so there needs a method to maintain oscillator energy. There must also be a method for counting oscillations in order to display the time kept by the oscillator. In mechanical clocks and watches, this has been achieved by the escapement which is the interface between the oscillator and the rest of the timekeeper. The principle of an escapement is illustrated in <FIG> and such devices are well known in the watch industry.

In the case of the present invention, two main methods are proposed to achieve this: without an escapement and with a simplified escapement.

In order to maintain energy to the isotropic harmonic oscillator, a torque or a force are applied, see <FIG> for the general principle of a torque T applied continuously to maintain the oscillator energy, and <FIG> illustrates another principle where a force FT is applied intermittently to maintain the oscillator energy. In practice, in the present case, a mechanism is also required to transfer the suitable torque to the oscillator to maintain the energy, and in <FIG> various crank embodiments according to the present invention for this purpose are illustrated. <FIG> and <FIG> illustrate escapement systems for the same purpose. All these restoring energy mechanisms may be used in combination with the various embodiments of oscillators and oscillators systems (stages etc.) described herein, for example in <FIG>, <FIG> (as the mechanism <NUM> illustrated in <FIG>), and <FIG>. Typically, in the embodiment of the present invention where the oscillator is used as a time base for a timekeeper, specifically a watch, the torque/force may by applied by the spring of the watch which is used in combination with an escapement as is known in the field of watches. In this embodiment, the known escapement may therefore be replaced by the oscillator of the present invention.

<FIG> illustrates the principle of a variable radius crank for maintaining oscillator energy. Crank <NUM> rotates about fixed frame <NUM> through pivot <NUM>. Prismatic joint <NUM> allows crank extremity to rotate with variable radius. Orbiting mass of time base (not shown) is attached to the crank extremity <NUM> by pivot <NUM>. Thus the orientation of orbiting mass is left unchanged by crank mechanism and the oscillation energy is maintained by crank <NUM>.

<FIG> illustrates a realization of variable radius crank for maintaining oscillator energy attached to the oscillator. A fixed frame <NUM> holds a crankshaft <NUM> on which maintaining torque M is applied. Crank <NUM> is attached to crankshaft <NUM> and equipped with a prismatic slot <NUM>'. Rigid pin <NUM> is fixed to the orbiting mass <NUM> and engages in the slot <NUM>'. The planar isotropic springs are represented by <NUM>. Top view and perspective exploded views are shown in this <FIG>.

<FIG> illustrates a flexure based realization of a variable radius crank for maintaining oscillator energy. Crank <NUM> rotates about fixed frame (not shown) through shaft <NUM>. Two parallel flexible rods <NUM> link crank <NUM> to crank extremity <NUM>. Pivot <NUM> attaches the mechanism shown in <FIG> to an orbiting mass. The mechanism is shown in neutral singular position in this <FIG>.

<FIG> illustrates another embodiment of a flexure based realization of variable radius crank for maintaining oscillator energy. Crank <NUM> rotates about fixed frame (not shown) through shaft <NUM>. Two parallel flexible rods <NUM> link crank <NUM> to crank extremity <NUM>. Pivot <NUM> attaches mechanism shown to orbiting mass. Mechanism is shown in flexed position in this <FIG>.

<FIG> illustrates an alternate flexure based realization of variable radius crank for maintaining oscillator energy. Crank <NUM> rotates about fixed frame <NUM> through shaft. Two parallel flexible rods <NUM> link crank <NUM> to crank extremity <NUM>. Pivot <NUM> attaches mechanism to orbiting mass <NUM>. In this arrangement the flexible rods <NUM> are minimally flexed for average orbit radius.

<FIG> illustrates an example of a completely assembled isotropic oscillator <NUM>-<NUM> and its energy maintaining mechanism according to the present invention. More specifically, a fixed frame <NUM> is attached to the ground or to a fixed reference (for example the object on or in which the oscillator is mounted) by three rigid feet <NUM> and top frame 140a. First compound parallel spring stage <NUM> holds second parallel spring stage <NUM> moving orthogonally to said spring stage <NUM>. Compound parallel spring <NUM> is attached rigidly to stage <NUM>. Fourth compound parallel spring stage <NUM> holds third parallel spring stage <NUM> moving orthogonally to spring stage <NUM>. Outer frames of stages <NUM> and <NUM> are connected kinematically in the x and y directions by L-shaped brackets <NUM> and <NUM> as well as by notched leaf springs <NUM>. The two outer frames of stages <NUM> and <NUM> constitute the orbiting mass of the oscillator while stages <NUM>-<NUM> are attached together and fixed to feet <NUM> and the orbiting mass moves therefore relatively to stages <NUM>-<NUM>. Alternatively, the moving mass may be formed by stages <NUM>-<NUM> and in that case the stages <NUM> and <NUM> are fixed to the feet <NUM>.

Bracket <NUM> mounted on the orbiting mass holds the rigid pin <NUM> (illustrated in <FIG> and <FIG>) on which the maintaining force is applied for example a torque or a force, by means identical or equivalent to the ones described above with reference to <FIG>.

Each stage <NUM>-<NUM> may be for example made as illustrated in <FIG> or in <FIG> discussed later herein in more details. Accordingly, the description of these figures applies to the stages <NUM>-<NUM> illustrated in these <FIG>. As will be described hereunder, to compensate, the stages <NUM> and <NUM> (respectively <NUM> and <NUM>) are identical but placed with a relative rotation (in particular of <NUM>°) to form the XY planar isotropic springs discussed herein.

<FIG> shows the same embodiment of <FIG>, and shows the rigid pin <NUM> mounted rigidly on the orbiting masses (stages <NUM> and <NUM>, for example as mentioned hereabove) and engages into slot <NUM> which acts as the driving crank and maintains the oscillation. The other parts are numbered as in <FIG> and the description of this figure applies correspondingly. The crank system used may be the one illustrated in <FIG> and described hereabove.

<FIG> illustrates the stages <NUM>-<NUM> of the embodiment of <FIG> and <FIG> without crank system <NUM>-<NUM> and using the reference numbers of <FIG>.

<FIG> illustrates the stages <NUM>-<NUM> of the embodiment of <FIG> without stage <NUM> and using the reference numbers of <FIG>.

<FIG> illustrates the stages <NUM>-<NUM> of the embodiment of <FIG> without stage <NUM> using the reference numbers of <FIG>.

<FIG> illustrates the stage <NUM> of <FIG> without stage <NUM> using the reference numbers of <FIG>.

Typically, each stage <NUM>-<NUM> may be made in accordance with the embodiments described later in the present specification in reference to <FIG>. Indeed, stage <NUM> of <FIG> comprises parallel springs 131a to 131d which hold a mass 131e and the springs and masses of said <FIG> may correspond to the ones of <FIG>.

To construct the oscillator of <FIG>, as mentioned above, stages <NUM> and <NUM> are placed with a relative rotation of <NUM>° between them, and their mass 131e-132e are attached together (see <FIG>). This provides a construction equivalent to the one of <FIG> described later with two parallel springs in each direction XY.

Stages <NUM> and <NUM> are attached as stages <NUM>-<NUM> and placed in a mirror configuration over stages <NUM>-<NUM>, stage <NUM> comprising as stages <NUM> and <NUM> springs 133a-133d and a mass 133e. The position of stage <NUM> rotated by <NUM>° with respect to stage <NUM> as one can see in <FIG>. The frames of stages <NUM> and <NUM> are attached together such that they will not move relatively one to another.

Then, as illustrated in <FIG>, fourth stage <NUM> is added with a <NUM>° relative rotation with respect to stage <NUM>. Stage <NUM> also comprise springs 134a-134d and mass 134e. Mass 134e is attached to mass 133e and the two stages <NUM> and <NUM> a linked together via brackets <NUM>, <NUM> to form the orbiting mass while stages <NUM> and <NUM> which are attached together are fixed to the frame <NUM>, 140a.

As illustrated in <FIG>, the mechanism for applying a maintaining force or torque is placed on top of the stages <NUM>-<NUM> and comprises the pin <NUM> and the crank system <NUM>, <NUM> which for example the system described in <FIG>, the pin <NUM> of <FIG> corresponding to pin <NUM> of <FIG>, the crank <NUM> corresponding to crank <NUM> and slot <NUM>' to slot <NUM>.

Of course, the stages <NUM>-<NUM> of <FIG> may be replaced by other equivalent stages having the XY planar isotropy in accordance with the principle of the invention, for example, one may use the configurations and exemplary embodiments of <FIG> to realize the oscillator of the present invention.

The XY isotropic harmonic oscillators can be generalized by replacing X translation and Y translation by other motions, in particular, rotation. When expressed as generalized coordinates in Lagrangian mechanics, the theory is identical and the mechanisms will have the same isotropic harmonic properties as the translational XY mechanisms.

<FIG> shows an example of an XY isotropic harmonic oscillator with generalized coordinates X a rotation and Y a rotation: On the fixed base <NUM> are attached two immobile beams <NUM> which support a rotating cage <NUM> via jewelled bearings at <NUM> and a spiral spring <NUM>. Inside the cage <NUM> is a balance wheel allowed to rotate and attached via a balance staff (not shown) which rotates on jewelled bearings <NUM>. To the balance wheel is attached a spiral spring <NUM> which provides a restoring force to the circular oscillation of the balance wheel around its axis. The spiral spring provides a restoring force to the rotation of the cage <NUM> around its neutral position where the balance wheel axis is perpendicular to the base <NUM>. The moment of inertia of the balance wheel assembly including the cage is such that the natural frequencies of the balance wheel and spring <NUM> is the same as that of the cage and balance wheel and spring <NUM>. The oscillations of the balance wheel model the isotropic harmonic oscillator and for small amplitudes of oscillations the mass <NUM> on the balance wheel moves in a unidirectional orbit approximating an ellipse as shown in <FIG>. This mechanism has the advantage of being insensitive to linear acceleration and gravity, as opposed to the standard translational XY isotropic oscillator. Its properties are.

<FIG> shows that a pin placed on the balance wheel in <FIG> has a roughly elliptical orbit on a sphere, allowing this mechanism to be maintained by a rotating crank as with the XY translational isotropic harmonic oscillators. The figure describes the motion of the mass <NUM> of <FIG> as the balance and cage oscillate. The sphere <NUM> represents the space of all possible positions of the mass <NUM> for arbitrarily large oscillations of the balance wheel and cage. Shown in the figure is the situation for a small oscillation in which the mass <NUM> moves along a periodic orbit <NUM> around its neutral point <NUM>. The angular motion of the mass <NUM> is always in the same angular direction and does not stop.

<FIG> shows that if the X and Y angles are graphed on a plane, then the same elliptical orbit is recovered as in the X and Y translational case. The figure describes the angular parameters of the mechanism of <FIG>. The mass <NUM> represents the mass <NUM> of <FIG>. The angle theta represents the angle of rotation of the balance wheel of <FIG> around its axis, with respect to its neutral position and the angle phi represents the angle of rotation of the cage <NUM> of <FIG> around its axis, with respect to its neutral position. In the theta-phi coordinate system, the mass <NUM> moves in the periodic orbit <NUM> around its neutral point <NUM>. The orbit <NUM> is a perfect ellipse and following Newton's result, all such orbits will have the same period.

<FIG> shows an example of an XY isotropic harmonic oscillator with X a translation and Y a rotation. It can be seen that a pin on the balance wheel has a roughly elliptical orbit, so this mechanism can be maintained by a rotating crank as with the XY translational isotropic harmonic oscillators. To the fixed base <NUM> are attached two vertical immobile beams <NUM>. At the top of the two beams <NUM> is a horizontal beam (transparent here), to which is attached a collet holding a cylindrical spring <NUM>. The bottom of the cylindrical spring <NUM> is attached via a collet to the cage <NUM>, allowing the cage to translate vertically via two grooves <NUM> on each of the vertical posts <NUM>, the grooves hold the cage axes <NUM>. The cylindrical spring <NUM> provides a linear restoring force to produce translational oscillation of the cage. The cage <NUM> contains a spiral spring <NUM> attached to a balance wheel <NUM>. The spiral spring provides a restoring torque to the balance wheel which causes it to have a isotropic oscillation. The frequency of the translational oscillation of the cage <NUM> is designed to equal the frequency of the angular oscillation of the balance wheel <NUM>, for small amplitudes the balance weights <NUM> move in a unidirectional rotation approximating an ellipse. If x represent the vertical displacement of the cage with respect to its neutral point and theta the angle of the balance wheel with respect to its neutral angle, then x, theta represent generalised coordinates of the mechanism's state and describe an ellipse in state space, as shown in <FIG> with x replacing phi. Its properties are.

The advantage of using an escapement is that the oscillator will not be continuously in contact with the energy source (via the gear train) which can be a source of chronometric error. The escapements will therefore be free escapements in which the oscillator is left to vibrate without disturbance from the escapement for a significant portion of its oscillation.

The escapements are simplified compared to balance wheel escapements since the oscillator is turning in a single direction. Since a balance wheel has a back and forth motion, watch escapements generally require a lever in order to impulse in one of the two directions.

The first watch escapement which directly applies to our oscillator is the chronometer or detent escapement [<NUM>, <NUM>-<NUM>]. This escapement can be applied in either spring detent or pivoted detent form without any modification other than eliminating passing spring whose function occurs during the opposite rotation of the ordinary watch balance wheel, see [<NUM>, Figure 471c]. For example, in <FIG> illustrating the classical detent escapement, the entire mechanism is retained except for Gold Spring i whose function is no longer required.

Bouasse describes a detent escapement for the conical pendulum [<NUM>, <NUM>-<NUM>] with similarities to the one presented here. However, Bouasse considers that it is a mistake to apply intermittent impulse to the conical pendulum. This could be related to his assumption that the conical pendulum should always operate at constant speed, as explained above.

Embodiments of possible detent escapements for the isotropic harmonic oscillator are shown in <FIG>.

<FIG> illustrates a simplified classical detent watch escapement for an isotropic harmonic oscillator. The usual horn detent for reverse motion has been suppressed due to the unidirectional rotation of the oscillator.

<FIG> illustrates an embodiment of a detent escapement for translational orbiting mass. Two parallel catches <NUM> and <NUM> are fixed to the orbiting mass (not shown but illustrated schematically by the arrows forming a circle, reference <NUM>) so have trajectories that are synchronous translations of each other Catch <NUM> displaces detent <NUM> pivoted at spring <NUM> which releases escape wheel <NUM>. Escape wheel impulses on catch <NUM>, restoring lost energy to the oscillator.

<FIG> illustrates an embodiment of a new detent escapement for translational orbiting mass. Two parallel catches <NUM> and <NUM> are fixed to the orbiting mass (not shown) so have trajectories that are synchronous translations of each other Catch <NUM> displaces detent <NUM> pivoted at spring <NUM> which releases escape wheel <NUM>. Escape wheel impulses on catch <NUM>, restoring lost energy to the oscillator Mechanism allows for variation of orbit radius. Side and top views shown in this <FIG>.

<FIG> illustrates examples of compliant XY-stages of the prior art.

The conical pendulum is a pendulum rotating around a vertical axis, that is, perpendicular to the force of gravity, see <FIG> The theory of the conical pendulum was first described by Christiaan Huygens see references [<NUM>] and [<NUM>] who showed that, as with the ordinary pendulum, the conical pendulum is not isochronous but that, in theory, by using a flexible string and paraboloid structure, can be made isochronous.

However, as with cycloidal cheeks for the ordinary pendulum, Huygens' modification is based on a flexible pendulum and in practice does not improve timekeeping. The conical pendulum has never been used as a timebase for a precision clock.

Despite its potential for accurate timekeeping, the conical pendulum has been consistently described as a method for obtaining uniform motion in order to measure small time intervals accurately, for example, by Defossez in his description of the conical pendulum see reference [<NUM>, p.

Theoretical analysis of the conical pendulum has been given by Haag see reference [<NUM>] [<NUM>, p. <NUM>-<NUM>] with the conclusion that its potential as a timebase is intrinsically worse than the circular pendulum due to its inherent lack of isochronism.

The conical pendulum has been used in precision clocks, but never as a time base. In particular, in the <NUM>'s, William Bond constructed a precision clock having a conical pendulum, but this was part of the escapement, the timebase being a circular pendulum see references [<NUM>] and [<NUM>, p. <NUM>-<NUM>].

Our invention is therefore a superior to the conical pendulum as choice of time base because our oscillator has inherent isochronism. Moreover, our invention can be used in a watch or other portable timekeeper, as it is based on a spring, whereas this is impossible for the conical pendulum which depends on the timekeeper having constant orientation with respect to gravity.

Governors are mechanisms which maintain a constant speed, the simplest example being the Watt governor for the steam engine. In the 19th Century, these governors were used in applications where smooth operation, that is, without the stop and go intermittent motion of a clock mechanism based on an oscillator with escapement, was more important than high precision. In particular, such mechanisms were required for telescopes in order to follow the motion of the celestial sphere and track the motion of stars over relatively short intervals of time. High chronometric precision was not required in these cases due to the short time interval of use.

An example of such a mechanism was built by Antoine Breguet, see reference [<NUM>], to regulate the Paris Observatory telescope and the theory was described by Yvon Villarceau, see reference [<NUM>], it is based on a Watt governor and is also intended to maintain a relatively constant speed, so despite being called a regulateurisochrone (isochronous governor), it cannot be a true isochronous oscillator as described above. According to Breguet, the precision was between <NUM> seconds/day and <NUM> seconds/day, see reference [<NUM>].

Due to the intrinsic properties of harmonic oscillators following from the wave equation, see Section <NUM>, constant speed mechanisms are not true oscillators and all such mechanisms have intrinsically limited chronometric precision.

Governors have been used in precision clocks, but never as the time base. In particular, in <NUM> William Thomson, Lord Kelvin, designed and built an astronomical clock whose escapement mechanism was based on a governor, though the time base was a pendulum, see references [<NUM>] [<NUM>, p. <NUM>-<NUM>] [<NUM>, p. <NUM>-<NUM>]. Indeed, the title of his communication regarding the clock states that it features "uniform motion", see reference [<NUM>], so is clearly distinct in its purpose from the present invention.

There have been at least two continuous motion wristwatches in which the mechanism does not have intermittent stop & go motion so does not suffer from needless repeated accelerations. The two examples are the so-called Salto watch by Asulab, see reference [<NUM>], and Spring Drive by Seiko, see reference [<NUM>]. While both these mechanism attain a high level of chronometric precision, they are completely different from the present invention as they do not use an isotropic oscillator as a time base and instead rely on the oscillations of a quartz tuning fork. Moreover, this tuning fork requires piezoelectricity to maintain and count oscillations and an integrated circuit to control maintenance and counting. The continuous motion of the movement is only possible due to electromagnetic braking which is once again controlled by the integrated circuit which also requires a buffer of up to ±<NUM> seconds in its memory in order to correct chronometric errors due to shock.

Our invention uses a mechanical oscillator as time base and does not require electricity or electronics in order to operate correctly. The continuous motion of the movement is regulated by the isotropic oscillator itself and not by an integrated circuit.

In some embodiments some already discussed above and detailed hereunder, the present invention was conceived as a realization of the isotropic harmonic oscillator for use as a time base. Indeed, in order to realize the isotropic harmonic oscillator as a time base, there requires a physical construction of the central restoring force. One first notes that the theory of a mass moving with respect to a central restoring force is such that the resulting motion lies in a plane. It follows that for practical reasons, that the physical construction should realize planar isotropy. Therefore, the constructions described here will mostly be of planar isotropy, but not limited to this, and there will also be an example of <NUM>-dimensional isotropy. Planar isotropy can be realized in two ways: isotropic isotropic springs and translational isotropic springs.

Isotropic isotropic springs have one degree of freedom and rotate with the support holding both the spring and the mass. This architecture leads naturally to isotropy. While the mass follows the orbit, it rotates about itself at the same angular velocity as the support. This leads to a spurious moment of inertia so that the mass no longer acts as a point mass and the departure from the ideal model described in Section <NUM> and therefore to a theoretical isochronism defect.

Translational isotropic springs have two translational degrees of freedom in which the mass does not rotate but translates along an elliptical orbit around the neutral point. This does away with spurious moment of inertia and removes the theoretical obstacle to isochronism.

As already discussed above, a rotating turntable <NUM> on which is fixed a spring <NUM> of rigidity k with the spring's neutral point at the center of rotation of the turntable is illustrated in <FIG>. Assuming a massless turntable and spring, a linear central restoring force is realized by this mechanism. However, given the physical reality of the turntable and spring, this realization has the disadvantages of having significant spurious mass and moment of inertia.

A rotating cantilever spring <NUM> supported in a cage <NUM> turning axially is illustrated in <FIG>, discussed above. This again realizes the central linear restoring force but reduces spurious moment of inertia by having a cylindrical mass and an axial spring. Numerical simulation shows that divergence from isochronism is still significant. A physical model has been constructed, see <FIG>, where vertical motion of the mass has been minimized by attaching the mass to a double leaf spring producing approximately linear displacement instead of the approximately circular displacement of the single spring of <FIG>. The data from this physical model is consistent with the analytic model.

Note that gravity does not affect the spring when it is in the axial direction However, these inventions have the disadvantage of having the spring and its support both rotating around their own axes, which introduces spurious moment of inertia terms which reduce the theoretical isochronism of the model. Indeed, considering the point mass of mass m and then including an isotropic support of moment of inertia <NUM>/<NUM> and constant total angular momentum L, then if friction is ignored, the equations of motion reduce to <MAT>.

This equation can be solved explicitly in terms of Jacobi elliptic functions and the period expressed in terms of elliptic integrals of the first kind, see [<NUM>] for definitions and similar applications to mechanics. A numerical analysis of these solutions shows that the divergence from isochronism is significant unless the moment of inertia <NUM>/<NUM> is minimized.

in this section we will describe the background leading to our principal invention of isotropic springs. From now on and unless otherwise specified, "isotropic spring" will denote "planar translational isotropic spring".

The invention is based on compliant XY-stages, see references [<NUM>, <NUM>, <NUM>, <NUM>] and <FIG> illustrating examples of the prior art. Compliant XY-stages are mechanism with two degrees of freedom both of which are translations. As these mechanisms comprise compliant joints, see reference [<NUM>], they exhibit planar restoring forces so can be considered as planar springs.

in the literature Simon Henein, see reference [<NUM>,p. <NUM>, <NUM>], has proposed two XY-stages which exhibit planar isotropy. The first one, illustrated in <FIG> comprises two serial compliant four-bar <NUM> mechanisms, also called parallel arms linkage, which allows, for small displacements translations in the X and Y directions The second one, illustrated in <FIG> comprises four parallel arms <NUM> linked with eight spherical joints <NUM> and a bellow <NUM> connecting the mobile platform <NUM> to the ground The same result can be obtained with three parallel arms linked and with eight spherical joints and a bellow connecting the mobile platform to the ground.

Isotropic springs are one object of the present invention and they appear most suitable to preserve the theoretical characteristics of the harmonic oscillator are the ones in which the central force is realized by an isotropic spring, where the term isotropic is again used to mean "same in all directions.

As described above, the simplest version is given in <FIG>. In this figure, two springs <NUM>, <NUM> SX and Sy of rigidity k are placed that spring <NUM> SX acts in the horizontal x-axis and spring <NUM> Sy acts in the vertical y-axis.

There is a mass <NUM> attached to both these springs and having mass m. The geometry is chosen such that at the point (<NUM>, <NUM>) both springs are in their neutral positions.

One can now show that this mechanism exhibits isotropy to first order, see <FIG>. Assuming now a small displacement d r = (dx, dy), then up to first order, there is a restoring force FX in the x direction of -k dx and a restoring force Fy in y direction of -k dy. This gives a total restoring force <MAT> and the central linear restoring force of Section <NUM> is verified. It follows that this mechanism is, up to first order, a realization of a central linear restoring force, as claimed.

In these realizations, gravity affects the spring in all directions as it changes the effective spring constant. However, the spring does not rotate around its own axis, minimizing spurious moments of inertia, and the central force is directly realized by the spring itself. We now list which of the theoretical properties of Section <NUM> hold for these embodiments (up to first order).

Since a timekeeper needs to be very precise, at least <NUM>/<NUM> for <NUM> second/day accuracy, an isotropic spring realization must itself be quite precise. This is the subject of embodiments of the present invention.

Since the invention closely models an isotropic spring and minimizes the isotropy defect, the orbits of a mass supported by the invention will closely model isochronous elliptical orbits with neutral point as center of the ellipse. <FIG> is basic illustration of the principle of the present invention (see above for its detailed description).

The principle exposed hereunder by reference to <FIG> may be applied to the stages <NUM>-<NUM> illustrated in <FIG> and described above as possible embodiments of said stages as has been detailed above.

The idea of combining two springs is refined by replacing linear springs with parallel springs <NUM>, <NUM> as shown in <FIG> forming a spring stage <NUM> holding orbiting mass <NUM>. In order to get a two degrees of freedom planar isotropic spring, two parallel spring stages <NUM>, <NUM> (as shown in <FIG>, each with parallel springs <NUM>, <NUM>, <NUM> and <NUM>) are placed orthogonally, see <FIG> and <FIG>.

We now list which of the theoretical properties of Section <NUM> hold for these embodiments.

This model has two degrees of freedom as opposed to the model of Section <NUM> which has six degrees of freedom. Therefore, this model is truly planar, as is required for the theoretical model of Section <NUM>. Finally, this model is insensitive to gravity when its plane is orthogonal to gravity.

We have explicitly estimated the isotropy defect of this mechanism and we will use this estimate to compare with the compensated mechanism isotropy defect.

The presence of intermediate blocks leads to reduced masses which are different in different directions. The ideal mathematical model of Section <NUM> is therefore no longer valid and there is a theoretical isochronism defect. The example shown in <FIG> minimizes this difference. The aim is to minimize reduced mass isotropy by stacking two identical in plane orthogonal parallel spring stages of <FIG> which are rotated by <NUM> degrees with respect to each other (angles of rotation about the z-axis).

In <FIG> a first plate <NUM> is mounted on top of a second plate <NUM>. Blocks <NUM> and <NUM> of first plate <NUM> are fixed onto blocks <NUM> and <NUM> respectively of second plate <NUM>. In the upper two figures the grey shaded blocks <NUM>, <NUM> of first plate and <NUM> of second plate <NUM> have a y-displacement corresponding to the y-component displacement of the orbiting mass <NUM>, while the black shaded blocks <NUM> of the first plate <NUM> and <NUM>, <NUM> of the second plate <NUM> remain immobile. In the lower figure, the grey shaded blocks <NUM>, <NUM> of first <NUM> and <NUM> of second plate <NUM> have an x-displacement corresponding to the x-component displacement of the orbiting mass <NUM> while the black shaded blocks <NUM>, <NUM>, <NUM> of the first <NUM> and second <NUM> plates remain immobile. Since the first and second plates <NUM>, <NUM> are identical, the sum of the masses of <NUM>, <NUM> and <NUM> is equal to the sum of the masses of <NUM>, <NUM> and <NUM>. Therefore, the total mobile mass (grey blocks <NUM>, <NUM>, <NUM>) is the same for displacements in x and in y directions, as well as in any direction of the plane.

As a result of the construction, the reduced mass in the x and y directions are identical and therefore the same in every planar direction, thus in theory minimizing reduced mass isotropy defect.

The goal of this mechanism is to provide an isotropic spring stiffness. Isotropy defect, that is, the variation from perfect spring stiffness isotropy, will be the factor minimized in our invention. Our inventions will be presented in order of increasing complexity corresponding to compensation of factors leading to isotropy defects.

This embodiment is shown in <FIG> with a top view given in <FIG>. Using compound parallel spring stages instead of simple parallel spring stages results in rectilinear movement at each stage. The principal cross-coupling effects leading to isotropy defects are therefore suppressed.

In particular, <FIG> and <FIG> illustrate an embodiment of an in plane orthogonal compensated parallel spring stages according to the invention. Fixed base <NUM> holds first pair of parallel leaf springs <NUM> connected to intermediate block <NUM>. Second pair of leaf springs <NUM> (parallel to <NUM>) connect to second intermediate block <NUM>. Intermediate block <NUM> holds third pair of parallel leaf springs <NUM> (orthogonal to springs <NUM> and <NUM>) connected to third intermediate block <NUM>. Intermediate block <NUM> holds parallel leaf springs <NUM> (parallel to springs <NUM>) which are connected to orbiting mass <NUM> or alternatively to a frame holding the orbiting mass <NUM>.

An alternative embodiment to the in plane orthogonal compensated parallel spring stages is given in <FIG>.

Instead of having the sequence of parallel leaf springs <NUM>, <NUM>, <NUM>, <NUM> as in <FIG>, the sequence is <NUM>, <NUM>, <NUM>, <NUM>.

In a specific example computed, the in-plane orthogonal non-compensated parallel spring stages mechanism has a worst case isotropy defect of <NUM>%. On the other hand, for the compensated mechanism, worst case isotropy is <NUM>%. The compensated mechanism therefore reduces the worst case isotropy stiffness defect by a factor of <NUM>.

A general estimate depends on the exact construction, but the above example estimate indicates that the improvement is of two orders of magnitude.

The presence of intermediate blocks leads to reduced masses which are different for different angles. The ideal mathematical model of Section <NUM> is therefore no longer valid and there is a theoretical isochronism defect. The example shown in <FIG> minimizes this difference. The aim is to minimize reduced mass isotropy by stacking two identical in plane orthogonal compensated parallel spring stages which are rotated <NUM> degrees with respect to each other (angles of rotation about the z-axis).

Accordingly, <FIG> discloses an embodiment minimizing the reduced mass isotropy defect.

A first plate <NUM> is mounted on top of a second plate <NUM> and the numbering has the same significance as in <FIG>. Blocks <NUM> and <NUM> of first plate <NUM> are fixed onto blocks <NUM> and <NUM> respectively of second plate <NUM>. In the upper figure the grey shaded blocks <NUM>, <NUM> of first plate <NUM> and <NUM>, <NUM>, <NUM>, <NUM> of second plate <NUM> have an x-displacement corresponding to the x-component displacement of the orbiting mass while the black shaded blocks <NUM>, <NUM>, <NUM> of the first plate <NUM> and <NUM> of the second plate <NUM> remain immobile. In the lower figure, the grey shaded blocks <NUM>, <NUM>, <NUM>, <NUM> of first plate <NUM> and <NUM> of second plate <NUM> have a y-displacement corresponding to the y-component displacement of the orbiting mass while the black shaded block <NUM> of the first plate <NUM> and <NUM>, <NUM>, <NUM> of the second plate <NUM> remain immobile.

As a result of this embodiment, the reduced mass in the x and y directions are identical and therefore identical in every direction, thus in theory minimizing reduced mass isotropy defect.

We now list which of the theoretical properties of Section <NUM> hold for this embodiment.

Another out of plane orthogonal compensated isotropic spring example is illustrated in <FIG>.

A fixed base <NUM> holds first pair of parallel leaf springs <NUM> connected to intermediate block <NUM>. Second pair of leaf springs <NUM> (parallel to <NUM>) connect to second intermediate block <NUM>. Intermediate block <NUM> holds third pair of parallel leaf springs <NUM> (orthogonal to springs <NUM> and <NUM>) connected to third intermediate block <NUM>. Intermediate block <NUM> holds parallel leaf springs <NUM> (parallel to <NUM>) which are connected to orbiting mass <NUM> (or alternatively frame holding the orbiting mass <NUM>).

We now list which of the theoretical properties of Section <NUM> hold for this example.

We can reduce the isotropy defect by making a copy of the isotropic spring and stacking the copy on top of the original, with a precise angle offset.

<FIG> illustrates a parallel assembly of two identical XY parallel spring oscillators for amelioration of the stiffness isotropy. The first XY parallel spring stage oscillator (upper stage on <FIG>) comprises a fixed outer frame <NUM>, a first pair of parallel leaf springs <NUM> and <NUM>, an intermediate block <NUM>, a second pair of parallel leaf springs <NUM> and <NUM>, and a mobile block <NUM> on which the orbiting mass (not shown on the figure) is to be rigidly mounted. The second XY parallel spring stage (lower stage on <FIG>) is identical to the first. Both stages are mounted together by rigidly attaching <NUM> to <NUM> and <NUM> to <NUM>. The second XY parallel spring stage is rotated <NUM> degrees around the Z axis with respect to the first one (the figure shows that indexing-notch A on <NUM> is opposite to indexing-notch A in <NUM>). Since the isotropy defect of a single stage is periodic, stacking two stages in parallel with the correct angular offset (in this case <NUM> degrees) leads to anti-phase cancellation of the defect. Shims <NUM> and <NUM> are used to separate slightly the two stages and avoid any friction between their mobile parts. The stiffness isotropy defect of the complete assembly is significantly smaller (typically a factor <NUM> to <NUM>) than that of a single XY parallel spring stage. The stiffness isotropy can be further improved by stacking more than two stages rotated by angles smaller than <NUM> degrees. It is possible to invert the mechanism, i.e. to attach <NUM>, <NUM> and <NUM> to the fixed base and mount the orbiting mass onto the outer frames <NUM>, <NUM> and <NUM> with no changes in the overall behavior. Its properties are.

<FIG> illustrates a parallel assembly of two identical XY compound parallel spring oscillators for amelioration of the stiffness isotropy. The first XY compound parallel spring stage (upper part on figure <NUM>) comprises a fixed outer frame <NUM> connected to a mobile block <NUM> via two perpendicular compound parallel spring stages mounted in series. The orbiting mass (not shown on the figure) is to be rigidly mounted onto the mobile block <NUM>. The second XY compound parallel spring stage (lower part on figure <NUM>) is identical to the first. It comprises a fixed outer frame <NUM> connected to a mobile rigid block <NUM> via two perpendicular compound parallel spring stages mounted in series. Both stages are mounted together by rigidly attaching <NUM> onto <NUM> and <NUM> onto <NUM>. The second XY parallel spring stage is rotated <NUM> degrees around Z with respect to the first one (the figure shows that the indexing-notch A on <NUM> is rotated <NUM> degrees with respect to indexing-notch A in <NUM>). Since the isotropy defect of a single stage is periodic, stacking two stages in parallel with the correct angular offset (in this case <NUM> degrees) leads to anti-phase cancellation of the defect. Shims <NUM> and <NUM> are used to separate slightly the two stages and avoid any friction between the mobile parts. The stiffness isotropy defect of the complete assembly is significantly smaller (typically a factor <NUM> to <NUM>) than that of a single XY compound parallel spring stage. Note <NUM>: The stiffness isotropy can be further improved by stacking more than two stages rotated by angles smaller than <NUM> degrees. Note <NUM>: It is possible to invert the mechanism, i.e. to attach <NUM>, <NUM> and <NUM> to the fixed base and mount the orbiting mass onto the outer frames <NUM>, <NUM> and <NUM> with no changes in the overall behavior. Its properties are.

Typically, the embodiments illustrated in <FIG> and <FIG> are applicable to the constructions and embodiments described hereinabove and illustrated in <FIG> and <FIG> which comprise similar stages. Also, in relation to these embodiments, stacks comprising several stages (two or more) may be formed by stacking them on top of each other, each stage having an angular offset for example <NUM>°, <NUM>°, <NUM>° or other values or even a combination thereof with respect to its neighboring stage, according to the principle described hereabove. Such combination of stages oriented with different angles allow reduction or even cancellation of the isotropy defect of the oscillator.

<FIG> illustrates a serial assembly of two identical XY parallel spring oscillators for amelioration of the stiffness isotropy. The first XY parallel spring stage oscillator (lower stage on <FIG>) comprises a fixed outer frame <NUM>, a first pair of parallel leaf springs <NUM>, an intermediate block <NUM>, a second pair of parallel leaf springs <NUM>, and a mobile block <NUM> on which the second XY parallel spring stage (upper stage on <FIG>) is rigidly mounted. This second stage is identical to the first one. Both stages are mounted together by rigidly attaching <NUM> to <NUM> via a shim <NUM> creating a gap between the two stages. The second stage is rotated <NUM> degrees around the Z axis with respect to the first one (the figure shows that indexing-notch A on <NUM> is opposite to indexing-notch A in <NUM>). The mobile mass of the oscillator is the block <NUM> (this block is made out of dense material whereas all the other mobiles blocks are made of low density material). Since the isotropy defect of a single stage is periodic, stacking two stages serially with the correct angular offset (in this case <NUM> degrees) leads to anti-phase cancellation of the defect. The stiffness isotropy defect of the complete assembly is significantly smaller (typically a factor <NUM> to <NUM>) than that of a single XY parallel spring stage. The stiffness isotropy can be further improved by stacking more than two stages rotated by angles smaller than <NUM> degrees. Its properties are.

<FIG> illustrates a serial assembly of two identical XY compound parallel spring oscillators for amelioration of the stiffness isotropy. The first XY parallel spring stage oscillator (lower stage on <FIG>) comprises a fixed outer frame <NUM>, and a mobile block <NUM> on which the second XY compound parallel spring stage (upper stage on <FIG>) is rigidly mounted. This second stage is identical to the first one. Both stages are mounted together by rigidly attaching <NUM> to <NUM> via a shim <NUM> creating a gap between the two stages. The second stage is rotated <NUM> degrees around the Z axis with respect to the first one (the figure shows that indexing-notch A on <NUM> is shifted with respect to indexing-notch A in <NUM>). The mobile mass of the oscillator is the block <NUM> (this block is made out of dense material whereas all the other mobiles blocks are made of low density material). Since the isotropy defect of a single stage is periodic, stacking two stages serially with the correct angular offset (in this case <NUM> degrees) leads to anti-phase cancellation of the defect.

The stiffness isotropy defect of the complete assembly is significantly smaller (typically a factor <NUM> to <NUM>) than that of a single XY parallel spring stage. The stiffness isotropy can be further improved by stacking more than two stages rotated by angles smaller than <NUM> degrees. Its properties are.

In order to place the new oscillator in a portable timekeeper, it is necessary to address forces that could influence the correct functioning of the oscillator. These include gravity and shocks.

The first method to address the force of gravity is to make a planar isotropic spring which when in horizontal position with respect to gravity does not feel its effect as described above.

However, this is adequate only for a stationary clock. For a portable timekeeper, compensation is required. This can be achieved by making a copy of the oscillator and connecting both copies through a ball or universal joint as described above in reference to <FIG>. In the realization of <FIG>, the center of gravity of the entire mechanism remains fixed. One uses the oscillator of Section <NUM>.

We now list which of the theoretical properties of Section <NUM> hold for this embodiment.

Linear shocks are a form of linear acceleration, so include gravity as a special case. Thus, the mechanism of <FIG> also compensates for linear shocks, see description above.

Effects due to angular accelerations can be minimized by reducing the distance between the centers of gravity of the two masses as shown in <FIG> by modifying the mechanism of the previous section shown in <FIG>. Precise adjustment of the distance I shown in <FIG> separating the two centers of gravity allows for a complete compensation of angular shocks including taking account the moment of inertia of the lever itself. Another example is shown in <FIG>, where two XY oscillators are coupled via a crankshaft similar to a bicycle crankset and bottom bracket, with the cranks impulsing each XY oscillator at possibly different radii. More precisely, <FIG> illlustrate a dynamically balanced angularly coupled double oscillator. The orbiting masses <NUM> and <NUM> of two planar oscillators are coupled by a double crank (similar to a bicycle crankset) comprising an upper crank <NUM>, a lower crank <NUM> and their shaft <NUM> (similar to a bicycle bottom bracket). Crank arm <NUM> contains a slot allowing a pin rigidly connected to mass <NUM> to slide in this slot. Similarly, mass <NUM> is rigidly connected to a pin sliding in a slot on crank <NUM>. Shaft <NUM> is driven by a gear <NUM> which is itself driven by a gear <NUM>, which in turn is driven by a gear <NUM>. This arrangement forces both masse <NUM> and <NUM> to orbit at <NUM> degrees from each other (angular coupling). The radial positions of the two masses are independent (no radial coupling). The full system thus behaves as a three degrees of freedom oscillator. The fixed frame <NUM> and <NUM> of the upper and lower oscillators are attached to a common fixed frame <NUM>. Its properties are.

Another example is given in <FIG>, where two XY oscillators are coupled via a ball joint so that the radii and amplitudes are the same for each XY oscillator. More precisely, <FIG> and 50B illustrate a dynamically balanced angularly and radially coupled double oscillator based on two planar oscillators. Orbiting masses <NUM> and <NUM> of two planar oscillators <NUM> and <NUM> are coupled by a coupling bar <NUM> connected to the fixed frame <NUM> by a ball joint <NUM>. The two extremities of <NUM> slide axially into two spheres <NUM> and <NUM> forming ball joint articulations with respect to <NUM> and <NUM> respectively. This kinematic arrangement results in an angular and radial coupling of both oscillators. The full system thus behaves a two degree of freedom oscillator. The fixed frames <NUM> and <NUM> of the upper and lower oscillators are attached to a common fixed frame <NUM>. Its properties are.

Another embodiment is given in <FIG> where the dynamic balancing is achieved via levers having flexure pivots, with lever lengths chosen with ratios eliminating undesirable force. More precisely, <FIG> illustrates a dynamically balanced isotropic harmonic oscillator: The orbiting mass <NUM> (M) in mounted onto a frame <NUM>. The frame <NUM> is attached to the fixed base <NUM> via two parallel spring stages mounted in series at <NUM> degrees: <NUM> and <NUM> provide a degree-of-freedom in the Y direction, and <NUM> and <NUM> provide a degree-of-freedom in the X direction. <NUM> is an intermediate mobile block. Additionally, <NUM> is connected to an X compensating mass <NUM> (m) moving in opposite direction for all movements in the X direction of <NUM>, and to a Y direction compensating mass <NUM> moving in opposite direction for all movements in the Y direction. The inversion mechanism is based on a leaf spring <NUM> connecting the main mass <NUM> to a rigid lever <NUM>. The lever pivots with respect to the fixed base thanks to a flexure-pivot comprising two leaf springs <NUM> and <NUM>. The X direction compensating mass <NUM> is mounted onto the opposite end of the lever. The lever lengths are chosen to have the particular ratio OA/OB = m/M, so that linear acceleration in the XY plane produce no torque on the pivot O. An identical mechanism <NUM> to <NUM> is used to balance the main mass <NUM> dynamically for acceleration in the Y direction. The overall mechanism is thus highly insensitive to linear accelerations in the range of small deformations. A rigid pin <NUM> is attached to <NUM> and engages into the driving crank (not shown in the figure) maintaining the orbiting motion. Note: all parts except the masses <NUM>, <NUM> and <NUM> are made out of a low-density material, for example aluminum alloy or silicon.

A three dimensional translational isotropic spring is illustrated in <FIG>. Three perpendicular bellows <NUM> connect to translational orbiting mass <NUM> to fixed base <NUM>. Using the argument of section <NUM>, see <FIG> above, this mechanism exhibits three dimensional isotropy up to first order. Unlike the two-dimensional constructions illustrated in <FIG>, the bellows <NUM> provide a <NUM> degree-of-freedom translational suspension making this a realistic working mechanism insensitive to external torque. Its properties are.

By adding a radial display to isotropic spring embodiments described herein, it is possible to constitute an entirely mechanical two degree-of-freedom accelerometer, for example, suitable for measuring lateral g forces in a passenger automobile.

In an another application, the oscillators and systems described in the present application may be used as a time base for a chronograph measuring fractions of seconds requiring only an extended speed multiplicative gear train, for example to obtain <NUM> frequency so as to measure <NUM>/<NUM>th of a second. Of course, other time interval measurement is possible and the gear train final ratio may be adapted in consequence.

In a further application, the oscillator described herein may be used as a speed governor where only constant average speed over small intervals is required, for example, to regulate striking or musical clocks and watches, as well as music boxes. The use of a harmonic oscillator, as opposed to a frictional governor, means that friction is minimized and quality factor optimized thus minimizing unwanted noise, decreasing energy consumption and therefore energy storage, and in a striking or musical watch application, thereby improving musical or striking rhythm stability.

Claim 1:
A mechanical X-Y planar isotropic harmonic oscillator comprising a fixed frame (<NUM>,140a),
a first stage (<NUM>) comprising parallel springs (131a to 131d) holding a mass (131e);
a second stage (<NUM>) comprising parallel springs (132a to 132d) holding a mass (132e),
a third stage (<NUM>) comprising parallel springs (133a to 133d) holding a mass (133e) and
a fourth stage (<NUM>) comprising parallel springs (134a to 134d) holding a mass (134e),
wherein said stages are successively located above each other such that the second stage (<NUM>) is above the first stage (<NUM>), the third stage (<NUM>) is above the second stage (<NUM>) and the fourth stage (<NUM>) is above the third stage (<NUM>);
wherein each stage is arranged with a relative anticlockwise <NUM>° rotation with respect to the stage above which it is placed,
wherein the masses (131e, 132e) of the first and second stages are attached to each other and the masses (133e, 134e) of the third and fourth stages are attached to each other;
wherein two of said stages (<NUM>,<NUM> or <NUM>,<NUM>) are attached together to form an orbiting mass and two other of said stages (<NUM>,<NUM> or <NUM>,<NUM>) are attached to said fixed frame
wherein the two stages forming the orbiting mass are the first (<NUM>) and fourth stage (<NUM>) and wherein the two stages attached to the fixed frame are the second (<NUM>) and the third stage (<NUM>)
or wherein the two stages forming the orbiting mass are the second (<NUM>) and third stage (<NUM>) and
wherein said two stages attached to the fixed frame are the first (<NUM>) and the fourth stage (<NUM>).