The need to measure the passage of time with increasing accuracy has required the man to design and build more and more complex systems able to partition the time in constant intervals and as short as possible. The first examples of instruments built for this purpose date back to 3000 BC. with the introduction of the solar sundial that exploited the shadow projected by a stick. Further developments took place with the invention of the hourglass and then of the water clocks which were adjusted so that one drop fell at each fixed time interval and the elapsed time could be measured by filling the container.
Before arriving at modern electronic and atomic clocks, the invention which revolutionized the method of measuring time was the mechanical clock in the Late Middle Ages. First pendulums loaded with weights and tails were used, but they had to remain fixed. Only in the seventeenth century the pendulum was replaced by an escapement mechanism consisting of a balance wheel driven by a spring to regulate the rotational motion imposed by a second spring. This mechanism allowed the creation of mobile clocks, such as pocket and wristwatches.
The problems afflicting the regularity of a mechanical watch with an escapement mechanism are numerous. The precision of construction is essential for the correct meshing of the parts, but it is not sufficient to guarantee high repeatability to the watch. Disturbing quantities such as temperature can cause a change in the inertia of moving parts and therefore a change in the swinging period of the balance wheel. Lubrication is also an important parameter since a high friction leads to a reduction in the period of oscillation and, due to the reduced powers involved, also the blockage of the mechanism itself.
Objective of the study
In this treatment we will study the regularity of a balance wheel -anchor-escapement system according to the spring stiffness, the damping and the presence of impacts between the parts.
Modeling with multibody software will also make it possible to verify the effectiveness of the part geometry during meshing and force.
Modelling of the system
The mechanical system studied here consists of a flywheel, or balance wheel, of an anchor and an escapement wheel. The purpose of this mechanism is to adjust the rotation of the escapement wheel by dividing each revolution into a certain number of steps that must be performed at regular intervals. The following figure shows the system:
In a mechanical watch, the escape wheel draws energy from the barrel which consists of a preloaded spring. In the absence of any locking system the wheel would turn free until the end of the energy supplied by the spring and without any type of rotation speed control. It would therefore not be possible to correlate this rotation with the passage of time. The role of the anchor is to block the rotation of the escapement by acting on its teeth. This is in fact equipped with two teeth that intercept, according to the angular position of the anchor itself, one of the teeth of the escapement preventing rotation. To be able to make a new shot to the wheel it is necessary to disengage the anchor allowing the rotation. For this purpose the balance wheel is inserted which is equipped with a pin which impacts against the fork of the anchor, generating the movement necessary to overcome the resistance of the wheel which is now free to rotate. This operation must be carried out cyclically with each shot and it is essential that each step is characterized by the same interval of time. By connecting a spring to the balance wheel, the period of oscillation is determined and therefore, in theory and in the absence of losses, the impact between the pin on the balance wheel and the anchor is carried out at regular intervals.
It is clear, however, that the mechanical energy of the balance wheel can not be conserved after each step because the impact absorbs energy by having to rotate the anchor, equipped with inertia, and to overcome the force generated by the escapement on the anchor to allow the defusing. The balance wheel must therefore be able to take at least the same amount of energy transferred in the impact with each shot to ensure correct disengagement even in the next step. This energy is provided by the same escapement through the anchor. The operating steps are described below to better understand.
At the first step the balance wheel is loaded and its pin, indicated by the red arrow, is 90 ° from the line passing through the wheel centers. The verse of incipient rotation is the one marked in green. At this moment the anchor and the escapement are stopped because their teeth are in contact.
In the second step the pin comes to impact against the fork of the anchor, the point of impact is indicated by the red arrow. This contact sets the anchor in motion in the direction indicated by the yellow arrow, bringing the two teeth to slide between them.
The third step shows the disengagement point between the anchor tooth and that of the escapement wheel. From here on the two faces of the teeth can slide between them allowing the exchange of energy through a pulse from the escapement to the anchor.
The left image shows how the sliding between the faces of the teeth produces a normal force to them and consequently a moment around the axis of the anchor, highlighted in yellow. At this moment the energy flows from the escapement to the anchor. The image on the right, on the other hand, shows how the transfer of energy brings the anchor in contact with the pin of the balance, but this time the contact is the anchor, in this way the balance starts to receive energy.
At step 5, as shown in the figure on the left, the transfer of energy from the escapement to the anchor stops, which abuts against the pin which prevents further rotation. The contact between the anchor and the balance wheel is lost and, while the pin continues its stroke again reloaded by the received impulse, the new escapement tooth rests against the shoulder of the anchor and ends the rotation of the escapement. Subsequently, the balance wheel will invert the direction of rotation and the steps previously seen will be repeated in the same way, but in the opposite direction.
This video shows the sequence just described:
The geometry of the parts is essential for the correct functioning of the mechanism. In the present discussion it was decided not to treat the processes for obtaining the geometry as it is preferred to emphasize the dynamic aspects that regulate the operation. The following figure shows the main dimensions, according to . This is a pocket movement where the masses and forces involved are reduced as much as possible for two reasons: allowing portability and minimizing losses in order to keep the power reserve longer. The outer diameter of the balance wheel is 6 mm, while the primary diameter of the escapement wheel is 7.5 mm and is equipped with 15 teeth. The thickness is equal to 0.2 mm for escapement and anchor, while for the balance wheel it is 0.4 mm in order to considerably increase the inertia.
The choice of the design parameters was performed taking into consideration the state of the art of watchmaking. Typically, brass wheels are used since it offers good machinability and above all low friction coefficients with respect to steel or other commonly used alloys . The brass density is equal to 8545 kg / m^3, from which the value of the moment of inertia of the balance wheel is obtained with respect to the axis of rotations equal to 1.6289E-10 kg m ^ 2. From this information it is possible to derive the spring stiffness necessary to guarantee the desired oscillation period. Typically, wristwatches work at 28,800 alternations / hour, that is to say 8 alternations per second, according to WatchTime Magazine . So we choose to impose this frequency on the bar. From the differential equation of the second homogeneous order which describes the motion of the system:
It is possible to obtain the natural pulsation:
Where K is the stiffness of the spring and I the moment of inertia of the balance wheel. The pulsation can be expressed as:
With f frequency in Hz, T period in seconds. Indicating with n the number of shots that we want to be executed in a second, it is possible to derive the period T as:
From which, by inverting the natural pulsation formula, we obtain the value of the stiffness K as a function of the moment of inertia of the balance wheel and of the number of alternations per second:
Introducing the value of I obtained from ADAMS and n equal to 8 gives a stiffness equal to:
In reality the spring will be subject to a certain damping and this must be kept in mind in the modeling. The value of the critical damping is then obtained from the formula:
Considering a damping ratio equal to 10%, the damping coefficient value is:
It is therefore desired to simulate the system when the values of stiffness and damping are those assigned and verify that there is regular running and at most quantify the actual period of oscillation which will obviously be different from that predicted due to contacts between the parts.
The system consists of four parts that have been drawn to the CAD and then saved as parasol and imported into ADAMS. The part that acts as a frame is constituted by the pins in which the balance wheel, the anchor and the escape wheel will be lodged, plus two stop pins to stop the anchor in the positions of maximum rotation. This part was then fixed to the ground by a fixed joint type constraint. Subsequently the escapement wheel was inserted which was linked to the chassis by means of a revolute joint. Mass properties have been given to the wheel, choosing brass as the material. The anchor has been imported and connected to a loom by a revolute joint and has been assigned brass as a material. The last element inserted is the balance wheel. This too was created in brass and connected to the frame by a revolute joint, but taking care that the pin was at 90 ° from the center line in order to have enough energy for the first shot. The resulting model is as follows:
A design variable was created with regard to stiffness and a rotational spring-damper was then created to which the stiffness and value of the damping coefficient was assigned. A constant torque equal to -5.46e-6 Nm was applied to provide energy to the escapement. In reality, this moment is transmitted by the spring of the barrel which generally has a 40-hour power reserve and a Maltese Cross mechanism which allows the spring not to discharge completely and then come to a stretch where the answer is not linear. Since the simulation time does not exceed 20 seconds, the transmitted force can be considered constant and the value has been chosen taking into account the lever effect generated by the anchor and then refined iteratively so that the pulse provided does not allow to the balance pin to complete a full turn and bump against the outer surface of the fork. All contacts that occur during operation have been defined. As described in the declaration of the steps, the contacts are as follows:
– Balance-anchor contact via the pin;
– Contact between the anchor and the limit switch pins belonging to the frame;
– Contact between anchor and exhaust.
All contacts have been defined as contacts between parasol, which is the most accurate method and provides the best solution.
Simulations and discussion
The first simulation performed was aimed at checking the periodicity of the motion defined as the spring stiffness and the rotor inertia. The balance wheel has been left to rotate freely, without any contact or damping. The result is visible in the figure:
It is noted that 4 complete oscillations were obtained in one second, ie 8 shots per second as obtained analytically. Once the goodness of the imposed stiffness has been ascertained, the damping of 10% of the critical stiffness has been added. The result is the following:
A complete oscillation is still performed every quarter of a second, but after 6 oscillations we can consider the balance practically stationary. By introducing the contact between the balance wheel and the anchor, a loss of energy is expected of the balance wheel which is used to transfer momentum to the anchor. It is possible to quantify this loss by calculating the residual potential energy of the balance wheel after a complete oscillation performed first without contact and then with contact with the anchor.
The time course of the angular position of the balance wheel with the presence of the contact with the anchor is:
The absolute value of the peaks is slightly lower than in the previous case, while it is noted that the oscillations are completed over a period of more than a quarter of a second. This inconvenience could generate irregularities in travel. The energy stored by the spring is calculated with the following formula:
Where = 0 when the balance pin is in the center line. The loss of energy after complete oscillation is calculated with:
Only the first three oscillations are considered since in the subsequent contacts the contact between the pin and the anchor is continuous and therefore they are not significant. The initial energy, ie with = 90 °, is equal to 1.27e-7 J. At the end of the first oscillation there is an energy loss equal to:
- 2.78e-8 J considering the isolated balance;
- 3.13e-8 J considering the contact between the balance wheel and the anchor.
The energy lost in the event that there is contact between the barbell and the anchor is more than 12% compared to the case of an isolated balance wheel. This means that on average, since there are two contacts at each oscillation, there is a loss of energy 6% higher than in the case of an isolated balance wheel.
The simulation of the complete system of all the contacts is then carried out, that is, balance-anchor and anchor-escapement, for 2.5 s. The angular position of the bar is:
Looking at the chart you can get two important information:
- The system is stable, the temporal evolution of the angular position of the balance always follows the same maximums and lows;
- The period of oscillation is different from the expected one of 0.25 s.
By sampling the last 8 periods (the first 2 are transitory) we obtain an average period equal to 0.2519 s which corresponds to a systematic error of about -12 min / day, evidently too high for a clock for daily use. In order to correct this error, it is possible to intervene on the stiffness of the spring K, increasing its value to decrease the period of oscillation.
A first test was performed by increasing the stiffness of 1%, obtaining an average period of 0.2506 s, corresponding to an error of about -4 min / day.
A further increase in stiffness equal to 1.5% of the initial rigidity completely reduces the error and the angular position of the balance wheel results:
It is noted that exactly 4 oscillations are necessary, ie 8 alternations, to scan the second.
Previous simulations have shown that it is necessary to increase the K-stiffness by 1.5% compared to the theoretical one to compensate for the loss of energy due to the collisions between the parts.
Finally, an analysis was carried out considering the increase in the moment of inertia equal to 0.1% as a consequence of a variation in temperature. The result of the analysis is an error of about 1 min / day. This data highlights the influence of temperature on the characteristics of a mechanical movement and is the main reason for irregularity in running of modern mechanical clocks. Observing the trend of the speed of time it is possible to make important considerations on the functioning of the mechanism. This speed varies during an oscillation as shown in the figure:
From the instant t = 0.15 s the speed increases in form until the moment of contact between pin and anchor. At this time the speed tends to decrease very quickly and then increase in form in an equally rapid time following the sliding between the tooth of the anchor and the escapement that transfers energy to the balance wheel. From here on, the speed returns to decrease, but starting from a higher value.
The regularity of a mechanical clock depends on a large number of parameters which can not be easily calculated in an analytical form. The analysis with the software ADAMS has allowed to quantify the loss of energy associated with mechanical shocks that occur periodically between the parts of the movement.
Once the balance wheel has been dimensioned with the values of the moment of inertia I and of the torsional stiffness K of the spring in order to obtain a certain number of alternations per second, a coefficient of damping of the same spring has been considered equal to 10% of the critical one . The loss of energy associated with the impact between the balance and the anchor was verified, the only passive element beyond the frame. The result was a loss of more than 6% with each alternation compared to the free balance case. Because of this, the oscillation period is greater than the design time and it is therefore necessary to intervene by increasing the rigidity of the torsion spring. Considering the complete system under the hypothesis of spring damping and collisions between the parts, the simulation has shown how the loss of energy translates into a clock error of the order of 12 minutes / day. By imposing a value of the spring stiffness 1.5% higher than the initial one, in the hypothesis of constancy of the moment applied to the exhaust, the running error has been completely avoided.
The influence of temperature on running regularity was then verified by assuming an increase in the moment of inertia of the balance equal to 0.1% and an increase in the period of oscillation was found to entail an error equal to about 1 min / day.
 “AN ANALYSIS OF A LEVER ESCAPEMENT” by H.R. Playtner
 “OROLOGIAIO RIPARATORE” by D. de Carle
 WatchTime Magazine: https://www.watchtime.com/reference-center/glossary/vibration-vph/