updated on June, 2019
The Tourbillon is an addition to the mechanics of a watch escapement and it was patented by Abraham-Louis Breguet in 1801. The Tourbillon improves the accuracy of mechanical watches, negating the effects of gravity. Besides nowadays a quartz watch can be much more accurate than any other mechanical watch, the Tourbillon is still an object of study thanks to its complication and fascination. Maisons of quality horology includes tourbillon in their timepieces, making them very high quality valuable and precious.
There are many types of tourbillon, nevertheless they all have the same function. In the following treatment I analyze a tourbillon carrousel.
The objective of that project is to simulate a real tourbillon with the multibody software MSC Adams, demonstrating how it works and the valuable accuracy it takes.
The modelling problem
The project started from a CAD of a Tourbillon carrousel , that I modify with the software Solidworks according to the geometrical issues of watchmaking . As shown in the figure below the mechanism consists of a classic swiss lever escapement. The complication added by a tourbillon consists in a cage containing all the other bodies and rotating with a constant period, usually about 60 seconds. For further information about the escapement mechanism check the reference .
- Balance wheel
- Escapement wheel
- Escapement wheel pinion
- Escapement wheel pivot
- Escapement anchor
- Escapement anchor pivot
- Internal fourth gear
The joints adopted are shown in the figure below. They seem to be more than the right number only because the software divides bodies when importing the CAD model, which would be a unique body.
The torsional spring has been set with MSC Adams tool and two design variables have been set to modify the spring’s stiffness and damping coefficients. Timing is provided by the escapement and the balance wheel which are integral to the cage but able to rotate about their axis. The internal fourth gear provides the torque coming from the mainspring and assumed constant. The balance wheel has been unbalanced with a small cube on the more external circumference as you can see in the figure below.
Contacts also have been set for the following connections:
- Escapement wheel and escapement anchor
- Escapement anchor and pin
- Escapement anchor and cage
while the connection between the internal fourth gear and the escapement pinion have been set as a coupler, as a matter of fact only torque must be transmitted and no vibration nor impact forces. In the following picture you can see the contacts’ settings:
I preferred to increase the stiffness of contacts between the escapement wheel and the anchor to provide lower penetration between bodies, besides all the other contacts were set with default parameters.
The essential problem that the tourbillon means to avoid is the effect of gravity on an imperfect balance wheel when the watch is in vertical position: if the center of mass of the balance wheel isn’t located on the axle of rotation of the system, the period of the oscillations changes because gravity accelerates or decelerates the unwinding of the torsional spring and so the watch’s accuracy gets lower. In the video linked below the same ring with an unbalanced mass and a preload torsional spring is shown in different position. The result of the presence of gravity is clear, you can see it also in the graphs below: the period of oscillation of the ring is very different depending on the initial position of the center of mass, besides the system is really the same! Remember that the accuracy in horology consists in the right timing of the watch. Modern watch usually performs 28800 alternations/hour, that is 8 alternations/second, that is 4 oscillation/second.
The idea to solve that problem is that over a period of rotation of the cage the unbalanced wheel changes position over 360 degrees. In that way, sometimes the period of oscillation is higher, sometimes it’s lower, but in a complete cycle of the cage the temporary errors compensate for themselves. Besides tourbillon’s cage usually complete a cycle in about 60 seconds, I preferred to reduce that cycle at only 6 seconds because of simulation’s time.
The demonstration of the valuable effect of the tourbillon consists in three simulations:
- The watch timing in absence if gravity without the rotation of the cage
- The watch timing in presence if gravity without the rotation of the cage
- The watch timing in presence if gravity with the rotation of the cage
First of all, it’s necessary to design the escapement mechanism of the current watch, besides presence of gravity. That step is usually very difficult because little deviation of the geometry or of the spring stiffness leads to high deviation of the timing of the watch. You can calculate the stiffness of the torsional spring with the following differential equation:
that leads to the following:
where I is the moment of inertia of the oscillating bodies with respect to the axle of rotation (that MSC Adams provides you) and n is the number of the alternations/second. By the way I preferred calculate the exact torsional stiffness by imposing to the only spring-balance wheel system a constant rotational displacement of 4/3*pi*(1-cos(8*pi*time)), in way to simulate 4 oscillations/second and measured the torque of the revolute joint of the axle as shown in the figure below: T=1.2774*10^4 Nm. Then I calculate the rotational stiffness of the spring with the following:
and then the damping coefficient, with damping ratio of 0.1, as:
In the first graph you can see the torque the axle must provide to get the displacement set. Then in the second graph you can see two simulations: the oscillations of the system spring-balance wheel without any contact, to verify the precision of the stiffness calculated, and the oscillations of the entire system (remember this step meant to design the torsional spring only, without any consideration about gravity).
You can easily notice that the period of the oscillations is not correct: that’s because the contacts between the pin and the anchor leads to a loss of energy and the period increases. So, I corrected the spring stiffness in way to provide the exact rotation of the balance wheel. I also design the torque of the internal fourth gear to provide a rotation of the balance wheel equal to 240 degrees. At the end I obtained K=3.0838*10^-5 Nm/rad and a mainspring torque T=0.01099Nm.
Simulations and analysis of results
Once every geometrical issues and design parameters have been set, I run the simulations for the three cases. Convergence has been reached with following solver settings:
The simulations’ time is 6,1 seconds, in that way I could compare the three cases observing when the 24th oscillation of the balance wheel ends. In macroscopic scale the results are pretty the same:
But zooming on the precise values it’s possible to observe relevant differences. In this treatment I always check if the rotation of the balance wheel is perfectly correct, that means the period must be exactly 0,25s to provide 4 oscillations/second and to do so, I observe where the slope changes from negative to positive. Very little errors in a period means huge errors in a month: for example, an oscillation period equal to 0,251s instead of 0,25s corresponds to an overall error of more than 1minute/day, that means more than 30minutes/month. In the first graph you can see the perfect oscillation of the case with no gravity, besides on the second graph you can see an error of 0,0050 seconds because of the influence if gravity. And then, the simulation with gravity and the rotation of the cage activated.
In the following link the animation of the tourbillon carrousel simulated:
Thanks to the three simulations I could verify that the tourbillon increases the accuracy of the watch, or better it negates the effect of gravity. It’s also interesting to observe Breguet’s valuable physical insight:
|Expected period’s end||Effective period’s end|
observing the various periods’ end of the balance wheel’s rotation, you can notice that, indeed, during a cycle of the cage’s rotation, the delays and the anticipations compensate for themselves.
 D. De Carle, “Orologiaio riparatore”, 1985