8. Conclusion

8.1 Significance of the obtained research results

Doctoral dissertation "Nonlinear dynamics of clock mechanisms" is dedicated to the analysis of all those phenomena that significantly disturb and disturb the functioning of clock mechanisms as linear dynamical systems. Namely, if these phenomena and influences were ignored or excluded from the analysis, the work of the clock mechanism would be mathematically described by the linear oscillation theory. Such a linear dynamics can not, either qualitatively or quantitatively, define the cause of any change in the equilibrium of the timing of the timer, and hence the method for correcting such a change. Accordingly, the significance of the achieved results in this dissertation is directly derived from the study and analysis of clock mechanisms as non-linear dynamic systems. In this doctoral dissertation the following results were achieved:

• Using the theory of perturbation by the perturbation method of the double time, as well as by the method of averaging by Krylov and Bogoliubov, the theory of errors of the average impulse mechanisms has been carried out. General analytical formulas have been performed in an integral form that quantitatively determine the change in the angular oscillation frequency oscillator frequency under the influence of the forced moment of force by which the impulse mechanism acts on the oscillator. These formulas can be applied to all types of impulse mechanisms and get an operational relation to the errors of each individual break.
• The error theory of the impulse mechanisms has shown that the change in the angular oscillation oscillator oscillator frequency is due to the phase difference between the forced force of the momentum, which the impulse mechanism acts on the oscillator and the angular oscillator oscillator speed. Depending on the sign of the phase difference, the classification of all the impulse mechanisms on the tachyron and bradichrons was performed.
• Apart from the influence of the phase difference between the forced force of the force by which the impulse mechanism acts on the oscillator and the angular oscillator oscillator speed, it has been shown that the frequency of the oscillator depends on the amplitude of the oscillations, the coefficient of damping and the intensity of the forced moment of force. These facts are typically the consequence of the nonlinear dynamics of the clock mechanisms because they do not arise from the linear oscillation theory, but from the perturbation account used in this dissertation.
• Operating error formulas for two types of pulse-deflection mechanisms in the oscillator oscillator oscillator oscillator oscillator: reversible twist and quiet lumbar interference. By applying these formulas, numerical values ​​can be obtained by changing the angular oscillation oscillator oscillator frequency under the influence of the force of force by which the impulse mechanism acts on the oscillator.
• Derived operational formulas for faults of pulse-pulse mechanisms are checked by computer simulations and oscillation oscillation oscillator analyzes (spiral spring balance). Simulations were performed using specially created models of breakers and oscillators, using the "Event Based Motion Study" method. The results of these simulations showed a high degree of agreement with the results obtained from the theoretical operational formulas.
• With the application of the theory of precoureration and this method of double the time, a formula for the circular error of the physical pendulum was made, that is, a formula that quantitatively describes the influence of the amplitude on the oscillation period of the physical pendulum.
• The external influences that affect the clock oscillator are analyzed and their frequency is changed, respectively, the oscillation period: thermal expansion, aerostatic thrust, resistance and density of air. For the external effects shown, appropriate mathematical expressions are given so that they can see to what extent they affect the timer's travel.
• The theoretical analysis was carried out and the calculations of the compensation of the thermal dilatations of the pendulum were carried out. This calculation, which involves changing the moment of the inertia of the physical pendulum with a change in temperature, allows almost perfectly neutralizing the effect of thermal dilatations on the oscillation period of the pendulum.
• An iterative process algorithm for compensating for thermal expansion of the physical pendulum using the SolidWorks application is also listed. In the above application, using the 3D model of the physical pendant, it is possible to compensate for thermal dilatations with the desired desired accuracy.
• A suggested method for compensating the effect of air density change on the change in the oscillation period of the pendulum.
• The dissertation explains the work of the pragmatic impulses and emphasizes the constructively-geometric origin of their influence on the oscillator oscillation period. A physical interpretation of the phenomenon of change in the angular frequency of the clock oscillator is explained by the physical force induced by the forced moment of the force of the average impulse mechanism.
• The classification of a mean-impulse mechanism is outlined in which the characteristics of the most significant impulse mechanisms are highlighted.
• Using the Solidworks software program, a fully functional 3D model of the watch mechanism was created. This 3D model provides a clear insight into all subassemblies of the clock mechanism and provides the possibility of simulating and analyzing timer operation with real parameters.

The scientific contributions of this doctoral dissertation can be seen through:

• The dissertation contributes to a better understanding of the theory of perturbation through its application on the practical example of clock mechanisms
  
• The use of perturbation methods is a contribution to theoretical mechanics and theory of mechanisms
  
• By creating a functional 3D model where it is possible to simulate the operation of the mechanism in sufficient realistic conditions to make a clear comparison with the results obtained by analytical formulas
  
• Using advanced 3D modeling methods and inventive use of specially defined iterative procedures for computer modeling of the oscillator with the aim of compensating for its thermal dilatations
  
• Using special methods of computer simulation that are not based on time as an independent variable, but on predefined events, that is, the event based motion study.
  
• The results of scientific research in this doctoral dissertation are important for the qualitative and quantitative understanding of the nonlinear dynamics of the clock mechanisms, so they can be used as practical recommendations for more efficient design of mechanical timers, which leads to a more precise and accurate measurement of the flow of time.
  
• The results of this dissertation can be used to design sub-assemblies of impulse mechanisms and oscillators, as the most important parts of watch mechanisms.
  

8.2 A review of the starting hypotheses

According to all of the previous chapters of this dissertation, the studies carried out showed the utmost justification of the initial hypotheses given in the introductory chapter.

The first hypothesis: using the perturbation method, it is possible to derive analytical expressions for changing the frequency of forced oscillations of the clock oscillator due to its interaction with the impulse mechanism.

Chapter 3 gives theoretical bases for the use of the perturbation account method - the method of averaging by the Krylov method and Bogoliubov and by the method of multiple scales or the time scale. By these methods, analytical expressions were obtained for determining the frequency change of forced silenced oscillations of the clock mechanism during interaction with the impulse mechanism. The results of this analysis are universally applicable to all other types of impulse mechanisms, regardless of whether they have a balancing wheel or a pendulum as an oscillator.

Another hypothesis: Analytical expressions for the change of the oscillator's own frequency due to its interaction with the impulse mechanism can be verified by computer simulation of compulsory muffled oscillations. It is also assumed that this verification will confirm the accuracy of the expressions obtained by the theory of the perturbation account.

Creating a 3D model in the Solidworks software package enabled These obtained analytical expressions were checked in chapters 5.3 and 7.2 by simulating the clock mechanism and the results of these simulations are tabulated.

Third hypothesis: It is possible to carry out analytical procedures for compensating for the thermal dilation of the clock oscillator. Thermal expansion of the clock oscillator is one of the most influential clockwork disorders. Section 4.3 gives a description of the effect of temperature on the physical pendulum and a numerical example of an uncompensated pendulum presented. Apart from the influence of the temperature on the oscillator, they are affected by the aerostatic thrust, resistance and density of the air so that their influence is described. The order of the magnitude of their disorder is given in relation to the temperature, in that way the temperature dilation is shown to be a more serious disorder than the other influences.

Fourth hypothesis: Iterative procedure for approximate compensation of heat dilatations of the clock oscillator (pendulum) can be formulated and successfully applied using the appropriate computer application for 3D modeling.

In section 4.4, a pendulum compensation calculation based on temperature changes that affect the equilibrium of the clock travel is performed. These compensations are based on the choice of the material having the smallest linear coefficient of temperature dilation and their interconnection, so that when the temperature changes, the effects of the dilation are interrupted by each other. In addition to temperature, both the neutralization and the pendulum compensation are shown in the change in the pendulum density.

Fifth hypothesis: it is possible to synthesize and generate a fully functional computer model of the clock mechanism and perform a successful computer simulation of its work.

Simulated work of the computer 3D model of clock mechanism was made, which fully satisfied the geometrical and functional criteria of each part of this circuit. The settings within the simulation themselves proved to be realistic enough that these results can be taken for comparison with the analytical results obtained, as confirmed in Chapter 7.

8.3 Future research

The dissertation can serve as a basis for further scientific research and development of new methods of scientific research. Research in the field of nonlinear dynamics as well as in the field of study of clock mechanisms are distinguished in several directions, and some of them are:

• Creation of functional 3D models of all the most important types of impulse mechanisms (angle reversal, Harrison's grasshopper, Graham's peace stop, Denison gravitational, free lumbar spacer and anglick's and chronometric impulse mechanism)
  
• Simulation of the work and analysis of the movement of all the above types of impulse mechanisms with the aim of discovering and defining their characteristic impulse functions. Defining the functional dependence of the moment of force by which the impulse mechanism acts on the oscillator from the angular oscillator and oscillation phase.
  
• Experimental checks of the possibility of barometric compensation or compensation of the influence of the change in the density of the air on the oscillator oscillation period. Detecting and defining the influence of viscous and dynamic resistance (pressure resistance) on the energy losses of the oscillator
  
• Computer simulation of the operation of the epicycle winding mechanism
  
• Experimental checks and tests by computer simulations of the influence of non-linear spiral spring characteristics on the period of oscillation of the balance point
  
• Theoretical analysis, experimental checks and checks by computer simulations of faults of the average impulse mechanisms in non-stationary oscillator oscillation regimes.