EP2613206B1 - Hairspring with two spiral springs with improved isochronism - Google Patents

Description

Field of the invention

The invention relates to a hairspring used to form a balance resonator-hairspring whose curvature allows development with a substantially immobile center of mass.

Background of the invention

The documents EP 2 184 652 , EP 2 196 867 and EP 2 105 807 explain how to manufacture curve-raising hairsprings in micro-machinable materials using three parts, two parts or in one piece respectively.

It is known to apply the Phillips criteria to determine the theoretical curvature of a terminal curve. However, the Phillips criteria are in fact an approximation which does not necessarily give satisfaction if an even smaller rate deviation is desired.

Summary of the invention

The object of the present invention is to overcome all or part of the aforementioned drawbacks by proposing a hairspring respecting predetermined conditions capable of reducing the displacement of the center of mass of the hairspring in contraction and in expansion.

où :

• n est un nombre entier supérieur ou égal à zéro ;
• P ⁽ⁿ⁾ est le moment du spiral d'ordre n ;
• L est la longueur du spiral ;
• sⁿ représente l'abscisse curviligne le long du spiral à la puissance d'ordre n ;
• x (s) est la paramétrisation du spiral par son abscisse curviligne, afin de réduire les déplacements de son centre de masse lors de ses contraction et expansion et en ce que chaque ressort-spiral comporte en outre au moins deux contrepoids afin de compenser le balourd formé par la masse de l'attache et personnaliser la pente d'anisochronisme du spiral, lesdits au moins deux contrepoids étant symétriques selon ladite même droite que les courbures , deux contrepoids (8, 8') étant situés à côté de l'attache et deux autres contrepoids étant situés du côté opposé à l'attache afin de minimiser le balourd.

To this end, the invention relates to a hairspring comprising a first hairspring whose curve extends in a first plane and whose internal turn comprises a ferrule, a second hairspring whose curve extends in a second plane parallel to the first plane, a clip securing the external turn of the first hairspring to the external turn of the second hairspring to form a double hairspring in series, characterized in that the curve of the first hairspring and the curve of the second hairspring each have a continuously variable pitch and are symmetrical with respect to a straight line parallel to the first and second planes and passing through the median plane of projection of the attachment, each curve being such that the moment of the hairspring of order n defined by the following relation is substantially zero: $${\overrightarrow{P}}^{\left(\mathrm{not}\right)}=\frac{\mathrm{not}+1}{L^{\mathrm{not}+1}}{\displaystyle \underset{0}{\overset{L}{\int }}\mathit{ds}\cdot s^{\mathrm{not}}\cdot \overrightarrow{x}}\left(s\right)$$

or :

• n is an integer greater than or equal to zero;
• P ⁽ⁿ⁾ is the moment of the balance spring of order n;
• L is the length of the hairspring;
• s ⁿ represents the curvilinear abscissa along the hairspring to the power of order n;
• x (s) is the parametrization of the hairspring by its curvilinear abscissa, in order to reduce the displacements of its center of mass during its contraction and expansion and in that each hairspring-spring also comprises at least two counterweights in order to compensate for the unbalance formed by the mass of the attachment and customize the anisochronism slope of the hairspring, said at least two counterweights being symmetrical along said same straight line as the curvatures, two counterweights (8, 8') being located next to the attachment and two other counterweights being located on the opposite side to the attachment in order to minimize unbalance.

• chaque contrepoids est sensiblement en forme de H, les bras parallèles du H étant sensiblement parallèles à la courbure locale du ressort-spiral auquel il est associé ;
• les faces principales de l'attache sont sensiblement parallèles à ladite droite de symétrie ;
• la spire interne du deuxième ressort-spiral comporte un dispositif de décalage agencé pour s'attacher à un piton dans le plan du deuxième ressort-spiral ;
• le dispositif de décalage comporte une pièce en prolongement de la spire interne du deuxième ressort-spiral, ladite pièce étant plus rigide que ledit deuxième ressort-spiral afin de ne pas fournir de couple élastique ;
• la pièce en prolongement est reliée à la spire interne à l'aide d'un coude sensiblement en forme de U ;
• la pièce en prolongement est venue de matière avec le deuxième ressort-spiral ;
• la pièce en prolongement est rendue plus rigide par une épaisseur au moins trois fois supérieure par rapport à celle dudit deuxième ressort-spiral ;
• la pièce peut être partiellement ajourée pour diminuer sa masse ;
• la spire interne du premier ressort-spiral comporte un dispositif d'agrandissement de la virole dans le plan du premier ressort-spiral ;
• le dispositif d'agrandissement comporte un bourrelet prolongeant la spire interne du premier ressort-spiral, ledit bourrelet étant plus rigide que ledit premier ressort-spiral afin de ne pas fournir de couple élastique ;
• le bourrelet est sensiblement en forme de U ;
• le bourrelet est venu de matière avec le premier ressort-spiral ;
• le spiral est formé en silicium ;
• le spiral comporte au moins une partie recouverte de dioxyde de silicium afin de limiter sa sensibilité aux variations de température et aux chocs mécaniques.

In accordance with other advantageous characteristics of the invention:

• each counterweight is substantially H-shaped, the parallel arms of the H being substantially parallel to the local curvature of the hairspring with which it is associated;
• the main faces of the fastener are substantially parallel to said line of symmetry;
• the inner turn of the second hairspring includes a shift device arranged to attach to a peak in the plane of the second hairspring;
• the shift device comprises a part extending from the internal turn of the second balance-spring, said part being more rigid than said second balance-spring so as not to supply an elastic torque;
• the extension part is connected to the internal turn by means of a substantially U-shaped bend;
• the extension piece is made in one piece with the second hairspring;
• the extension piece is made more rigid by a thickness at least three times greater than that of said second hairspring;
• the part can be partially perforated to reduce its mass;
• the inner coil of the first hairspring includes a device for enlarging the ferrule in the plane of the first hairspring;
• the enlargement device comprises a bead extending the internal turn of the first hairspring, said bead being more rigid than said first hairspring so as not to provide elastic torque;
• the bead is substantially U-shaped;
• the bead came from material with the first hairspring;
• the hairspring is made of silicon;
• the hairspring includes at least one part covered with silicon dioxide in order to limit its sensitivity to temperature variations and mechanical shocks.

Consequently, advantageously according to the invention, a hairspring can be manufactured respecting predetermined conditions in order to reduce the displacement of the center of mass of the hairspring in contraction and in expansion. This small or negligible displacement advantageously makes it possible to reduce the antinode of the anisochronism curves to a value substantially equal to or less than 0.5 sj ^-1 . Moreover, advantageously according to the invention, the anisochronism slope of the hairspring can be personalized in order to compensate for that given by the delay in the escapement.

Furthermore, the invention relates to a resonator for a timepiece comprising a balance, characterized in that the resonator comprises a hairspring conforming to one of the preceding variants, cooperating with the balance.

Brief description of the drawings

• les figures 1 et 2 sont des schémas destinés à expliquer les raisonnements suivis ;
• les figures 3 à 5 sont des exemples de calcul de courbures à 2,3 spires respectant respectivement les équations des moments jusqu'à l'ordre 2, 3 et 4 ;
• les figures 6 à 8 sont des exemples de calcul de courbures à 5,3 spires respectant respectivement les équations des moments jusqu'à l'ordre 2, 3 et 4 ;
• les figures 9 et 10 sont des représentations en perspective d'un spiral selon l'invention ;
• la figure 11 est une représentation vue de dessus du spiral des figures 9 et 10.

Other features and advantages will emerge clearly from the description given below, by way of indication and in no way limiting, with reference to the appended drawings, in which:

• the figures 1 and 2 are diagrams intended to explain the reasoning followed;
• the figures 3 to 5 are examples of calculation of curvatures with 2.3 turns respectively respecting the equations of moments up to order 2, 3 and 4;
• the figures 6 to 8 are examples of calculation of curvatures with 5.3 turns respectively respecting the equations of moments up to order 2, 3 and 4;
• the figures 9 and 10 are representations in perspective of a hairspring according to the invention;
• the figure 11 is a representation seen from above of the balance spring figures 9 and 10 .

Detailed Description of Preferred Embodiments

Variations in the rate of a mechanical watch relative to its theoretical frequency are mainly due to the escapement and the balance resonator - hairspring. There are two types of rate variations depending on whether they are generated by the amplitude of oscillation of the balance wheel or by the position of the watch movement. This is why, for anisochronism tests, a watch movement is tested in six positions, 2 horizontal (dial up and down) and 4 vertical (stem being turned 90° from a position the top). From the six distinct curves obtained, the maximum deviation between them, also called “belly”, is determined, expressing the maximum rate variation of the movement in seconds per day (sj ^-1 ).

The escapement induces a rate variation depending on the amplitude of the balance wheel, which is difficult to adjust. Consequently, the hairspring is generally adapted so that its variation as a function of the same amplitude is substantially opposite to that of the escapement. In addition, the hairspring is adapted so that its variation is minimal between the four vertical positions.

The necessary adaptations of the hairsprings have attempted to be posed mathematically in order to determine the ideal curvatures by calculation. Geometric conditions have been stated in particular by MM. Phillips and Grossmann in order to build a satisfactory hairspring, that is to say one whose center of mass remains on the axis of the balance wheel. However, current conditions are rough approximations. In fact, as very small displacements of the center of mass can generate large rate variations, the rate variations obtained by following the current geometric conditions are often disappointing.

This is why, advantageously according to the invention, new conditions are presented below in order to obtain better rate variation results than by current geometric conditions, in particular those laid down by MM. Phillips and Grossmann.

où :

• n est un nombre entier supérieur ou égal à zero ;
• L est la longueur du spiral ;
• sⁿ représente l'abscisse curviligne le long du spiral à la puissance d'ordre n ;
• x (s) est la paramétrisation du spiral par son abscisse curviligne.

We define a "moment of the hairspring of order n", P ⁽ⁿ⁾ , by the following formula: $${\overrightarrow{P}}^{\left(\mathrm{not}\right)}=\frac{\mathrm{not}+1}{L^{\mathrm{not}+1}}{\displaystyle \underset{0}{\overset{L}{\int }}\mathit{ds}\cdot s^{\mathrm{not}}\cdot \overrightarrow{x}\left(s\right)}$$

or :

• n is an integer greater than or equal to zero;
• L is the length of the hairspring;
• s ⁿ represents the curvilinear abscissa along the hairspring to the power of order n;
• x ( s ) is the parametrization of the hairspring by its curvilinear abscissa.

Thus, in order to obtain an immobile center of mass, it is necessary, for each order n, that the moment of the hairspring P ^{( n )} is zero. Not all orders can be calculated since there are an infinite number of them. Thus, the greater the number of orders for which relation (1) zero is respected, the more the amount of displacement of the center of mass will be reduced.

In the example shown in figure 1 , eight orders of moment of the hairspring are represented by points which allow, by a parametrization using a polynomial comprising at least as many coefficients as orders (in our case at least eight), to define a theoretical curvature "ideal".

In order to apply these conditions of the zero hairspring moment, we start from a hairspring of the type of figures 9 to 11 , that is to say, a hairspring 1 comprising a first hairspring 3 whose curve extends in a first plane, a second hairspring 5 whose curve extends in a second plane parallel to the first plane . Each end of hairspring 3, 5 is preferably secured by a clip 4 to form a double hairspring in series.

As explained above, the manufacture of such a hairspring is possible by the processes explained in the documents EP 2 184 652 , EP 2 196 867 and EP 2 105 807 from micro-machinable materials such as silicon respectively using three-part, two-part or in one piece. Of course, such a hairspring can be made from other processes and/or other materials.

In order to simplify the calculations, the curve of the first hairspring 3 and the curve of the second hairspring 5 preferably each have a continuously variable pitch and are symmetrical with respect to a straight line A parallel to the first and second planes passing by the center of the median plane P of projection of the attachment 4 and the center of the balance shaft.

$$P_y^{\left(1\right)}=2P_y^{\left(0\right)}$$

$$P_x^{\left(2\right)}=3P_x^{\left(1\right)}$$

$$P_y^{\left(3\right)}=4P_y^{\left(2\right)}-8P_y^{\left(0\right)}$$

$$P_x^{\left(4\right)}=5P_x^{\left(3\right)}-20P_x^{\left(1\right)}$$

$$P_y^{\left(5\right)}=6P_y^{\left(4\right)}-40P_y^{\left(2\right)}+96P_y^{\left(0\right)}$$

$$P_x^{\left(6\right)}=7P_x^{\left(5\right)}-70P_x^{\left(3\right)}+336P_x^{\left(1\right)}$$

Consequently, by way of example, for each hairspring 3, 5, the first seven orders must respect the following relationships: $$P_x^{\left(0\right)}=0$$

$$P_{\mathrm{there}}^{\left(1\right)}=2P_{\mathrm{there}}^{\left(0\right)}$$

$$P_x^{\left(2\right)}=3P_x^{\left(1\right)}$$

$$P_{\mathrm{there}}^{\left(3\right)}=4P_{\mathrm{there}}^{\left(2\right)}-8P_{\mathrm{there}}^{\left(0\right)}$$

$$P_x^{\left(4\right)}=5P_x^{\left(3\right)}-20P_x^{\left(1\right)}$$

$$P_{\mathrm{there}}^{\left(5\right)}=6P_{\mathrm{there}}^{\left(4\right)}-40P_{\mathrm{there}}^{\left(2\right)}+96P_{\mathrm{there}}^{\left(0\right)}$$

$$P_x^{\left(6\right)}=7P_x^{\left(5\right)}-70P_x^{\left(3\right)}+336P_x^{\left(1\right)}$$

As explained above, the more the number of relations (2)-(8) are respected, the more the displacement of the center of mass of balance spring 1 will be limited. For comparison, the Phillips conditions approximate relation (2), that is, a first-order approximation. An application of relations (2)-(5) is shown in figure 2 which is a partial and enlarged view of the figure 1 .

Using a parametrization as explained above, it is possible to define a wide variety of spring-spring curves according to the chosen inertia of the balance wheel, the material, the section and the length of the balance spring but also the coefficients parametrization polynomials. It is also possible to choose particular solutions by limiting for example the number of orders and/or the number of turns.

Simulations of possible curves are shown in figures 3 to 8 . Thus, to form the picture 3 , the parametrization was limited to relations (2) to (4) with a hairspring with 2.3 turns and a parametrization polynomial of degree 2. The figure 4 corresponds to the parametrization with a polynomial of degree 3 from relations (2) to (5) always limiting the winding to 2.3 turns. Finally, the figure 5 corresponds to the parametrization with a polynomial of degree 4 from relations (2) to (6) by limiting the winding to 2.3 turns. The figures 6 to 8 correspond to the same criteria respectively as the figures 3 to 5 but increasing the winding from 2.3 turns to 5.3 turns. We realize that there is an infinity of solutions of curve while respecting the relations (2)-(8) stated.

As shown in figures 9 to 11 , the end 6 of the spiral spring 3 is connected to a ferrule 10 in one piece and the end 7 of the spiral spring 5, which is opposite to the attachment 4, is arranged to cooperate with a peak (not shown ). Moreover, as visible to figures 9 to 11 , the main faces 11, 12 of the attachment 4 are substantially parallel to the line of symmetry A.

In the particular case of figures 9 to 11 in which the hairspring 1 is formed from three parts as explained in the document EP 2 184 652 , in addition to respecting the most relations (2)-(8), it then becomes necessary to compensate for the unbalance generated by the attachment 4, that is to say to compensate for the mass of the attachment 4 with respect to its distance from the axis of the balance wheel.

Thus, as illustrated in figures 9 to 11 , each spiral spring 3, 5 comprises, preferably, at least two counterweights 8-9, 8'-9' which are symmetrical along the same line A as the curvatures in order to compensate for the imbalance formed by the mass of the attachment 4 and customize the anisochronism slope of hairspring 1. Preferably, the masses of the counterweights 8, 8' and 9, 9' are substantially equal and their sum is greater or less than that of attachment 4 depending on the difference distance between, on the one hand, the attachment 4 and the balance shaft, and, on the other hand, the counterweights 8, 8 ', 9, 9' and said balance shaft.

Indeed, it has been shown empirically that two single counterweights on the opposite side of the attachment did not allow the rate variation belly to be lowered below 1.4 sj ^-1 . This is due to the fact that even if the counterweights perfectly balance the unbalance for an angle of rotation of the shell of 0°, this is no longer the case when the shell rotates through a certain angle because the radial distance of the attachment 4 does not vary in the same way as the radial distance of the counterweights 9, 9'.

This is why, in order to better balance the unbalance over a usual angle of rotation range of 0° to substantially 300°, at least two additional counterweights 8, 8' are added by placing them at other places on the spiral springs 3, 5. Thus, it was found that four counterweights 8, 8', 9, 9' all aligned on axis A as seen at figure 11 , two 8, 8' next to the attachment 4 and two 9, 9' on the side opposite the attachment 4 make it possible to optimize the unbalance by making it almost zero regardless of the angle of the ferrule 10.

Preferably according to the invention, each counterweight 8, 8', 9, 9' is substantially H-shaped, the parallel arms of the H being substantially parallel to the local curvature of the spiral spring 3, 5 with which it is associated. As visible to figures 9 to 11 , it is noted that these H shapes provide a local extra thickness of each hairspring 3, 5 which locally increases their stiffness.

Thus, advantageously according to the invention, the stiffnesses and/or the masses provided locally by the counterweights 8, 8', 9, 9' and the attachment 4 are used to modify the anisochronism slope of the hairspring 1.

A simulation of the antinode and the anisochronism slope of the hairspring 1 according to the invention was carried out by varying the length of the attachment 4 or of the counterweights 8, 8', 9, 9' along the hairspring 1: Attachment 4 Counterweight 9, 9' Counterweight 8, 8' Slope Stomach [mm] [mm] [mm] [s/d/100°] [s/d] 0.22 0.1 0.1 -8.54 0.26 0.1 0.1 0.1 -7.51 0.35 0.22 0.04 0.22 -12.97 1.14 0.22 0.22 0.04 -7.88 0.54 0.22 0.44 0.22 -5.49 0.22 0.22 0.33 0.22 -5.05 0.29 0.22 0.33 0.25 -4.3 0.49

The length represents the length of the portion of balance spring 1 which is stiffened by attachment 4 or counterweight 8, 8', 9, 9'. For the simulation, a balance wheel inertia amounting to 2.5 mg.cm ² and a silicon hairspring 1 with a section of 0.033 mm×0.1 mm and a length L of 45 mm were chosen.

It can be seen that by decreasing the length of the attachment 4, the anisochronism slope tends to straighten while the belly remains advantageously less than 0.4 sj ^-1 . Moreover, by decreasing the length of the counterweights 9, 9', the anisochronism slope tends to straighten while the antinode advantageously remains less than 0.3 sj ^-1 . Finally, by increasing the length of the counterweights 8, 8', the anisochronism slope tends to straighten while the antinode advantageously remains less than 0.5 sj ^-1 .

Of course, the mass of the attachment 4 and therefore of the counterweights 8, 8', 9, 9' can also be modified to adapt the anisochronism slope: Attachment 4 Counterweight 9, 9' Counterweight 8, 8' Slope Attachment 4 Stomach [mm] [mm] [mm] [mm ³ ] [s/d/100°] [s/d] 0.22 0.1 0.1 0.016 -8.54 0.26 0.22 0.1 0.1 0.008 -5.58 0.56 0.22 0.1 0.1 0.002 -4.58 0.53

It is then seen that by reducing the mass of the attachment 4, the anisochronism slope tends to straighten while the belly remains advantageously less than 0.6 sj ^-1 . Consequently, advantageously according to the invention, the anisochronism slope of balance spring 1 can be personalized in order to compensate for that given by the delay in the escapement.

Of course, the present invention is not limited to the example illustrated but is susceptible to various variants and modifications which will become apparent to those skilled in the art. In particular, other framing criteria can be provided, such as for example a limitation of the ratio between the inner radius and the outer radius so that the ends of the hairsprings are not too close to the point of origin where the the balance shaft.

In addition, advantageously according to the invention, the internal turn 7 of the second hairspring 5 preferably comprises a shift device 13 arranged to be attached to a stud (not shown) in the plane of the second hairspring 5. The offset device 13 is particularly useful to prevent the particular shapes of the hairspring 1 from making it impossible to mount it because of the proximity of its free end 7 with respect to the balance staff.

As visible to figures 9 to 11 , the offset device 13 comprises a part 14 in extension of the internal turn 7 of the second hairspring 5. Preferably, the part 14 is more rigid than the second hairspring 5 so as not to provide elastic torque to the balance resonator - hairspring. In addition, the part 14 is preferably made more rigid by a greater thickness, such as for example at least three times greater, relative to the thickness of the second hairspring 5, that is to say the width of its blade . It is therefore understood that the shape of the part 14 is partly adapted according to the curvature of the turns of the second hairspring 5 so that no contact takes place.

Moreover, according to a particular alternative, it is preferred that the part 14 be made in one piece with the second hairspring 5 and, preferably, that the latter has a height substantially equal to that of the part 14, that is that is to say that the latter is contained in the same plane.

The extension piece 14 is also preferably connected to the internal coil 7 of the second spiral spring 5 by means of a substantially U-shaped elbow 15 in order to further limit the supply of a any elastic couple. It will be understood that the extension piece 14 and the bend 15 make it possible to virtually bring the fixed point formed by the peak (not shown) closer to the end 7 of the hairspring 1.

In addition, the part 14 includes a recess 16, blind or through, of substantially asymmetrical section and intended to cooperate with the peak (not shown). Finally, as seen in figures 9 to 11 , part 14 can be partially perforated by holes 19 to reduce its mass and thus make its weight less penalizing when mounting balance spring 1.

Similarly, the internal turn 6 of the first hairspring 3 comprises a device 17 for enlarging the ferrule 10 in the plane of the first hairspring 3. The enlargement device 17 is particularly useful to prevent particular shapes of the hairspring 1 make it impossible to mount it because of the proximity of its free end 6 to the balance staff. It is therefore understood that without the enlargement device 17, the ferrule 10 would necessarily have a smaller diameter because of the proximity of the internal turn 6.

Preferably, the enlargement device 17 comprises a bead 18 extending the inner turn 6 of the first hairspring 3, the bead 18 being more rigid than the first hairspring 3 so as not to provide elastic torque. In addition, the bead 18 is preferably made more rigid by a thickness greater than the thickness of the first hairspring 3, that is to say the width of its blade.

In addition, according to a particular alternative, it is preferred that the bead 18 be substantially U-shaped. Finally, the bead 18 is preferentially integral with the first hairspring 3.

Consequently, advantageously according to the invention, a hairspring can be manufactured respecting predetermined conditions in order to reduce the displacement of the center of mass of the hairspring in contraction and in expansion. This small or negligible displacement advantageously makes it possible to reduce the antinode of the anisochronism curves to a value substantially equal to or less than 0.5 sj ^-1 . Moreover, advantageously according to the invention, the anisochronism slope of the hairspring can be personalized in order to compensate for that given by the delay in the escapement.

Finally, the configuration of figures 9 to 11 defines an axis A of very robust symmetry which makes it possible to minimize the chronometric defects induced by disturbances in the direction orthogonal to the axis A. It is therefore understood that it is possible to maximize the manufacturing precision in the direction attachment - counterweight, i.e. the A axis, as the only critical direction instead of the usual two.

Of course, the present invention is not limited to the example illustrated but is susceptible to various variants and modifications which will become apparent to those skilled in the art. In particular, the counterweights 8, 8', 9, 9' can be of different shape and/or geometry without departing from the scope of the invention. It is also possible to increase their number and/or distribute them differently, that is to say in particular that the counterweights 8, 8', 9, 9' are not necessarily symmetrical along the line A like the curvatures .

Thus, by way of example, it is perfectly possible, for each hairspring 3, 5, to add two additional counterweights, that is to say to have four counterweights, in order to distribute them substantially at 90° one another.

Furthermore, when the hairspring is made of silicon, it can be at least partially covered with silicon dioxide in order to make it less sensitive to temperature variations and to mechanical shocks. It is then understood that the variation in section of the counterweights 8, 8', 9, 9' also modifies the local thermal compensation of the hairspring.

Finally, it can be envisaged to also compensate for the unbalance induced by the bead 18 by adding an additional counterweight on the opposite side of the shroud 10.

Claims (16)

1. Balance spring (1) including a first hairspring (3), the curve of which extends in a first plane and the inner coil (6) of which includes a collet (10), a second hairspring (5), the curve of which extends in a second plane parallel to the first plane, an attachment member (4) securing the outer coil of the first hairspring (3) to the outer coil of the second hairspring (5) so as to form a dual balance spring (1) in series, characterised in that the curve of the first hairspring (3) and the curve of the second hairspring (5) each have a continuously variable pitch and are symmetrical relative to a straight line (A) parallel to the first and second planes and passing through the median plane of projection (P) of the attachment member (4), each curve being such that the moment of the balance spring of order n defined by the following relation is substantially zero: $${\overrightarrow{P}}^{\left(n\right)}=\frac{n+1}{L^{n+1}}{\displaystyle \underset{0}{\overset{L}{\int }}\mathit{ds}\cdot s^n}\cdot \overrightarrow{x}\left(s\right)$$

   

   where:

   - n is an integer greater than or equal to zero;

   - P ⁽ⁿ⁾ is the moment of the balance spring of order n;

   - L is the length of the balance spring;

   - sⁿ represents the curvilinear abscissa along the balance spring to the power of n;

   - x (s) is the parameterisation of the balance spring by the curvilinear abscissa thereof.

   so as to reduce the displacements of the centre of mass thereof during contraction and expansion and in that each hairspring (3, 5) further includes at least two counterweights (8, 8', 9, 9') so as to compensate for the unbalance formed by the mass of the attachment member (4) and personalise the anisochronism slope of the balance spring (1), said at least two counterweights (8, 8', 9, 9') being symmetrical along said same straight line (A) as the curves, two counterweights (8, 8') being located beside the attachment member (4) and two other counterweights (9, 9') being located on the side opposite the attachment member (4) so as to minimise the unbalance.
2. Balance spring (1) according to the preceding claim, characterised in that each counterweight (8', 8', 9, 9') is substantially H-shaped, the parallel arms of the H being substantially parallel to the local curvature of the hairspring (3, 5) with which said counterweight is associated.
3. Balance spring (1) according to any of the preceding claims, characterised in that the main faces (11, 12) of the attachment member (4) are substantially parallel to said line of symmetry (A).
4. Balance spring (1) according to any of the preceding claims, characterised in that the inner coil (7) of the second hairspring (5) includes a shifting device (13) arranged to be attached to a balance spring stud in the plane of the second hairspring (5).
5. Balance spring (1) according to the preceding claim, characterised in that the shifting device (13) comprises a piece (14) extending from the inner coil (7) of the second hairspring (5), said piece being more rigid than said second hairspring to avoid providing elastic torque.
6. Balance spring (1) according to the preceding claim, characterised in that the extension piece (14) is connected to the inner coil via a substantially U-shaped bend (15).
7. Balance spring (1) according to claim 5 or 6, characterised in that the extension piece (14) is integral with the second hairspring (5).
8. Balance spring (1) according to any of claims 5 to 7, characterised in that the extension piece (14) is made more rigid by a thickness that is at least three times greater than that of said second hairspring.
9. Balance spring (1) according to any of claims 5 to 8, characterised in that the piece (14) may be pierced to reduce the mass thereof.
10. Balance spring (1) according to any of the preceding claims, characterised in that the inner coil (6) of the first hairspring (3) includes a device (17) for enlarging the collet (10) in the plane of the first hairspring (3).
11. Balance spring (1) according to the preceding claim, characterised in that the enlarging device (17) includes a flange (18) extending the inner coil (6) of the first hairspring (3), said flange being more rigid than said first hairspring to avoid providing any elastic torque.
12. Balance spring (1) according to the preceding claim, characterised in that the flange (18) is substantially U-shaped.
13. Balance spring (1) according to claim 11 or 12, characterised in that the flange (18) is integral with the first hairspring (3).
14. Balance spring (1) according to any of the preceding claims, characterised in that it is formed from silicon.
15. Balance spring (1) according to the preceding claim, characterised in that it includes at least one part coated with silicon dioxide so as to limit the sensitivity thereof to temperature variations and mechanical shocks.
16. Resonator for a timepiece including a balance spring (1) according to any one of the preceding claims and a balance, characterised in that the balance cooperates with said balance spring (1).

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