1. Introduction

1.1 The subject of research

Mechanisms of timers represent a real treasury of both theoretical and practical knowledge and skills from almost all fields of mechanical engineering, in particular theoretical mechanics and theories of mechanisms. Despite the fact that, in spite of this, the principles of their work, their constructive and functional characteristics, geometry, kinematics, analysis and synthesis are little studied in these disciplines, they can be the subject of numerous interesting and fruitful research. The results of such research can be significant, not only as recommendations for the correct synthesis of clock mechanisms, but also for theoretical disciplines such as kinematics, nonlinear dynamics, in particular the theory of nonlinear oscillations, mechanism theory, the science of machine elements and machine materials, etc. What's more, the results of these research can be extended and generalized to other fields of science, such as the theory of automatic control, the theory of deterministic chaos, celestial mechanics, the mathematical basics of the theory of perturbation account, and so on.

The fact is that clock mechanisms belong to areas of precision mechanics, which with a high degree of uniformity of their movement measures the flow of time [1], [2], and [3]. The slightest disturbances in the uniformity of their movement are the source of unacceptable inaccuracy of these mechanical instruments. For this reason, the correct mathematical and physical description of the functioning of clock mechanisms can not be achieved correctly by neglecting small disturbances by higher order linearizations of differential equations and approximations common to the classical approach in the study of mechanisms [4]. Such approximations would eliminate the essence of the dynamics of their work. In accordance with the prominent facts [5] and [6], the subject of the study of this work are precisely small disorders of higher order that impair the uniformity of the clock mechanism and thus generate errors in the measurement of the flow of time. In order to understand the effects of these small clock disturbances, it is necessary to carry out an analysis of its mechanism, which includes the classification of sub-assemblies and the functional analysis of each sub-assembly of the clock mechanism.

Essentially each clock mechanism consists of five sub-assemblies: a drive mechanism, a transmission mechanism, an oscillator, a movement regulator or a pulse mechanism and a cursor.

The focus of research on this dissertation will be the non-linear dynamic properties of the oscillators [7] and [8] and the impulse mechanisms [1], [9] and [10] as functionally the most important sub-assemblies of the clock mechanism. The characteristics of these sub-assemblies have a decisive influence on the behavior of the clock mechanism as a whole on the process of measuring the flow of time. A hypothetical view is that forced silenced oscillations of a clock mechanism such as a pendulum [11] or a spiral spring balancer [12] and [13] do not have a constant frequency, but that it suffers small disturbances [5] due to interaction with a mean-impulse mechanism. It is also assumed that the key factor of this disorder is the phase difference between the angular oscillation speed and the force of force that periodically affects the oscillator. A hypothetical view is that it is possible to carry out an analytical-algebraic expression for changing the frequency of forced oscillations of the clock oscillator using a perturbation account [14] and [2]. It is known that the piano watch is a physical pendulum whose oscillation period is not constant, but depends on the amplitude [8] and [15]. In addition, neither the spiral spring at the balance point of the timer has a constant stiffness coefficient [12], but it is a function of the angle of rotation of the oscillator. In accordance with the above, it is hypothesized that the intensity of the forced moment, the attenuation factor, that is, the quality factor of the oscillator and the oscillation amplitudes, have a significant effect on the uniformity of the clock travel. As the own frequency of oscillation of the clock oscillator (pendulum [7] or balancing point) depends on the geometric characteristics, the distribution of its masses and the moment of the inertia of the oscillator, it is reasonable to assume that the thermal dilatations [16] have a significant influence on the process of time measurement itself. It is also assumed that these thermal dilatations can be compensated by reducing their harmful effect or possibly eliminating them altogether [17].

One of the research goals in this doctoral dissertation is, first of all, to determine one or more of the suitable methods of perturbation [14] and [18] which would lead to general analytical formulas for changing the frequency of the oscillator due to its interaction with the impulse mechanism. The assumption is that these analytic expressions will be able to derive the use of commutative meanings by Krylov and Bogoliubov [14] and [18], as well as the method of multiple scales and the time scale [18].

Among the more important aims of this study is that the obtained general-formula formulates to a few typical types of impulse-based mechanisms and, if necessary, to classify them, depending on whether they increase or decrease their own frequency of the clock oscillator. Furthermore, it is important to verify the validity of these formulas by computer simulation and motion analysis, and thus verify not only the obtained analytical expressions, but also the methods of the perturbation account [14] and [18] used. In addition to the formal-mathematical and quantitative description, one of the goals of this paper is the physical and qualitative clarification of the phenomenon described above. One of the major research goals in this dissertation is the study of the effect of thermal dilatations on the frequency stability of the clock oscillator oscillations [16] and [17]. Accordingly, it is necessary to find suitable analytical and numerical methods, as well as computer modeling and simulation procedures to compensate for these harmful effects. One of the final goals of this doctoral dissertation is the synthesis and construction of a fully functional 3D computer clock model, including computer simulation and analysis of its work [19] and [20].

1.2 Scientific methods of research and expected results

In the process of realization of scientific results, in this doctoral dissertation the following general scientific methods were applied:

• methods of differential and integral calculus,
  
• numerical methods,
  
• methods of computer modeling of forms,
  
• methods of computer simulation and motion analysis.
  

In the process of realization of scientific results, in this doctoral dissertation the following special scientific methods were applied:

• the method of perturbation account
  
• the method of averaging according to the metric of the perturbation account
  
• the method of multiple scales or the time scale,
  
• Computer simulation method "Event based motion study" in the SolidWorks application.
  

1.3 Hypotheses

The linear oscillation theory shows that the frequency of compulsory damped oscillations is constant if the attenuation coefficient is constant. Empirical facts oppose the results of the linear theory of oscillations and find that the frequency of forced oscillations can be changed due to the interaction of the oscillator with the impulse mechanism. In this dissertation, it is assumed that the frequency of the clock oscillator (pendulum or balance point) depends on the phase difference between the angular oscillation speed and the forced force of the force that periodically affects the oscillator. Moreover, starting from the fact that the clock frequency oscillator's own frequency depends on its moment of inertia, it is assumed that the temperature dilatations have a significant effect on the said frequency. The assumption is that these influences can be reduced or completely eliminated by a suitable oscillator design. According to the prominent, the key starting hypotheses in this dissertation read as follows:

• Using the perturbation method it is possible to perform analytical expressions for changing the frequency of forced oscillations of the clock oscillator due to its interaction with the impulse mechanism.
  
• 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 forced silenced oscillations. It is also assumed that this verification will confirm the accuracy of the expressions obtained by the theory of the perturbation account.
  
• It is possible to perform analytical procedures for compensating for the thermal dilation of the clock oscillator.
  
• Iterative procedure for approximate compensation of the heat dilatations of the clock oscillator (pendulum) can be formulated and successfully using the appropriate computer application for 3D modeling. The methods of Krylov and Bogoliubov,
  
• It is possible to achieve synthesis and generate a fully functional computer model of the clock mechanism and perform a successful computer simulation of his work.
  

1.4 Displays the chapters

Doctoral dissertation "Non-linear dynamics of clock mechanisms" is divided into 9 parts: introduction, 6 chapters and conclusion.

Chapter 2 "Theory of Time Measurement" shows a brief history of time measurement, in line with the modern definitions of the time measurement unit. It is shown that man recognized the need to have information about time in prehistory, observing the cycles of the Sun and the Moon. Then the first civilizations of ancient Egypt and Mesopotamia developed calendars. These calendars were used to predict the date of seasonal astronomical events essential for successful organization of the state and society. These civilizations gave the first division to sixty parts that remained to this day (clock is 60 minutes, minutes 60 seconds). Later, the civilizations of ancient Greece and Rome, are improving methods and devices for measuring the flow of time, so that the first solar and water clocks appear. The Middle Ages records the emergence of the first primitive mechanical monitors made by monks, of which the first records of the description of mechanical watches come to us. In the XVII century, the development of precise measurement of the flow of time begins and the transformation era begins - from timers that were so large that they could only be placed in towers to mechanisms that reduce the clock so that it can be worn on the hand. At the end of the chapter we give an overview of the development of a unit of time - from the first civilizations (who took the day as a natural and basic unit) to the modern age - where we can define a second over the radiation period in the atom of a cesium.

Chapter 3, "Theoretical Basics of the Perturbation Account," depicts the purpose and meaning of the technique of double-time conditions, which become apparent when the crucial deficiency of the method of regular perturbations is discovered. Since the main topic of this dissertation is precisely the analysis of errors of the average impulse mechanisms, these disorders will be described later in the specific formulas for calculating the change of the clock travel. The intention is to carry out the analysis using the theory of perturbations, and this, inter alia, by the method of double time scale.

First, in this chapter, the essence of the method mentioned will be summarized. Therefore, we will demonstrate an attempt to approximately solve the linear differential equation of the second order with constant coefficients by the method of regular perturbations. After explaining the perturbation technique of double the time, the essence of the Krilov and Bogoljub methods is explained briefly. In this way, control can be made of the results obtained and possibly compare both methods and give a critical overview of the benefits and difficulties of their applications.

Chapter 4, "Oscillator", describes the first of the two most important satellites subassembly. The oscillator can be constructed as a physical pendulum or as a balancing wheel. The physical pendulum represents a body swinging around a point located outside the center of equilibrium, while the balancing wheel is a massive body, a circle around the fixed axis, which passes through its center of mass and performs oscillations under the action of the elastic restitution moment of force. In addition to describing the functionality of these two oscillator performances, the calculation of the circular error by the double-time method is given. A circular error is a change in the pendulum period caused by a change in the amplitude. This chapter also gives an overview of the oscillator's disturbances that occur as a result of external influences, such as temperature, aerostatic thrust, resistance and air density. The influence of temperature is considered to be the most significant, and it manifests itself through the propagation and collecting of the pendulum materials, and this affects the periodic oscillations. The aerostatic thrust, in accordance with the Archimedes Law, reduces the weight of the pendulum for the weight of the pendulous air. The physical pendulum of the timer, with the exception of the restitution gravitational force, acts also with the force of the air resistance, due to which the pendulum loses energy. The change in air density caused by a change in pressure, humidity, and / or a change in air temperature has a direct impact on the overall energy of the pendulum that oscillates, the other energy parameters, and the timing of the timer.It is, in essence, caused by a change in the dynamic resistance acting on the pendulum. These changes can be classified as the effects of the environment in which the pendulum moves and are often considered to be of lowerer influences than the temperature. This chapter presents their mathematical model of the impact of aerostatic thrust, as well as the mathematical model of resistance and density of air. With the development of mechanical timers, solutions for compensating oscillator errors were developed as well. Most attention is devoted to temperature compensation and given the calculation and example of the pendulum, which is temperature compensated. The impact of air (resistance, density and aerostatics) is negligible, although with precision clocks (astronomical clocks, chronometers, high-quality public and tower timers), this error is noticeable and can become impermissibly large,in the case of long periods of extremely high or low atmospheric pressure. Therefore, the basic budget of this error is given and its order of magnitude is shown.

Chapter 5, "Spit-impulse mechanisms", is a key sub-assembly of every mechanical clock, because it maintains and counts the oscillator oscillations and thus measures the flow of time. With its functions this mechanism introduces disruptions into the oscillatory process so oscillations are no longer their own, but are forced with a frequency that is susceptible to change. Therefore, the process of measuring time itself distorts the accuracy of this measurement. The occurrence of an average impulse mechanism changes the oscillation oscillator oscillation time, and thus the timing of the timer, is called a fault of the average impulse mechanism or a short error of the breaker. Within the classification, three types of a mean-impulse mechanism are shown: a boiler-backed, impulse, pulse mechanism, a calm, pulse-free mechanism and a free-flow-impulse mechanism. Using the appropriate momentum interaction diagrams,the constructive and dynamic characteristics of the obstacles are shown qualitatively, with particular reference to the errors of the oscillation period they generate. After classifying the impulse-based mechanisms, a general formula for the error of the average-impulse mechanism was derived using the double-time scale (scale) of the perturbation method. A general formula for the error of the average-impulse mechanism was also developed, using the technique of averaging by the Krylov and Bogoliubov methods. In both cases, only those clock mechanisms that incorporate spiral springs with the balance point as an oscillator were considered, but it should be emphasized that the results of these analyzes are universally applicable to all other types of impulse mechanisms, including those that are installed in stationary clamps with pendulum .It starts from the fact that the balancing wheel of the clock mechanisms performs compulsory muffled oscillations, and it is assumed that the attenuation is due to the viscous moment of force proportional to the angular oscillation velocity. The identity of all approximate solutions and full compliance with the estimation of the order of the size of the approximation error and the order of the size of the time interval are shown, on which these approximations apply to both methods.

Chapter 6, "Other parts of the clock", completes the whole story related to sub-clocks of the clock. The chapter provides descriptions of the three remaining parts of the watch: remontoire, transmission group and winding mechanism. Remontoire is the mechanism most commonly seen with towers in towers. The most important purpose of this mechanism is to provide a secondary, uniform and constant winding source for the pulse mechanism, thus ensuring the accuracy of the clock.The principle of operation of the remontoire as well as its basic parts is shown. The parameters of the remontoire are given, which will be used in Chapter 7, "Model of the whole hour". The transmission group represents a group of gears that are paired to transfer the drive in one direction, from this loop to the drill to the average impulse mechanism, and in the second direction the uniform intervals of the impulse mechanism are converted into seconds, minutes and hours. The winding mechanism is shown as a separate sub-assembly interlaced with a transmission group. By showing its components, as well as the principle of winding and maintaining the moment of the clock, a story about all sub-assemblies of the clock will be made, which will be made as a 3D model, and on which mathematical models are tested in the previous chapters.The parameters of the remontoire are given, which will be used in Chapter 7, "Model of the whole hour". The transmission group represents a group of gears that are paired to transfer the drive in one direction, from this loop to the drill to the average impulse mechanism, and in the second direction the uniform intervals of the impulse mechanism are converted into seconds, minutes and hours. The winding mechanism is shown as a separate sub-assembly interlaced with a transmission group. By showing its components, as well as the principle of winding and maintaining the moment of the clock, a story about all sub-assemblies of the clock will be made, which will be made as a 3D model, and on which mathematical models are tested in the previous chapters.The parameters of the remontoire are given, which will be used in Chapter 7, "Model of the whole hour". The transmission group represents a group of gears that are paired to transfer the drive in one direction, from this loop to the drill to the average impulse mechanism, and in the second direction the uniform intervals of the impulse mechanism are converted into seconds, minutes and hours. The winding mechanism is shown as a separate sub-assembly interlaced with a transmission group. By showing its components, as well as the principle of winding and maintaining the moment of the clock, a story about all sub-assemblies of the clock will be made, which will be made as a 3D model, and on which mathematical models are tested in the previous chapters.and in the second direction, the uniform intervals of the mean-impulse mechanism turn into seconds, minutes and hours. The winding mechanism is shown as a separate sub-assembly interlaced with a transmission group. By showing its components, as well as the principle of winding and maintaining the moment of the clock, a story about all sub-assemblies of the clock will be made, which will be made as a 3D model, and on which mathematical models are tested in the previous chapters.and in the second direction, the uniform intervals of the mean-impulse mechanism turn into seconds, minutes and hours. The winding mechanism is shown as a separate sub-assembly interlaced with a transmission group. By showing its components, as well as the principle of winding and maintaining the moment of the clock, a story about all sub-assemblies of the clock will be made, which will be made as a 3D model, and on which mathematical models are tested in the previous chapters.as well as the principle of winding and maintaining the moment of the clock, the story about all sub-assemblies of the clock that will be made as a 3D model and on which the mathematical models are tested in the previous chapters are rounded up.as well as the principle of winding and maintaining the moment of the clock, the story about all sub-assemblies of the clock that will be made as a 3D model and on which the mathematical models are tested in the previous chapters are rounded up.

Chapter 7, "3D model of the clock mechanism and simulation of work", gives an overview of the results that are checking the accuracy of the formula for faults of the impulse mechanisms in the quasi-oscillation mode of oscillation of the balance point, carried out using the theory of the perturbation account. At the beginning, a 3D model of the whole clock was created in the SolidWorks application and the execution process was simulated. The watchmaker clock was made as a watch with a pendulum. In addition to the simulation of his work, a simulation was performed with a watchmaker, which uses a balanced spring instead of the pendulum. On this type of timer, at the very beginning of the simulation process, the oscillation period, or the frequency of the free-damped oscillations of the balance point, which is built into both oscillator assemblies and wires (balancing point and free overvoltage circuit and balancing point and reverse rotation), were measured.All results are tabulated. If only the dynamic characteristics are taken into account, all the impulse mechanisms can be divided into three large groups: bradycrons, tachycrons and isohrons. A detailed analysis of the oscillator oscillation oscillator oscillator also shows the physical cause - the errors of the mean-impulse mechanisms are not the result of a linear theory of oscillation, but precisely the nonlinear dynamics of the clock mechanism, which in this dissertation was mathematically treated by the methods of the perturbation account.A detailed analysis of the oscillator oscillation oscillator oscillator oscillator also shows the physical cause - the errors of the mean-impulse mechanisms are not the result of the linear oscillation theory, but precisely the nonlinear dynamics of the clock mechanisms, which is mathematically processed by the methods of the perturbation account in this dissertation.A detailed analysis of the oscillator oscillation oscillator oscillator oscillator also shows the physical cause - the errors of the mean-impulse mechanisms are not the result of the linear oscillation theory, but precisely the nonlinear dynamics of the clock mechanisms, which is mathematically processed by the methods of the perturbation account in this dissertation.