Balance-hairspring oscillator for a timepiece

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 (S1-S4;
• S1′-S4′) 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 (B1-B4;
• B1′-B4′) 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 (S1-S4;
• S1′-S4′) 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, Ms 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, θ0 is the amplitude of the balance with respect to its equilibrium position, φ is the phase, yg 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, Mb is the mass of the balance, d is the radial position of the center of gravity of the balance, Jb is the moment of inertia of the balance, ωo is the natural angular frequency of the oscillator, J1 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.