5. Forced-impulse mechanisms

5.1 General principle

The impulse-impulse mechanism is a key part of every mechanical clock, because it maintains and counts oscillations of the oscillator and thus measures the flow of time. But at the same time, with its functions this mechanism introduces disruptions into the oscillatory process so that oscillations are no longer their own, but forced ones, 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 impulse mechanism or a short-fault error. Qualitative and quantitative analysis of this phenomenon is the topic of this chapter of the doctoral dissertation.

The errors of the average-impulse mechanisms were first noticed empirically, by watchmakers and clockwork constructors. Namely, by the late seventeenth and early eighteenth centuries, it was noticed that only some types of opponents had the ability to compensate for the piercing circular error. It has also been observed that, during the fall of the torque of the timer, some interruptions cause prolongation, and the other shortening of the oscillation period of the pendulum or the balance point. In other words, it was noticed that with the decrease in the drive torque some obstacles tend to slow down, and others, paradoxically, accelerate the timer's travel. The first mathematical analysis of the errors of the mean-impulse mechanisms was accomplished by the Royal astronomer George Biddell Airy (1801-1892) in 1826 [66]. Airy carried out the equations of the change in the period and the oscillation amplitude of the pendulum, solving the differential equations of the motion of its gravity under the influence of forces that carry out small disturbances of the gravitational restitution force. At Harrison's seminar of the British Horror Institute held in 1988, Airy's equations were rated perhaps excessively strictly as "simply wrong". Although they have certain uncertainties and inconsistencies (for example, they neglect energy dissipation and do not apply to the stationary state of oscillation), they at least qualitatively correctly describe the errors of some types of impulse mechanisms. During the nineteenth century these problems were mostly dealt with by British mathematicians, astronomers and watchmakers. The most famous among them is certainly Edmund Beckett Denison, who published in several editions of a comprehensive study on watch mechanisms, in which, among other things, he explains in detail the constructive and dynamic properties of the impulse mechanisms. He uses the Airy equations, elaborates, complements, interprets and, through concrete numerical examples, demonstrates changes in the oscillation period of the pendulum under the influence of the impulses of some of the most well-known types of average mechanisms. Particularly interesting is his minuscule analysis of the influence of the geometrical characteristics of the obstacles on the disturbance of the course of the clock, as well as the possibility of compensating this disturbance by the effects of other sub-assemblies of the timers. In addition to theoretical, Denison's study of breaks also has practical significance, since its conclusions, warnings and recommendations can be of great use to watchmakers and clock designers. Philip Woodward (Philip Woodward, 1919-) is of particular importance from contemporary horologists and watchmakers who wrote about the properties of the pragmatic impulses. In one collection [67] of his scientific papers in the field of horology, he publishes five scientific discussions on the breaks, in which, with quite a few mathematics, he gives a new and original view of the theory of the average impulse mechanisms. In the case of a damped oscillation of a pendulum with harmonic coercion, and in a stationary oscillation mode, Woodward discovers the physical cause of the change in the oscillation period, or the error of the breakthrough. Pointing out that the precise phase difference between the force of the force of the overcurrent and the viscous resistance force is the cause of the errors of the average impulse mechanisms, introduces the concept of the phase center, defines the so-called. tangential rule, explains how a tangent rule can be derived from the aforementioned Airy's equations and demonstrates that the change in oscillation amplitude influences the size of the error of the average impulse mechanisms. At the end of his study on the hurdles, he presents the parameters, the course and the results of a series of computer simulations of the period disturbances and the amplitude of the damped and forced oscillations of the pendulum to which, besides the gravitational restitution force and the resistance forces proportional to the angular velocity of the pendulum, also act the short-term forces of the coarse impulse mechanism. Impulse impediments are so decentered to operate in a specific angular segment after passing the pendulum through the equilibrium position. The results of these simulations generally agree well with theoretical calculations and empirically reveal the significance of the impact of an error on the accuracy of the timer. The idea that Woodward has fully succeeded, without using a complex mathematical apparatus, to explain himself the essence of this complex phenomenon is quite justified. Peter Hoying's work [68] analyzes the dynamics of forced and subdued oscillations of the pendulum and is particularly interesting because it uses a perturbation account in solving the differential equation of motion. He performs the equations for the amplitude and oscillation phase, gives the term for the piercing circular error, as well as the equation of error of the mean-impulse mechanism in the integral form. The author solves this integral equation in the case of an idealized model of an average impulse mechanism and, at the end of the paper, communicates the results of the computer simulation of the influence of the forced moment of the force of the adopted model of the breakthrough on the change of the pendulum oscillation period. This work is significant and interesting primarily because it uses the technique of the perturbation account by the Krylov and Bogoliubov method in the approximative analysis of the nonlinear oscillations of the pendulum. However, certain omissions are quite obvious in the content of this paper. First of all, members in most of the derived equations are dimensionally incoherent, or disagreeable in units of measure. Why did this author corrupt this error and why he did not correct it, this will not be discussed in this place. Second, Hoying analyzes examples of two abstract obstructions, but does not establish a connection with specific types of these mechanisms, and hence the work has more theoretical and less practical significance. In other words, from the presented mathematical models, it is not possible to find out what the nature of the error is generated by some typical impulse mechanisms, such as: the anchor of the waves with and without backward movement, the chronometric impediment or the free breakthroughs, such as, for example, the Swiss lantern. Thirdly, the author performs a computer simulation of the transition of the oscillator (pendulum) from a non-stationary to a new quasi-stationary state, describes the change of the error of the breaks numerically and graphs, but neither does it, nor solves the differential equation of that transition. The notion that Hoying was supposed to solve this differential equation is quite justified, at least for the cases of abstract protections adopted, not only because there are exact solutions for them, but also because the main characteristic of his work is the representation of the fundamental mathematics behind the above mentioned problem . This would make his analysis more consistent and more complete. The paper [46] is devoted to the evolution of the impulse mechanisms through history, explains the gradual improvement of the constructive and dynamic characteristics of the most common types of breaks. It is precisely the authors who take the wrong impulse mechanism to be the key criterion for assessing the quality of their properties. Using the appropriate diagram, work qualitatively describes the constructive and dynamic characteristics of the obstacles, with particular reference to the errors of the oscillation period they generate. What is the main characteristic of this paper, and this is the analysis of the physical essence of the error of the breakthrough, Airy and Hoying's analysis barely mention. What works of Airy and Hoying have an abundance, and that is mathematics, this work is just missing. The work is interesting because, by introducing the concepts of bradycronism, isohronism and tachyronism, it explains the nature and sign of the error for mostly all known types of impulse mechanisms. Of the papers dealing with the experimental determination of the error of the mean-impulse mechanism, it is necessary to mention and briefly discuss the work [69] of George Feinstein. The author presents the results of the experimental measurement of the error of the wrists and the piercing circular errors in order to determine the "ideal location" of the effect of the pulse effect on the pendulum by performing muffled and forced oscillations. The most important result of this work is the confirmation that the piercing circular error can be neutralized by adjusting the phase angle, direction and intensity of the impulse impedance. It is interesting to note that what Feinstein discovered by his modern experiments was known to the old watchmakers who, more than two hundred years ago, found that an ancestral reverse anchor can, in particular circumstances, annul the piercing circular error.

The aim is to analyze in this dissertation the period of oscillation of the clock oscillators caused by their impulse mechanisms, using the theory of perturbation, by the method of a double scale of time and the technique of averaging by the Krylov and Bogoliubov methods. Also, one of the goals of this doctoral dissertation is to deal with this problem, not only by using mathematical formalism, but also by describing and clarifying its physical essence. All theoretical considerations will be concretized through examples of some typical impulse mechanisms.

Unfortunately, in the literature in the Serbian language, the topic related to errors of the average impulse mechanisms is not sufficiently addressed. Therefore, many terms and expressions used in this dissertation do not have an appropriate expression in the Serbian language and therefore appear in their original form, or, as with the case of the impulse mechanisms, they form as a description of their functionalities.

5.2. Classification of the impulse mechanisms

5.2.1. Backward rotary regulators

The speed controller type "verge & foliot" (Figure 7), discussed in Chapter 2, is conditioned by the huge amplitude oscillations of the pendulum that go over ± 50°. These huge amplitudes significantly increase the impact of the circular error (chapter 4.2) on the total drop of isohronism. The need for a smaller pendulum vibration amplitude was the main reason for the introduction of a new type of regulator in the construction of the timer. These are backyard or anchor regulators, whose construction was explained by English watchmaker William Clement and British scientist Robert Hook in 1670. That same year, watchmaker Jozef Knib built the first watch with the "Clement-Huke" boiler in the tower hour of the "Vadham" college in Oxford. The basic geometry and function of the reciprocating wheel controller is shown in Figure 27, and in Figure 28 and corresponding diagrams of momentary interactions from which the views on the indicators of the evolution of its chloro characteristics are derived.

Fig. 27 An anchor bolt with reverse spin

Figure 28 Temporal interaction diagrams for the boiler-backed pulse mechanism

The diagram in Figure 28 has the following labels:

$\,\,G · l · \sin φ$ - shows the moment of restitution force of the pendulum gravity, where the $G$ - weight is pendulum,
$\,\,M$ - moment of the impulse impulse function of the travel regulator i
$\,\,Ω$ - is the force of resistance generated by the regulator and acts on the pendulum.

All three marks are given in relation to the point of suspension and depending on the angle of deflection of the pendulum $φ$. When moving the pendulum from the left amplitude position "1", through the equilibrium "0" to the "2" position, the moment $M$ moves in the direction of its rotation. At the point "2", the moment changes direction and packs the pendulum to the right amplitude position "3", and the center wheel falls backward. The cycle is now repeated, but from right to left, through points 3-0-4-5 = 1. The moment of the frictional force $Ω$ always has a direction opposite to the direction of rotation of the pendulum. In the phases 1-0-2 and 3-0-4, the regulator achieves a pulse, and in the phases 2-3 and 4-5 = 1, the average function with a back twist. From the above it can be concluded that there is no constructive separation of the impulse and the average function. Both are realized on the same impulse-average surfaces of the anchor palette, only at different stages of the oscillation period of the pendulum. The interaction between the travel controller and the oscillator is continuous. The pendulum does not vibrate freely at any time, but under the direct and constant influence of the momentum force. When moving the pendulum from left to right, the moments $M$ and $G · l · \sin φ$ are in the zones 1-0 and 2-3 are direct, which causes a tachyron effect, and mutually opposite in the phase 0-2, which is bradichrono. In the second semiperiod, when moving the pendulum from right to left, the moments $M$ and $G · l · \sin φ$ are in the zones 3-0 and 4-5 = 1 dichotomous, which is tachyronous, and mutually opposite in the phase 0-4, which is bradycrono . The momentum effect of the dry friction is integral isohron in each half-period. Overall, tachychronism prevails, which means that any increase in the torque leads to a reduction in the oscillation period of the pendulum and the acceleration of the travel of the timer. All of the above mentioned characteristics of the roller-back wheel regulator are horo logically bad, which is quite obvious by the fact that they were created at the very beginning of their long evolution. However, in relation to the "verge & foliot", these mechanisms marked the progress as they reduced the pendulum amplitude from ± 50° to ± (5° -10°), and in proportion to the reduction in amplitude and the impact of the circular error. It should also be noted that the pendulum coupled with this type of regulator is relatively constant and independent of the change in the drive torque. From this phenomenon, the hororicologists have long gone the wrong conclusion about a high degree of isohronism of the regulator with a return twitch. However, as one single moment of force performs a pulse and the average function, it can be shown that the amplitude is stable, but not the oscillation period constant. The constructive and dynamic separation of these two functions was the most important goal of further improvement of the corner regulators.

5.2.2. Quiet-pulse mechanism

The constructive solution for the separation of the impulse from the average function of the wheel regulators was carried out by Tomas Tompion according to the idea of mathematics and astronomer Richard Taunley in 1675. George Gray, Tompion's pupil and the once-great Grand Master of the Honored Guild of Watchmakers in London, perfected this invention in 1715 and enabled his massive application. It is about quiet beat-escapement (nem. Ruhereibende Hemmungen) whose shape and function is shown in Figures 29 and 30.

In contrast to the backward rotary angle controller, where the impulse and average function take place on the same surface, in this case, the pallets have clearly separated the average of the impulse surfaces. The catchment surfaces are parts of circular cylinders with an axis passing through the roofing point as shown in Figure 29. This constructive feature annulates the moment of reaction force of the average function relative to the point of suspension, thus completely eliminating the return stroke of the center point. The impulse surfaces shown in Figure 30 are flat and adjusted to transmit force only during the direct pendulum transition through the equilibrium position. Figure 31 gives the diagrams of momentary interactions, based on which the conclusions about the sound characteristics of the quiet regulators will be presented. The symbols $G · l · \sin φ$, $M$ and $Ω$ have the meanings already explained in the preceding chapter in the wheel regulators. When moving the pendulum from the left amplitude position "1" to the equilibrium "0", the moment $M$ only works in the phase 2-0-3, and in the other half-period, when the pendulum moves from position "4" to "0", in phase 5-0-6 and then the regulator performs a pulse function. The impulse function is viewed as a contact of the teeth of the average point and impulse surface of the pallet. In other phases, the regulator performs a normal function. The moment of the frictional force $Ω$ is present almost continuously and directed always opposite to the direction of rotation of the pendulum. If it would be possible to perfect an ankle and an average point, the diagrams $M$ and $Ω$ would be symmetrical in relation to the equilibrium position "0", and the regulator isohron. In order to ensure safe clamping of the anchor palette and the average point, in spite of the inevitable manufacturing errors, it is necessary to reduce the radius of the cylindrical average surface of the left (inlet) pallet and to increase the right (output). This change, although only slightly, leads to an effective reduction of the impulse surface to wet with a tooth of the average point in front of the equilibrium position, both on the input and on the output bar of the boiler.

The described change in geometry has an impact on the dynamic characteristics of the walk regulation. In the first half-period, the effect of the moment $M$ is shorter in the phase 2-0 than in 0-3, and in the second half-period, $M$ appears shorter in the phase 5-0 than in 0-6. Since the total simultaneous effect of the moment $M$ and $G · l · \sin φ$ (in phases 2-0 and 5-0) is shorter compared to the mutually opposite (in phases 0-3 and 0-6), the bradychronic effect prevails.

Figure 29 Swing function on the right and left pallets

Figure 30 Impulse function on the right and left pallets

It is similar to the momenta of the friction force $Ω$. Due to the constructive gap between the pallet area and the tooth point, the average functions do not start the activity immediately after the termination of the impulses in points 3 and 6, but in 3 'and 6', so that the total effect of the moments $Ω$ and $G · l · \sin φ$ in phases 0-4 and 0-7 = 1) is shorter in relation to the mutually opposite (in stages 1-0 and 4-0).

Figure 31 Diagrams of momentary interactions of peaceful, impulse mechanisms

Both moments, both $M$ and $Ω$, act essentially bradycron, which means that any increase in the torque causes an increase in the oscillation period of the pendulum and slowing down the clock speed. As the growth of the drive force is always to some extent offset by the waste of energy generated by the simultaneous increase in friction force on pallets, the oscillation amplitude of the pendulum can only be slightly modified. In the case where the drive moment is constant, the bradycron effect of the eventual increase in the friction coefficient is always partially compensated by the coupled tachyronism of the circular error caused by the simultaneous decrease in oscillation amplitude. This behavior is typical of peaceful, impulse mechanisms. In addition to the well-analyzed Graham's quiet walk-through regulator, shown in Figures 29 and 30, many other impulse mechanisms belong to the same class. Among those that are built into stationary (tower and wall) timepieces: Amman-Lepo's (Amant-Lepaute 1741, 1750), Brokó (Achille Brocot 1849), a mechanism with needle pallets; as well as the names of quiet regulators for mobile (hand and pocket) watches: Tompion's cylinder (Figures 12 and 32) (Tompion 1695), duplex (Pierre Le Roy 1748) and "virgule" (Lépine Jean Antoine 1780).

The aforementioned clock mechanisms have the same or very similar characteristics, superior to the counter-rotating regulators: the impulse function, constructively and dynamically separated from the average, lasts briefly, only during the direct transition of the pendulum through the equilibrium position. More subtle, external compulsion now weakens his own oscillations. The amplitude angle of the pendulum vibrations is small and is about ± (20-30), and the influence of the circular error is negligible. Bradyhronism is becoming almost imperceptible through quality production. These constructive improvements influence that timers with quiet regulators show a daily error of only ± (3-5) seconds. The disadvantages of this solution are reflected in the fact that from the momentary interaction diagram, the oscillator, by means of the momentum force, continues to be under the continuous influence of the drive torque. Any change in either the driving forces or the coefficient of friction on the impulse and the average pallet surface affects stochasticly complicated to change the amplitude and oscillation period of the pendulum. The intention of the horologist and watchmaker to eliminate this effect or to minimize it to a minimum led to the so-called invention. free-of-charge impulse mechanisms.

Figure 32 Cylindrical calm overtones

5.2.3 Free-of-charge impulse mechanisms

The energy losses of the oscillator imply the need for the impulse function of the regulator, and the regulation of the speed itself and the effect of its average functions. If the impulse and average function are directly supplied by the drive power of the timer, then each drive change causes the variation of the impulse, the oscillator energy, and the frequency standard. The invention of free-flow-impulse mechanisms originates precisely from the idea that the impulse and the average function, or completely free of the direct influence of the drive, or that the oscillator itself longer free from any influence of the regulator. The realization of the first principle led to the construction of the so-called. gravity-based pulse mechanisms, and from the other, the technical solutions of chronometric regulators and the so-called. English and Swiss free ancestors with an anchor. The first gravity walker was constructed by English watchmakers Tomas Madge (Thomas Mudge, 1715 - 1794) and Alexander Cumming (1732-1184) in 1766. The invention is perfected by Henry Kather (Henry Kather, 1777-1835) around 1830, and Dž. M. Bloxam (James Mackenzie Bloxam) around 1850. However, the realization of these conceptual solutions was hampered by the unstable behavior of their average function, known as "approximate tripping," or "galoppieren". This significant problem was finally solved by the great British horologist and lawyer Edmund Beckett Denison (Edmund Beckett Denison, 1816-1905). In order to control his ideas, Denison carried out a series of experiments with new impulse mechanisms, which lasted continuously from 1854 until 1859, when the construction of the tower of St. Stephen was completed. The first attempt at installing a three-legged, peaceful pacing with direct action on the pendulum did not give the expected accuracy of the walk, so it was soon discarded. Experiments were continued with four-crore and double triangular gravity-impulse mechanisms, of which the latter showed excellent results, and finally it was incorporated into the clock in 1859. The invention of this famous "double three-stroke gravity interference" shown in Figure 33, achieved exceptional accuracy and reliability in the operation of the Big Ben clock mechanism, which still regulates the journey of the Great Westminster Clock. For stationary timers of high precision, Denison constructed another, the so-called. A "four-wire gravity prevention" on which the work of free-flow-impulse mechanisms will be presented.

In Fig. 33. the main constructive elements of the mentioned prevention are shown, and in Fig. 34 and the corresponding interaction diagrams. (The labels $G · l · \sin φ$, and $Ω$ have the meanings already explained in the previous interaction diagrams, and $M$ is the moment of the gravitational force of the impulse pallets). Observing the movement of the pendulum from the left amplitude position "1", through the equilibrium "0", to the position "2", it can be seen that the left impulse pallet is in contact with the pendulum over the impulse needle. On that occasion, the left pulse pallet announces a pulse pulse at the expense of reducing its gravitational potential energy. It's been done during that time and the average function because one arm of the mean point is blocked on the right pulse pallet. In the "2" position, the left impulse pallet, stopped by the stop, interrupts the contact with the pendulum and completes the impulse function. The pendulum then in the "3" position accepts and raises the right impulse pallet to the right amplitude position "4", whereby the moment $M$ changes the direction of the action. At the same time, the pendulum moves the arm of the center point from the base and, by breaking the average function, allows the rotation of the center point and the camshaft. At this moment, the short-term moment of the friction force Ω occurs.

Figure 33 Denison's four-wire gravity barrier

Figure 34 Diagrams of momentary interactions of Denison gravity with the impulse mechanisms

While the pendulum with the right impulse pallet moves from position "3" to "4", the camshaft of the center point contacts the left impulse pallet, raises the palette to the "6" position and thus communicates a certain gravitational potential energy. That energy amount will be handed over to the pendant a little later, in the moment of execution of the impulse function. Turning the camshaft and lifting the left pallet stops at a time when the arm of the center point is again blocked in contact with the left impulse pallet. The cycle is now being repeated in the second semester of oscillation of the pendulum, through the positions 4-0-5-6-7 = 1. The fact that the moment $M$ and $G · l · \sin φ$ is simultaneously in the stages 1-0, 3-4, 4-0 and 6-7 = 1, and the opposite only in the phases 0-2 and 0-5, so the tachyron effect is prevailing. Since the short-term effect of the moment $Ω$ (friction force on the resistors) is simultaneously with the moment $G · l · \sin φ$ immediately behind positions 3 and 6, then the friction, although only slightly, nevertheless contributes to tachyronism. Thus, both moments, both $M$ and $Ω$, act integral tachyron, which means that any increase in the momentum force of the weight of the impulse pallets and the friction force on the pockets will cause a decrease in the oscillation period of the pendulum and the acceleration of the clock. However, as in this case, the moment $M$ is not propelled, but it is precisely from the influence of the drive completely free, the pendulum vibration period of the pendulum is guaranteed from any alterations in the drive power. What's more, from the fact that the moment M is working tachihrono proishodi and the ability to fine-tune the timing of the clock. To this end, pulse pallets are sometimes placed on pulse pallets with weights whose displacement is achieved by the moment of their weight, the intensity of the impulse, and thus the amplitude of the oscillation of the pendulum. One drawback of this type of gravity regulators needs to be emphasized. The moment of the friction force $Ω$ still depends on the drive, because its change changes the intensity of the pressure of the arm of the center point on the base. However, as this effect is short-lived and, due to the length of the arm, of very low intensity, it can be neglected. Finally, it should be emphasized that, although in 1986, this defect was completely removed by the constructor James Arnfield by the patent of the so-called. isodynamic gravity impedances. It is necessary to briefly discuss the hororological characteristics of the so-called. English and Swiss free antenna overcurrents, as well as chronometric regulators, shown in Figures 35 and 36.

By careful observation of these images, one can easily see the basic principles of functioning of the mechanisms under consideration. The idea is simple and common to all: most of the oscillatory oscillator oscillator oscillator is completely free from any influence of the regulator. Namely, in free-impulse mechanisms with anchor anchors are adjusted to oscillate an isohrono with a very large amplitude angle of ± 270°, and that the clamping of an anomaly with the anchor is carried out by ± 15°, which means that the oscillator freely oscillates more than 88.88% of the duration of its oscillation period. In chronometric regulators, this ratio is even more favorable (95.5%) [46].

Figure 35 Switzerland free lumbar impedance

An English free ankle barrier. shown in Figure 34, characterized by pointed tooths of the average point, was constructed by Englez Tomas Madge in 1757, and perfected by French watchmakers Brege (Abraham-Louis Bréguet, 1747-1823) and Robin (Robert Robin, 1742-1999. ). Swiss freeway regulator with anchor, which differs from the English variant only in the shape of a tooth of the average point, was created around 1910, and because of its simplicity, today it has the widest use in the mechanisms of hand and pocket watches. Both the English and Swiss free barriers have a bradychronous effect on the movement of the clock mechanism for the same reasons already explained for the Graham's breakthrough.

Figure 36 Free English lumbar interference

Except for being distinguished by all the features of free-flow-impulse mechanisms, chronometric walk regulators have another, valuable property: the lucid constructive solution eliminates the necessity of lubrication. If not, the stability of the chronometer's course would be partly endangered by changes in the characteristics of the oil that changes its viscosity over time due to the oxidation and accumulation of impurities. The first free chronometric walker was made by French watchmaker Pierre Leroy in 1748. The invention, shown in the figure. 37, were perfected by English watchmakers John Arnold in 1779 and Tomas Ernso in 1783, which enabled the mass production of naval chronometers. For their choral discoveries, John Arnold and Tomas Ernso received great and well-deserved public recognition because it turned out that the technical solutions that were just embedded in the chronometric average pulse mechanisms literally saved tens of thousands of lives of seafarers and passengers at sea.

Figure 37 Chronometric Z-I mechanism

5.3 Error of the average - impulse mechanisms - definition and mathematical models of error

5.3.1. The double time method

In this chapter of the dissertation, a general formula for the error of the mean-impulse mechanisms will be derived using the double-time scale (scale) of the perturbation method. Only those clock mechanisms which include spiral springs with a balance point as an oscillator are discussed, as in FIG. 38 but, it should be emphasized that, regardless of this, the results of this analysis will be universally applicable to all other types of impulse mechanisms, including those that are installed in stationary clocks with a pendulum.

Figure 38 Constructive-geometrical origin of the error of the free Swiss lumbar interference

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 a viscous moment of force proportional to the angular velocity of the oscillation $𝑊 = -𝑐φ̇.$ It is also assumed that the mean - pulse mechanism acts on the balancing wheel with a forced moment of force which is a periodic function of only the angular (generalized) coordinates $φ$. In accordance with the fact that the clock mechanism oscillator always oscillates in the resonance with the force of force of the mean - impulse mechanism, it is assumed that the angular frequency of the force of force is equal to the angular frequency of its own oscillations of the balance point.

Under these assumptions, the differential equation of compulsory damped oscillations of the balance point at the level $𝑂𝑥𝑦$ about the axis is:, with the corresponding initial conditions, is given by the expression:

$$𝐽𝜑̈ + 𝑐φ̇ + 𝑘φ = 𝑀 (φ),\tag{5.1}$$

wherein:

$φ$ - the angle between the vertical fixed axis and the pendulum axis measured in the vertical plane $𝑂𝑥𝑦$; generalized coordinate which determines the position of the oscillator (balance point) Fig. 22,
$𝐽$ - the moment of the inertia of the balance point, that is, the oscillator,
$𝑐$ - constant viscosity resistance, respectively, viscous moment of force $𝑊 = -𝑐φ̇$,
$𝑘$ - coefficient of stiffness of the spiral spring,

$𝑀(φ)$ force force, which is a periodic function of time $𝑀(φ(𝑡))$, angular frequency $ω_0$. The initial conditions are: $φ(0) = Φ_0, φ̇ (0) = 0$.

The equation (5.1) can be transformed into the equation:

$$𝜑̈ + 2ξω_0φ̇ + ω_0^2φ = μ(φ),\tag{5.2}$$

wherein:

$ω_0 = \sqrt{𝑘/𝐽}$ angular frequency of its own oscillations,
$ξ = 𝑐/(2𝐽ω_0)$ attenuation coefficient, $ξ ∈ (0,1), ξ≪1$,
$μ(φ) = 𝑀(φ)/𝐽$ specific forced moment of force.

Since the attenuation $ξ$ is low, the specific force coefficient $μ(φ)$ is a small disorder of the free-damped oscillations of the balance point. To make this property prominent and make the budget flow clearer, the function $μ_1(φ)$ is formally introduced so that

$$μ (φ) = ξ ∙ μ_1 (φ).\tag{5.3}$$

In accordance with the two-time time-scales method whose principles are summarized in Chapter 3, the independent variable $𝑡$ is replaced by two variable $𝑡_1$ and $𝑡_2$,

$$𝑡_1 = 1 ∙ 𝑡; 𝑡_2 = ξ ∙ 𝑡, ξ∈ (0,1).\tag{5.4}$$

Admits that, during the execution of the perturbation account, they are mutually independent. The constant $ξ$ is the attenuation factor. The physical meaning of this procedure is based on the fact that the rate of change in the amplitude of the forced, poorly suppressed oscillations of the balance of the clock mechanism is considerably lower than the angular frequency of its oscillations. The impression is that within the oscillatory oscillator the process runs two different time scales. In this sense, the time coordinate $𝑡_1$ describes the flow of "regular", and $𝑡_2$ "slow" time.

The approximate solution $φ(𝑡_1, 𝑡_2, ξ)$ of the differential equation (5.2) is sought in the form of an initial or zero solution of assemblies $φ_0(𝑡_1, 𝑡_2)$ and first order correction $ξφ_1(𝑡_1, 𝑡_2)$, as shown by the expression:

$$φ(𝑡_1, 𝑡_2, ξ) ≈ φ_0 (𝑡_1, 𝑡_2) + ξφ_1 (𝑡_1, 𝑡_2). \tag{5.5}$$

The initial conditions $φ(0) = Φ_0, φ̇ (0) = 0$ are formulated for both functions $φ_0(𝑡_1, 𝑡_2)$ and $φ_1(𝑡_1, 𝑡_2)$, the terms:

$$φ_0 (0,0) = Φ_0, φ̇_0 (0,0) = 0, φ_1 (0,0) = 0, φ̇_1 (0,0) = 0. \tag{5.6}$$

In accordance with the expressions (5.4), the first and second copies of the angular coordinates $φ$ by time are given by the following formulas, respectively:

$$φ̇ = \frac{𝜕φ}{𝜕𝑡_1} ∙ \frac{𝜕𝑡_1}{𝜕𝑡} + \frac{𝜕φ}{𝜕𝑡_2} ∙ \frac{𝜕𝑡_2}{𝜕𝑡} = \frac{𝜕φ}{𝜕𝑡_1} + ξ\frac{𝜕φ}{𝜕𝑡_2}, (5.7)$$ $$𝜑̈ = \frac{𝜕φ̇}{𝜕𝑡_1} + ξ\frac{𝜕φ̇}{𝜕𝑡_2} = \frac{𝜕^2φ}{𝜕𝑡_1^2} + 2ξ\frac{𝜕^2φ}{𝜕𝑡_1𝜕𝑡_2} + ξ^2\frac{𝜕^2φ}{𝜕𝑡_1^2} ≅ \frac{𝜕^2φ}{𝜕𝑡_1^2} + 2ξ\frac{𝜕^2φ}{𝜕𝑡_1𝜕𝑡_2}. \tag{5.8}$$

In the expression (5.8) the member containing $ξ^2$ is neglected as a small value of higher order. When the expressions (5.3), (5.5), (5.7) and (5.8) are included in the differential equation (5.2), after ignoring the members containing $ξ^2$, the following differential equation is obtained:

$$\frac{𝜕^2φ_0}{𝜕𝑡_1^2} + ω_0^2φ_0 + ξ \Big(\frac{𝜕^2φ_1}{𝜕𝑡_1^2} + ω_0^2φ_1 + 2\frac{𝜕^2φ_0}{𝜕𝑡_1𝜕𝑡_2} + 2ω_0\frac{𝜕φ_0}{𝜕𝑡_1}\Big) = ξ ∙ μ_1 (φ). \tag{5.9}$$

In order for the left side of the equation (5.9) to be equal to zero, and in accordance with the fact that the attenuation factor $ξ$ is small but different from zero $ξ ≠ 0$, it is necessary that the equations:

$$\frac{𝜕^2φ_0}{𝜕𝑡_1^2} + ω_0^2φ_0 = 0; \Big(φ_0 = Φ_0, \frac{𝜕φ_0}{𝜕𝑡_1} = 0, \,\,\text{for}\,\, 𝑡_1 = 𝑡_2 = 0\Big) \tag{5.10}$$ $$\begin{split}&\frac{𝜕^2φ_1}{𝜕𝑡_1^2} + ω_0^2φ_1 = μ_1(φ) - 2\frac{𝜕^2φ_0}{𝜕𝑡_1𝜕𝑡_2} - 2ω_0\frac{𝜕φ_0}{𝜕𝑡_1}; \\ &\Big(φ_1 = 0, \frac{𝜕φ_1}{𝜕𝑡_1} = -\frac{𝜕φ_0}{𝜕𝑡_2}, \,\,\text{for}\,\, 𝑡_1 = 𝑡_2 = 0\Big)\end{split} \tag{5.11}$$

simultaneously satisfied. With each of these equations, specific initial conditions are given, according to their general formulation (5.6) and in accordance with expression (5.5).

Differential equations in the system of equations (5.10) - (5.11) are solved successively. It is necessary to notice the fact that the differential equation (5.10) is only formally partial, and in essence it represents an ordinary homogeneous second-order equation with constant constants, with a variable $𝑡_1$. The solution of the equation (5.10) is called initial or zero $φ_0 (𝑡_1, 𝑡_2)$ and is corrected by the first order correction $ξφ_1 (𝑡_1, 𝑡_2)$. In this corrective term, the function $φ_1 (𝑡_1, 𝑡_2)$ represents the solution of the differential equation (5.11), which depends on the previously obtained function $φ_0 (𝑡_1, 𝑡_2)$. One solution of the differential equation (5.10) is given by the expression:

$$𝜑_0(𝑡_1) = 𝛷\sin(𝜔_0𝑡_1 + 𝛾),𝛷 = 𝑐𝑜𝑛𝑠𝑡, 𝛾 = 𝑐𝑜𝑛𝑠𝑡, \tag{5.12}$$

which describes the free harmonic oscillations by the independent variable $𝑡_1$ and in which the constant $γ$ is the angle of the phase difference. It was pointed out and discussed in Chapter 2 of this thesis that the solution $φ_0 (𝑡_1)$ is limited to $φ_0(𝑡_1) = 𝑂(1)$ at an arbitrary long time interval $𝑡_1 ∈ [0, ∞)$.

$$𝜑_0(𝑡_1, 𝑡_2) = 𝛷(𝑡_2)\sin(𝜔_0𝑡_1+𝛾(𝑡_2))=𝛷\sin 𝜓.\tag{5.13}$$

For the sake of conciseness, the phase angle (phase) oscillation was introduced $ψ = ω_0𝑡_1 + γ (𝑡_2)$. The partial derivative of the function (5.13) by the variable $𝑡_1$ is given by the formula:

$$\frac{𝜕𝜑_0}{𝜕𝑡_1} = 𝛷(𝑡_2)𝜔_0\cos(𝜔_0𝑡_1+𝛾(𝑡_2)) = 𝛷𝜔_0\cos 𝜓,\tag{5.14}$$

and the partial derivative of the function (5.14) by the variable $𝑡_2$ is given by the formula:

$$\begin{split}\frac{𝜕^2𝜑_0}{𝜕𝑡_1𝜕𝑡_2} &= \frac{𝜕𝛷(𝑡_2)}{𝜕𝑡_2}𝜔_0\cos(𝜔_0𝑡_1+𝛾(𝑡_2))− 𝛷(𝑡_2)𝜔_0\sin(𝜔_0𝑡_1+𝛾(𝑡_2))\frac{𝜕𝛾(𝑡_2)}{𝜕𝑡_2} \\ &= \frac{𝜕𝛷}{𝜕𝑡_2}𝜔_0\cos 𝜓−𝛷\frac{𝜕𝛾}{𝜕𝑡_2}𝜔_0\sin 𝜓.\end{split}\tag{5.15}$$

The specific force of force $μ(φ)$ as well as $μ_1(φ)$ functions are angular coordinates $φ$ and, in accordance with the expressions (5.13), can be formally represented as the function of the amplitude $Φ$ and the total phase $ψ$:

$$𝜇(𝜑) = 𝜇(𝛷(𝑡),𝜓(𝑡)) = 𝜇(𝛷,𝜓) = 𝜉∙𝜇_1(𝛷(𝑡),𝜓(𝑡)) = 𝜉∙𝜇_1(𝛷,𝜓)\tag{5.16}$$

When the expressions (5.14) and (5.15) are included in the differential equation (5.11), the following equation is obtained:

$$\frac{𝜕^2𝜑_1}{𝜕𝑡_1^2} + 𝜔_0^2𝜑_1 = 𝜇_1(𝛷,𝜓) + 2𝛷\frac{𝜕𝛾}{𝜕𝑡_2}𝜔_0\sin 𝜓 − 2(\frac{𝜕𝛷}{𝜕𝑡_2}+𝛷𝜔_0)𝜔_0\cos 𝜓.\tag{5.17}$$

It has already been emphasized that the force compulsive moment $𝑀(φ)$ is a periodic function of time $𝑡, 𝑀(φ(𝑡))$, angular frequency $ω_0$, which is equal to the angular frequency of its own oscillations of the balance point. From this fact, it follows directly that the function $μ_1(Φ, ψ)$ is periodic with an angular frequency $ω_0$ and can be developed into a Fourier (Jean-Baptiste Joseph Fourier, 1768-1830) trigonometric order. This procedure must be carried out in order to establish and perceive the appearance of those members of the resonant coercion that generate the basic harmonics of the periodic force of force.

The periodic, integral function $𝑓(𝑥)$, the period $2π$, in the segment $[0, 2π]$, can be represented by the Fourier trigonometric order

$$𝑓(𝑥)=\frac{𝐴_0}{2} + \sum_{𝑛=0}^∞ 𝐴_𝑛\cos(𝑛𝑥) + \sum_{𝑛=0}^∞ 𝐵_𝑛\sin(𝑛𝑥),\tag{5.18}$$

in which the coefficients $𝐴_0$, $𝐴_𝑛$ and $𝐵_𝑛$ are called the Fourier coefficients of the function $𝑓(𝑥)$ and are determined by the following relations (Euler - Fourier formulas):

$$ \begin{split}&𝐴_0 = \frac{1}{𝜋}\int^{2𝜋}_{0}𝑓(𝑥)𝑑𝑥, \\ &𝐴_𝑛 = \frac{1}{𝜋}\int^{2𝜋}_{0}𝑓(𝑥)\cos(𝑛𝑥)𝑑𝑥, \\ &𝐵_𝑛 = \frac{1}{𝜋}\int^{2𝜋}_{0}𝑓(𝑥)\sin(𝑛𝑥)𝑑𝑥.\end{split} \tag{5.19}$$

In accordance with the formulas (5.18) and (5.19), the function $μ_1(Φ, ψ)$ can be developed into the Fourier trigonometric series

$$𝜇_1(𝛷,𝜓)= \frac{𝑎_0(𝛷)}{2} + \sum_{𝑛=0}^{∞} 𝑎_𝑛(𝛷)\cos(𝑛𝜓) + \sum_{𝑛=0}^{∞} 𝑏_𝑛(𝛷)\sin(𝑛𝜓),\tag{5.20}$$

in which the coefficients $𝑎_0$, $𝑎_𝑛$ and $𝑏_𝑛$ are determined by the following expressions:

$$\begin{split}𝑎_0(𝛷)&=\frac{1}{𝜋}\int^{2𝜋}_{0}𝜇_1(𝛷,𝜓)𝑑𝜓,\\ 𝑎_𝑛(𝛷)&=\frac{1}{𝜋}\int^{2𝜋}_{0}𝜇_1(𝛷,𝜓)\cos(𝑛𝜓)𝑑𝜓,\\ 𝑏_𝑛(𝛷)&=\frac{1}{𝜋}\int^{2𝜋}_{0}𝜇_1(𝛷,𝜓)\sin(𝑛𝜓)𝑑𝜓.\end{split}\tag{5.21}$$

If the fundamental harmonics $𝑎_1(Φ)\cos ψ$ and $𝑏_1(Φ)\sin ψ$ are distinguished in Fourier's development of the function $μ_1(Φ, ψ)$ in the trigonometric order, the expression (4.21) reads as follows:

$$\begin{split} 𝜇_1(𝛷,𝜓) &= \frac{𝑎_0(𝛷)}{2} + 𝑎_1(𝛷)\cos 𝜓 + 𝑏_1(𝛷)\sin 𝜓 \\ &+ \sum_{𝑛 = 1}^{∞}𝑎_𝑛(𝛷)\cos(𝑛𝜓)+\sum_{𝑛 = 1}^{∞}𝑏_𝑛(𝛷)sin(𝑛𝜓).\end{split} \tag{5.22}$$

In Chapter 3 of this dissertation, it is explained that members of resonant coercion, as functions of time $𝑡_1$, cause the differential equation solution (5.23) to contain secular members by the variable $𝑡_1$. By eliminating these resonant coercive members in the differential equation (5.23), the secular members as the function of the variable $𝑡_1$ are eliminated from the solution $φ_1 (𝑡_1, 𝑡_2)$, thus ensuring that $φ_1 (𝑡_1, 𝑡_2)$ is limited by the variable $𝑡_1$ at the time interval $𝑡_1 ∈ [0 , ∞)$. Annihilation of the members of the resonant coercion in the equation (5.23) is done by the following expressions:

$$𝑎_1(𝛷)−2\Big(\frac{𝜕𝛷}{𝜕𝑡_2}+𝛷𝜔_0\Big)𝜔_0 = 0,\tag{5.24}$$ $$𝑏_1(𝛷)+2𝛷\frac{𝜕𝛾}{𝜕𝑡_2}𝜔_0 = 0.\tag{5.25}$$

In accordance with the relations (5.21), the terms $𝑎_1(Φ)$ and $𝑏_1(Φ)$ in the expressions (5.24) and (5.25) are determined from the following formulas:

$$𝑎_1(𝛷)=\frac{1}{𝜋} \int_{0}^{2𝜋}𝜇_1(𝛷,𝜓)\cos 𝜓\,𝑑𝜓,\tag{5.26}$$ $$𝑏_1(𝛷)=\frac{1}{𝜋} \int_{0}^{2𝜋}𝜇_1(𝛷,𝜓)\sin 𝜓\,𝑑𝜓.\tag{5.27}$$

In accordance with the relation (5.3) and replacing the expression (5.26) in (5.24), we obtain:

$$\frac{𝜕𝛷}{𝜕𝑡_2} = −𝜔_0𝛷 + \frac{1}{2𝜋𝜉𝜔_0}\int_{0}^{2𝜋}𝜇(𝛷,𝜓)\cos 𝜓\,𝑑𝜓.\tag{5.28}$$

In accordance with the relation (5.3) and replacing the expression (5.27) in (5.25), we obtain:

$$\frac{𝜕𝛾}{𝜕𝑡_2} = −\frac{1}{2𝜋𝜉𝜔_0𝛷}\int_{0}^{2𝜋}𝜇(𝛷,𝜓)\sin 𝜓\,𝑑𝜓.\tag{5.29}$$

Equations (5.28) and (5.29) are a system of differential equations that are solved simultaneously. In order to determine their solutions it is necessary to first solve the specified integrals in the oscillation phase $ψ$ which is a function of variable $𝑡_1$. Since the oscillation amplitude $Φ$ is a function of time $𝑡_2$, the integrals are solved assuming that $Φ = 𝑐𝑜𝑛𝑠𝑡$, at the interval $ψ∈⌈0,2π⌉$. The specific moment of momentum $μ = μ (Φ, ψ)$ is a periodic function of time whose amplitude, in general, is also variable with time. Intermediate, as assumed for the oscillation amplitude, and for the amplitude of the drive momentum, it is assumed that it is constant within the integration bounds $ψ ∈ ⌈0,2π⌉$. The assumption is justified because, if the torque amplitude changes at all, then this change takes place so slowly that it can be neglected at the time interval of the oscillation period. When these integrals are solved, a solution of the differential equation system (5.28) - (5.29) is reached, in which neither the oscillation amplitudes $Φ$ nor the specific torque amplitudes are anymore constants, but the quantities that depend on time $𝑡$.

Let the solution of the integral in the equation (5.28) of the function $𝐹(Φ)$, then the differential equation (5.28) reads:

$$\int_{0}^{2𝜋}𝜇(𝛷,𝜓)\cos 𝜓 𝑑𝜓=𝐹(𝛷).\tag{5.30}$$ $$\frac{𝜕𝛷}{𝜕𝑡_2} = −𝜔_0𝛷 + \frac{𝐹(𝛷)}{2𝜋𝜉𝜔_0}.\tag{5.31}$$

The solution of the differential equation (5.31) is a function $Φ = Φ(𝐶, 𝑡_2)$, in which $𝐶$ is the integration constant. At the starting point $𝑡_2 = 0, Φ(𝐶, 0) = Φ_0$, from which the integration constant $𝐶$ is determined as the oscillation amplitude function $𝐶 = A (Φ_0)$.

Let the solution of the integral in the equation (5.29) of the function $G(Φ)$:

$$\int_{0}^{2𝜋}𝜇(𝛷,𝜓)\sin 𝜓 𝑑𝜓=𝐺(𝛷).\tag{5.32}$$

When the equation (4.31) includes the solution of the differential equation (5.31) $Φ = Φ(𝐶, 𝑡_2) = Φ(A(Φ_0), 𝑡_2)$ and the integral solution (5.32) $G(Φ) = 𝐺(Φ (A(Φ_0), 𝑡_2))$, the following equation is obtained:

$$\frac{𝜕𝛾}{𝜕𝑡_2} = −\frac{𝐺(𝛷(А(𝛷0),𝑡2))}{2𝜋𝜉𝜔0𝛷(А(𝛷_0),𝑡_2)}.\tag{5.33}$$

The solution of the differential equation (5.33) is the function $γ(𝑡_2) = γ_1(𝑡_2) + γ_0$, in which $γ_0$ is the integration constant (angle of the initial phase difference). In accordance with the initial condition $𝜕φ_0/𝜕𝑡_1 = 0$, for $𝑡_1 = 𝑡_2 = 0$ specified in relations (5.10), the expression (5.14) reads:

$$\frac{𝜕𝜑_0(0,0)}{𝜕𝑡_1} = 𝛷(0)∙ 𝜔_0\cos(𝛾_1(0) + 𝛾_0) = 0,\tag{5.34}$$

from which immediately follows $γ_0 = π/2 - γ_1(0)$.

This equation is completely defined, ie, the zero solution (5.13) of the differential equation (5.10) which, when the solution of the differential equation (5.31) $Φ = Φ (A(Φ_0), 𝑡_2)$ and the solution of the equation (5.33) $γ (𝑡_2) = γ_1(𝑡_2) - γ_1(0) + π/2$ reads:

$$𝜑_0(𝑡_1, 𝑡_2)=𝛷(А(𝛷_0), 𝑡_2)∙\cos(𝜔_0𝑡_1+𝛾(𝑡_2)−𝛾_1(0)).\tag{5.35}$$

By passing to the "regular time" coordinate $𝑡$, and in accordance with relations (5.4), the null solution (5.35) formally reads:

$$𝜑_0(𝑡) = 𝛷(А(𝛷_0),𝜉𝑡)∙\cos(𝜔_0𝑡+𝛾(𝜉𝑡)−𝛾_1(0)).\tag{5.36}$$

These are the exhausted possibilities of further solving the differential equation (5.11), so that the function $φ_1(𝑡_1, 𝑡_2)$ remains indeterminate. The approximate solution of the equation (5.5) $φ(𝑡) ≈ φ_0(𝑡)$ is defined only as the zero "improved" approximation (5.36) of the perturbation order. However, in spite of the fact that no analytic expression for the function $φ_1(𝑡_1, 𝑡_2)$ is defined, the elimination process of secular members ensures that it is limited by the variable $𝑡_1$ at the time interval $𝑡_1 ∈ [0, ∞)$. But, as the procedure of solving the differential equation (5.11) did not eliminate secular members from the function $φ_1(𝑡_1, 𝑡_2)$ according to the variable $𝑡_2$, the function $φ_1(𝑡_1, 𝑡_2)$ can only be claimed to be limited $φ_1(𝑡_1, 𝑡_2) = O(1)$ by the variable $𝑡_2$, at some final time interval $𝑡_2 ∈ [0, 𝑡_𝑀]$. (For function $φ_0 (𝑡_1, 𝑡_2)$ it has already been shown that $φ_0(𝑡_1, 𝑡_2) = 𝑂(1)$ is bounded according to the variable $𝑡_1$, at arbitrary long time interval $𝑡_1 ∈ [0, ∞)$.) By switching to the regular time coordinate, $𝑡_2 = ξ ∙ 𝑡$, it follows that the function $φ_1(𝑡)$ is bounded $φ_1(𝑡) = 𝑂(1)$ at the final time interval $𝑡 ∈ [0, 𝑡_𝑀/ξ]$ which for the order of the dimension factor $ξ$ is longer than the interval $𝑡 ∈ [0, 𝑡_𝑀]$. From here it follows implicitly that $ξφ_1(𝑡_1, 𝑡_2) = O(ξ)$, at $𝑡 ∈ [0, 𝑡_𝑀/ξ], 𝑡 = 𝑂 (1/ξ)$. For the fixed value of the damping factor $ξ$, the conclusions are correct for each other value of the number $ξ_1 ≤ ξ$, that is, the approximation is more accurate and valid for a longer time interval, if the attenuation factor is lower. Based on the above, it follows:

$$𝜑_0(𝑡_1,𝑡_2)= 𝑂(1),\,\text{for}\,\, 𝑡_1 ∈ [0, 𝑡_𝑀/𝜉_1]\,\,\text{and for each}\,\, 𝜉_1 ≤ 𝜉, 𝜉∈(0,1).\tag{5.37}$$

In accordance with the analysis and conclusions in Chapter 2 of this dissertation, for the error of approximation, which is calculated as the difference of the correct solution and its zero approximation, it is valid:

$$\big|𝜑(𝑡,𝜉) − 𝜑_0(𝑡,𝜉_𝑡)\big| = \big|𝜉𝜑_1(𝑡,𝜉_𝑡)\big| = О(𝜉).\tag{5.38}$$

The relation (5.38) means that the order of the size of the remainder of the perturbation approximation is equal to the order of the size of the first excluded member $ξφ_1(𝑡_1, 𝑡_2)$ of the asymptotic functional order, which approximates the exact solution of the equation (5.10). From here it follows immediately that there are constants $𝑐$, $ξ_1$ and $𝑡_𝑀$ such that the exact solution $φ(𝑡, ξ)$ satisfies the following condition:

$$|𝜑(𝑡,𝜉)−𝜑_0(𝑡,𝜉_𝑡)|≤𝑐∙𝜉,\,\,\text{for}\,\, 𝑡∈[0,𝑡_𝑀/𝜉_1]\,\,\text{and for each}\,\, 𝜉_1 ≤ 𝜉,𝜉∈(0,1).\tag{5.39}$$

This statement confirms that the approximate solution $φ_0(𝑡)$ asymptotically approaches the exact solution $φ(𝑡)$ at a time interval $𝑡 ∈ [0, 𝑡_𝑀/ξ]$.

Since this dissertation is dedicated to the errors of the average - impulse mechanisms, it is necessary to discuss relations in more detail (5.28) and (5.29). In accordance with the relation $𝑡_2 = ξ ∙ 𝑡$, expression (4.30) becomes:

$$𝛷̇=\frac{𝑑𝛷}{𝑑𝑡}=−𝜉𝜔_0𝛷 + \frac{1}{2𝜋𝜔_0} \int_{0}^{2𝜋} 𝜇(𝛷,𝜓)\cos 𝜓 𝑑𝜓,\tag{5.40}$$

and the term (5.29)

$$𝛾̇=\frac{𝑑𝛾}{𝑑𝑡} = −\frac{1}{2𝜋𝜔0𝛷}\int_{0}^{2𝜋}𝜇(𝛷,𝜓)\sin 𝜓 𝑑𝜓.\tag{5.41}$$

The formula (5.40) is a general-order relation that defines the dependence of the rate of change in the amplitude of the forced damped oscillations of the balance point from the amplitude of the oscillations and the force of force acting on the balance wheel. It has already been emphasized that this is a differential equation of the first order whose solution for the given initial conditions gives the oscillation amplitude in the function of time $Φ = Φ(𝑡)$. Formula (5.41) is a relation, also in the general form, which defines the dependence of the rate of change of the phase difference γ of the forced damped oscillations of the balance point from the forced moment of force and the amplitude of its oscillations. And it was said to represent a differential equation of the first order whose solution, according to the same initial conditions, determines the dependence of the phase difference of the oscillation $γ$ from the time $γ = γ(𝑡)$. The formula (5.41) is of special significance because it defines the change in the angular frequency $ω_0$ of the oscillator oscillation oscillator under the influence of the forced moment of force by which the impulse mechanism acts on the oscillator. This change in the angular frequency $ω_0$ is called the error of the mean - pulse mechanism (English: Escapement error; German: Die Hemmungsfehler;) and represents the phenomenon to which this dissertation is dedicated.

The expressions under the integrals in the formulas (5.40) and (5.41) are functions of the phase coordinates $ψ$ and the oscillation amplitude $Φ$ as a constant parameter, and the integration bounds are $0 → 2π$. These terms can also be defined as functions of the angular coordinates $φ$ and the oscillation amplitude $Φ$ (also as a constant parameter), due to which the integrals become curvilinear along the closed curve, with the bounds of integration $(-Φ) → (+ Φ) → (-Φ)$. This coordinate transformation can be derived from the relation (5.35). As the first time derivative of the function $φ(𝑡) ≈ φ_0(𝑡)$ is:

$$𝜑̇=𝛷̇(𝛷_0,𝜉𝑡)𝜉∙\sin 𝜓 + 𝛷\cos 𝜓 𝜓̇ = 𝛷̇ 𝜉\sin 𝜓 + 𝛷\cos 𝜓 𝜓̇,\tag{5.42}$$

its total differential 𝑑φ is given by the following relation:

$$𝑑𝜑=𝜉\sin 𝜓 𝑑𝛷 + 𝛷\cos 𝜓 𝑑𝜓.\tag{5.43}$$

But, in accordance with the fact that $ξ ∈ (0,1), ξ≪1$, one can accept the approximation that $ξ𝑑Φ ≈ 0$ is a small value of a higher order. Then the relation (4.45) becomes:

$$𝑑𝜑 ≈ 𝛷\cos 𝜓 𝑑𝜓.\tag{5.44}$$

In accordance with expression (5.44), the formulas for transforming the coordinates of the oscillation phase $ψ$ in the angular coordinate $φ$ are as follows:

$$\begin{split}&𝜑 = 𝛷\sin 𝜓;\,\,\,\,\,\,\,\, 𝑑𝜑 = 𝛷\cos 𝜓𝑑𝜓; \\ &\sin 𝜓=\frac{𝜑}{𝛷};\,\,\,\,\,\,\,\, \cos 𝜓 = ±\sqrt{1−\Big(\frac{𝜑}{𝛷}\Big)^2} = ±\frac{1}{𝛷}\sqrt{𝛷^2−𝜑^2}; \\ &𝑑𝜓 = \frac{𝑑𝜑}{𝛷\cos 𝜓} = ±\frac{𝑑𝜑}{\sqrt{𝛷^2−𝜑^2}}\end{split}\tag{5.45}$$

The approximation (5.44) is consistent with the fact that the relations (5.45) are used for the transformation of the coordinates in the integrals (5.30) and (5.32) by the variable $ψ$, in which the oscillation amplitude is a constant parameter $Φ = 𝑐𝑜𝑛𝑠𝑡$, on the segment $ψ ∈ [0 , 2π]$, that is, within the bounds of integration.

When this coordinate transformation is performed, the expression (5.40) becomes the following formula:

$$\dot Φ = \frac{𝑑𝛷}{𝑑𝑡} = −𝜉𝜔_0𝛷+\frac{1}{2𝜋𝜔_0𝛷}\oint_{𝛷} 𝜇(𝜑) 𝑑𝜑.\tag{5.46}$$

The formula (5.46), as well as (5.40), determines the dependence of the rate of change in the amplitude of the forced damped oscillations of the balance point from the amplitude of the oscillations and the force of force acting on the balance wheel. As in the case of the relation (5.40), and after solving the given integral, the expression (5.46) becomes a differential equation of the first order, whose solution describes the functional dependence of the amplitude from the time $Φ = Φ(𝑡)$.

By passing to the generalized angular coordinate $φ$, the expression (5.41) becomes the following formula:

$$𝛾̇=\frac{𝑑𝛾}{𝑑𝑡}=−\frac{1}{2𝜋𝜔_0𝛷^2}\oint_{𝛷}\frac{𝜇(𝜑)𝜑 𝑑𝜑}{\sqrt{𝛷^2−𝜑^2}}.\tag{5.47}$$

The formula (5.47), as well as (5.41), determines the dependence of the rate of change in the phase difference $γ$ of the forced damped oscillations of the balance point from the amplitude of the oscillations and the force of force acting on the balance wheel. As in the case of expression (5.41), and after solving the given integral, the expression (5.47) becomes a differential equation of the first order, whose solution defines the function of the phase difference from the time $γ = γ(𝑡)$. The formula (5.47) describes the change in the angular frequency $ω_0$ of the oscillator oscillation oscillator under the influence of the forced force of the force. It has already been emphasized that this change in the angular frequency $ω_0$ is called the error of a mean - impulse mechanism.

If the dissipation energy of an oscillator is not equal to the energy it receives from the impulse mechanism, the oscillator is in a non-stationary regime described by differential equations (5.46) and (5.47) respectively (5.40) and (5.41). The non-stationary oscillations of the balance point appear whenever they change, as functions of time $𝑡$, the amplitude of the periodic function of the drive moment $𝑀(φ)$, the coefficient of damping $ξ$ and the moment of inertia of the balance point $𝐽$. If the dissipation energy of the oscillator is equal to the energy it receives from the impulse mechanism, the oscillator is in the quasi-oscillation oscillation mode. The term "quasi-stationary" was used instead of "stationary" because, between each two impulse impedances, the amplitude of the oscillator is nevertheless slightly decreasing. However, in accordance with the fact that the amplitude drop is annihilated by each impulse of the impedance, and this is already during a period of oscillation, it can be assumed that, over longer time intervals, the average amplitude value is constant. In addition, the driving torque of the clutch mechanism also decreases with time, and strictly speaking, the balance wheel is never in the steady-state oscillation mode. But, as the torque gradient is small, it can be ignored, or it can be declared "quasi-stationary". It's about the usual mode of operation of every clock mechanism, in which $\dot Φ = 0$, $Φ = 𝑐𝑜𝑛𝑠𝑡$ or $Φ$ changes so little over time that this change can be ignored. From the formula (5.40), the term for the amplitude of the oscillation of the balance point in the quasi-stationary regime follows directly:

$$𝛷=\frac{1}{2𝜋𝜉𝜔_0^2}\int_{0}^{2𝜋} 𝜇(𝛷,𝜓)\cos𝜓 𝑑𝜓 \tag{5.48}$$

By substituting the expression (5.48) in (5.41), we obtain the following formula for the error of the impulse mechanism in the quasi-oscillation mode of the oscillation of the balance point:

$$𝑅=\frac{𝑑𝛾}{𝑑𝑡} = −𝜉𝜔_0∙\int_{0}^{2𝜋}𝜇(𝛷,𝜓)\sin 𝜓 𝑑𝜓\int_{0}^{2𝜋}𝜇(𝛷,𝜓)\cos 𝜓 𝑑𝜓\tag{5.49}$$

If instead of the damping factor $ξ$ is used the oscillator quality factor $𝑄 = 1/(2ξ)$, formula 5.49 becomes:

$$𝑅=\frac{𝑑𝛾}{𝑑𝑡} = −\frac{𝜔_0}{2𝑄}∙\int_{0}^{2𝜋}𝜇(𝛷,𝜓)\sin 𝜓 𝑑𝜓\int_{0}^{2𝜋}𝜇(𝛷,𝜓)\cos 𝜓 𝑑𝜓\tag{5.50}$$

The expressions under the integrals in the formulas (5.48), (5.49) and (5.50) are functions of the $ψ$ and oscillation amplitude $Φ$. In these terms, the transformation of the coordinates (5.25) can be performed, so that they are defined as functions of the angular coordinates $φ$ and the oscillation amplitude $Φ$. In accordance with the fact that in the stationary clock mode $Φ̇ = 0$, the amplitude of the oscillation of the balance point is constant $Φ = 𝑐𝑜𝑛𝑠𝑡$, from formula (5.46) directly follows:

$$𝛷^2=\frac{1}{2𝜋𝜉𝜔_0^2}\oint_{𝛷}𝜇(𝜑)𝑑𝜑\tag{5.51}$$

By substituting the expression (5.51) in (5.47), we obtain the following formula for the error of the impulse mechanism in the quasi-oscillation mode of the oscillation of the balance point:

$$𝑅=\Big(\frac{𝑑𝛾}{𝑑𝑡}\Big)_{𝑆𝑅}=−𝜉𝜔_0∙\oint_{𝛷}\frac{𝜇(𝜑)𝜑 𝑑𝜑}{\sqrt{𝛷^2−𝜑^2}}\Big/ \oint_{𝛷}𝜇(𝜑)𝑑𝜑\tag{5.52}$$

If instead of the damping factor $ξ$ is used the oscillator quality factor $𝑄 = 1/(2ξ)$, formula 5.52 becomes:

$$𝑅=\Big(\frac{𝑑𝛾}{𝑑𝑡}\Big)_{𝑆𝑅}=−\frac{𝜔_0}{2𝑄}∙\oint_{𝛷}\frac{𝜇(𝜑)𝜑 𝑑𝜑}{\sqrt{𝛷^2−𝜑^2}}\Big/ \oint_{𝛷}𝜇(𝜑)𝑑𝜑\tag{5.52}$$

It is necessary to emphasize again that the expressions (5.40) and (5.46), (5.41) and (5.47), (5.48) and (5.51), (5.49) and (5.52), as well as (5.50) and (5.53) formally different but essentially equivalent. In the first group, the integration formula is done by the coordinates of the total oscillation phase $ψ$, and in the second by the angular coordinate $φ$. Thus, the given formula pairs describe the same phenomena, only in different coordinate systems. Which expressions will be used to solve specific problems, this depends primarily on the degree of their formal complexity and, in essence, is a matter of free choice.

5.3.2 Perturbation method of averaging over Krylov and Bogoliubov

In this chapter of the dissertation, a general formula for the error of the average impulse mechanisms will be derived, using the technique of averaging by the Krylov and Bogoliubov methods. It should be noted that only those clock mechanisms that incorporate spiral springs with the balance point as an oscillator are considered. 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 a viscous moment of force proportional to the angular velocity of the oscillation $𝑊 = -𝑐φ̇ $. It is also assumed that the mean - pulse mechanism acts on the balancing wheel with a forced moment of force which is a periodic function of only the angular (generalized) coordinates $φ$. Under these assumptions, the differential equation of compulsory damped oscillations of the balance point with the corresponding initial conditions is given by the following expressions:

$$𝐽𝜑̈+𝑐𝜑̇+𝑘𝜑=𝑀(𝜑),\tag{5.54}$$

wherein:

- $𝐽$ the moment of the inertia of the balance point, that is, the oscillator,
- $𝑐$ constant viscosity resistance, respectively, viscous moment of force $𝑊 = −𝑐𝜑̇$,
- $𝑘$ coefficient of stiffness of the spiral spring,
- $𝑀$ (φ) the forced moment of force.

It is accepted that the force of force is a periodic function of the angular frequency $ω$. The initial conditions are: $φ(0) = Φ_0, φ̇ (0) = 0$.

The differential equation (5.54) can be transformed into the equation:

$$𝜑̈ + 2ξω_0𝜑̇ + ω_0^2φ = μ (φ),(5.55)$$

wherein:

$ω0 = √𝑘/𝐽$ angular frequency of own oscillations,
$ξ = 𝑐 /(2𝐽ω_0)$ attenuation coefficient, $ξ∈ (0,1), ξ≪1$,
$μ(φ) = 𝑀(φ)/𝐽$ specific forced moment of force

If $ξ = 0$ and $μ(φ) = 0$, that is, if there is neither attenuation nor external coercion, the equation (5.54) describes the free oscillations of the balance point whose solution is given by the expression:

$$φ = Φ\sin (ω_0𝑡 + γ),\tag{5.56}$$

in which $Φ$ is constant amplitude, and $γ$ is the constant angle of the phase difference of oscillation. By equating the expression (5.56) with time, the equation is obtained

$$φ̇ = Φω_0\cos (ω_0𝑡 + γ).\tag{5.57}$$

For starting initial conditions

$$𝑡 = 0, φ = Φ_0, φ̇ = 0,\tag{5.58}$$

in the equation (5.56), the values for constant $Φ$ and $γ$ are determined.

If $ξ ≠ 0$ and $μ (φ) ≠ 0$, the equation solution (5.55) is sought in the following form:

$$φ = Φ (𝑡) \sin (ω𝑡 + γ (𝑡)),\tag{5.59}$$

in which amplitudes $Φ (𝑡)$ and phase difference $γ (𝑡)$ are not constants, but functions from time $𝑡$. As pointed out in Chapter 3 of this dissertation, the relation (5.59) is called the zero approximation solution of the differential equation (5.55). The parameter $ω$ is the angular frequency of the force of force, which in general does not have to be equal to the angular frequency of its own oscillations $ω_0$. It should be noted that the oscillators of the timers have a slight damping or that $ξ$ is 1.

Differentiation of the expression (5.59) by time gives the expression

$$𝜑̇ = Φ̇ (𝑡) \sin (ω𝑡 + γ (𝑡)) + Φ (𝑡) \cos (ω𝑡 + γ (𝑡)) ∙ (ω +𝛾̇(𝑡)).\tag{5.60}$$

For the sake of conciseness, $Φ (𝑡) = Φ$ will be typed, and the amplitude $Φ$ is a function of time $t$. By introducing the phase oscillation angle marker

$$ψ = ψ (𝑡) = ω𝑡 + γ (𝑡),\tag{5.61}$$

the expression (5.60) is transformed into the following expression:

$$φ̇ = 𝛷̇\sin ψ + Φω\cos ψ + Φγ̇\cos ψ.\tag{5.62}$$

Since two unknown functions of time $t$ have been introduced, it is necessary to introduce some suitable restriction, that is, an additional condition. In Chapter 3, it is explained that an additional condition (3.42) is introduced, which, in the case of equation resolution (5.55), is given by the following expression:

$$φ̇ = Φω\cos ψ.\tag{5.63}$$

The meaning of this term will be explained and discussed later. Applying the conditions (5.63) to the equation (5.62) leads to the expression

$$Φ̇\sin ψ + Φγ̇\cos ψ = 0.\tag{5.64}$$

Differentiating the expression (5.63) by time $t$ gives the following expression:

$$𝜑̈ = \dot Φω\cos ψ-Φω^2\sin ψ-Φω𝛾̇\sin ψ.\tag{5.65}$$

The specific force of the force $μ(φ)$ is angular coordinates $φ$ and, in accordance with the expressions (5.61) and (5.59), can be formally represented as a function of the amplitude $Φ$ and the total phase $ψ$:

$$μ(φ) = μ(Φ(𝑡), ψ(𝑡)) = μ (Φ, ψ).\tag{5.66}$$

By inserting the expressions (5.59), (5.63), (5.65) and (5.66) into the differential equation (5.55), the following equation is derived:

$$\dot Φω \cos ψ - Φω^2\sin ψ- Φω\dot γ \sin ψ + 2ξω_0ωΦ\cos ψ + ω_0^2Φ\sin ψ = μ(Φ, ψ).\tag{5.67}$$

If, by applying the conditions (5.64), in equation (5.67), γ is eliminated, the equation is obtained:

$$Φ̇ = -\frac{ω_0^2-ω^2}{ω}\sin ψ\cos ψ-2ξω_0Φ\cos^2ψ + \frac{μ(Φ, ψ)}{ω}\cos ψ.\tag{5.68}$$

If, using the conditions (5.64), in equation (5.67) it is eliminated $\dot Φ$, the equation is obtained:

$$γ̇ = -Φ\frac{ω_0^2-ω^2}{ω}\sin^2ψ + 2ξω_0\sin ψ \cos ψ - \frac{μ(Φ, ψ)}{Φω}\sin ψ.\tag{5.69}$$

Equations (5.68) and (5.69) are still equivalent equations (5.54). Since the functions $Φ$ and $γ$ slowly change with time (since $ξ ≪ 1$), the right sides of the equations (5.68) and (5.69) can be approximately replaced by the mean values ​​over a time interval $𝑡 ∈ ⌈0, 2π/ω_0⌉$ respectively, respectively, at the appropriate interval oscillation phases $ψ∈⌈0,2π⌉$. The approximation is carried out by assuming that the functions $Φ$ and $γ$ at that time interval are constant, that is, independent of time. This is the essence of the perturbation technique of averaging by the Krylov method and Bogoliubov, which is explained in chapter 3 of this dissertation. At the interval (0, 2π), the mean value of the function $Φ̇ (ψ)$ is given by the integral

$$Φ̇ _{𝑆𝑅} = 12𝜋\int_{0}^{2𝜋}Φ̇ (𝜓)𝑑𝜓,\tag{5.70}$$

and the functions $γ(ψ)$, integral

$$γ̇_{𝑆𝑅} = \frac{1}{2π}\int_{0}^{2π}γ̇ (ψ)𝑑ψ.\tag{5.71}$$

In accordance with formula (3.47), by replacing the expression (5.15) for $Φ̇ $ in (5.17) we obtain the integral:

$$𝛷̇_{𝑆𝑅} = \frac{1}{2𝜋} \int_{0}^{2𝜋}\Big(−\frac{𝜔02−𝜔2}{𝜔} \sin 𝜓 \cos 𝜓 − 2𝜉𝜔_0𝛷\cos^2𝜓 + \frac{𝜇(𝛷,𝜓)}{𝜔}\cos 𝜓\Big)𝑑𝜓,\tag{5.72}$$

in accordance with formula (3.48), substituting the expression (5.16) for $γ̇ $ in (5.18), integral:

$$𝛾̇_{𝑆𝑅}=\frac{1}{2𝜋}\int_{0}^{2𝜋}(−\frac{𝜔_0^2−𝜔^2}{𝜔}\sin^2𝜓+2𝜉𝜔_0\sin 𝜓\cos 𝜓−\frac{𝜇(𝛷,𝜓)}{𝛷𝜔}\sin 𝜓)𝑑𝜓.\tag{5.73}$$

Solving these specified integrals is reduced to the following terms:

$$\begin{split}&\int_{0}^{2𝜋}\sin 𝜓\cos 𝜓𝑑𝜓 = −\frac{1}{2}\cos^2𝜓\Big|_{0}^{2𝜋} = 0, \\ &\int_{0}^{2𝜋}\sin^2 𝜓 𝑑𝜓 = \frac{1}{2}(𝜓−\sin 𝜓\cos 𝜓)\Big|_{0}^{2𝜋} = 𝜋, \\ &\int_{0}^{2𝜋}\cos^2 𝜓 𝑑𝜓 = \frac{1}{2}(𝜓+\sin 𝜓\cos 𝜓)\Big|_{0}^{2𝜋} = 𝜋.\end{split}\tag{5.74}$$

As the clock mechanism oscillator always oscillates in the resonance with the force of force of the mean - pulse mechanism, the angular frequency of the forced moment of force and its own oscillations of the balance point are the same: $ω = ω_0$. In accordance with this fact and the above terms, the integral (5.73) becomes:

̇$$\dot Φ_{𝑆𝑅}=\Big(\frac{𝑑𝛷}{𝑑𝑡}\Big)_{𝑆𝑅}=−𝜉𝜔_0𝛷+\frac{1}{2𝜋𝜔_0}\int_{0}^{2𝜋}𝜇(𝛷,𝜓)\cos 𝜓𝑑𝜓\tag{5.75}$$

and the integral (5.74) converts to the formula:

$$𝛾̇_{𝑆𝑅} = \Big(\frac{𝑑𝛾}{𝑑𝑡}\Big)_{𝑆𝑅} = −\frac{1}{2𝜋𝜔_0𝛷}\int_{0}^{2𝜋}𝜇(𝛷,𝜓)\sin 𝜓𝑑𝜓\tag{5.76}$$

Formula (5.75) is an expression in the general form which determines the dependence of the rate of change in the amplitude of the forced damped oscillations of the balance point from the amplitude of these oscillations and the force of force acting on the balance wheel. This is a differential equation of the first order whose solution, for given initial conditions (5.58), gives an oscillation amplitude in the function of time $Φ = Φ(𝑡)$. The formula (5.76) is also the expression in the general form, which determines the dependence of the rate of change of the phase difference $γ$ of the forced damped oscillations of the balance point from the forced moment of force and the amplitude of its oscillations. It represents the differential equation of the first order whose solution is in accordance with the same initial conditions (5.58), determines the dependence of the phase oscillation difference $γ$ from the time $γ = γ(𝑡)$. The formula (5.76) has a special significance because it defines the change of the angular frequency $ω_0$ of the oscillator oscillation oscillator under the influence of the forced moment of force by which the impulse mechanism acts on the oscillator. This change in the angular frequency $ω_0$ is called the error of the mean - pulse mechanism (English: Escapement error; German: Die Hemmungsfehler;) and represents the phenomenon to which this dissertation is dedicated.

In order to find the solution of the system of differential equations (5.75) and (5.76), it is necessary to solve the specified integrals first. In accordance with the method of averaging by Krylov and Bogoliubov, the integrals are solved assuming that the oscillation amplitude is a constant parameter $Φ = 𝑐𝑜𝑛𝑠𝑡$, at the interval $ψ ∈ ⌈0, 2π⌉$. The specific moment of momentum $μ = μ (Φ, ψ)$ is a periodic function of time whose amplitude, in general, is also variable with time. However, as assumed for the oscillation amplitude, and for the amplitude of the drive momentum, it is assumed that it is constant within the integration bounds $ψ ∈ ⌈0, 2π⌉$. The assumption is justified and completely in accordance with the Krylov and Bogoliubov method, because if the amplitude of the drive moment changes at all, then this change takes place so slowly (because $ξ≪1$) that it can be ignored at the time interval of the oscillation period. When the integrals are resolved, the solution of the differential equations system (5.75) - (5.76) is solved, in which neither the oscillation amplitudes $Φ$ nor the specific torque amplitudes are anymore constants, but the quantities that depend on time $𝑡$. The solutions of these equations are represented by the functions $Φ = Φ(𝑡)$ and $γ = γ(𝑡)$ which describe the nonstationary oscillation mode of the balance point.

Expressions under the integrals in the formulas (5.75) and (5.76) are functions of the phase $ψ$ and oscillation amplitude $Φ$ as a constant parameter, and the integration bounds are $0 → 2π$. These terms can also be defined as functions of the angular coordinates $φ$ and the oscillation amplitude $Φ$ (also as a constant parameter), due to which the integrals become curvilinear along the closed curve, with the bounds of integration $(-Φ) → (+ Φ) → (-Φ)$. This coordinate transformation can be performed directly from the conditions (5.63):

$$𝜑̇ = 𝛷𝜔 \cos 𝜓 ⟹ 𝑑𝜑 = 𝛷\cos 𝜓𝑑𝜓;\tag{5.77}$$

In accordance with expression (5.24), the formulas for transforming the coordinates of the oscillation phase $ψ$ in the angular coordinate $φ$ are:

$$\begin{split} &𝜑 = 𝛷\sin 𝜓;\,\,\,\, 𝑑𝜑 = 𝛷\cos 𝜓𝑑𝜓; \\ &\sin 𝜓 = \frac{𝜑}{𝛷};\,\,\,\, \cos 𝜓= ±\sqrt{1−\Big(\frac{𝜑}{𝛷}\Big)^2} = ±\frac{1}{𝛷}\sqrt{𝛷^2−𝜑^2}; \\ &𝑑𝜓 = \frac{𝑑𝜑}{𝛷\cos 𝜓} = ±\frac{𝑑𝜑}{\sqrt{𝛷^2 − 𝜑^2}}\end{split} \tag{5.78}$$

The concept of introducing the expression (5.63), as a special condition in the Krylov and Bogoliubov method, is considered in the formulas for the transformation of the coordinates (5.78). Namely, the exact term for the total differential 𝑑𝜑 is:

$$𝑑𝜑 = \sin 𝜓𝑑𝛷 + 𝛷\cos 𝜓𝑑𝜓.\tag{5.79}$$

But, as already explained, in accordance with the perturbation method of Krylov and Bogoliubov, the assumption is that the integral of the oscillation amplitudes is a constant parameter $Φ = 𝑐𝑜𝑛𝑠𝑡$, within the bounds of the integral $ψ ∈ [0,2π]$. It follows from this assumption that $𝑑Φ = 0$, that is, $𝑑𝜑 = Φ\cos ψ𝑑ψ$, in the segment $ψ ∈ [0,2π]$.

When this coordinate transformation is performed, the expression (5.75) becomes the following formula:

$$𝛷̇_{𝑆𝑅} = \Big(\frac{𝑑𝛷}{𝑑𝑡}\Big)_{𝑆𝑅} = −𝜉𝜔_0𝛷 + \frac{1}{2𝜋𝜔_0𝛷}\oint_{𝛷} 𝜇(𝜑)𝑑𝜑.\tag{5.80}$$

Formula (5.80), as well as formula (5.75), determines the dependence of the velocity of change in the amplitude of the forced damped oscillations of the balance point from the amplitude of these oscillations and the force of force acting on the balance wheel. As in the case of the expression (5.75), and after solving the given integral, the expression (5.80) becomes the differential equation of the first order, whose solution represents the functional dependence of the amplitude from the time $Φ = Φ(𝑡)$.

By passing to the generalized angular coordinate $φ$, the expression (5.76) becomes the following formula:

$$𝛾̇_{𝑆𝑅} = \Big(\frac{𝑑𝛾}{𝑑𝑡}\Big)_{𝑆𝑅} = −\frac{1}{2𝜋𝜔_0𝛷^2}\oint_{𝛷} \frac{𝜇(𝜑)𝜑 𝑑𝜑}{\sqrt{𝛷^2−𝜑^2}}.\tag{5.81}$$

The formula (5.81), as well as (5.76), determines the dependence of the rate of change in the phase difference $γ$ of the forced damped oscillations of the balance point from the oscillation amplitude and the forced force of the force acting on the balance wheel. As in the case of expression (5.76), and after solving the given integral, the expression (5.81) becomes a differential equation of the first order, whose solution defines the function of the phase difference $γ$ from the time $γ = γ(𝑡)$. The formula (5.81) defines the change in the angular frequency $ω_0$ of the oscillator oscillation oscillator under the influence of the force forced moment. It has already been emphasized that this change in the angular frequency $ω_0$ is called the error of the mean-impulse mechanism.

If the dissipation energy of an oscillator is not equal to the energy it receives from the impulse mechanism, the oscillator is in a non-stationary regime described by differential equations (5.75) and (5.76) respectively (5.80) and (5.81). The non-stationary oscillations of the balance point occur whenever they change as time functions $𝑡$, torque amplitudes $𝑀(φ)$, attenuation coefficient $ξ$ and the moment of inertia of the balance point $𝐽$. If the dissipation energy of the oscillator is equal to the energy it receives from the impulse mechanism, the oscillator is in the quasi-oscillation oscillation mode. The term "quasi-stationary" was used instead of "stationary" because, between each two impulse impedances, the amplitude of the oscillator is nevertheless slightly decreasing. However, as this fall of the amplitude is annihilated by each impulse of the waves, and this already during a period of oscillation, it can be assumed that, over longer time intervals, the average amplitude value is constant. In addition, the driving torque of the watch mechanism also drops, and strictly speaking, the balance wheel is never in the steady-state oscillation mode. But, since the fall of the torque is very slow and insignificant, it can be neglected, or it can be declared "quasi-stationary". It's about the usual mode of operation of each clock mechanism, in which $Φ̇_{𝑆𝑅} = 0, Φ_{𝑆𝑅} = 𝑐𝑜𝑛𝑠𝑡$ or $Φ$ changes so little over time that this change can be ignored. From formula (5.75), immediately follows the expression for the amplitude of the oscillation of the balance point in the quasi-stationary regime:

$$𝛷=\frac{1}{2𝜋𝜉𝜔_0^2}\int_{0}^{2𝜋} 𝜇(𝛷,𝜓)\cos 𝜓𝑑𝜓\tag{5.82}$$

Replacing the expression (5.82) in (5.76) gives the following formula for the error of the mean-impulse mechanism in the quasi-oscillation mode of the oscillation of the balance point:

$$𝑅=\Big(\frac{𝑑𝛾}{𝑑𝑡}\Big)_{𝑆𝑅} = −𝜉𝜔_0∙\int_{0}^{2𝜋}𝜇(𝛷,𝜓)\sin 𝜓𝑑𝜓 \bigg/ \int_{0}^{2𝜋}𝜇(𝛷,𝜓)\cos 𝜓𝑑𝜓\tag{5.83}$$

If instead of the damping factor $ξ$ is used the oscillator quality factor $𝑄 = 1/(2ξ)$, formula 5.83 becomes:

$$𝑅=\Big(\frac{𝑑𝛾}{𝑑𝑡}\Big)_{𝑆𝑅} = −\frac{𝜔_0}{2𝑄}∙\int_{0}^{2𝜋}𝜇(𝛷,𝜓)\sin 𝜓𝑑𝜓 \bigg/ \int_{0}^{2𝜋}𝜇(𝛷,𝜓)\cos 𝜓𝑑𝜓\tag{5.83}$$

The expressions under integrals in the formulas (5.82), (5.83) and (5.84) are functions of the $ψ$ and oscillation amplitude $Φ$. In these expressions, the coordinate transformation (5.78) can be performed, so that they are defined as functions of the angular coordinates $φ$ and the oscillation amplitude $Φ$. In accordance with the fact that in the stationary clock mode $Φ̇_{𝑆𝑅} = 0$, the amplitude of the oscillation of the balance point is constant $Φ_{𝑆𝑅} = 𝑐𝑜𝑛𝑠𝑡$, from formula (5.80) directly follows:

$$𝛷^2=\frac{1}{2𝜋𝜉𝜔_0^2}\oint_{𝛷}𝜇(𝜑)𝑑𝜑\tag{5.85}$$

By substituting the expression (5.85) in (5.81), we obtain the following formula for the error of the average - impulse mechanism in the quasi-oscillation mode of the oscillation of the balance point:

$$𝑅=\Big(\frac{𝑑𝛾}{𝑑𝑡}\Big)_{𝑆𝑅} = −𝜉𝜔_0∙\oint_{𝛷}\frac{𝜇(𝜑)𝜑 𝑑𝜑}{\sqrt{𝛷^2−𝜑^2}} \bigg/ \oint_{𝛷}𝜇(𝜑)𝑑𝜑 \tag{5.86}$$

If instead of the damping factor $ξ$ is used the oscillator quality factor $𝑄 = 1/(2ξ)$, formula 5.86 becomes: $$𝑅=\Big(\frac{𝑑𝛾}{𝑑𝑡}\Big)_{𝑆𝑅} = −\frac{𝜔_0}{2𝑄}∙\oint_{𝛷}\frac{𝜇(𝜑)𝜑 𝑑𝜑}{\sqrt{𝛷^2−𝜑^2}} \bigg/ \oint_{𝛷}𝜇(𝜑)𝑑𝜑 \tag{5.87}$$

It is emphasized once again that the expressions (5.75) and (5.80), (5.76) and (5.81), (5.82) and (5.85), (5.83) and (5.86), as well as (5.84) and (5.87) different, but essentially equivalent. In the first group, the integration formula is done by the coordinates of the total oscillation phase $ψ$, and in the second by the angular coordinate $φ$. Thus, the given formula pairs describe the same phenomena, only in different coordinate systems. Which expressions will be used to solve specific problems, this depends primarily on the degree of their formal complexity and, in essence, is a matter of free choice.

It is also necessary to discuss the formation of the zero approximation solution (5.59) of the differential equation (5.57) for the given initial conditions (5.58). Let the integral solution be in the equation (5.75), that is, in the equation (5.80) of the function $𝐹(Φ)$:

$$\int_{0}^{2𝜋} 𝜇(𝛷,𝜓)\cos 𝜓𝑑𝜓 = \oint_{𝛷} \frac{𝜇(𝜑)𝑑𝜑}{𝛷} = 𝐹(𝛷).\tag{5.88}$$

Then the differential equation (5.22) and (5.27) read as follows:

$$𝛷̇_{𝑆𝑅}=\Big(\frac{𝑑𝛷}{𝑑𝑡}\Big)_𝑆𝑅 = −𝜉𝜔_0𝛷 + \frac{𝐹(𝛷)}{2𝜋𝜔_0}.\tag{5.89}$$

The solution of the differential equation (5.36) is the function $Φ = Φ (𝐶, 𝑡)$, in which $𝐶$ is the integration constant. At the starting point $𝑡 = 0, Φ(𝐶, 0) = Φ_0$, from which the integration constant $𝐶$ is determined as the oscillation amplitude function $𝐶 = A (Φ_0)$.

Let the integral solution be in the equation (5.76) and (5.81) of the function $G(Φ)$:

$$\int_{0}^{2𝜋} 𝜇(𝛷,𝜓)\sin𝜓 𝑑𝜓= \oint_{𝛷} \frac{𝜇(𝜑)𝜑 𝑑𝜑}{𝛷\sqrt{𝛷^2−𝜑^2}} = 𝐺(𝛷).\tag{5.90}$$

When the solution of the differential equation (5.89) $Φ(𝐶, 𝑡) = Φ(A (Φ0), 𝑡)$ and the solution of the integral (5.90) $G(Φ) = 𝐺(Φ ( A (Φ_0), 𝑡))$, the following equation is obtained:

$$𝛾̇_{𝑆𝑅} = \Big(\frac{𝑑𝛾}{𝑑𝑡}\Big)_{𝑆𝑅} = −\frac{𝐺(𝛷(А(𝛷_0),𝑡))}{2𝜋𝜔_0𝛷(А(𝛷_0),𝑡)}.\tag{5.91}$$

The solution of the differential equation (5.91) is the function $γ(𝑡) = γ_1 (𝑡) + γ_0$, in which $γ_0$ is the integration constant (angle of the initial phase difference). In accordance with the relation (5.63), the initial condition $φ̇ (0) = 0$ is:

$$𝜑̇(0) = 𝛷(А(𝛷_0),0)∙𝜔_0\cos(𝛾_1(0)+𝛾_0)=0,\tag{5.92}$$

from which immediately follows $γ_0 = π/2-γ_1(0)$.

When the integration integrals obtained are assigned to the relation (5.59), the zero approximation solution $φ(𝑡)$ can be written in the form:

$$𝜑(𝑡)=𝛷(А(𝛷_0),𝑡)∙\cos(𝜔_0𝑡+𝛾_1(𝑡)−𝛾_1(0)).\tag{5.93}$$

The error of the approximate solution is calculated as the difference of the exact solution $φ_𝑇(𝑡, ξ)$ of the differential equation (5.57) and its zero approximation $φ(𝑡, ξ)$. The magnitude of this error, which evaluates the accuracy of approximation,

$$\big|𝜑_𝑇(𝑡,𝜉)−𝜑(𝑡,𝜉)\big| = О(𝜉),\tag{5.94}$$

is equal to the order of the size $O(ε)$ of the first excluded member in the perturbation asymptotic development of the exact solution. In Chapter 3 of this thesis it was pointed out that Krylov and Bogoliubov have proved that there are constants $𝑐, ε_1$ and $𝑡_𝑀$ such that the error of approximation (5.94) satisfies the following condition:

$$\big|𝜑_𝑇(𝑡,𝜉)−𝜑(𝑡,𝜉)\big| ≤ 𝑐∙𝜉,\,\,\text{for}\,\, 𝑡∈[0,𝑡_𝑀/𝜉]\,\,\text{and for each}\,\,𝜉_1 ≤ 𝜉,\,\,𝜉∈(0,1).\tag{5.95}$$

The approximation validity interval $𝑡∈ [0, 𝑡_𝑀/ξ]$, as well as in the double time scale method, has been extended to the order of magnitude of the attenuation coefficient $ξ$. For the fixed value of the parameter $ξ$, these conclusions are also valid for each other value of the parameter $ξ_1≤ ξ$, respectively, the approximation is more accurate and valid for a longer time interval, if the number is less, respectively, the attenuation is weaker and the energy dissipation is lower. The position (5.95) confirms that the approximate solution $φ(𝑡, ξ)$ asymptotically approaches the exact solution $φ_𝑇(𝑡, ξ)$ at the time interval $𝑡∈ [0, 𝑡𝑀/ξ]$. This confirms the correctness of the perturbation method of Krylov and Bogoliubov as one possible heuristic procedure in the approximate solution of differential equations of the form (5.10).

The relationship between the perturbation method of Krylov and Bogoliubov and the two time conditions described in Chapter 3 of this dissertation will be discussed. First of all, the identity of all approximate solutions and the full consensus of the estimation of the size of the error of approximation and the order of the size of the time interval, on which this approximation is valid by both methods, is noted. In particular, the differential equations (5.75) and (5.40), (5.80) and (5.46) are identical, whose solutions determine the dependence of the rate of change in the amplitude of the forced damped oscillations of the balance point from the oscillation amplitude and the forced force of the force acting on the balance wheel. Also identical are the differential equations (5.76) and (5.41), (5.81) and (5.47), whose solutions describe the dependence of the rate of change in the phase difference of the forced damped oscillations of the balance point from the amplitude of the oscillations and the force of force acting on the balance wheel. It has already been explained and emphasized that the relations (5.76) and (5.41), (5.81) and (5.47) describe the change in the angular frequency $ω_0$ of the oscillator oscillation oscillator under the influence of the force forced moment, which is called the error of the mean - impulse mechanism. The approximate solutions (5.93) and (5.36) of the differential equation of oscillation of the clock mechanism oscillator are also mutually consistent, as well as estimates of the order of the size of the error approximation (5.95) and (5.39) respectively the order of the time interval in which these approximations are valid.

5.4 Operational formula for faults of the impulse mechanism

In this chapter of the thesis, the accuracy of the formula for errors of the mean-impulse mechanisms in the quasi-oscillation regime of oscillation of the balance point will be checked, which are carried out using the theory of the perturbation account. In particular, using the formulas (5.82) and (5.83) respectively (5.84), the angular frequency $ω_0$ oscillations of the balance point (oscillator) were calculated under the influence of the forced moment of force generated by two different models of the impulse mechanisms. Then, 3D models of the balance point and corresponding impulse mechanisms were simulated, simulations of their work were created and, on the basis of the results of these simulations, certain changes in the angular frequency of the $ω_0$ oscillator were also made. By comparing theoretical results with the results of the simulation, the accuracy of the formulas (5.82), (5.83) and (5.84) is checked. Balance point models and pulse mechanisms, as well as the process of simulation and motion analysis, were achieved using the SolidWorks 2016 application.

Taking into account only the dynamic characteristics, all the beam-pulse mechanisms can be divided into three large groups: bradycrons, tachycrons and isochrons. Bradychrone strains prolong, tachycrons shorten, and isochrons do not change the period of their own oscillations of the balance point. In brachyron, obstructions include numerous types of impulse mechanisms that are installed as stationary timepieces (for example, Grejemova and Amman-Lepotova peaceful restraint), as well as watches (for example, duplex and cylindrical respectively free - Swiss and English lumbar .) As already explained, their common feature is to generate a periodic moment of a phase that is delayed by the angular speed of rotation of the oscillator (balance point or pendulum). In other words, bradycrone interruptions supply the oscillator with energy asymmetrically with respect to the equilibrium position in such a way that half the total amount of energy per oscillator half-life is passed to the oscillator only after its passage through the equilibrium position. The tachyron wounds include all of the oblique-immanent back-twist mechanisms, of which the best known (or anchor) Klement-Hukov regulator and Harrison's "grasshopper", as well as crown-point wrenches that were built into pocket watches until mid-19th. century. In tachyron wrenches, all gravitational, impulse mechanisms are also provided, which are supplied with large, stationary and tower watches, among which is certainly the most famous Denison double-triangular gravity regulator of the walk. Their common feature is to generate a periodic moment of momentum that is in phase with respect to the angular speed of rotation of the oscillator. In other words, the tachyron wires supply the oscillator with energy asymmetrically with respect to the equilibrium position in such a way that half the total amount of energy per half-life of oscillation is delivered to the oscillator before its passage through the equilibrium position. In isochronous impediments, there are few chronometers (Arnold and Ernsho's breakthroughs) that impulse the balance point in only one half-period of oscillation of the balance point. As this impulse can be centered centrally so that it is in phase with the angular velocity oscillation of the balance point, their main feature is to supply the oscillator with energy symmetrically with respect to the equilibrium position in such a way that half the total amount of energy per oscillator half-circle is passed to the oscillator precisely at the moment of his passing through the equilibrium position. In this chapter of the dissertation, two mathematical models of impulse mechanisms were created and presented. The first is the model of a free breakthrough that passes on to the short-time impulse oscillator in each half-period of its oscillation with a bradycar effect on the course of the clock mechanism. The second model mathematically describes a reverse twist to prevent intermittent interaction with the oscillator and produces a tachyron effect on the timer stroke.

The free obstacle, as already pointed out, acts on the balancing wheel with a discrete, periodic moment of force that phase late for the angular velocity of the balance point. Figure 39 shows the angular velocity $𝜑$ of the balance point and the specific momenta of the force $μ$, both in the function of the oscillation phase $ψ$. The specific momentum $μ(ψ)$, constant intensity $μ_0$, phase late at angular velocity $φ̇ (ψ)$ in such a way that it acts on the balance wheel only at intervals $[ψ_1 + 𝑘π, ψ_2 + 𝑘π], 𝑘 = 0,1,2,3 ...$ and alternately changes direction. In this case, it was assumed that the intensity of the specific moment of the force $μ$ is constant, and in practice it can be variable as a specific function $μ = μ(φ)$ of each particular mechanism. Assuming that the amplitude during the oscillation period is constant, the diagram (40) shows the phase diagram $(φ̇, φ) = (φ̇ (ψ), φ(ψ))$ of the forced damped oscillations of the balance point on which the moments of the free percussion force act.

Figure 39 Diagram of angular velocity $φ$ balance point and specific moment of momentum $μ(ψ)$ in the function of oscillation phase $ψ$

Figure 40 Phase diagram of forced balancing oscillations of the balance point on which the moments of the free percussion force

As shown in the phase diagram, this type of a mean impulse mechanism interacts with the oscillator in two narrow segments of the coordinates of the angular displacement of the balance point $φ∈ [φ_0-α, φ_0 + α]$ and $φ∈ [-φ_0-α, -φ_0 + α]$, and acts on it with constant moments of force, the same intensity, but the opposite direction. The angles $φ_0$ and $α$ are constructive characteristics of the average impulse mechanism. The relationship between the angular coordinates and the oscillation phase is given by the following relations:

$$\begin{split}𝜓_1 &= 𝑎𝑟𝑐𝑠𝑖𝑛 {𝜑_0−𝛼}{𝛷}, \\ 𝜓_2 &= 𝑎𝑟𝑐𝑠𝑖𝑛 \frac{𝜑_0+𝛼}{𝛷}.\end{split}\tag{5.96}$$

If the oscillation amplitude is constant and the impulse of the impedance is short-lived ($α≪1$), then the following approximation can be accepted:

$$𝜓_0 = \frac{𝜓_1 + 𝜓_2}{2} = \frac{1}{2}\Big(𝑎𝑟𝑐𝑠𝑖𝑛 \frac{𝜑_0 − 𝛼}{𝛷} + 𝑎𝑟𝑐𝑠𝑖𝑛 \frac{𝜑_0 + 𝛼}{𝛷}\Big),\tag{5.97}$$

which means that the angular center of the impulse approximately corresponds to the phase.

The formula, derived in chapter 5, which defines the error of the average impulse mechanism in the quasi-oscillation mode of the oscillation of the balance point, is:

$$𝑅=\Big(\frac{𝑑𝛾}{𝑑𝑡}\Big)_{𝑆𝑅} = −\frac{𝜔_0}{2𝑄}∙\oint_{𝛷} 𝜇(𝜑)𝜑𝑑𝜑 \bigg/ \sqrt{𝛷^2−𝜑^2} \oint_{𝛷}𝜇(𝜑)𝑑𝜑,\tag{5.98}$$

in which the $Φ$ quasistation amplitude of the oscillator is determined by the following relationship:

$$𝛷_2 = \frac{𝑄}{𝜋𝜔_0^2}\oint_{𝛷}𝜇(𝜑)𝑑𝜑.\tag{5.99}$$

Therefore, in formula (5.98) it is necessary to determine the solution of the following integral:

$$\oint_{𝛷} \frac{𝜇(𝜑)𝜑𝑑𝜑}{\sqrt{𝛷^2−𝜑^2}} = 𝜇_0 \int_{𝜑_0−𝛼}^{𝜑_0+𝛼}\frac{𝜑𝑑𝜑}{+\sqrt{𝛷^2−𝜑^2}} − 𝜇_0 \int_{−𝜑_0+𝛼}^{−𝜑_0-𝛼} \frac{𝜑𝑑𝜑}{-\sqrt{𝛷^2−𝜑^2}},\tag{5.100}$$

and it reads:

$$\oint_{𝛷} \frac{𝜇(𝜑)𝜑 𝑑𝜑}{\sqrt{𝛷^2−𝜑^2}} = 2𝜇_0\Big(\sqrt{𝛷^2−(𝜑_0−𝛼)^2}−\sqrt{𝛷^2−(𝜑_0+𝛼)^2}\Big).\tag{5.101}$$

In the formulas (5.98) and (5.99) it is necessary to determine the solution of the following integral:

$$\oint_{𝛷} 𝜇(𝜑)𝑑𝜑,\tag{5.102}$$

which reads as follows:

$$\oint_{𝛷} 𝜇(𝜑)𝑑𝜑 = \int_{𝜑_0−𝛼}^{𝜑_0+𝛼} 𝜇_0𝑑𝜑 − \int_{−𝜑_0+𝛼}^{−𝜑_0−𝛼} 𝜇_0𝑑𝜑 = 4𝜇_0𝛼\tag{5.103}$$

In accordance with the solution (5.101) of the integral (5.100) and the solution (5.103) of the integral (5.102), the operational formula on the basis of which the free-break error can be effectively calculated is given by the following expression

$$𝑅=\Big(\frac{𝑑𝛾}{𝑑𝑡}\Big)_{𝑆𝑅} = −\frac{𝜔_0}{4𝑄}∙\frac{\big(\sqrt{𝛷^2−(𝜑_0−𝛼)^2}−\sqrt{𝛷^2−(𝜑_0+𝛼)^2}\big)}{𝛼}.\tag{5.104}$$

As already emphasized, it determines the change in the angular frequency of its own oscillations of the balance point under the bradycar effect of the force of force generated by a free average pulse mechanism. In accordance with the solution (5.103) of the integral (5.102), the operational formula on the basis of which the amplitude can be calculated in the quasi-oscillation regime of the oscillation of the balance point is:

$$𝛷=\frac{2}{𝜔_0}\sqrt\frac{𝑄𝜇_0𝛼}{𝜋}\tag{5.105}$$

Formula (5.104) is checked directly, and formula (5.105) indirectly, results of simulation and analysis of the movement of the corresponding 3D model of the free average pulse mechanism generated by the application SolidWorks 2016.

The reciprocating impulse mechanism, as already pointed out, acts on the balance wheel with a continuous, periodic moment of a force that is phase-ahead in relation to the angular velocity of the balance point. Figure 41 shows the angular velocity $𝜑̇ $ of the balance point and the specific torque $μ$, both in the function of oscillation phase $ψ$. The specific momentum $μ(ψ)$, constant intensity $μ_0$, acts on the balance wheel at intervals $\big[\big(\frac{π}{2}-ψ_0\big) + 𝑘π, \big(\frac{3π}{2}-ψ_0\big) + 𝑘π\big], 𝑘 = 0,1,2,3 ...$ and alternately it changes direction. The intensity of the specific moment of force $μ$ is chosen to be constant in this case, and in practice it can also be a variable and represents the specific function $μ = μ(φ)$ of the very mechanism itself. Assuming that the amplitude during the oscillation period is constant, the diagram (42) shows the phase diagram $(φ̇, φ) = (φ̇ (ψ), φ (ψ))$ of the forced damped oscillations of the balance point on which the momentum forces of the mean-impulse reverse twitch mechanism.

Figure 41 Diagram of angular velocity φ ̇ of the balance point and the specific moment of the force μ (ψ), both in the function of oscillation phase ψ

Figure 42 Phase diagram of compulsory damped oscillations of the balance point on which the forces of the ZI mechanism of the reverse twisting force

As shown in the phase diagram, this type of pulse mechanism is continuously in interaction with the oscillator in the following segments of the coordinates $φ$ of the angular displacement of the balance point:

$$\begin{split}&(−𝛷)→(+𝜑_М),𝜇_0>0, \\ &(+𝜑_М)→(+𝛷),𝜇_0<0, \\ &(+𝛷)→(−𝜑_М),𝜇_0<0, \\ &(−𝜑_М)→(−𝛷),𝜇_0>0\end{split} \tag{5.106}$$

and acts on it with moments of force of constant intensity, but of varying direction. The angular coordinate interval $φ∈ (-φ_M, + φ_M)$ is a constructive characteristic of all reversible twisting pulses and represents the angle of pivoting of the pallet of an average impulse mechanism with the teeth of the average point. It corresponds to the minimum amplitude of oscillation of the balance point or pendulum, which ensures the continuous operation of the breaker. If the amplitude of the oscillator would be smaller than the patch angle, the obstruction could not function and the clock mechanism would stop.

Therefore, in order to determine the error of a normal pulse mechanism with a return twist, it is necessary to determine in the formula (5.98) the solution of the following integral:

$$\begin{split}\oint_{𝛷} \frac{𝜇(𝜑)𝜑 𝑑𝜑}{\sqrt{𝛷^2−𝜑^2}} &= +𝜇_0\int_{−𝛷}^{+𝜑_М} \frac{𝜑𝑑𝜑}{+\sqrt{𝛷^2−𝜑^2}} − 𝜇_0 \int_{+𝜑_М}^{+𝛷} \frac{𝜑 𝑑𝜑}{+\sqrt{𝛷^2−𝜑^2}} \\ &− 𝜇_0\int_{+𝛷}^{−𝜑_М} \frac{𝜑 𝑑𝜑}{−\sqrt{𝛷^2−𝜑^2}} + 𝜇_0\int_{−𝜑_М}^{−𝛷} \frac{𝜑 𝑑𝜑}{−\sqrt{𝛷^2−𝜑^2}} \end{split} \tag{5.107}$$

and it reads:

$$\oint_{𝛷}𝜇(𝜑)𝜑𝑑𝜑\sqrt{𝛷^2−𝜑^2} = −4𝜇_0\sqrt{𝛷^2−𝜑_М^2}.\tag{5.108}$$

In the formulas (5.98) and (5.99) it is necessary to determine the solution of the following integral:

$$\oint_{𝛷} 𝜇(𝜑) 𝑑𝜑 = +𝜇_0\int_{−𝛷}^{+𝜑_М}𝑑𝜑 − 𝜇_0\int_{+𝜑_М}^{+𝛷}𝑑𝜑 − 𝜇_0\int_{+𝛷}^{−𝜑_М}𝑑𝜑 + 𝜇_0\int_{-𝛷}^{-𝜑_М}𝑑𝜑,\tag{5.109}$$

which reads as follows:

$$\oint_{𝛷} 𝜇(𝜑)𝑑𝜑 = 4𝜇_0𝜑_М\tag{5.110}$$

In accordance with the solution (5.108) of the integral (5.107) and the solution (5.110) of the integral (5.109), the operational formula on the basis of which it is possible to calculate the error of the average impulse mechanism with a return twist is given by the following expression:

$$𝑅=\Big(\frac{𝑑𝛾}{𝑑𝑡}\Big)_{𝑆𝑅} = +\frac{𝜔_0}{2𝑄}∙\frac{\sqrt{𝛷^2−𝜑_М^2}}{𝜑_М}.\tag{5.111}$$

As already emphasized, the relation (5.111) determines the change in the angular frequency of its own oscillations of the balance point under the tachyron effect of the forced moment of force generated by a normal impulse mechanism with a back twist. From the same formula (5.111), it immediately assumes that the reversible twisted-line operation works correctly if the oscillation amplitude $Φ$ is greater than or equal to the angle of $φ_M$ patching. Only in the limit case $Φ = φ_M$ this type of breaker works without reversion and does not generate an error, that is, change its own frequency of the oscillator. The fault error exists if $Φ > φ_M$ and then the oscillator works with a back twist. In practice, an oscillator coupled with an average impulse mechanism with reverse twitching is always supplied with so much energy that it is $Φ > φ_M$, which ensures a safe and steady travel of the clock mechanism. In accordance with the solution (5.110) of the integral (5.109), the operational formula on the basis of which the amplitude can be calculated in the quasi-oscillation mode of the oscillation of the balance point is:

$$𝛷 = \frac{2}{𝜔_0}\sqrt{\frac{𝜑_М𝑄𝜇_0}{𝜋}}\tag{5.112}$$

The formula (5.111) is checked directly, and the formula (5.112) indirectly, the results of the simulation and the analysis of the movement of the corresponding 3D model of the average impulse mechanism with the back twist generated by the SolidWorks 2016 application.