II. MARCO TEÓRICO
2.2. Bases teóricas
2.2.1. Estabilización de suelos
Simulations are carried out based on the model presented in the previous section. The parameters of the escapement are summarized in Table 3.1.
Figures 3.5–3.7 show the time history of the angular displacement, velocity, and acceleration of the balance wheel, pallet fork, and the escape wheel. From the figures it is seen that the motion of the balance wheel is mainly determined by the supplementary arc. However, the push steps change the magnitude
TABLE 3.1 The Simulation Parameters
r1 1.381 mm r9 1.84 mm Je 1 × 10−8 kg.m2
Balance Wheel Displacement versus t
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Balance Wheel Velocity versus t
(a) (b)
Balance Wheel Acceleration versus t
(c)
&
θb θ&&b
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3-14 Mechatronic Systems: Devices, Design, Control, Operation and Monitoring
of oscillation until it is stabilized. There are three impulses that occur within a half period. For the pallet fork, the motion follows a step function. The angle of displacement changes between two extremes as determined by the banking pins. When the displacement reaches the extremes, it remains there for a period of time. From Figure 3.6(c) it is clear that four impulses incur in a half-period. The last one takes place when the pallet fork impacts the guard pin. The angle of the escape wheel changes step by step.
The width of the steps is determined by the dynamic system and hence changes in a small neighborhood of a fixed value. The small variation, however, has a significant effect as seen in Figure 3.7(c).
Figure 3.8 shows the velocity of the pallet fork and the escape wheel in a half-period. From the figure it is seen that the first shock and the fifth shock are critical for the pallet fork, whereas the second and the fourth shocks are critical for the escape wheel.
Figure 3.9 shows the relationship between the displacement and velocity. It is seen that the motion is almost periodic. It should be pointed out that, if the dynamics are not considered, the motion will be exactly periodic. However, owing to the dynamics, the initial condition is different each time. It is the escapement mechanism that constrains the dynamic effect, thereby making the escapement accurate in timekeeping. This is shown in Figure 3.9(b). It is seen that regardless of the variation, the pallet fork always comes to the exact position with exact velocity. This is very important in a watch mechanism.
Based on the simulation of 10 periods, the standard deviation of the half-period is found to be 0.0002781 s. The standard deviation of the amplitude is 1.3685 degrees.
FIGURE 3.6 Motion of pallet fork: (a) θp versus t, (b) versus t, (c) versus t.
Pallet Fork Displacement versus t
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Pallet Fork Velocity versus t
(a) qp versus t (b) q.p versus t
Pallet Fork Acceleration versus t
&
θp θ&&p
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Swiss Lever Escapement Mechanism 3-15
It should be emphasized here that the use of the impulsive differential equation is necessary. The use of ordinary differential equations has led to mathematical and numerical difficulty. Therefore, it is suggested that any mechanisms involving impacts or abrupt changes should be modeled using impulsive differential equations.
3.5 Conclusion
From the study presented in the chapter, the following conclusions can be made:
1. The Swiss level escapement is a complex mechanical system that can be treated as a mechatronic device. It is necessary to use impulsive differential equations to model this system.
2. The Swiss level escapement mechanism has a complex pattern of motion. It can be regarded as a type of mechanical switch. With an applied torque, it can self-start and run almost periodically.
Within a half period, the balance wheel experiences three shocks, whereas the pallet fork and the escape wheel experience four shocks.
3. Owing to the dynamics of the system, the motion of the balance wheel is near periodic, and the variation is noticeable. It is the pallet fork that constrains the effect of the dynamics, making the escapement accurate for the purpose of timekeeping.
FIGURE 3.7 Motion of escape wheel: (a) θe versus t, (b) versus t, (c) versus t.
Escape Wheel Acceleration versus t
(c)
Escape Wheel Displacement versus t
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Escape Wheel Velocity versus t
(a) (b)
&
θe θ&&e
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3-16 Mechatronic Systems: Devices, Design, Control, Operation and Monitoring
FIGURE 3.8 Critical shocks: (a) shocks on pallet fork, (b) shocks on escape wheel.
FIGURE 3.9 Phase diagram: (a) for balance wheel, (b) for pallet fork.
1.15 1.2 1.25
Escape Wheel Velocity versus t
5th shock
Pallet Fork Velocity versus t
1st shock 5th shock
Balance Wheel Displacement versus Velocity
–15 –10 –5 0 5 10 15
Escape Wheel Displacement versus Velocity
(a) (b)
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Swiss Lever Escapement Mechanism 3-17
The impulsive differential equation is a powerful tool. It can be applied to model other escapements, mechanisms, or mechatronic devices that involve shocks.
Acknowledgment
This project is supported by the Hong Kong Watch Manufacturers Association (HKWMA), Federation of Hong Kong Watch Industry and Trade (FHKWIT), and Hong Kong Innovation and Technology Commission (ITC).
References
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