Lors d’une augmentation de la fréquence de l’organe réglant, il est nécessaire d’optimiser l’énergie. C’est premièrement en augmentant le facteur de qualité de l’oscillateur qu’il est possible d’avoir un gain en puissance. Nous avons donc décidé de commencer par travailler sur des oscillateurs sans pivots, car ceux-ci entrainent des pertes, notamment en position verticale où l’amplitude du balancier spiral chute. Etant donné qu’à haute fréquence l’oscillateur devient nettement plus rigide et doit fonctionner à faible amplitude, la suppression de pivots prend tout son sens. Le dimensionnement d’un résonateur horloger est un système vibratoire qui tire le maximum d’avantages de part sa configuration. Le facteur de qualité doit être le plus grand possible, la fréquence la plus stable possible, et les modes parasites qui peuvent perturber le mouvement de l’oscillateur doivent être évités. La création d’un résonateur est donc complexe et les possibilités sont très vastes. En regardant un peu dans la littérature scientifique on observe une multitude de formes et de types de résonateurs selon les applications visées. L’oscillateur mécanique le plus simple connu est la lame vibrante, et peut d’ailleurs être modélisé comme un oscillateur à un degré de liberté. Selon sa section, ses propriétés vibratoires changent légèrement; on passe d’une lame, à une poutre, à un fil ou même une plaque. Ensuite sa géométrie complexifie déjà son mouvement vibratoire: il suffit de comparer les modes vibratoires d’une simple lame droite avec la même lame courbée ou carrément enroulée en un spiral. Un organe réglant horloger sera évidement un résonateur complexe, constitué de plusieurs parties, lame couplée, masse répartie en différentes zones. C’est donc déjà un objet qui nécessite d’être modélisé comme un système à plusieurs degrés de liberté. Nos recherches Résonique visant à explorer les possibilités de développer des organes réglants horlogers hautes fréquences, nous avons testé les idées suivantes.

Résonateur Croix: quatre lames perpendiculaires couplées par une bague et quatre masselottes situées en bout de lame. Le principe est similaire à une lame vibrante, cependant le système impose une symétrie vibratoire qui tend à réduire les faibles variations de fréquence selon son orientation par rapport au champs de pesanteur. Les quatre masselottes sont en rotation en phase pour le premier mode vibratoire. Remarquons que bien que l’anneau de couplage ne se déforme que très faiblement, il sera aussi sollicité par le mouvement des quatre lames et participe ainsi à la rigidité des quatre bras.

Anneau oscillant: une bague circulaire qui se déforme en ellipse, avec quatre masselottes, et quatre bras de fixation. Les quatre masselottes auront un mouvement linéaire en opposition de phase pour le premier mode. Ce principe garde une double symétrie pour éviter les défauts de pesanteur, et permet de réduire les efforts sur la fixation. En effet le mouvement principal oscillant reste contenu dans la bague, les quatre lames servent uniquement à maintenir le résonateur en place et ne vibre que faiblement. Ce qui permet de réduire les pertes et donc d’augmenter le facteur de qualité. Remarquons que ce type de résonateur nous a permis d’observer d’autres modes vibratoires, qui ont un intérêt certain, et peuvent servir comme premier mode vibratoire par un redimensionnement de sa géométrie.

Anneau Oscillant, fréq. de 867 Hz, échappement a impulsion 1/8 ème de cycle

Double spiral inversé: deux lames couplées et agencées en parfaite symétrie, avec des masselottes au bout. Comme sur le principe du diapason, les bras vibrent en opposition de phase sur la simple excitation d’une lame. Ceci permet un gain supplémentaire en facteur de qualité. Effectivement l’énergie vibratoire contenue dans un bras est transmise à l’autre par faible couplage ce qui permet de la contenir et de la faire circuler plutôt que d’être absorbée par le support de fixation.

Double Spiral, fréq. De154 Hz, échappement a impulsion 1/2 de cycle

Différents styles de résonateurs ont été qualifiés. Leur facteur de qualité dépasse ceux des meilleurs balanciers spiraux. Nous avons pu mesurer des facteurs de qualité entre 500 et 2000 pour des fréquences entre 100 et 1000 Hz. Nous poursuivons notre recherche exploratoire de différents types de résonateurs avant d’entamer la phase d’optimisation d’une configuration précise.

L’échappement est le mécanisme placé entre la source d’énergie et l’organe réglant. Il a pour but de compter les oscillations de l’organe réglant ainsi que d’entretenir celui-ci. Son fonctionnement est optimal s’il transmet l’énergie sur une partie précise et fine de la course de l’oscillateur, généralement au plus proche du point de repos, on parle alors d’échappement libre. La recherche de l’échappement perturbant le moins possible le balancier spiral dans sa course, a mené à l’utilisation de l’échappement à ancre suisse et de l’échappement à détente. Malgré de nombreux développements, l’échappement reste une pièce qui a un très faible rendement, env. 35%.

Afin de réaliser notre premier échappement Résonique nous sommes partis sur les exigences suivantes : créer l’échappement le plus simple et efficace possible, adapté à des fréquences plus élevées, fonctionnant pour un oscillateur vibrant à des amplitudes nettement plus faibles, et atteignant un rendement supérieur. Il nous a paru donc nécessaire de réduire les problèmes de chocs et de trouver un système d’échappement continu ou celui-ci ne fait plus d’aller retour. Cela nous a orienté vers l’utilisation d’un échappement de type magnétique où les forces sont transmises sans contact.

Les premiers échappements Résonique présentés par De Bethune étaient constitués d’une roue faite dans un matériau ferromagnétique avec un profil de dent optimisé afin d’avoir un échange de force entre la roue d’échappement et les aimants fixés sur l’oscillateur de forme sinusoïdale. Ce type d’échappement présente, si l’on applique un moment de force, une plage de couple où la vitesse de rotation de la roue se synchronise à la fréquence de vibration du résonateur, et permet ainsi d’entretenir ses oscillations et de mesurer le temps. Bien que cette méthode permette une parfaite synchronisation entre la vitesse de l’échappement et les oscillations du résonateur, il s’est avéré qu’elle tend à trop accompagner l’oscillateur dans ses déplacements et crée donc des perturbations sur l’isochronisme de celui-ci. Nos premiers échappements nous ont permis d’entretenir des oscillateurs à hautes fréquences avec un rendement nettement plus élevé qu’un échappement standard, soit environ 70%.

Nous avons donc travaillé sur la recherche d’un échappement magnétique qui transmet cycliquement l’énergie sous forme d’une impulsion. Nous avons développé une roue avec un profil spécifique qui transfert l’énergie sur une part plus courte du mouvement de l’oscillateur. La force est transmise par impulsion, comme c’est le cas pour un échappement à ancre standard. Ce profil permet de réduire la durée de contact (magnétique) entre les deux organes et réduit donc nettement les perturbations de l’isochronisme du résonateur, tout en gardant un rendement élevé.

Nous avons même été plus loin en testant les possibilités d’entretenir un oscillateur sur une part réduite de son cycle, c’est-à-dire à une vitesse inférieure à un multiple entier près de sa fréquence. Ceci garantit une liberté de mouvement sur une période encore plus grande du résonateur. C’est le principe de l’échappement à détente qui nous en a donner l’idée: il transfert son énergie au balancier spiral deux fois moins souvent qu’un échappement à ancre ce qui lui confère des qualités chronométriques élevées. Cette possibilité a été envisagée car l’amélioration du facteur de qualité de nos oscillateurs réduit la proportion de perte d’amplitude en fonction du nombre d’oscillations libres. Par exemple si on donnait une impulsion à un balancier spiral de bonne qualité Q=250 que toutes les 2 oscillations, en équivalence on pourrait donner une force que toutes les 12 périodes à un oscillateur de facteur de qualité Q=1500.

Ces recherches nous ont permis d’améliorer considérablement les problèmes de variation de marche. L’échappement magnétique à impulsion reste un peu plus délicat pour maintenir la synchronisation. Forts de nos connaissances et expériences, nous sommes maintenant en phase de tester des échappements plus proches de ceux que l’ont connait en horlogerie: des échappements qui garantissent la synchronisation tout en laissant l’oscillateur libre. Le dispositif sera un peu plus complexe mais reste dans l’optique d’être le plus simple et efficace. Nous en reparlerons sur ce blog.

Comparaison entre l’échappement à impulsion cyclique et l’échappement sinusoïdale à 8 dents

-How stable is the magnetism over the years?
Thanks to recent materials such as neodymium magnets, demagnetization over time or under external “daily” magnetic fields is insignificant. In a normal magnetic environment with temperature bellow 80°C (176°F) the magnets manufacturer is announcing 5% of loss over a century. De Bethune is taking a 25% safety margin in terms of magnetic variation that should cover 500 years of demagnetization.

- What is the impact of temperature variation on magnets ?
Magnetic field of permanent magnets is indeed changing with temperature. Variation is low and reversible as long as you stay bellow maximum working temperature and bellow Curie temperature where magnetization would disappear. Today, magnets used by De Bethune specification is 100°C (212°F) maximum working temperature, 300°C (572°F) Curie temperature and magnetic variation of -0.1% per °C above 20°C (68°F). A temperature of 70°C would lower magnetic field by 5% which is still in the tolerance threshold.

- Does magnetic field associated to the magnets affect the rest of the mechanism?
The magnets that are used today are among the smallest available on the market. The magnetic field has a very short range and the effects on the rest of the mechanism are insignificant, even considering long term usage.

- Wouldn’t a strong magnetic field damage the magnets?
Yes, if very strong. But we could consider as well that any other watch would be severely damaged under the same circumstances. Nevertheless, for specific conditions we could still protect the mechanism with some kind of Faraday shield using mu metal.

In order to sustain a high frequency resonator, the traditional swiss lever escapement had to be rethought. Increasing regulating organ’s frequency requires apparently more energy to sustain the oscillator. Four key points related to reduce energy consumption: Reduce oscillator’s weight or inertia, Reduce oscillator’s amplitude, Increase oscillator’s Q factor, Improve energy transmission’s efficiency in the gear train and escapement.

Classical watch power train.

Classical swiss lever escapement would consume far too much energy at high frequencies. Consider that the lever and the whole gear train up to the barrel are accelerated and stopped at each oscillation of the regulating organ. Reducing weight might improve the efficiency, however each acceleration and stop requires energy. At high frequency it will present a lot of waste. This is not even taking into account shocks and frictions, bringing down the energy efficiency to an insignificant level.

Résonique watch power train

The Résonique had to reinvent the escapement. An escapement where energy is no more transmitted in tiny intermittent impulses but in a continuous sinusoidal form. No more abrupt mechanical stops but a smooth and continuous transmission will definitely improve efficiency. One of the De Bethune’s discoveries is the use of a magnetic escapement that allows such a sinusoidal transmission.

Proposed solution

A mechanical power source is transmitting torque to the wheel train ended by a magnetic rotor. The rotor is transmitting energy to the resonator via permanent magnets. Therefore, wheel train’s rotational speed is synchronized according to the resonator’s natural frequency. As the regulating organ, the resonator determines the time unit. Hands being attached to the gear train, their speed are controlled by this accurate time division.

Résonique principle diagram

The trio model rotor, resonator, magnets brings the escapement function, energy transmission to the timekeeping element and oscillations counting. The rotor is transmitting energy to the resonator via permanent magnets. The resonator vibrates and resonates with the rotor spinning thus controlling the wheel train’s speed. The rotor’s speed is getting synched by the resonator’s oscillations, a continuous counting is carried out. For every rotor’s tooth passing, there is one oscillation. The resonator’s vibrations are sustained to its natural frequency as long as the main power source is providing sufficient energy. Oscillations’ amplitude can be considered as constant during total operating time.

To meet high frequency’s constraints and needs, the classical balance-spring has been replaced by acoustic resonators. No more pivots, affecting Quality factor improvements with frequencies increases. The developed resonators only have a few fixed or pinned points; they are stiffer and therefore allow the use of their natural modal shape. Having short amplitude brings a much higher Q factor and better regulating performances. Keeping amplitude short means, firstly, remaining in the elastic regime and therefore preserves isochronous factor, secondly, reduce air resistance proportional to speed and distance travelled. Regulating organs developed by De Bethune are called acoustic or sonic resonators because they are in the audio frequencies range (20 Hz – 20 kHz). They can be highly varied in terms of shape and vibrating mode as long as they are balanced, having a high Q factor and having a stable natural frequency.

Two resonators. A rotational mode at 360 Hz on the left picture, an elliptic mode at 890 Hz on the right.

On December 8th 2011, during the first public presentation, De Bethune demonstrated a 926 Hz, 6’667’200 vph prototype. The 22 teeth rotor turns at a rate of 2525.5 RPM. Each tooth operates as a magnetic pole released with each resonator’s oscillation. Such a rotational speed is not that high compared to some other mechanical applications. Obvious limitations in term of frequency increases will come from the number of magnetic poles that can be fitted on the rotor, as well as rotational speed limits before it induces mechanical wear.

926 Hz prototype. Left picture is showing both magnetic rotor and resonator in static state. On the right, both elements synchronized, in running condition.

926 oscillations per second is actually among the highest frequency for a regulating organ powered with pure mechanical energy. However, the goal is not to beat frequency records but experiment and test a technology to its physical limits. Résonique open a new field of unmatched frequencies. It might probably allow, later on, meeting the accuracy level of quartz movements in wearing conditions. Two developments axis are to be considered: testing higher frequencies on one hand and achieve Résonique wristwatches with lower frequencies on the other, frequencies greater than 100 Hz but with more than 24 hours autonomy.

Although humanity has had the sufficient mathematical knowledge for over 2000 years, mechanical science had to wait until 17th century to finally develop. Before Galileo was born (1564) nobody had ever consider using numbers, weight and measurement to build an accurate piece of science. The birth of the XVII th century science’s was founded on the Platon’s view of a world based on mathematical structures.
La création de la nouvelle science du XVIIème fut fondée sur l’idée platonique du monde basé sur des structures mathématiques.
Without Archimedes, there wouldn’t have been Galileo or Descartes, and without statics there wouldn’t have been dynamics. “Classical“ mechanics based on materials science developed since 17th century is studying systems through the concepts of statics and dynamics.

Horology is definitely part of the « classical » mechanics appeared at the beginning of the 17th century. Creating instruments was a necessity to experiment and evaluate scientific principles (concept) and their practical applications. “Classical” is used to avoid confusion with recent sciences such as relativity or quantum mechanics born during the 20th century.

Oscillator’s history

For over the last three centuries, mechanical watches have been using balance-spring as regulating organ. End of 17th century, after abbot of Hautefeuille attempted several times to link a balance to a linear spring in order to get, a natural frequency, Christian Huyguens, in 1675 improves the principle introducing the spiral spring associated to a balance wheel. This oscillator shows a great advantage being, as the pendulum, isochronous.

Common regulating organ, escapement wheel, lever and balance-spring

Many escapements and balance-spring have been tested over the centuries. New materials and new manufacturing process made improvements possible in term of accuracy, miniaturization and frequency. Technical researches lead to better compensation for external disturbances such as temperature, gravity force, and accelerations.

However there hasn’t been any conceptual change regarding the regulating organ, the balance-spring sustained by a lever has remained the only option. Today’s wristwatches are facing strong constraints that did not exist while they were still in the pocket like untidy and vivid position changes. Physical laws are rather simple on this: in order to reduce the impact of those disturbances, it is needed to reduce weight as much as possible and increase frequency.

With De Bethune first self-compensating silicone balance-spring and its 72’000 vph silicone escapement, demonstrated in Baselworld 2006, it became obvious that a balance-spring with lever escapement would hardly exceed such a limit (10 Hz 72’000 vph), mainly for reliability and mechanical resistance to the wear over time.

De Bethune regulating organ evolution since 2004

De Bethune took up the challenge to revolutionize timekeepers’ mechanism introducing an audio frequency watch with a magnetic escapement.

Magnetic escapement is not a new concept since it appeared in 1927, some watchmakers have been exploring this field like Frank Clifford. In clocks, the use of a vibrating cantilever beam as a regulating organ is not new either. The first use appeared in 1866 invented by Louis-François Breguet (1804-1883) and came back about a century later with the tuning forks electrical watches. Studies related to synchronization between a tuning fork and a balance-spring date from the 80’s.

De Bethune’s new idea is to have an acoustic oscillator resonating thanks to pure mechanical power in order to regulate a wrist watch movement.

– Improving a watch’s chronometric performance is directly related to increasing the time base Q factor and the latter is largerly determined by its natural frequency. Increasing the frequency allows a higher Q factor.

– Statistically, in relation to disturbance induced timing variations proportional to the duration of the oscillation, the shorter the oscillation’s period is, i.e the higher frequency, the smaller the timing variations will be.

– A resonator’s balancing default will induce perturbations in wearing condition inversely proportional to the square of the frequency. Therefore, the faster the frequency is, the smaller the impact of unbalanced resonator.

The sonic oscillator replaces the usual balance-spring oscillator in mechanical watches. The balance-spring is a resonator in which the inertia part (balance wheel) is separated from the oscillator (hairspring). The standard hairspring is very flexible, not free-standing and therefore, necessitates guiding through the pivots, which induces friction. Résonique proposes a different approach where inertia and spring are merged into one single part. Rigid and free-standing, the resonator is used with its natural frequency and normal modes shape determined by the attachment points, geometry and the material’s specifications. Accuracy does not require a wide amplitude with these oscillators. Today’s resonators developed by De Bethune cover a wide frequency range from 20 Hz to 20 kHz.

Various resonators and rotors

The characteristics of mechanical resonators are related to their geometry, the material’s elastic deformation property, and mass or inertia. As per equation (1), frequency depends on the global stiffness and equivalent mass. A high Q factor ensures a very stable frequency. The oscillator’s power also depends on those properties as well as oscillations’ amplitude, as per equation (2). The resonator’s regulating power is proportional to the oscillator’s power. The greater the power, the lower will be the impact of disturbances  on the oscillator.

(1) Frequency (2) Power Translation Rotation
(3) Q factor (4) Dissipated power

High frequency strategy

From equation (2), we can determine that higher frequency has a very strong effect on the oscillator’s power. In order to remain in an acceptable energy range, De Bethune’s strategy is to considerably reduce the oscillator’s amplitude. This is the reason why De Bethune is developing low amplitude oscillators. Both the Q factor and isochronism are better at low amplitude. If the oscillator’s Q factor could be improved, proportionally so could the power, and therefore the power required to sustain the oscillator remains equivalent as per equation (3). For a given power consumption level, such a regulating organ will run at a higher frequency, higher Q factor, and higher regulating power. These combined improvements will greatly improve chronometric performance.

Example

Let’s consider a specific case comparing a classical 4 Hz balance-spring escapement with a Résonique 800 Hz escapement. Frequency is multiplied by 200. From equation (2) we can determine that power will be multiplied by 8’000’000 times if amplitude remains the same. If we reduce amplitude by dividing it by 894, the oscillator’s power is now 10 times greater than the balance-spring model. If we consider that the Q factor is also multiplied by 10, we end up with the same power consumption for regulating power 10 times greater, as per equation (4).

There is still a wide field of research on the resonator to find the ideal geometry and the best compromise between power and Q factor according to the frequency.

The Résonique concept has been simulated. A set of two differential equations coupled by magnetic forces is enough to mathematically demonstrate the possibility of having the gear train’s rotational speed synchronized with a sustained resonator. Both of these equations are based on Newton’s second law linking the acceleration of a body with his mass m and the net force it is subject to.

Equation (1) determines the gear train’s angular motion. The first term, on the left side of the equation represents the rotor’s angular acceleration and the gear train’s total inertia. The second term is the torque of the barrel, modelled as decreasing linearly. The third term is the friction torque, which is proportional to the rotational speed, due to viscous interaction. Finally, the last term is the magnetic torque, which depends on the location of the oscillator and the angular location of the rotor.

Equation (2) determines the oscillator’s motion. The first term, on the left side represents the oscillator’s acceleration and equivalent mass. The second term is the elastic force. The third term is the friction forces, which are proportional to the speed, due to viscous interaction. Finally, the last term is the magnetic force, which depends on the location of the oscillator and the angular location of the rotor.

Both these equations are coupled by magnetic forces, that depend on the rotor and the magnet geometry. They are modelled by the finite element method. The concept was tested by a numerical analysis with finite difference method. The simulation demonstrated that a phase synchronisation between oscillator and rotor can be achieved depending on the values chosen as parameters.

After a transient phase, the rotor’s speed and oscillator’s amplitude become stationary. The oscillator’s amplitude compensates for the barrel’s decreasing torque, so that synchronisation can occurs over a relatively large energy variation. Random perturbations were added to check synchronisation in more realistic conditions. The simulations accomplished, were sufficient to confirm our hypothesis, and to finely model finely working prototypes.

Differential equations

I_1: moment d’inertie des rouages [kg ∙ m2] θ: angle parcouru par le rotor magnétique [rad] k_1: constante de couple du barillet [N ∙ m] θ_0: angle de parcours total du rotor [rad] η_1: coefficient de frottement [N ∙ m ∙ s] M(θ,x): moment de force magnétique [N ∙ m] m: masse équivalente [kg] x: déplacement de l’oscillateur [m] k_2: constante de raideur [N/m] η_2: coefficient d’amortissement [Ns/m] F(θ,x): force magnétique sur l’oscillateur [N]

Magnetic force

Magnetic forces were simulated using the finite elements method. This helped determining the size of the rotor and the permanent magnets with regard to the primary aim: An energy transmission curve with a sinusoidal form when the rotor is turning at constant speed. The secondary aim was to have a strongly localized magnetic field in order not to disturb the rest of the mechanism.

Resonator

Resonators were also simulated using the finite elements method, adding several analytical models in order to evaluate damping, mostly due to thermal flow, aerodynamics frictions, energy absorption by the binding plate and interfering vibrations. Simulation allowed testing various vibration modes in order to determine optimal frequencies and power levels. Simulation also allowed testing of different materials and to select those offering the best Q factor.

Harmonic oscillator: An oscillator where motion can be described by a sinusoidal function and where frequency depends on its physical characteristics.

Resonator: A device that vibrates at some frequencies through resonance.

Resonance: A physical phenomenon occurring with a system susceptible to specific frequencies. A resonator can cumulate energy if it comes periodically with a frequency close to the system’s natural frequency.

Frequency: The number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency. The period T is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency. Hertz (Hz) is the unit commonly associated where 5 Hz means 5 oscillations per second. In horology we also use vph (vibration per hour) where 1 Hz is the equivalent of 7200 vph.

Q factor or quality factor: A dimensionless parameter that describes how under-damped an oscillator or resonator is, or equivalently, characterizes a resonator’s bandwidth relative to its center frequency. Higher Q indicates a lower rate of energy loss relative to the stored energy of the oscillator; the oscillations die out more slowly. Sinusoidally driven resonators having higher Q factors resonate with greater amplitudes but have a smaller range of frequencies around that frequency for which they resonate.

Power: The rate at which energy is transferred, used or transformed.

Synchronization: Timekeeping which requires the coordination of events to operate a system in unison.