Balance-hairspring oscillator for a timepiece
In an oscillator for a timepiece including a balance and a hairspring, the balance lacks equilibrium, such that: the curves for running of the oscillator owing to weight of the hairspring as a function of the oscillation amplitude of the balance in at least four vertical positions of the oscillator spaced by 90° each pass through 0 at an oscillation amplitude of the balance between 200° and 240°; and between oscillation amplitudes of 150° and 280°, curves representing the running of the oscillator owing to lack of equilibrium in the balance as a function of the oscillation amplitude in the vertical positions each has an average slope of opposite sign to the average slope of the corresponding curve among the curves representing the running of the oscillator owing to the weight of the hairspring. A reduction in the running discrepancies between the vertical positions can thus be achieved.
1. An oscillator for a timepiece, comprising:
a balance ( 1 ); and
a hairspring ( 3 ; 3 ′), the balance having a lack of equilibrium,
wherein the lack of equilibrium in the balance and the geometry of the hairspring are such that
a) first curves (S 1 -S 4 ; S 1 ′-S 4 ′) representing the running of the oscillator owing to the weight of the hairspring as a function of the oscillation amplitude of the balance in at least four vertical positions of the oscillator spaced apart by 90° each pass through a value zero of running at an oscillation amplitude of the balance between 200° and 240°; and
b) between the oscillation amplitude of 150° and the oscillation amplitude of 280°, second curves (B 1 -B 4 ; B 1 ′-B 4 ′) representing the running of the oscillator owing to the lack of equilibrium in the balance as a function of the oscillation amplitude of the balance in said vertical positions of the oscillator each have an average slope of opposite sign to the average slope of a corresponding curve among said first curves (S 1 -S 4 ; S 1 ′-S 4 ′) representing the running of the oscillator owing to the weight of the hairspring,
wherein said first curves are obtained by the following formula:
μ
(
θ
0
)
=
-
8
6
4
0
0
.
M
s
.
L
E
.
I
.
g
.
1
2
.
π
.
θ
0
2
∫
0
2.
π
δ
y
g
(
θ
(
φ
)
)
δ
θ
.
θ
(
φ
)
.
d
φ
and said second curves are obtained by the following formula:
μ
(
θ
0
)
=
8
6
4
0
0
.
M
b
.
g
.
d
J
b
.
ω
0
2
.
J
1
(
θ
0
)
θ
0
.
cos
(
ϕ
+
β
)
where μ is the running, M s is the mass of the hairspring, L is the length of the hairspring, E is the Young's modulus of the hairspring, I is the second moment of area of the hairspring, g is the gravitational constant, θ is the elongation of the balance with respect to its equilibrium position, θ 0 is the amplitude of the balance with respect to its equilibrium position, φ is the phase, y g is the ordinate of the center of gravity of the hairspring in a coordinate system (O, x, y) where the y axis is opposite to gravity, M b is the mass of the balance, d is the radial position of the center of gravity of the balance, J b is the moment of inertia of the balance, ω o is the natural angular frequency of the oscillator, J 1 is the Bessel function of order 1 , β is the angular position of the center of gravity of the balance with respect to an impulse pin of the balance and ϕ is the angular position of the impulse pin with respect to the direction of gravity.
2. The oscillator as claimed in claim 1 , wherein the geometry of the hairspring is such that said first curves (S 1 -S 4 ; S 1 ′-S 4 ′) representing the running of the oscillator owing to the weight of the hairspring each pass through the value zero at an oscillation amplitude of the balance between 210° and 230°.
3. The oscillator as claimed in claim 2 , wherein the geometry of the hairspring is such that said first curves (S 1 -S 4 ; S 1 ′-S 4 ′) representing the running of the oscillator owing to the weight of the hairspring each pass through the value zero at an oscillation amplitude of the balance between 215° and 225°.
4. The oscillator as claimed in claim 3 , wherein the lack of equilibrium in the balance and the geometry of the hairspring are such that the average slope of each curve among said second curves (B 1 -B 4 ; B 1 ′-B 4 ′) representing the running of the oscillator owing to the lack of equilibrium in the balance has substantially the same absolute value as the average slope of the corresponding curve among said first curves (S 1 -S 4 ; S 1 ′-S 4 ′) representing the running of the oscillator owing to the weight of the hairspring, in the range of oscillation amplitudes of 150° to 280°.
5. The oscillator as claimed in claim 2 , wherein the lack of equilibrium in the balance and the geometry of the hairspring are such that the average slope of each curve among said second curves (B 1 -B 4 ; B 1 ′-B 4 ′) representing the running of the oscillator owing to the lack of equilibrium in the balance has substantially the same absolute value as the average slope of the corresponding curve among said first curves (S 1 -S 4 ; S 1 ′-S 4 ′) representing the running of the oscillator owing to the weight of the hairspring, in the range of oscillation amplitudes of 150° to 280°.
6. The oscillator as claimed in claim 2 , wherein the inner turn of the hairspring ( 3 ; 3 ′) has a stiffened portion ( 3 d ) and/or is shaped as a Grossmann curve.
7. The oscillator as claimed in claim 1 , wherein the lack of equilibrium in the balance and the geometry of the hairspring are such that the average slope of each curve among said second curves (B 1 -B 4 ; B 1 ′-B 4 ′) representing the running of the oscillator owing to the lack of equilibrium in the balance has substantially the same absolute value as the average slope of the corresponding curve among said first curves (S 1 -S 4 ; S 1 ′-S 4 ′) representing the running of the oscillator owing to the weight of the hairspring, in the range of oscillation amplitudes of 150° to 280°.
8. The oscillator as claimed in claim 1 , wherein the lack of equilibrium in the balance and the geometry of the hairspring are such that the maximum discrepancy in the running of the oscillator owing to the lack of equilibrium in the balance and to the weight of the hairspring between said vertical positions in the range of oscillation amplitudes of 150° to 280° is less than 4 seconds/day.
9. The oscillator as claimed in claim 1 , wherein the distance (R) between the inner end ( 3 a ) of the hairspring ( 3 ′) and the center of rotation (O) of the hairspring ( 3 ′) is greater than 500 μm.
10. The oscillator as claimed in claim 1 , wherein the imbalance of the balance is greater than 0.5 μg·cm.
11. The oscillator as claimed in claim 1 , wherein the inner turn of the hairspring ( 3 ; 3 ′) has a stiffened portion ( 3 d ) and/or is shaped as a Grossmann curve.
12. The oscillator as claimed in claim 11 , wherein the outer turn of the hairspring ( 3 ; 3 ′) has a stiffened portion ( 3 c ).
13. The oscillator as claimed in claim 1 , wherein the hairspring has a stiffness and/or a pitch which vary continuously over at least several turns.
14. The oscillator as claimed in claim 1 , wherein the lack of equilibrium in the balance and the geometry of the hairspring are such that the maximum discrepancy in the running of the oscillator owing to the lack of equilibrium in the balance and to the weight of the hairspring between said vertical positions in the range of oscillation amplitudes of 150° to 280° is less than 2 seconds/day.
15. The oscillator as claimed in claim 1 , wherein the lack of equilibrium in the balance and the geometry of the hairspring are such that the maximum discrepancy in the running of the oscillator owing to the lack of equilibrium in the balance and to the weight of the hairspring between said vertical positions in the range of oscillation amplitudes of 150° to 280° is less than 1 second/day.
16. The oscillator as claimed in claim 1 , wherein the lack of equilibrium in the balance and the geometry of the hairspring are such that the maximum discrepancy in the running of the oscillator owing to the lack of equilibrium in the balance and to the weight of the hairspring between said vertical positions in the range of oscillation amplitudes of 150° to 280° is less than 0.7 seconds/day.
17. The oscillator as claimed in claim 1 , wherein the distance (R) between the inner end ( 3 a ) of the hairspring ( 3 ′) and the center of rotation (O) of the hairspring ( 3 ′) is greater than 600 μm.
18. The oscillator as claimed in claim 1 , wherein the distance (R) between the inner end ( 3 a ) of the hairspring ( 3 ′) and the center of rotation (O) of the hairspring ( 3 ′) is greater than 700 μm.
19. The oscillator as claimed in claim 1 , wherein the imbalance of the balance is greater than 1 μg·cm.
20. The oscillator as claimed in claim 3 , wherein the inner turn of the hairspring ( 3 ; 3 ′) has a stiffened portion ( 3 d ) and/or is shaped as a Grossmann curve.