EP2138912B1 - Horological hairspring with concentric development - Google Patents

Description

The present invention relates to a clockwork hairspring, that is to say a spiral-shaped spring intended to be mounted on the shaft of a beam to form with the latter the regulating member of a movement of mechanical watchmaking.

A known problem of the usual spirals is that they deform eccentrically relative to the balance shaft during their oscillations in the movement, which makes the regulating organ anisochronous and therefore impairs the accuracy of operation of the movement.

There are currently two types of spirals designed to remedy this problem, that is to say, allow the turns to remain concentric during the oscillations of the spiral: Breguet spirals, as described in traditional timepieces, and flat spirals with variable rigidity as described in the Swiss patent CH 327 796 , the British patent 697,864 and European patent applications EP 1 445 670 , EP 1 473 604 and EP 1 605 323 . The Breguet spirals comprise a spiral flat central part and an end curve which emerges from the plane of the central part in order to reduce the center of gravity of the spiral to the geometric axis of rotation of the balance. These spirals have the disadvantage of a large footprint in height. The flat spirals with variable rigidity are located in a single plane and comprise one or more stiffened turn portions by folding or by material supply, or are made monolithically by micro-fabrication techniques such as DRIE (ion etching) processes. deep reactive) or LIGA (X-ray lithography, electroplating, forming) so as to have a variable coil thickness. Folding operations or supply of material are difficult to achieve and imprecise. Micro-manufacturing techniques require great know-how and are expensive to implement.

The present invention aims to provide a flat watch winder whose turns are deformed concentrically during its oscillations and is easy to manufacture.

To this end is provided a hairspring for a watch movement, comprising a flat central part comprising turns and extending from an inner point to an external point and means for maintaining the concentric turns during the oscillations of the hairspring in the movement, characterized in that said means comprise at least first and second flexible end circumferential curves located in the plane of the central portion and extending disjointly from the outer point to one or more respective end points.

By the term "flexible" is meant that at least a portion of each terminal peripheral curve deforms during oscillations of the spiral so as to move the outer point. The terminal peripheral curves are blades or portion of the blade that terminate the hairspring, or the end points being intended to be rigidly connected to a frame of the movement.

The hairspring according to the invention can be manufactured according to traditional methods and in a material. It does not require to vary its blade section in a particular manner, as in the prior art.

In particular embodiments, the first and second terminal peripheral curves extend to respective separate endpoints.

The first and second terminal peripheral curves may each have a general shape in an arc.

Advantageously, at least one of the first and second terminal peripheral curves comprises a bend at its end joined to the outer point to prevent this end peripheral curve is touched by another part of the spiral during the expansions of the latter.

In particular embodiments, the first end peripheral curve is in the extension of the central portion. Preferably, in this case, the first end peripheral curve is in one piece with the central portion and the second end peripheral curve is an insert attached by one of its ends to the outer point. The second terminal peripheral curve may extend in a direction opposite to that of the first terminal peripheral curve, or in the same direction. The first end peripheral curve may be in an arc and the second end peripheral curve may consist of a bend at its end joined to the outer point and a main portion in an arc of a circle between the bend and the end point of the second terminal peripheral curve.

In other embodiments, the first and second terminal peripheral curves are in the general form of circular arcs having the same center and the same radius and extend in two opposite directions from the outer point.

The central portion and the first and second terminal peripheral curves preferably each have a constant section over their entire length. The central portion and the first and second terminal peripheral curves may have the same section.

The present invention also provides a clockwork comprising a frame, a rocker shaft pivotally mounted in the frame and a spiral te! that defined above mounted on the balance shaft by means of a ferrule, the said point (s) terminal (s) (terminals) of the spiral being fixed (s) to the frame.

• la figure 1 est une vue en coupe axiale d'un organe régulateur d'un mouvement d'horlogerie mécanique ;
• les figures 2 à 4 sont des vues de dessus d'un spiral selon un premier mode de réalisation de l'invention, respectivement dans des états de repos, d'expansion et de contraction ;
• les figures 5 à 7 sont des vues de dessus d'un spiral selon un deuxième mode de réalisation de l'invention, respectivement dans des états de repos, d'expansion et de contraction ;
• les figures 8 à 10 sont des vues de dessus d'un spiral selon un troisième mode de réalisation de l'invention, respectivement dans des états de repos, d'expansion et de contraction ;
• les figures 11 à 13 sont des vues de dessus d'un spiral selon un quatrième mode de réalisation de l'invention, respectivement dans des états de repos, d'expansion et de contraction.

Other features and advantages of the present invention will appear on reading the following detailed description given with reference to the accompanying drawings in which:

• the figure 1 is an axial sectional view of a regulating member of a mechanical clockwork movement;
• the Figures 2 to 4 are top views of a hairspring according to a first embodiment of the invention, respectively in states of rest, expansion and contraction;
• the Figures 5 to 7 are top views of a hairspring according to a second embodiment of the invention, respectively in states of rest, expansion and contraction;
• the Figures 8 to 10 are top views of a hairspring according to a third embodiment of the invention, respectively in states of rest, expansion and contraction;
• the Figures 11 to 13 are top views of a hairspring according to a fourth embodiment of the invention, respectively in states of rest, expansion and contraction.

In what follows, the term "center of the spiral" its geometric center, consisting of the center of the landmark in which is defined the spiral in the rest position. The numerical values that will be given will correspond to the state of rest of the spiral. The radii of curvature of the blades or portions of the blade will be measured on the neutral fiber thereof.

As shown in figure 1 , in a mechanical clockwork movement, the hairspring 1 has its inner end which is fixed to the periphery of a shell 2 mounted on the shaft 3 of a rocker 4 and its outer end which is fixed to a fixed part of movement, typically the rooster 5, via a peg 6, the center of the hairspring being located on the geometric axis 3a of rotation of the balance. If one is placed in a theoretical configuration where the balance shaft 3, and thus the ferrule 2 which is integral therewith, are free to move in the radial plane (plane perpendicular to the axis 3a), and if 'We apply to this shaft 3 a pair of pure forces, that is to say forces producing a moment but whose resultant is zero, then the center of the spiral (or the balance shaft) will move towards the fixed outer end if the moment has the effect of contracting the hairspring and in the opposite direction to the fixed outer end if the moment has the effect to enlarge the spiral. During these movements, the turns will remain concentric. This is illustrated in figures 3 and 4 of the patent application EP 1 473 604 . The movements of the center of the spiral can be calculated by simulation.

In practice, the balance shaft 3 is of course not free to move in the radial plane because its pivots 7 rotate in bearings 8 respectively formed in the plate 9 and the cock 5 of the movement. During oscillations of the balance spring-balance control member, the bearings 8 exert reaction forces on the pivots 7 of the balance shaft 3, forces that hold the balance shaft 3 and therefore the center of the balance spring 1. C This is why in practice a spiral deforms eccentrically during its oscillations.

If, instead of fixing the outer end of the hairspring, it is made movable and if it is arranged to move this outer end during the oscillations of the hairspring in a manner which corresponds to the inverse of the displacement that would undergo the center of the hairspring. spiral if this center was free in the radial plane and if the outer end was fixed, then the deformations of the spiral will remain concentric. This is what the present invention proposes to do.

For this purpose, the present invention provides a flat hairspring comprising a central spiral portion extending from an inner point to an outer point. The inner point is a junction point to a ferrule for mounting the spring on a balance shaft in a frame typically formed by the plate and the cock of the movement or by a tourbillon cage, the balance spring and the balance constituting the body movement regulator. The outer point is a junction point between the central portion and a movable fastener. The movable attachment member is shaped to allow the outer point to move relative to the frame in a manner that substantially corresponds to the reciprocal of the displacement that would undergo the center of the spiral if the center was free in the radial plane and if the outer point was fixed in relation to the frame, in other words in a manner that cancels the reaction forces exerted by the bearings on the pivots of the balance shaft, so that the latter is only subjected to a pure couple.

The figures 2 , 5 , 8 and 11 show four different embodiments of the spiral according to the invention. In these figures, the spiral central part is designated by the reference sign 10, the inner point by the reference sign 11, the outer point by the reference sign 12, the mobile fastening member by the reference sign 13 and the center of the spiral by the reference sign O, this center also constituting the center of a geometric reference (O, x, y) fixed relative to the frame and being located on the geometric axis of rotation of the balance and of the ferrule.

The central spiral portion 10 is the same in these four embodiments. It has a classical Archimedean spiral shape, made by winding a blade. In the rest position, the inner point 11 and the outer point 12 are on the axis (O, x). In these four embodiments, the movable attachment member 13 consists of two flexible terminal peripheral curves 14, 15. These terminal peripheral curves 14, 15 both start from the external point 12 and extend without touching each other until at respective end points 16, 17 separated from each other. Each end peripheral curve 14, 15 is a blade or blade portion that terminates the hairspring. The end points 16, 17 are attachment points intended to be fixed to the frame, for example by means of respective pitons, as for the outer end of a traditional hairspring. The terminal peripheral curves 14, 15 deform during the oscillations of the balance-sprung regulator member so as to move the outer point 12 as indicated above to allow the turns of the central portion 10 to remain concentric. These concentric deformations of the turns are shown in figures 3 and 4 for the first embodiment, figures 6 and 7 for the second embodiment, to figures 9 and 10 for the third embodiment and to figures 12 and 13 for the fourth embodiment, in which the hairspring has been shown in states of expansion and contraction.

In the first embodiment ( figure 2 ), the terminal peripheral curves 14, 15, designated here more particularly by the reference signs 14a, 15a, are each generally arcuate in shape, have different radii of curvature and centers of curvature, and extend in two opposite directions from the outer point 12 to the respective end points 16, 17, designated here more particularly by the reference signs 16a, 17a. The central portion 10 and the terminal peripheral curves 14a, 15a all have the same section, which remains constant over their entire length. The first end peripheral curve 14a is a blade or blade portion located in the extension of the last turn of the central portion 10, and is preferably made in one piece with the central portion 10. The first terminal peripheral curve 14a has the shape of an arc of circle along its entire length. The second end peripheral curve 15a is constituted by a blade whose end 18a opposite the end point 17a is bent and fixed by welding, gluing or other suitable method to the outer point 12 of the central portion 10. The bend defined by the end 18a has an arcuate shape of small radius. It removes the second terminal peripheral curve 15a from the last turn of the central portion 10 to prevent them from touching during expansions of the hairspring. The main portion 19a of the second terminal peripheral curve 15a, extending from the elbow 18a to the end point 17a, has an arcuate shape of greater radius. This radius as well as the radius of the first end peripheral curve 14a are chosen sufficiently large to avoid any contact between the last turn of the central portion 10, the first end peripheral curve 14a and the second terminal peripheral curve 15a during the expansions of the hairspring.

• Hauteur du spiral : 0,12 mm
• Distance entre le centre O et la fibre neutre du spiral au point intérieur 11 : 0,565 mm
• Nombre de spires de la partie centrale 10 : 14
• Pas de la spirale : 0,0954 mm
• Longueur de la partie centrale 10 : 108,416 mm
• Epaisseur de lame de la partie centrale 10 et des courbes périphériques terminales 14a, 15a: 0,02759 mm
• Rayon de courbure de la première courbe périphérique terminale 14a : 2,2 mm
• Coordonnées du centre de courbure de la première courbe périphérique terminale 14a : $$\mathrm{X}\mathrm{=}\mathrm{-}\mathrm{0}\mathrm{,}\mathrm{2999}\mathrm{mm}$$
  
  $$\mathrm{Y}\mathrm{=}\mathrm{-}\mathrm{0}\mathrm{,}0176\mathrm{mm}$$
  
• Etendue angulaire de la première courbe périphérique terminale 14a (mesurée depuis son centre de courbure) : 145° Rayon de courbure de la partie principale 19a de la deuxième courbe périphérique terminale 15a : 2,50266 mm
• Coordonnées du centre de courbure de la partie principale 19a de la deuxième courbe périphérique terminale 15a : $$\mathrm{X}\mathrm{=}\mathrm{-}\mathrm{0}\mathrm{,}4618\mathrm{mm}$$
  
  $$\mathrm{Y}\mathrm{=}\mathrm{-}\mathrm{0}\mathrm{,}123\mathrm{mm}$$
  
• Etendue angulaire de la partie principale 19a de la deuxième courbe périphérique terminale 15a (mesurée depuis son centre de courbure) : 266,76°
• Rayon de courbure du coude 18a : 0,12853 mm
• Coordonnées du centre de courbure du coude 18a : $$\mathrm{X}\mathrm{=}1\mathrm{,}899\mathrm{mm}$$
  
  $$\mathrm{Y}\mathrm{=}\mathrm{0}\mathrm{,}1285\mathrm{mm}$$
  
• Etendue angulaire du coude 18a (mesurée depuis son centre de courbure) : 95,62°

To produce the hairspring according to this first embodiment, the following values can be used:

• Height of the spiral: 0,12 mm
• Distance between the center O and the neutral fiber of the spiral at the inner point 11: 0.565 mm
• Number of turns of the central part 10: 14
• No spiral: 0.0954 mm
• Length of the central part 10: 108.416 mm
• Blade thickness of the central portion 10 and terminal peripheral curves 14a, 15a: 0.02759 mm
• Radius of curvature of the first terminal peripheral curve 14a: 2.2 mm
• Coordinates of the center of curvature of the first terminal peripheral curve 14a: $$\mathrm{X}\mathrm{=}\mathrm{-}\mathrm{0}\mathrm{,}\mathrm{2999}\mathrm{mm}$$
  
  $$\mathrm{Y}\mathrm{=}\mathrm{-}\mathrm{0}\mathrm{,}0176\mathrm{mm}$$
  
• Angular extent of the first end peripheral curve 14a (measured from its center of curvature): 145 ° Radius of curvature of the main part 19a of the second end peripheral curve 15a: 2.50266 mm
• Coordinates of the center of curvature of the main part 19a of the second terminal peripheral curve 15a: $$\mathrm{X}\mathrm{=}\mathrm{-}\mathrm{0}\mathrm{,}4618\mathrm{mm}$$
  
  $$\mathrm{Y}\mathrm{=}\mathrm{-}\mathrm{0}\mathrm{,}123\mathrm{mm}$$
  
• Angular extent of the main part 19a of the second terminal peripheral curve 15a (measured from its center of curvature): 266.76 °
• Radius of curvature of the elbow 18a: 0.12853 mm
• Coordinates of the center of curvature of the elbow 18a: $$\mathrm{X}\mathrm{=}1\mathrm{,}899\mathrm{mm}$$
  
  $$\mathrm{Y}\mathrm{=}\mathrm{0}\mathrm{,}1285\mathrm{mm}$$
  
• Angular extent of the elbow 18a (measured from its center of curvature): 95.62 °

In the second embodiment ( figure 5 ), the two terminal peripheral curves 14, 15, designated here more particularly by the markers 14b, 15b are each generally arcuate in shape, have different radii of curvature and centers of curvature, and extend in the same direction from the outer point 12 to the respective end points 16, 17 , designated here more particularly by the marks 16b, 17b. The central portion 10 has a constant section. The terminal peripheral curves 14b, 15b have the same constant section, which is different, in this case greater, than the section of the central portion 10. The difference in section (which is here a difference in thickness) between the curves 14b 15b and the central portion 10 is not visible in the drawings. The first end peripheral curve 14b is a blade located in the extension of the last turn of the central portion 10, and whose end other than the end point 16b is fixed by welding, gluing or other suitable method at the outer point 12. first end peripheral curve 14b is in an arc along its entire length. The second end peripheral curve 15b is located outside the first end peripheral curve 14b and is constituted by a blade whose end 18b other than the end point 17b is bent and fixed by welding, gluing or other suitable method to the point 12. The elbow that defines the end 18b is in a small radius arc. It moves the second terminal peripheral curve 15b away from the first end peripheral curve 14b to prevent these two curves from touching during expansions of the hairspring. The main portion 19b of the second terminal peripheral curve 15b, extending from the elbow 18b to the end point 17b, has a shape of a circular arc of greater radius. This radius and the radius of the first end peripheral curve 14b are chosen sufficiently large to avoid any contact between the last turn of the central portion 10, the first end peripheral curve 14b and the second terminal peripheral curve 15b during the expansions of the hairspring.

• Hauteur du spiral : 0,12 mm
• Distance entre le centre O et la fibre neutre du spiral au point intérieur 11 : 0,565 mm
• Nombre de spires de la partie centrale 10 : 14
• Pas de la spirale : 0,0954 mm
• Longueur de la partie centrale 10 : 108,416 mm
• Epaisseur de lame de la partie centrale 10 : 0,02759 mm
• Epaisseur de lame de chaque courbe périphérique terminale 14b, 15b : 0,03255 mm
• Rayon de courbure de la première courbe périphérique terminale 14b : 2,185 mm
• Coordonnées du centre de courbure de la première courbe périphérique terminale 14b : $$\mathrm{X}\mathrm{=}\mathrm{-}\mathrm{0}\mathrm{,}2849\mathrm{mm}$$
  
  $$\mathrm{Y}\mathrm{=}\mathrm{-}\mathrm{0}\mathrm{,}0175\mathrm{mm}$$
  
• Etendue angulaire de la première courbe périphérique terminale 14b (mesurée depuis son centre de courbure) : 155°
• Rayon de courbure de la partie principale 19b de la deuxième courbe périphérique terminale 15b : 2,4882 mm
• Coordonnées du centre de courbure de la partie principale 19b de la deuxième courbe périphérique terminale 15b : $$\mathrm{X}\mathrm{=}\mathrm{-}\mathrm{0}\mathrm{,}4224\mathrm{mm}$$
  
  $$\mathrm{Y}\mathrm{=}\mathrm{-}\mathrm{0}\mathrm{,}1226\mathrm{mm}$$
  
• Etendue angulaire de la partie principale 19b de la deuxième courbe périphérique terminale 15b (mesurée depuis son centre de courbure) : 194,13°
• Rayon de courbure du coude 18b : 0,16416 mm
• Etendue angulaire du coude 18b (mesurée depuis son centre de courbure) : 91,48°
• Coordonnées du centre de courbure du coude 18b : $$\mathrm{X}\mathrm{=}1\mathrm{,}9013\mathrm{mm}$$
  
  $$\mathrm{Y}\mathrm{=}\mathrm{-}\mathrm{0}\mathrm{,}1642\mathrm{mm}$$
  

To produce the hairspring according to this second embodiment, the following values can be used:

• Height of the spiral: 0,12 mm
• Distance between the center O and the neutral fiber of the spiral at the inner point 11: 0.565 mm
• Number of turns of the central part 10: 14
• No spiral: 0.0954 mm
• Length of the central part 10: 108.416 mm
• Blade thickness of the central portion 10: 0.02759 mm
• Blade thickness of each end peripheral curve 14b, 15b: 0.03255 mm
• Radius of curvature of the first terminal peripheral curve 14b: 2.185 mm
• Coordinates of the center of curvature of the first terminal peripheral curve 14b: $$\mathrm{X}\mathrm{=}\mathrm{-}\mathrm{0}\mathrm{,}2849\mathrm{mm}$$
  
  $$\mathrm{Y}\mathrm{=}\mathrm{-}\mathrm{0}\mathrm{,}0175\mathrm{mm}$$
  
• Angular extent of the first terminal peripheral curve 14b (measured from its center of curvature): 155 °
• Radius of curvature of the main part 19b of the second terminal peripheral curve 15b: 2.4882 mm
• Coordinates of the center of curvature of the main part 19b of the second terminal peripheral curve 15b: $$\mathrm{X}\mathrm{=}\mathrm{-}\mathrm{0}\mathrm{,}4224\mathrm{mm}$$
  
  $$\mathrm{Y}\mathrm{=}\mathrm{-}\mathrm{0}\mathrm{,}1226\mathrm{mm}$$
  
• Angular extent of the main part 19b of the second terminal peripheral curve 15b (measured from its center of curvature): 194.13 °
• Elbow curvature radius 18b: 0.16416 mm
• Angular extent of the elbow 18b (measured from its center of curvature): 91.48 °
• Coordinates of the center of curvature of the elbow 18b: $$\mathrm{X}\mathrm{=}1\mathrm{,}9013\mathrm{mm}$$
  
  $$\mathrm{Y}\mathrm{=}\mathrm{-}\mathrm{0}\mathrm{,}1642\mathrm{mm}$$
  

In the third embodiment ( figure 8 ), the two end peripheral curves 14, 15, designated here more particularly by the marks 14c, 15c, are each generally in the shape of an arc of a circle, have the same radius of curvature and the same center of curvature, and extend in two opposite directions from the outer point 12 to the respective end points 16, 17, designated here more particularly by the marks 16c, 17c. The central portion 10 and the terminal peripheral curves 14c, 15c have the same constant section. The terminal peripheral curves 14c, 15c are two blades whose end 18c, 19c opposite the end point 16c, 17c is bent and fixed by welding, gluing or other suitable method to the outer point 12 of the central portion 10. The elbows that define the ends 18c, 19c are each arcuate circle of small radius. These elbows move the terminal peripheral curves 14c, 15c away from the central portion 10 to prevent the last turn of the central portion 10 from touching the terminal peripheral curves 14c, 15c during expansions of the hairspring. The main part 20c, 21c of each end peripheral curve 14c, 15c, extending from the bend 18c, 19c to the end point 16c, 17c, is in a circular arc of greater radius. This radius is chosen large enough to prevent the last turn of the central portion 10 touches the terminal peripheral curves 14c, 15c during expansions of the spiral.

• Hauteur du spiral : 0,12 mm
• Distance entre le centre O et la fibre neutre du spiral au point intérieur 11 : 0,565 mm
• Nombre de spires de la partie centrale 10 : 14
• Pas de la spirale : 0,0954 mm
• Longueur de la partie centrale 10 : 108,416 mm
• Epaisseur de lame de la partie centrale 10 et des courbes périphériques terminales 14c, 15c : 0,02759 mm
• Rayon de courbure des parties principales 20c, 21c des courbes périphériques terminales 14c, 15c : 3,515 mm
• Coordonnées du centre de courbure des parties principales 20c, 21c des courbes périphériques terminales 14c, 15c : $$\mathrm{X}\mathrm{=}\mathrm{-}1\mathrm{,}4771\mathrm{mm}$$
  
  $$\mathrm{Y}\mathrm{=}0$$
  
• Etendue angulaire de la partie principale 20c de la première courbe périphérique terminale 14c (mesurée depuis son centre de courbure) : 115,83°
• Etendue angulaire de la partie principale 21c de la deuxième courbe périphérique terminale 15c (mesurée depuis son centre de courbure) : 235,17°
• Rayon de courbure des coudes 18c, 19c : 0,13522 mm Coordonnées du centre de courbure du coude 18c : $$\mathrm{X}\mathrm{=}1\mathrm{,}90\mathrm{mm}\mathrm{.}$$
  
  $$\mathrm{Y}\mathrm{=}\mathrm{-}\mathrm{0}\mathrm{,}135\mathrm{mm}$$
  
• Etendue angulaire du coude 18c (mesurée depuis son centre de courbure) : 92,29°
• Coordonnées du centre de courbure du coude 19c : $$\mathrm{X}\mathrm{=}1\mathrm{,}90\mathrm{mm}$$
  
  $$\mathrm{Y}\mathrm{=}\mathrm{0}\mathrm{,}135\mathrm{mm}$$
  
• Etendue angulaire du coude 19c (mesurée depuis son centre de courbure) : 92,29°

To produce the hairspring according to this third embodiment, the following values can be used:

• Height of the spiral: 0,12 mm
• Distance between the center O and the neutral fiber of the spiral at the inner point 11: 0.565 mm
• Number of turns of the central part 10: 14
• No spiral: 0.0954 mm
• Length of the central part 10: 108.416 mm
• Blade thickness of the central portion 10 and the terminal peripheral curves 14c, 15c: 0.02759 mm
• Radius of curvature of the main parts 20c, 21c of the terminal peripheral curves 14c, 15c: 3.515 mm
• Coordinates of the center of curvature of the main parts 20c, 21c of the terminal peripheral curves 14c, 15c: $$\mathrm{X}\mathrm{=}\mathrm{-}1\mathrm{,}4771\mathrm{mm}$$
  
  $$\mathrm{Y}\mathrm{=}0$$
  
• Angular extent of the main part 20c of the first end peripheral curve 14c (measured from its center of curvature): 115.83 °
• Angular extent of the main part 21c of the second terminal peripheral curve 15c (measured from its center of curvature): 235.17 °
• Curvature radius of the elbows 18c, 19c: 0.13522 mm Coordinates of the center of curvature of the elbow 18c: $$\mathrm{X}\mathrm{=}1\mathrm{,}90\mathrm{mm}\mathrm{.}$$
  
  $$\mathrm{Y}\mathrm{=}\mathrm{-}\mathrm{0}\mathrm{,}135\mathrm{mm}$$
  
• Angular extent of the elbow 18c (measured from its center of curvature): 92.29 °
• Coordinates of the center of curvature of the elbow 19c: $$\mathrm{X}\mathrm{=}1\mathrm{,}90\mathrm{mm}$$
  
  $$\mathrm{Y}\mathrm{=}\mathrm{0}\mathrm{,}135\mathrm{mm}$$
  
• Angular extent of the elbow 19c (measured from its center of curvature): 92.29 °

The fourth embodiment ( figure 11 ) is identical to the third embodiment with the difference that the terminal peripheral curves 14, 15, designated here more particularly by the marks 14d, 15d, have a radius curvature smaller than the curves 14c, 15c and the same constant section that is smaller than the section of the central portion 10. The section difference (which here is a difference in thickness) between the curves 14d, 15d and the part Central 10 is not visible in the drawings.

• Hauteur du spiral: 0,12 mm
• Distance entre le centre O et la fibre neutre du spiral au point intérieur 11 : 0,565 mm
• Nombre de spires de la partie centrale 10 : 14
• Pas de la spirale : 0,0954 mm
• Longueur de la partie centrale 10 : 108,416 mm
• Epaisseur de lame de la partie centrale 10 : 0,02759 mm
• Epaisseur de lame de chaque courbe périphérique terminale 14d, 15d : 0,01931 mm
• Rayon de courbure des parties principales 20d, 21d des courbes périphériques terminales 14d, 15d : 2,66 mm
• Coordonnées du centre de courbure des parties principales 20d, 21 d des courbes périphériques terminales 14d, 15d : $$\mathrm{X}\mathrm{=}\mathrm{-}\mathrm{0}\mathrm{,}6221\mathrm{mm}$$
  
  $$\mathrm{Y}\mathrm{=}0$$
  
• Etendue angulaire de la partie principale 20d de la première courbe périphérique terminale 14d (mesurée depuis son centre de courbure) : 140,4°
• Etendue angulaire de la partie principale 21 d de la deuxième courbe périphérique terminale 15d (mesurée depuis son centre de courbure) : 210,6°
• Rayon de courbure des coudes 18d, 19d : 0,13435 mm Coordonnées du centre de courbure du coude 18d : $$\mathrm{X}\mathrm{=}1\mathrm{,}90\mathrm{mm}$$
  
  $$\mathrm{Y}\mathrm{=}\mathrm{-}\mathrm{0}\mathrm{,}134\mathrm{mm}$$
  
• Etendue angulaire du coude 18d (mesurée depuis son centre de courbure) : 93,05°
• Coordonnées du centre de courbure du coude 19d : $$\mathrm{X}\mathrm{=}1\mathrm{,}90\mathrm{mm}$$
  
  $$\mathrm{Y}\mathrm{=}\mathrm{0}\mathrm{,}134\mathrm{mm}$$
  
• Etendue angulaire du coude 19d (mesurée depuis son centre de courbure) : 93,05°

The following values can be used to make the hairspring according to this fourth embodiment:

• Height of the spiral: 0,12 mm
• Distance between the center O and the neutral fiber of the spiral at the inner point 11: 0.565 mm
• Number of turns of the central part 10: 14
• No spiral: 0.0954 mm
• Length of the central part 10: 108.416 mm
• Blade thickness of the central portion 10: 0.02759 mm
• Blade thickness of each end peripheral curve 14d, 15d: 0.01931 mm
• Radius of curvature of the main portions 20d, 21d of the terminal peripheral curves 14d, 15d: 2.66mm
• Coordinates of the center of curvature of the main portions 20d, 21d of the terminal peripheral curves 14d, 15d: $$\mathrm{X}\mathrm{=}\mathrm{-}\mathrm{0}\mathrm{,}6221\mathrm{mm}$$
  
  $$\mathrm{Y}\mathrm{=}0$$
  
• Angular extent of the main part 20d of the first end peripheral curve 14d (measured from its center of curvature): 140.4 °
• Angular extent of the main part 21 d of the second terminal peripheral curve 15d (measured from its center of curvature): 210.6 °
• Radius of curvature of elbows 18d, 19d: 0,13435 mm Coordinates of center of curvature of elbow 18d: $$\mathrm{X}\mathrm{=}1\mathrm{,}90\mathrm{mm}$$
  
  $$\mathrm{Y}\mathrm{=}\mathrm{-}\mathrm{0}\mathrm{,}134\mathrm{mm}$$
  
• Angular extent of the elbow 18d (measured from its center of curvature): 93.05 °
• Coordinates of the center of curvature of the elbow 19d: $$\mathrm{X}\mathrm{=}1\mathrm{,}90\mathrm{mm}$$
  
  $$\mathrm{Y}\mathrm{=}\mathrm{0}\mathrm{,}134\mathrm{mm}$$
  
• Angular extent of the elbow 19d (measured from its center of curvature): 93.05 °

Alternatively, in the third and fourth embodiments, the terminal peripheral curves 14c, 15c, respectively 14d, 15d, could consist of a single continuous peripheral blade from the end point 16c, respectively 16d, to the end point 17c, 17d, and the zone of the outer point 12 of the central portion 10 could be bent radially outward to be fixed by welding, gluing or other suitable method to the peripheral blade. In another variant, it is possible to provide a peripheral blade portion in one piece with the central portion 10, a portion of a blade that would define, after the external point 12, a bend similar to the bend 18c of the third embodiment, which would then extend continuously. with a constant radius of curvature (such as the curve portions 20c, 21c) to its end which would be bent to form a bend similar to the bend 19c and fixed by any suitable method at the outer point 12. The portion of this blade portion device corresponding to the space between the end points 16c and 17c would be maintained in a pin fixed to the frame, thus defining two fitting points corresponding to the end points 16c and 17c.

Moreover, as it appears on figures 8 and 11 the end points 16c, 17c, respectively 16d, 17d, are close to one another and could be fixed to a common peak.

The embodiments described above and illustrated at figures 2 , 5 , 8 and 11 are not limiting. The equations governing the spiral according to the invention show that an infinity of solutions exist for the shape and dimensions of the terminal peripheral curves, as will now be explained.

$${\mathrm{F}}_{\mathrm{x}\mathrm{1}}\mathrm{+}{\mathrm{F}}_{\mathrm{x}\mathrm{2}}\mathrm{=}\mathrm{0}$$

$${\mathrm{F}}_{\mathrm{y}\mathrm{1}}\mathrm{+}{\mathrm{F}}_{\mathrm{y}\mathrm{2}}\mathrm{=}\mathrm{0}$$

If, in the frame (O, x, y), Δx and Δy denote the respective displacements in translation along the axis (O, x) and in translation along the axis (O, y) of the terminal peripheral curves 14 , 15 at the outer point 12, Δω by their displacement in rotation at the outer point 12 (angle between the axis (O, x) and the straight line passing through the center O and the outer point 12), by M _f the moment of the couple applied to the balance shaft, by M _f1 and M _f2 the respective moments of force experienced by the terminal peripheral curves 14, 15 at the outer point 12, by F _x1 and F _y1 the components of the force undergone by the first peripheral curve terminal 14 at the outer point 12, and F _x2 and F _y2 the components of the force undergone by the second terminal peripheral curve 15 at the outer point 12, then we have: $${\mathrm{M}}_{\mathrm{f}}\mathrm{=}{\mathrm{M}}_{\mathrm{f}\mathrm{1}}\mathrm{+}{\mathrm{M}}_{\mathrm{f}\mathrm{2}}$$

$${\mathrm{F}}_{\mathrm{x}\mathrm{1}}\mathrm{+}{\mathrm{F}}_{\mathrm{x}\mathrm{2}}\mathrm{=}\mathrm{0}$$

$${\mathrm{F}}_{\mathrm{there}\mathrm{1}}\mathrm{+}{\mathrm{F}}_{\mathrm{there}\mathrm{2}}\mathrm{=}\mathrm{0}$$

$$\mathrm{\Delta x}\mathrm{=}{\mathrm{a}}_{21}\mathrm{*}{\mathrm{M}}_{\mathrm{f}\mathrm{1}}\mathrm{+}{\mathrm{a}}_{22}\mathrm{*}{\mathrm{F}}_{\mathrm{x}\mathrm{1}}\mathrm{+}{\mathrm{a}}_{23}\mathrm{*}{\mathrm{F}}_{\mathrm{y}\mathrm{1}}\mathrm{=}{\mathrm{b}}_{21}\mathrm{*}{\mathrm{M}}_{\mathrm{f}\mathrm{2}}\mathrm{+}{\mathrm{b}}_{22}\mathrm{*}{\mathrm{F}}_{\mathrm{x}\mathrm{2}}\mathrm{+}{\mathrm{b}}_{23}\mathrm{*}{\mathrm{F}}_{\mathrm{y}\mathrm{2}}$$

$$\mathrm{\Delta y}\mathrm{=}{\mathrm{a}}_{31}\mathrm{*}{\mathrm{M}}_{\mathrm{f}\mathrm{1}}\mathrm{+}{\mathrm{a}}_{32}\mathrm{*}{\mathrm{F}}_{\mathrm{x}\mathrm{1}}\mathrm{+}{\mathrm{a}}_{33}\mathrm{*}{\mathrm{F}}_{\mathrm{y}\mathrm{1}}\mathrm{=}{\mathrm{b}}_{31}\mathrm{*}{\mathrm{M}}_{\mathrm{f}\mathrm{2}}\mathrm{+}{\mathrm{b}}_{32}\mathrm{*}{\mathrm{F}}_{\mathrm{x}\mathrm{2}}\mathrm{+}{\mathrm{b}}_{33}\mathrm{*}{\mathrm{F}}_{\mathrm{y}\mathrm{2}}$$

The sum of the forces experienced by the terminal peripheral curves 14, 15 is zero because it is desired that the spiral as a whole does not undergo any force transmitted by the ferrule. As a first approximation, we also have: $$\mathrm{\Delta \omega }\mathrm{=}{\mathrm{at}}_{\mathrm{11}}\mathrm{*}{\mathrm{M}}_{\mathrm{f}\mathrm{1}}\mathrm{+}{\mathrm{at}}_{\mathrm{12}}\mathrm{*}{\mathrm{F}}_{\mathrm{x}\mathrm{1}}\mathrm{+}{\mathrm{at}}_{\mathrm{13}}\mathrm{*}{\mathrm{F}}_{\mathrm{there}\mathrm{1}}\mathrm{=}{\mathrm{<figure-callout\; id="11"\; label="b"\; filenames="imgf0002.png,imgf0004.png"\; state="\{\{state\}\}">b</figure-callout>}}_{\mathrm{11}}\mathrm{*}{\mathrm{M}}_{\mathrm{f}\mathrm{2}}\mathrm{+}{\mathrm{<figure-callout\; id="12"\; label="b"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">b</figure-callout>}}_{\mathrm{12}}\mathrm{*}{\mathrm{F}}_{\mathrm{x}\mathrm{2}}\mathrm{+}{\mathrm{<figure-callout\; id="13"\; label="b"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">b</figure-callout>}}_{\mathrm{13}}\mathrm{*}{\mathrm{F}}_{\mathrm{there}\mathrm{2}}$$

$$\mathrm{Dx}\mathrm{=}{\mathrm{at}}_{21}\mathrm{*}{\mathrm{M}}_{\mathrm{f}\mathrm{1}}\mathrm{+}{\mathrm{at}}_{22}\mathrm{*}{\mathrm{F}}_{\mathrm{x}\mathrm{1}}\mathrm{+}{\mathrm{at}}_{23}\mathrm{*}{\mathrm{F}}_{\mathrm{there}\mathrm{1}}\mathrm{=}{\mathrm{b}}_{21}\mathrm{*}{\mathrm{M}}_{\mathrm{f}\mathrm{2}}\mathrm{+}{\mathrm{b}}_{22}\mathrm{*}{\mathrm{F}}_{\mathrm{x}\mathrm{2}}\mathrm{+}{\mathrm{b}}_{23}\mathrm{*}{\mathrm{F}}_{\mathrm{there}\mathrm{2}}$$

$$\mathrm{.delta.Y}\mathrm{=}{\mathrm{at}}_{31}\mathrm{*}{\mathrm{M}}_{\mathrm{f}\mathrm{1}}\mathrm{+}{\mathrm{at}}_{32}\mathrm{*}{\mathrm{F}}_{\mathrm{x}\mathrm{1}}\mathrm{+}{\mathrm{at}}_{33}\mathrm{*}{\mathrm{F}}_{\mathrm{there}\mathrm{1}}\mathrm{=}{\mathrm{b}}_{31}\mathrm{*}{\mathrm{M}}_{\mathrm{f}\mathrm{2}}\mathrm{+}{\mathrm{b}}_{32}\mathrm{*}{\mathrm{F}}_{\mathrm{x}\mathrm{2}}\mathrm{+}{\mathrm{b}}_{33}\mathrm{*}{\mathrm{F}}_{\mathrm{there}\mathrm{2}}$$

$$\mathrm{\Delta x}\mathrm{=}{\mathrm{a}}_{21}\mathrm{*}{\mathrm{M}}_{\mathrm{f}\mathrm{1}}\mathrm{+}{\mathrm{a}}_{22}\mathrm{*}{\mathrm{F}}_{\mathrm{x}\mathrm{1}}\mathrm{+}{\mathrm{a}}_{23}\mathrm{*}{\mathrm{F}}_{\mathrm{y}\mathrm{1}}\mathrm{=}{\mathrm{b}}_{21}\mathrm{*}\left({\mathrm{M}}_{\mathrm{f}}-{\mathrm{M}}_{\mathrm{f}1}\right)-{\mathrm{b}}_{22}\mathrm{*}{\mathrm{F}}_{\mathrm{x}1}-{\mathrm{b}}_{23}\mathrm{*}{\mathrm{F}}_{\mathrm{y}1}$$

$$\mathrm{\Delta y}\mathrm{=}{\mathrm{a}}_{31}\mathrm{*}{\mathrm{M}}_{\mathrm{f}\mathrm{1}}\mathrm{+}{\mathrm{a}}_{32}\mathrm{*}{\mathrm{F}}_{\mathrm{x}\mathrm{1}}\mathrm{+}{\mathrm{a}}_{33}\mathrm{*}{\mathrm{F}}_{\mathrm{y}\mathrm{1}}\mathrm{=}{\mathrm{b}}_{31}\mathrm{*}\left({\mathrm{M}}_{\mathrm{f}}-{\mathrm{M}}_{\mathrm{f}1}\right)-{\mathrm{b}}_{32}\mathrm{*}{\mathrm{F}}_{\mathrm{x}1}-{\mathrm{b}}_{33}\mathrm{*}{\mathrm{F}}_{\mathrm{y}1}$$

et par la suite : $${\mathrm{M}}_{\mathrm{f}\mathrm{1}}\mathrm{*}\left({\mathrm{a}}_{\mathrm{11}}\mathrm{+}{\mathrm{b}}_{\mathrm{11}}\right)\mathrm{+}{\mathrm{F}}_{\mathrm{x}\mathrm{1}}\mathrm{*}\left({\mathrm{a}}_{\mathrm{12}}\mathrm{+}{\mathrm{b}}_{\mathrm{12}}\right)\mathrm{+}{\mathrm{F}}_{\mathrm{y}\mathrm{1}}\mathrm{*}\left({\mathrm{a}}_{\mathrm{13}}\mathrm{+}{\mathrm{b}}_{\mathrm{13}}\right)\mathrm{=}{\mathrm{b}}_{\mathrm{11}}\mathrm{*}{\mathrm{M}}_{\mathrm{f}}$$

$${\mathrm{M}}_{\mathrm{f}\mathrm{1}}\mathrm{*}\left({\mathrm{a}}_{21}\mathrm{+}{\mathrm{b}}_{21}\right)\mathrm{+}{\mathrm{F}}_{\mathrm{x}\mathrm{1}}\mathrm{*}\left({\mathrm{a}}_{22}\mathrm{+}{\mathrm{b}}_{22}\right)\mathrm{+}{\mathrm{F}}_{\mathrm{y}\mathrm{1}}\mathrm{*}\left({\mathrm{a}}_{23}\mathrm{+}{\mathrm{b}}_{23}\right)\mathrm{=}{\mathrm{b}}_{21}\mathrm{*}{\mathrm{M}}_{\mathrm{f}}$$

$${\mathrm{M}}_{\mathrm{f}\mathrm{1}}\mathrm{*}\left({\mathrm{a}}_{31}\mathrm{+}{\mathrm{b}}_{31}\right)\mathrm{+}{\mathrm{F}}_{\mathrm{x}\mathrm{1}}\mathrm{*}\left({\mathrm{a}}_{32}\mathrm{+}{\mathrm{b}}_{32}\right)\mathrm{+}{\mathrm{F}}_{\mathrm{y}\mathrm{1}}\mathrm{*}\left({\mathrm{a}}_{33}\mathrm{+}{\mathrm{b}}_{33}\right)\mathrm{=}{\mathrm{b}}_{31}\mathrm{*}{\mathrm{M}}_{\mathrm{f}}$$

where the coefficients a _ij , b _ij are determined by the physical characteristics of the first terminal peripheral curve 14, respectively of the second terminal peripheral curve 15. By grouping these equations, we obtain: $$\mathrm{\Delta \omega }\mathrm{=}{\mathrm{at}}_{\mathrm{11}}\mathrm{*}{\mathrm{M}}_{\mathrm{f}\mathrm{1}}\mathrm{+}{\mathrm{at}}_{\mathrm{12}}\mathrm{*}{\mathrm{F}}_{\mathrm{x}\mathrm{1}}\mathrm{+}{\mathrm{at}}_{\mathrm{13}}\mathrm{*}{\mathrm{F}}_{\mathrm{there}\mathrm{1}}\mathrm{=}{\mathrm{<figure-callout\; id="11"\; label="b"\; filenames="imgf0002.png,imgf0004.png"\; state="\{\{state\}\}">b</figure-callout>}}_{\mathrm{11}}\mathrm{*}\left({\mathrm{M}}_{\mathrm{f}}\mathrm{-}{\mathrm{M}}_{\mathrm{f}\mathrm{1}}\right)\mathrm{-}{\mathrm{<figure-callout\; id="12"\; label="b"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">b</figure-callout>}}_{\mathrm{12}}\mathrm{*}{\mathrm{F}}_{\mathrm{x}\mathrm{1}}\mathrm{-}{\mathrm{<figure-callout\; id="13"\; label="b"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">b</figure-callout>}}_{\mathrm{13}}\mathrm{*}{\mathrm{F}}_{\mathrm{there}\mathrm{1}}$$

$$\mathrm{Dx}\mathrm{=}{\mathrm{at}}_{21}\mathrm{*}{\mathrm{M}}_{\mathrm{f}\mathrm{1}}\mathrm{+}{\mathrm{at}}_{22}\mathrm{*}{\mathrm{F}}_{\mathrm{x}\mathrm{1}}\mathrm{+}{\mathrm{at}}_{23}\mathrm{*}{\mathrm{F}}_{\mathrm{there}\mathrm{1}}\mathrm{=}{\mathrm{b}}_{21}\mathrm{*}\left({\mathrm{M}}_{\mathrm{f}}-{\mathrm{M}}_{\mathrm{f}1}\right)-{\mathrm{b}}_{22}\mathrm{*}{\mathrm{F}}_{\mathrm{x}1}-{\mathrm{b}}_{23}\mathrm{*}{\mathrm{F}}_{\mathrm{there}1}$$

$$\mathrm{.delta.Y}\mathrm{=}{\mathrm{at}}_{31}\mathrm{*}{\mathrm{M}}_{\mathrm{f}\mathrm{1}}\mathrm{+}{\mathrm{at}}_{32}\mathrm{*}{\mathrm{F}}_{\mathrm{x}\mathrm{1}}\mathrm{+}{\mathrm{at}}_{33}\mathrm{*}{\mathrm{F}}_{\mathrm{there}\mathrm{1}}\mathrm{=}{\mathrm{b}}_{31}\mathrm{*}\left({\mathrm{M}}_{\mathrm{f}}-{\mathrm{M}}_{\mathrm{f}1}\right)-{\mathrm{b}}_{32}\mathrm{*}{\mathrm{F}}_{\mathrm{x}1}-{\mathrm{b}}_{33}\mathrm{*}{\mathrm{F}}_{\mathrm{there}1}$$

and then : $${\mathrm{M}}_{\mathrm{f}\mathrm{1}}\mathrm{*}\left({\mathrm{at}}_{\mathrm{11}}\mathrm{+}{\mathrm{<figure-callout\; id="11"\; label="b"\; filenames="imgf0002.png,imgf0004.png"\; state="\{\{state\}\}">b</figure-callout>}}_{\mathrm{11}}\right)\mathrm{+}{\mathrm{F}}_{\mathrm{x}\mathrm{1}}\mathrm{*}\left({\mathrm{at}}_{\mathrm{12}}\mathrm{+}{\mathrm{<figure-callout\; id="12"\; label="b"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">b</figure-callout>}}_{\mathrm{12}}\right)\mathrm{+}{\mathrm{F}}_{\mathrm{there}\mathrm{1}}\mathrm{*}\left({\mathrm{at}}_{\mathrm{13}}\mathrm{+}{\mathrm{<figure-callout\; id="13"\; label="b"\; filenames="imgf0002.png,imgf0003.png"\; state="\{\{state\}\}">b</figure-callout>}}_{\mathrm{13}}\right)\mathrm{=}{\mathrm{<figure-callout\; id="11"\; label="b"\; filenames="imgf0002.png,imgf0004.png"\; state="\{\{state\}\}">b</figure-callout>}}_{\mathrm{11}}\mathrm{*}{\mathrm{M}}_{\mathrm{f}}$$

$${\mathrm{M}}_{\mathrm{f}\mathrm{1}}\mathrm{*}\left({\mathrm{at}}_{21}\mathrm{+}{\mathrm{b}}_{21}\right)\mathrm{+}{\mathrm{F}}_{\mathrm{x}\mathrm{1}}\mathrm{*}\left({\mathrm{at}}_{22}\mathrm{+}{\mathrm{b}}_{22}\right)\mathrm{+}{\mathrm{F}}_{\mathrm{there}\mathrm{1}}\mathrm{*}\left({\mathrm{at}}_{23}\mathrm{+}{\mathrm{b}}_{23}\right)\mathrm{=}{\mathrm{b}}_{21}\mathrm{*}{\mathrm{M}}_{\mathrm{f}}$$

$${\mathrm{M}}_{\mathrm{f}\mathrm{1}}\mathrm{*}\left({\mathrm{at}}_{31}\mathrm{+}{\mathrm{b}}_{31}\right)\mathrm{+}{\mathrm{F}}_{\mathrm{x}\mathrm{1}}\mathrm{*}\left({\mathrm{at}}_{32}\mathrm{+}{\mathrm{b}}_{32}\right)\mathrm{+}{\mathrm{F}}_{\mathrm{there}\mathrm{1}}\mathrm{*}\left({\mathrm{at}}_{33}\mathrm{+}{\mathrm{b}}_{33}\right)\mathrm{=}{\mathrm{b}}_{31}\mathrm{*}{\mathrm{M}}_{\mathrm{f}}$$

$${\mathrm{F}}_{\mathrm{x}\mathrm{1}}\mathrm{donc}{\mathrm{F}}_{\mathrm{x}\mathrm{2}}\mathrm{=}\mathrm{-}{\mathrm{F}}_{\mathrm{x}\mathrm{1}}$$

$${\mathrm{F}}_{\mathrm{y}\mathrm{1}}\mathrm{donc}{\mathrm{F}}_{\mathrm{y}\mathrm{2}}\mathrm{=}\mathrm{-}{\mathrm{F}}_{\mathrm{y}\mathrm{1}}$$

This system is balanced, and makes it possible to calculate for a torque moment M _f given and as a function of the coefficients a _ij , b _ij : $${\mathrm{M}}_{\mathrm{f}\mathrm{1}}\mathrm{therefore}{\mathrm{M}}_{\mathrm{f}\mathrm{2}}\mathrm{=}{\mathrm{M}}_{\mathrm{f}}\mathrm{-}{\mathrm{M}}_{\mathrm{f}\mathrm{1}}$$

$${\mathrm{F}}_{\mathrm{x}\mathrm{1}}\mathrm{therefore}{\mathrm{F}}_{\mathrm{x}\mathrm{2}}\mathrm{=}\mathrm{-}{\mathrm{F}}_{\mathrm{x}\mathrm{1}}$$

$${\mathrm{F}}_{\mathrm{there}\mathrm{1}}\mathrm{therefore}{\mathrm{F}}_{\mathrm{there}\mathrm{2}}\mathrm{=}\mathrm{-}{\mathrm{F}}_{\mathrm{there}\mathrm{1}}$$

$$\mathrm{\Delta x}\mathrm{=}{\mathrm{c}}_{\mathrm{x}}\mathrm{*}{\mathrm{M}}_{\mathrm{f}}$$

$$\mathrm{\Delta y}\mathrm{=}{\mathrm{c}}_{\mathrm{y}}\mathrm{*}{\mathrm{M}}_{\mathrm{f}}$$

For a given shape and dimensions for the terminal peripheral curves 14, 15, it is then possible to calculate the displacement of the outer point 12: $$\mathrm{\Delta \omega }\mathrm{=}{\mathrm{vs}}_{\mathrm{\omega }}\mathrm{*}{\mathrm{M}}_{\mathrm{f}}$$

$$\mathrm{Dx}\mathrm{=}{\mathrm{vs}}_{\mathrm{x}}\mathrm{*}{\mathrm{M}}_{\mathrm{f}}$$

$$\mathrm{.delta.Y}\mathrm{=}{\mathrm{vs}}_{\mathrm{there}}\mathrm{*}{\mathrm{M}}_{\mathrm{f}}$$

$${\mathrm{a}}_{\mathrm{12}}\mathrm{=}{\mathrm{a}}_{\mathrm{21}}\mathrm{=}\mathrm{-}\frac{\underset{\mathrm{0}}{\overset{\mathrm{L}}{\mathrm{\int }}}\left({\mathrm{Y}}_{\mathrm{0}}\mathrm{-}\mathrm{Y}\right)*\mathrm{dl}}{\mathrm{E}\mathrm{*}\mathrm{l}}$$

$${\mathrm{a}}_{13}\mathrm{=}{\mathrm{a}}_{31}\mathrm{=}\mathrm{-}\frac{\underset{\mathrm{0}}{\overset{\mathrm{L}}{\mathrm{\int }}}\left({\mathrm{X}}_{\mathrm{0}}\mathrm{-}\mathrm{X}\right)*\mathrm{dl}}{\mathrm{E}\mathrm{*}\mathrm{l}}$$

$${\mathrm{a}}_{22}\mathrm{=}\frac{\underset{\mathrm{0}}{\overset{\mathrm{L}}{\mathrm{\int }}}{\left({\mathrm{Y}}_{\mathrm{0}}\mathrm{-}\mathrm{Y}\right)}^2*\mathrm{dl}}{\mathrm{E}\mathrm{*}\mathrm{l}}$$

$${\mathrm{a}}_{23}\mathrm{=}{\mathrm{a}}_{32}\mathrm{=}\mathrm{-}\frac{\underset{\mathrm{0}}{\overset{\mathrm{L}}{\mathrm{\int }}}\left({\mathrm{X}}_{\mathrm{0}}\mathrm{-}\mathrm{X}\right)*\left({\mathrm{Y}}_{\mathrm{0}}\mathrm{-}\mathrm{Y}\right)*\mathrm{dl}}{\mathrm{E}\mathrm{*}\mathrm{l}}$$

$${\mathrm{a}}_{33}\mathrm{=}\frac{\underset{\mathrm{0}}{\overset{\mathrm{L}}{\mathrm{\int }}}{\left({\mathrm{X}}_{\mathrm{0}}\mathrm{-}\mathrm{X}\right)}^2*\mathrm{dl}}{\mathrm{E}\mathrm{*}\mathrm{l}}$$

où L est la longueur de la première courbe périphérique terminale 14 et ∫₀ ^L désigne l'intégrale le long de la première courbe périphérique terminale 14 depuis le point terminal 16 jusqu'au point extérieur 12, (X₀, Y₀) représente les coordonnées du point extérieur 12, (X, Y) représente les coordonnées du point courant, dl l'élément de longueur, E le module élastique du matériau formant la première courbe périphérique terminale 14 et I le moment d'inertie de la première courbe périphérique terminale 14. Les coefficients b_ij pour la deuxième courbe périphérique terminale 15 se calculent de manière similaire.where the coefficients c _ω , c _x , c _y depend on the coefficients a _ji , b _ij and are calculated from the equations indicated above. The coefficients a _ij for the first terminal peripheral curve 14 can be calculated according to the theory of elementary rotations: $${\mathrm{at}}_{\mathrm{11}}\mathrm{=}\frac{\underset{\mathrm{0}}{\overset{\mathrm{The}}{\mathrm{\int }}}\mathrm{dl}}{\mathrm{E}\mathrm{*}\mathrm{l}}$$

$${\mathrm{at}}_{\mathrm{12}}\mathrm{=}{\mathrm{at}}_{\mathrm{21}}\mathrm{=}\mathrm{-}\frac{\underset{\mathrm{0}}{\overset{\mathrm{The}}{\mathrm{\int }}}\left({\mathrm{Y}}_{\mathrm{0}}\mathrm{-}\mathrm{Y}\right)*\mathrm{dl}}{\mathrm{E}\mathrm{*}\mathrm{l}}$$

$${\mathrm{at}}_{13}\mathrm{=}{\mathrm{at}}_{31}\mathrm{=}\mathrm{-}\frac{\underset{\mathrm{0}}{\overset{\mathrm{The}}{\mathrm{\int }}}\left({\mathrm{X}}_{\mathrm{0}}\mathrm{-}\mathrm{X}\right)*\mathrm{dl}}{\mathrm{E}\mathrm{*}\mathrm{l}}$$

$${\mathrm{at}}_{22}\mathrm{=}\frac{\underset{\mathrm{0}}{\overset{\mathrm{The}}{\mathrm{\int }}}{\left({\mathrm{Y}}_{\mathrm{0}}\mathrm{-}\mathrm{<figure-callout\; id="2"\; label="Y"\; filenames="imgf0001.png"\; state="\{\{state\}\}">Y</figure-callout>}\right)}^2*\mathrm{dl}}{\mathrm{E}\mathrm{*}\mathrm{l}}$$

$${\mathrm{at}}_{23}\mathrm{=}{\mathrm{at}}_{32}\mathrm{=}\mathrm{-}\frac{\underset{\mathrm{0}}{\overset{\mathrm{The}}{\mathrm{\int }}}\left({\mathrm{X}}_{\mathrm{0}}\mathrm{-}\mathrm{X}\right)*\left({\mathrm{Y}}_{\mathrm{0}}\mathrm{-}\mathrm{Y}\right)*\mathrm{dl}}{\mathrm{E}\mathrm{*}\mathrm{l}}$$

$${\mathrm{at}}_{33}\mathrm{=}\frac{\underset{\mathrm{0}}{\overset{\mathrm{The}}{\mathrm{\int }}}{\left({\mathrm{X}}_{\mathrm{0}}\mathrm{-}\mathrm{<figure-callout\; id="2"\; label="X"\; filenames="imgf0001.png"\; state="\{\{state\}\}">X</figure-callout>}\right)}^2*\mathrm{dl}}{\mathrm{E}\mathrm{*}\mathrm{l}}$$

where L is the length of the first end peripheral curve 14 and ∫ ₀ ^L denotes the integral along the first end peripheral curve 14 from the end point 16 to the outer point 12, (X ₀ , Y ₀ ) represents the coordinates of the outer point 12, (X, Y) represents the coordinates of the current point, dl the element of length, E the elastic modulus of the material forming the first end peripheral curve 14 and I the moment of inertia of the first curve terminal device 14. The coefficients b _ij for the second terminal peripheral curve 15 are calculated in a similar manner.

$${\mathrm{\Delta y}}_{\mathrm{c}}\mathrm{=}\mathrm{\Delta y}\mathrm{+}\mathrm{\Delta \omega }\mathrm{*}{\mathrm{V}}_{\mathrm{x}}\mathrm{=}\left({\mathrm{c}}_{\mathrm{y}}+{\mathrm{c}}_{\mathrm{\omega }}\mathrm{*}{\mathrm{V}}_{\mathrm{x}}\right)\mathrm{*}{\mathrm{M}}_{\mathrm{f}}$$

If we call V the vector going from the outer point 12 to the center of the spiral (located on the geometric axis of rotation of the balance and the ferrule), of components V _x , V _y , and that we neglect the variation this V vector, Dx displacement _c, .delta.Y _c of the spiral from the center (this center is considered to be free, the reference (O, x, y) remaining fixed) due to the movement Dx, .delta.Y and Δω from the outer point 12 is given by the following equations: $${\mathrm{Dx}}_{\mathrm{vs}}\mathrm{=}\mathrm{Dx}\mathrm{-}\mathrm{\Delta \omega }\mathrm{*}{\mathrm{V}}_{\mathrm{there}}\mathrm{=}\left({\mathrm{vs}}_{\mathrm{x}}\mathrm{-}{\mathrm{vs}}_{\mathrm{\omega }}\mathrm{*}{\mathrm{V}}_{\mathrm{there}}\right)\mathrm{*}{\mathrm{M}}_{\mathrm{f}}$$

$${\mathrm{.delta.Y}}_{\mathrm{vs}}\mathrm{=}\mathrm{.delta.Y}\mathrm{+}\mathrm{\Delta \omega }\mathrm{*}{\mathrm{V}}_{\mathrm{x}}\mathrm{=}\left({\mathrm{vs}}_{\mathrm{there}}+{\mathrm{vs}}_{\mathrm{\omega }}\mathrm{*}{\mathrm{V}}_{\mathrm{x}}\right)\mathrm{*}{\mathrm{M}}_{\mathrm{f}}$$

où K est le coefficient de proportionnalité, calculé par simulation des déformations du type de celles illustrées aux figures 3 et 4 de la demande de brevet EP 1 473 604 . Cette dernière équation constitue une condition à respecter pour définir les courbes périphériques terminales 14, 15. Une autre condition est l'amplitude de déplacement du centre du spiral, en d'autres termes la longueur du segment de droite précité.The terminal peripheral curves 14, 15 must be chosen so that this displacement Δx _c , Δy _c is substantially equal to the inverse of the displacement that the center of the spiral would undergo under the effect of the moment M _f applied to the balance shaft if the center was free and the outer point 12 was fixed. According to a linear approximation, this last displacement can be represented by a line segment, in other words the component Δy _c can be considered as proportional to the component Δx _c . We thus have: $${\mathrm{vs}}_{\mathrm{there}}\mathrm{+}{\mathrm{vs}}_{\mathrm{\omega }}\mathrm{*}{\mathrm{V}}_{\mathrm{x}}\mathrm{=}\mathrm{K}\mathrm{*}\left({\mathrm{vs}}_{\mathrm{x}}\mathrm{-}{\mathrm{vs}}_{\mathrm{\omega }}\mathrm{*}{\mathrm{V}}_{\mathrm{there}}\right)$$

where K is the coefficient of proportionality, calculated by simulation of the deformations of the type illustrated in figures 3 and 4 of the patent application EP 1 473 604 . This last equation is a condition to be respected for defining the terminal peripheral curves 14, 15. Another condition is the amplitude of displacement of the center of the spiral, in other words the length of the aforementioned line segment.

The system of equations thus has two conditions and one has three deformation parameters, Δx, Δy and Δω. There is therefore an infinity of solutions for the terminal peripheral curves 14, 15.

To facilitate the manufacture of the hairspring, the terminal peripheral curves 14, 15 may be circular arcs, as in the embodiments described above and shown in the figures. The rays and angular expanses of these arcs of circle already bring four degrees of freedom. An infinity of solutions exist in this particular case too.

It has been found by the present inventors that the use of two terminal peripheral curves 14, 15 confers a certain rigidity on the zone of the external point 12, rigidity which allows a displacement of the outer point 12 in a range which corresponds to the natural displacement of the center of the spiral that we want to compensate. The use of a single non-rigidified terminal peripheral curve would cause the outer point 12 to move too much.

The results obtained with the spiral according to the invention in terms of concentricity are at least comparable to those obtained with a Breguet spiral or a spiral as described in the patent application. EP 1 473 604 . The hairspring according to the invention has the advantage of being simple to manufacture. It can indeed be achieved using traditional techniques such as rolling, welding, etc. and in one or more traditional materials such as steel. Moreover, since an infinite number of pairs of terminal peripheral curves may be suitable, the choice may be made according to the space available or other criteria relating to the arrangement of the components of the movement.

• le nombre de courbes périphériques terminales pourrait être supérieur à deux ;
• les points terminaux des courbes périphériques terminales (c'est-à-dire du spiral) pourraient être confondus en un seul point qui serait fixé au bâti par soudage ou autre procédé approprié ;
• la section, par exemple l'épaisseur, de l'une au moins de la partie centrale et des courbes périphériques terminales pourrait varier ;
• des formes simples autres qu'un arc de cercle peuvent être envisagées pour chaque courbe périphérique terminale, par exemple une succession d'arcs de cercle de rayons et/ou centres différents, un segment de droite ou des segments de droite définissant des coudes entre eux ;
• l'une au moins des courbes périphériques terminales pourrait comporter une partie rigide et une partie élastique, par exemple une partie principale rigide (par exemple rectiligne) et une partie terminale élastique définissant un point terminal destiné à être fixé à un piton ;
• la partie centrale 10 comporte des spires mais n'est pas nécessairement complètement en spirale. Elle pourrait par exemple comporter après la dernière spire et avant le point extérieur 12 une portion de lame qui s'écarte de la spirale, le point extérieur 12 constituant le point terminal de cette portion de lame et le point de départ des courbes périphériques terminales. En variante, le point extérieur 12 pourrait être sur la dernière spire de la partie centrale 10 et l'une des courbes périphériques terminales pourrait comporter juste après le point extérieur 12 une portion de spire constituant un prolongement exact de ladite dernière spire.

The invention has been described above by way of example only. It goes without saying that modifications could be made without departing from the scope of the claimed invention. For example :

• the number of terminal peripheral curves could be greater than two;
• the end points of the terminal peripheral curves (that is to say the spiral) could be merged into a single point which would be fixed to the frame by welding or other suitable method;
• the section, for example the thickness, of at least one of the central portion and the terminal peripheral curves could vary;
• simple shapes other than an arc of circle may be envisaged for each terminal peripheral curve, for example a succession of arcs of different radii and / or centers, a line segment or straight segments defining elbows between them ;
• at least one of the terminal peripheral curves could comprise a rigid portion and an elastic portion, for example a rigid main portion (for example rectilinear) and an elastic end portion defining an end point to be fixed to a peak;
• the central portion 10 has turns but is not necessarily completely spiral. It could for example comprise after the last turn and before the outer point 12 a blade portion that deviates from the spiral, the outer point 12 constituting the end point of this blade portion and the starting point of the terminal peripheral curves. Alternatively, the outer point 12 could be on the last turn of the central portion 10 and one of the terminal peripheral curves could comprise just after the outer point 12 a turn portion constituting an exact extension of said last turn.

Moreover, in a manner known per se, the central portion of the hairspring can form with the ferrule two separate parts assembled to one another by gluing, welding or other suitable method at the inner point, or a single piece.

Claims (13)

1. Balance spring for a timepiece movement, comprising a planar central part (10) comprising turns and extending from an inner point (11) to an outer point (12) and means (13) allowing the turns to be kept concentric during oscillations of the balance spring in the movement, characterised in that said means comprise at least first and second flexible, terminal, peripheral curves (14, 15) located in the plane of the central part (10) and extending in a disjointed manner from the outer point (12) up to one or several respective terminal point(s) (16, 17).
2. Balance spring as claimed in Claim 1, characterised in that the first and second terminal, peripheral curves (14, 15) extend up to separate respective terminal points (16, 17).
3. Balance spring as claimed in Claim 1 or 2, characterised in that the first and second terminal, peripheral curves (14, 15) each have a general circular arc shape.
4. Balance spring as claimed in any one of Claims 1 to 3, characterised in that at least one (15a; 15b; 14c, 15c; 14d, 15d) of the first and second terminal, peripheral curves (14, 15) comprises a bend (18a; 18b; 18c, 19c; 18d, 19d) at its end joined to the outer point (12) to prevent this terminal, peripheral curve from being contacted by another part of the balance spring during expansions thereof.
5. Balance spring as claimed in any one of Claims 1 to 4, characterised in that the first terminal, peripheral curve (14a; 14b) is in the extension of the central part (10).
6. Balance spring as claimed in Claim 5, characterised in that the first terminal, peripheral curve (14a; 14b) is integral with the central part (10) and the second terminal, peripheral curve (15a; 15b) is an insert attached by one of its ends to the outer point (12).
7. Balance spring as claimed in Claim 5 or 6, characterised in that the second terminal, peripheral curve (15a) extends in a direction opposite that of the first terminal, peripheral curve (14a).
8. Balance spring as claimed in Claim 5 or 6, characterised in that the second terminal, peripheral curve (15b) extends in the same direction as the first terminal, peripheral curve (14b).
9. Balance spring as claimed in any one of Claims 5 to 8, characterised in that the first terminal, peripheral curve (14a; 14b) is circular arc-shaped and the second terminal, peripheral curve (15a; 15b) is formed of a bend (18a; 18b) at its end joined to the outer point (12) and of a circular arc-shaped main part (19a; 19b) between the bend (18a; 18b) and the terminal point (17a; 17b) of the second terminal, peripheral curve (15a; 15b).
10. Balance spring as claimed in any one of Claims 1 to 4, characterised in that the first and second terminal, peripheral curves (14c, 15c; 14d, 15d) are in the general shape of circular arcs having the same centre and same radius and extend in two opposite directions from the outer point (12).
11. Balance spring as claimed in any one of Claims 1 to 10, characterised in that the central part (10) and the first and second terminal, peripheral curves (14, 15) each have a constant cross-section over their entire length.
12. Balance spring as claimed in Claim 11, characterised in that the central part (10) and the first and second terminal, peripheral curves (14a, 15a; 14c, 15c) have the same cross-section.
13. Timepiece movement comprising a frame (5, 9), a balance arbor (3) mounted in a pivoting manner in the frame, and a balance spring as claimed in any one of Claims 1 to 12 mounted on the balance arbor via a collet, said terminal point(s) (16, 17) of the balance spring being attached to the frame.

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