7. 3D model of clock mechanism and simulation of work

7.1 3D timer model

Previous chapters provide descriptions and calculations of individual parts of the clock mechanism. Chapter 2 shows the development of clock mechanisms throughout history, Chapter 4 provides insights into two types of oscillators used in watches. Chapter 5 "Spit-impulse mechanisms" shows the budget and gives the order of magnitude for the error of the average impulse mechanism. Also, Chapter 5 gives you the most commonly used time-impulse mechanism for timer design. In order to demonstrate that the error of the average-impulse mechanism corresponds to the size described in chapter 5, it is necessary to create an adequate model in which a simulation of the timer operation could be performed and measures of possible errors occurring. In works [71] and [72] it was shown that the simulation of the work of various mechanisms in the SolidWorks software package justified the accuracy and showed that it was possible to perform simulations.

The timer model used for simulations is shown in Figures 46-57. The need for modeling the timer is the statement in the desire to check, first of all, the geometric validity of all parts of the clock. Geometric characteristics are a direct product of the kinematic requirements of the clock itself. From the kinematic side, the clock is an oscillator that supplies the timer gear with a uniform motion through the impulse mechanism, while the gears transfer the potential energy of the weights into the kinetic energy. Referring to Figures 52, 53 and 54, a complete clock is shown, and in Figures 55-63, individual sub-assemblies are integrated into one whole.

The timetable consists of the following sub-folders:

- A fixed frame
- Klatno
- An impulse impulse mechanism - "Grasshopper"
- Portable groups
- Remontoire
- Winding mechanism.

A fixed frame (Figure 63) consists of two supports between which the letter is in the form of the letter "T". The ram is designed to allow the timer to hang onto the wall.

As an oscillator, for the purposes of this simulation, a pendulum is used (Figure 64) which has a oscillation period of 2 seconds. Only the pendulum is fitted with a temperature compensation system, as discussed in chapter 4.4.1. Temperature compensation of the pendulum. Also, the selection of the pendulum as an oscillator is done for practical reasons because it is easier to simulate than the balance wheel. The pendulum is hung about a stationary frame and hangs hanging on a feather.

Due to the property not requiring lubrication, the famous "Grasshopper" John Harrison (John Harrison, 1693-1776) was selected. Harrison used this impulse mechanism in the first marine chronometer "N1" and thus solved the "problem of longitude" [20]. In Figures 59 and 65 this impulse-mechanism is shown. The contact between the pallet and the average point was achieved using the "contact surface" command and real contact between the surfaces on the pallets and the pins at the average point is displayed. The pendulum is, as already mentioned, a pendulum whose oscillation period lasts 2 seconds. Accordingly, the average wheel has 30 trunks and in this way, the average wheel with the help of "Grasshopper" makes one turn in 60 seconds.

The transmission group (Figure 60) is made up of gears that are paired to transmit certain relationships from the average pulse mechanism to the time display subsystem. A more detailed view of this group can be found in chapter 6.2 "Transmission mechanism". In section 6.2, the module and the number of teeth of each gears of this group are given. Real-time simulation has shown that gear selection is good and that it meets the required requirements.

The timing winding mechanism supplies the entire clock mechanism with the required mechanical energy and is shown in Figure 61. As already mentioned in Chapter 6.3, the entire mechanism is designed to allow winding the timer while the timer is running, ie, without stopping the timer mechanism.

Remontoire is explained in detail in Section 6.1 and is an auxiliary source of energy for the pulse mechanism. In Figure 56, a remontoire is included in the timer.

A clock simulation was conducted during which all parts of the clock showed that they are geometrically correct and that the kinematic connections set in the SolidWorks software package correspond to the real conditions in which the clock operates. As a lack of these simulations, it can be stated that this software package does not have the ability to check the operation of the mechanism in conditions when the temperature and pressure change so that it remains the subject of future research.

Figure 52 Model clock

Picture 53 Model clock

Picture 54 Model of the whole hour

Figure 55 Clock model details

Fig. 56 Mechanisms within the timer

Fig. 57 Mechanisms within the timer

Fig. 58 Mechanisms within the timer

Figure 59 Remontoire and the mean-impulse mechanism

Figure 60 Portable group

Figure 61 Winding mechanism

Picture 62 Remontoire

Figure 63 Nepomic frame clock

Picture 64 Pendulum

Figure 65 Grasshopper Impulse Mechanism

7.2. Results of the simulation

7.2.1 3D models of obstacles and balancing circuits

As the error of the average-impulse mechanism is generated in the interaction of the average-impulse mechanism with an oscillator of the clock, it is necessary to create the whole set of the breakthrough and the balance point and perform an analysis of their movement as a dynamic whole. It is emphasized that the models are abstract, created to materialize in the previous chapter the derived mathematical formulas (5.104) and (5.111). Simulation and analysis of the movement of the circuit breaker and oscillator was performed using the "Event Based Motion Study" method using the SolidWorks 2016 application, which is based on controlling the state and activity of the mechanism with a group of specially selected sensors [73]. In order for these formulas to be correctly verified by analyzing the results of the computer simulation, it is necessary to first determine and adopt appropriate simulation parameters.

A model of the wheelbase assembly with spiral spring and a free-of-charge (such as Switzerland or England cranked), the error of which is described by formula (5.104), is shown in Figure 66.

Figure 66 Balancing wheel assembly model

As already emphasized, this type of welding acts on the balancing wheel with discrete impulses, whose parameters are: $φ_0 = + 12°, α = ± 3°$ and mean that the impedance acts as a moment of force on the oscillator in the segment of the angular coordinate $φ ∈ (+9°, +15°)$. It is assumed that the oscillation amplitude is $Φ = ±270°$. The balance wheel (1) has five different proximity sensors at position (2). The assembly of the pulse mechanism consists of two pallets, (3) and (4), and five activators of the proximity sensors (5) - (9). Four sensor activators (5) - (8) activate specially defined proximity sensors that can change and control the positions of the pallets (3) and (4). The proximity sensor activator at the position (9) detects the transition of the balance point through the equilibrium position $φ = 0°$.

The model of the wheelbase assembly with the spiral spring and the reverse twisting pulse mechanism (such as "grasshoper" or a crown wheel), the error of which is described by formula (5.111), is shown in Figure 67.

Figure 67 Balanced wheel assembly model

In this example, it is assumed that the pivot angle of the pallet with the teeth of the mean point is $φ_𝑀 = ±45°$ and the oscillation amplitude is $Φ = ±90°$. The balance wheel (1) carries three different proximity sensors at the same position (2). The welding assembly has two pallets (3) and (4), which have the function of an activator of the corresponding sensors. These activators activate two proximity sensors to control and change the positions of the pallets (3) and (4). Since the average impulse mechanism with reverse spinning constantly interacts with the oscillator and thus transfers to the balance point the external torque of the moving direction without interruption, the sensor at the position (2) activated by the actuators at positions (3) and (4), changes only the direction of the force . The proximity sensor activator at position (5) detects the transition of the balance point through the equilibrium position $φ = 0°$.

In both models described, the positions of the pallets (3) and (4) are changed by changing the appropriate mates of the SolidWorks application and controlled by the sensors located in the specially-defined position of the balance point which performs cyclic rotation.

The oscillator, spiral-spring balancing wheel, for both models of impulse motors, was created with the following characteristics:

- Material: beryllium bronze, density $ρ = 0.0083 g/𝑚𝑚^3$
- The moment of inertia of the balance point: $𝐽 = 180000.0 𝑔𝑚𝑚^2 = 1.8 ∙ 10^{-4}𝑘𝑔𝑚^2$
- Frequency of own damped oscillations of the balance point: $ν = 4s^{-1}$
- Coil spring constant: $𝑘 = 4π^2ν_0^2 ∙ 𝐽 = 0.11369784 𝑁𝑚/𝑟𝑎𝑑 = 1.98440171 𝑁𝑚𝑚/𝑑𝑒𝑔$
- Coefficient of attenuation: $c = 2π ∙ 𝐽 ∙ ν_0/𝑄 = 2.261947 ∙ 10-5 𝑁𝑚/(\frac{𝑟𝑎𝑑}{𝑠}) = 0.000394784 𝑁𝑚/= 𝑁𝑚𝑚/(\frac{𝑑𝑒𝑔}{𝑠})$
- oscillator quality factor $𝑄 = 2π ∙ 𝐽 ∙ ν_0/𝑐 = 200$
- The moment of force for the free breakthrough: $M = J ∙ μ_0 = J ∙ π^3 ∙ ν_0^2 ∙ Φ^2 / Q ∙ α = 0.189363273 𝑁𝑚 = 189.3632733 𝑁𝑚$
- moment of force with reverse spinning force: $M = J ∙ μ_0 = J ∙ π^3 ∙ ν_0^2 ∙ Φ^2 / Q ∙ φ𝑀 = 0.001402691Nm = 1.402690911 Nm$

The moment of free-force force momentum is calculated from the formula (5.105) based on the previously adopted stationary amplitude of the oscillation of the balance point $Φ = ± 270°$ and the angle $α = ± 3°$ as the constructive characteristics of the breakthrough. The moment of force for the reverse twisting force was calculated using the formula (5.112), based on the previously adopted stationary amplitude of the oscillation of the balance point $Φ = ± 90°$ and the coupling angle $φ_𝑀 = ± 45°$ which also represents the constructive characteristic of the breakthrough.

It is necessary to emphasize that the free interference acts on the balance wheel with short-term, discrete moments of force, and the reverse-spinning barrier is continuous.

7.2.2. Flow of simulation and analysis of oscillator motion

An analysis of the oscillator motion (balance point) was performed using the "Event Based Motion Study" method using the SolidWorks application. This method is based on the control of the flow of simulation to the characteristic events that occur during the movement of the mechanism and is complementary to the simulation based on the flow of time. Since the effect of the average impulse mechanism on the balancing wheel is determined and controlled by the movement of the balance point itself (the position of the oscillator is determined by the angular coordinate $φ$), it is appropriate that the simulation and analysis of oscillator motion is realized precisely by the "Event Based Motion Study" method. In particular, this method was used to analyze the complex dynamic behavior of the assemblies of two models of the breaker and the oscillator, and thus determined the numerical values ​​of the so- errors of the impulse mechanisms.

The parameter setting for the "Event Based Motion Study" of the free breaker and oscillator circuit is shown in Figure 68.

Figure 68 Parameters of "Event based motion study" of the free breaker and oscillator circuit

Firstly, it is necessary to acquire the main characteristic of proximity sensors that play a key role in controlling the dynamic behavior of the circuit. Each proximity sensor, in a previously defined manner, detects whether its distance is less than the minimum of the corresponding component of the sensor - activator. Thus, the pallets (3) and (4) shown in FIG. 66 have the role of a proximity sensor activator located at position (2). The pallets (3) and (4) in the upper position activate the sensor at the position of the balance point (2), which is triggered by the action of the external moments of the force. The pallet (3) activates the moment of force in the clockwise direction (negative, indirect direction), and the pallet (4) is in the opposite direction from the clockwise direction (positive, direct direction). The position of the pallet (upper or lower) is changed by the sensor at position (2), controlled by actuators at positions (5) - (8). In particular, the actuator (5) places the pallet (3) in the lower position, and the actuator (6) is in the same position in the upper position. Symmetrically, the activator (7) places the pallet (4) in the upper position, and the actuator (8) is in the same position in the lower position. During the oscillatory rotation of the balance point, the sensors located at the position (2) activate the action of the external forces of force always in the direction of the angular velocity of the balance point. In accordance with the geometry of the modeled average impulse mechanism, the external moments of the force act on the balancing wheel exclusively in the segment of the angular coordinate φ∈ (+ 9°, + 15°).

The setting of the parameters for the "Event Based Motion Study" of the circuit of a counter-rotational impulse mechanism is shown in Figure 69.

Figure 69 Parameters of the "Event Based Motion Study" of the back-twisted-pulse mechanism of the circuit

The pallets (3) and (4) shown in figure (67) function as proximity sensing proximity sensors located at position (2). In the upper position, they activate the sensor at the position (2) of the rotary balance point and change the direction of the external torque of the force. The pallet (3) changes the direction of the moment from negative to positive (from indirect to direct) and palette (4) from positive to negative (from direct to indirect). In addition, the same pallets (3) and (4) activate the sensors at the position (2), which alternates their position (from the upper to the lower and vice versa). Pallet (3) sets its own position in the bottom, a position the pallet (4) above. Similarly, the pallet (4) sets its own position in the lower, and the position of the pallet (3) in the upper one.

For both sets of balancing point with a helical spring and the shaped bar simulation is played motor winding a spiral spring of the oscillator (balancing of wheels) for a period of one second, while each of the oscillator does not achieve the angular movement equal to the corresponding stationary amplitude $φ = Φ$, which is a value obtained by calculation. Balancing wheel in conjunction with the free escapement agile engine constant angular velocity of 45 rpm, until a $φ = 270°$, a balance wheel escapement in conjunction with the kickback agile engine constant angular velocity of 15 rpm, while not reached $φ = 90°$. After that, the engines were disconnected, and each balance wheel started to perform compulsory muffled oscillations for 9 seconds and during this time performed 36 full oscillations.

Simulation and analysis of oscillator motion (modeled balance point), the measurement of the duration of the oscillation oscillation oscillations, and the duration of the oscillation period of forced forced oscillations were performed under the effects of the external torque of the force generating both the free break and the reverse twist. From the above data, a change in the period of oscillation of the balance point is determined by the forced forces of force that generate both models of impulse mechanisms. By comparing the results obtained with the theoretical calculation with the results of the simulation, the correctness of the formulas made using the perturbation account was checked.

7.2.3 Results of simulation and analysis of oscillator motion

At the very beginning of the simulation process in the SolidWorks application, the oscillation period was measured, ie, the frequency of the free-damped oscillations of the balance point, which was incorporated in both oscillator assemblies and breakers (Balance Point and Free Breakpoints and Balance Point and Reverse Circuit Breaker). All results are shown in the tables in tables from T4 to T12.

The frequency of free-damped oscillations of the balance point in the free-per-stake basis was obtained on the basis of the data shown in tables T4, T5 and T6. By measuring and calculating the mean time in which the balance wheel passes through the equilibrium position ($φ = 0°$, the activator of the sensor (9)), the duration of the 72 half-lengths is determined, that is, the 36th period of oscillation of the balance point and the determined mean value of the angular frequency of these oscillations. It states:

$$𝜔_{01} = 25.13266316 𝑟𝑎𝑑/𝑠,\tag{7.1}$$

Accordingly, the average value of the oscillation period:

$$𝑇_{01} = 0.25000078 𝑠.\tag{7.2}$$

The frequency of the free-damped oscillations of the balance point in the frame with the reversible twisted turn signal was obtained on the basis of the data shown in tables T7, T8 and T9. By measuring and calculating the mean time in which the balance wheel passes through the equilibrium position ($φ = 0°$, the activator of the sensor (5)), the duration of the 72 half-ferries, respectively, the 36th period of oscillation of the balance point and the determined mean value of the angular frequency of these oscillations is also determined. It states:

$$𝜔_{02} = 25.13262744 𝑟𝑎𝑑/𝑠,\tag{7.3}$$

Accordingly, the average value of the oscillation period:

$$𝑇_{02} = 0.25000113 𝑠.\tag{7.4}$$

The frequency of compulsory damped oscillations of the balance point in the frame with the free breakthrough and error of the breaks was obtained on the basis of the data shown in tables T13, T14 and T15. Determination of the mean time in which the balance wheel passes through the equilibrium position (φ = 0°, sensor activator (9)) determines the duration of the 72 half-circuits, ie, 36 periods of compulsory damped oscillations of the balance point within the free interval. The mean value of the angular frequency of these oscillations is determined, and it is:

$$𝜔_1 = 25.12986985 𝑟𝑎𝑑/𝑠,\tag{7.5}$$

Accordingly, the average value of the oscillation period:

$$𝑇_1 = 0.250028565 𝑠.\tag{7.6}$$

The error of the free average impulse mechanism is calculated by definition as the difference between the angular frequency of the forced and angular frequencies of the free-damped oscillations of the balance point:

$$𝑅_1 = 𝜔_1 − 𝜔_{01} = 25.12986985 − 25.13266316 = −0.00279331 𝑟𝑎𝑑/𝑠.\tag{7.7}$$

The numerical value of the error of the free-flow-impulse mechanism performed by the theory of perturbation (5.104) is calculated according to the adopted parameters of the breaker from the following expression:

$$𝑅_{1Т} = −\frac{25.13266316}{4∙200}∙\frac{(\sqrt{270^2−(12−3)^2}−\sqrt{270^2−(12+3)^2})}{3},\tag{7.8}$$

and amounts to:

$$𝑅_{1Т} = −0.002795454 𝑠 \tag{7.9}$$

The relative difference in the numerical error values of the free breakthrough obtained from the theoretical calculation and the computer simulation was calculated from the following expression:

$$𝛿_1 = \Big|\frac{𝑅_{1Т}−𝑅_1}{𝑅_{1Т}}\Big| = \Big|\frac{−0.002795454+0.00279331}{0.002795454}\Big|,\tag{7.9}$$

and amounts to:

$$𝛿_1 = 7.66959∙10^{−4} < 0.08% \tag{7.9}$$

It can be concluded that the obtained value (7.9) is acceptable and that this numerical result of the computer simulation carried out proved the validity of the formula for the error of the free average impulse mechanism that was performed using the theory of the perturbation account.

The frequency of compulsory damped oscillations of the balance point in the frame with the reverse twist and the fault of the breaker was obtained based on the data shown in T10, T11 and T12 tables. Determination of the mean time in which the balance wheel passes through the equilibrium position ($φ = 0°$, the sensor activator (9), p. 67) determined the duration of the 72 half-circuits, that is, 36 periods of forced balancing oscillations of the balance point in the free percussion. The mean value of the angular frequency of these oscillations is determined, and it is:

$$𝜔_2 = 25.2413366 𝑟𝑎𝑑/𝑠,\tag{7.10}$$

Accordingly, the average value of the oscillation period:

$$𝑇_2 = 0.24892443𝑠.\tag{7.11}$$

The error of a mean pulse mechanism with a back twist is calculated by definition as the difference between the angular frequency of forced and angular frequencies of the free-damped oscillations of the balance point:

$$𝑅_2 = 𝜔_2 − 𝜔_0^2= 25.2413366 − 25.13262744 = 0.10870916 𝑟𝑎𝑑/𝑠.\tag{7.12}$$

The numerical value of the error of the reverse twisted pulse mechanism, carried out by the theory of perturbation (5.111), is calculated according to the adopted parameters of the breakthrough from the following expression:

$$𝑅_{2Т} = +\frac{25.13262744}{2∙200}∙\frac{\sqrt{90^2−45^2}}{45}.\tag{7.13}$$

and amounts to:

$$𝑅_{2Т} = +0.108827469 𝑠 \tag{7.14}$$

The relative difference in the numerical values of errors in the reverse twist obtained by theoretical calculation and computer simulation were calculated from the following expression:

$$𝛿_2 = \Big|\frac{𝑅_{2Т} − 𝑅_2}{𝑅_{2Т}}\Big|=\Big|\frac{0.108827469−0.10870916}{0.108827469}\Big|,\tag{7.15}$$

and amounts to:

$$δ_2 = 0.001087 < 0.11%\tag{7.16}$$

It can be noted that the obtained value (7.16) is acceptable and that this numerical result of the computer simulation carried out proved the validity of the formula for the error of the counter-rotational impulse mechanism that was performed using the theory of the perturbation account.

Table 4 Results of simulation for damped free oscillations of the balance point in the frame with a reverse twist (1 sec-3 sec.)

Table 5 Results of simulation for damped free oscillations of the balance point in the frame with a reverse twist (4 sec-6 sec.)

Table 6 Results of simulation for damped free oscillations of the balancing point in the frame with the reverse twist (7 sec-9 sec.)

Table 7 Results of the simulation for the damped free oscillations of the balance point in the free overload (1 sec-3 sec.)

Table 8 Results of the simulation for the damped free oscillations of the balance point in the frame with the free breakthrough (4 sec-6 sec)

Table 9 Results of the simulation for the damped free oscillations of the balance point in the free overload (7 sec-9 sec.)

Table 10 Results of the simulation for damped forced balancing points in the frame with a reverse twist (1 sec-3 sec.)

Table 11 Results of the simulation for damped forced balancing points in the frame with a reverse twist (4 sec-6 sec.)

Table 12 Results of the simulation for damped forced balancing points in the frame with a reverse twist (7 sec-9 sec.)

Table 13 Results of the simulation for the damped forced balancing of the balance point within a free breakthrough (1 sec-3 sec.)

Table 14 Results of the simulation for the damped forced balancing of the balance point in the frame with the free breakthrough (4 sec-6 sec.)

Table 15 Results of the simulation for the damped forced balancing of the balance point within a free-range interval (7 sec-9 sec.)

7.3 Physical interpretation of the error of the average impulse mechanisms

In the head (5.3), the theory of errors of the mean-impulse mechanisms was formulated and explained using a perturbation account, both by the procedure of double the time conditions, and by the method of Krylov and Bogoliubov. Formulas in a general integral form are defined which define the change in the angular frequency of oscillation of the clock mechanism oscillator under the influence of the forced moment of force by which the impulse mechanism acts on the oscillator. As already explained, this change in the angular frequency is called the error of the mean - impulse mechanism and represents a non - linear phenomenon to which this dissertation is partly dedicated. Both of these perturbation methods have led to identical formulas, and this consensus was certainly the first check of their validity. Further, in Chapter (5.4), using general formulas in an integral form, mathematical expressions for calculating errors generated by two specific types of impulse mechanisms are derived. The correctness of both these mathematical expressions and those general integral formulas was verified by computer simulation and the analysis of the motion of the 3D model of the corresponding average impulse mechanisms. The numerical results of these simulations showed a high degree of agreement with the results of the theoretical numerical calculations, which confirmed the correctness of mathematical expressions performed using the perturbation account. The rest is yet to expose, explain, and comment on the physical interpretation of the phenomenon of change in the angular frequency of the clock oscillator induced by the forced moment of the force of the mean-impulse mechanism.

The quasistation mode of compulsory damped oscillations of the clock oscillator (balancing point or pendulum) is considered, in which the dissipation energy of an oscillator is equal to the energy it receives from the impulse mechanism. As already explained in the previous chapter (5.3) of this dissertation, the differential equation of compulsory damped oscillations of the balance point (oscillator) is given by the expression:

$$𝐽𝜑̈+𝑐𝜑̇+𝑘𝜑=𝑀(𝜑),\tag{7.17}$$

wherein:

- $φ = φ(𝑡)$ of the angular coordinate defining the motion of the oscillator,
- $𝐽$ the moment of the inertia of the balance point, that is, the oscillator,
- $𝑐$ constant viscosity resistance, respectively, viscous moment of force $𝐷 = -𝑐φ̇$,
- $𝑘$ coefficient of stiffness of the spiral spring,
- $𝑀(φ)$ force force, which is a periodic function of angular coordinates $φ$, that is, time $𝑡$, $𝑀(φ(𝑡))$, resonance with oscillator oscillations (balance point).

The equation (7.17) can be transformed into the equation:

$$𝜑̈ + 2ξω_0𝜑 + ω_0^2φ = μ(φ),\tag{7.18}$$

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 force of force which, as well as $𝑀(φ)$, is a periodic function of angular coordinates $φ$, that is, time $𝑡$, in resonances with oscillator oscillations.

In accordance with the theory presented in Chapter 3 and the corresponding initial conditions $φ(0) = 0, φ̇ (𝑡) = φ̇_0$, the solution of the differential equation (7.18) for the quasistationary regime of the oscillation of the balance point can be described by the harmonic function:

$$𝜑(𝑡)=𝛷\sin(𝜔_0𝑡+𝛾(𝑡));\,\,\,\,𝜑(𝜓) = 𝛷\sin 𝜓,\tag{7.19}$$

in which $Φ$ amplitude, $γ(𝑡)$ phase difference oscillator oscillation, and $ψ$ oscillation phase $ψ = ω_0𝑡 + γ(𝑡)$. In the quasistation mode, the amplitude constant is $Φ = 𝑐𝑜𝑛𝑠𝑡$, and $γ(𝑡)$ describes the phenomenon of change in the angular frequency of the clock oscillator induced by the forced moment of the force of the impulse mechanism. In accordance with the relation (5.63), the angular velocity of the oscillations is given by the following expression:

$$𝜑̇(𝑡)=𝜔_0𝛷\cos(𝜔_0𝑡+𝛾(𝑡));\,\,\,\, 𝜑̇(𝜓) = 𝜔_0𝛷\cos 𝜓.\tag{7.20}$$

As already mentioned, the specific force force $μ$ is a periodic function of the angular coordinates $φ$, that is, time $𝑡$, and is always resonant with oscillations oscillations. It is accepted, without reducing the generality of this analysis, that the specific forced moment of momentum $μ$ is the harmonic function of the oscillation phase $ψ, μ = μ(ψ)$ which, in relation to the angular velocity of the oscillation $φ̇ (ψ)$, is phase-shifted to the angle $δ$. If the phase difference is positive $δ > 0, μ(ψ)$ is phase-delayed for $φ̇ (ψ)$ and is described by the following function:

$$𝜇 = 𝜇(𝜓) = 𝜇_0\cos(𝜓−𝛿),\tag{7.21}$$

in which $μ_0$ is amplitude. According to the derived relation (5.84), the error of the mean-impulse mechanism acting on the oscillator by the forced moment of force $𝑀(ψ) = 𝐽 ∙ μ(ψ)$ is calculated according to the following integral formula:

$$𝑅 = \frac{𝑑𝛾}{𝑑𝑡} = −\frac{𝜔_0}{2𝑄}∙\int_{0}^{2𝜋}𝜇(𝛷,𝜓)\sin 𝜓𝑑𝜓 Big/ \int_{0}^{2𝜋} 𝜇(𝛷,𝜓)\cos 𝜓𝑑𝜓.\tag{7.22}$$

The solution of the integral in the formula for the fraction of formula (7.22) is given by the following expression:

$$\int_{0}^{2𝜋} 𝜇(𝛷,𝜓)\sin 𝜓𝑑𝜓 = 𝜇_0∙\int_{0}^{2𝜋}\cos(𝜓−𝛿)\sin 𝜓𝑑𝜓 = 𝜋\sin 𝛿,\tag{7.23}$$

and the solution of the integral in the fraction of the fraction of formula (7.22) by the expression:

$$\int_{0}^{2𝜋} 𝜇(𝛷,𝜓)\cos 𝜓𝑑𝜓 = 𝜇_0∙\int_{0}^{2𝜋}\cos(𝜓−𝛿)\cos 𝜓𝑑𝜓 = 𝜋\cos 𝛿.\tag{7.24}$$

In accordance with solutions (7.23) and (7.24), the error of the average impulse mechanism in case the phase difference is positive $δ > 0$ is given by the following relation:

$$𝑅 = \frac{𝑑𝛾}{𝑑𝑡} = −\frac{𝜔_0}{2𝑄}∙tg𝛿.\tag{7.25}$$

In this case, the balance wheel oscillates with an angular frequency $ω$ which is less than $ω_0$:

$$𝜔 = 𝜔_0−𝑅.\tag{7.26}$$

From the relation (7.26) it follows immediately that in this case oscillation period is extended $𝑇 = 2π/(ω_0-𝑅)$, and the timer travel slows down. This is a bradyron effect of an error of a mean impulse mechanism.

If the phase difference is negative $δ < 0, μ(ψ)$ is phase-leading in relation to $φ̇ (ψ)$ and is described by the following function:

$$𝜇 = 𝜇(𝜓) = 𝜇_0\cos(𝜓+𝛿).\tag{7.27}$$

In this case ($δ < 0$), the formula (7.25) changes the sign, and the relation that defines the error of the mean pulse mechanism is:

$$𝑅=\frac{𝑑𝛾}{𝑑𝑡} = +\frac{𝜔_0}{2𝑄}∙tg𝛿.\tag{7.28}$$

Thus, in this case ($δ < 0$), the balancing wheel oscillates with an angular frequency $ω$ greater than $ω_0$:

$$ω = ω_0 + 𝑅.\tag{7.29}$$

From the relation (7.29) it follows immediately that in this case the period of scaling $𝑇 = 2π / (ω0 + 𝑅)$ is shortened, and the timer speed is accelerated. This is a tachyron effect of the error of a mean impulse mechanism. In the boundary isohronom case, when the phase difference does not exist $δ = 0, μ(ψ)$ is in phase with the angular oscillation speed $φ̇ (ψ)$, so the error of the mean pulse mechanism does not exist $𝑅 = 0$.

The restoration moment of the spiral spring force, the moment of the viscous friction force and the forced moment of force of the average impulse mechanism, operates on the timer oscillator. In the steady-state oscillation mode, according to (7.19), the restoration moment of the spiral spring force can be described by the following expression:

From the relation (7.29) it follows immediately that in this case the period of scaling $𝑇 = 2π / (ω_0 + 𝑅)$ is shortened, and the timer speed is accelerated. This is a tachyron effect of the error of a mean impulse mechanism. In the boundary isohronom case, when the phase difference does not exist $δ = 0, μ(ψ)$ is in phase with the angular oscillation speed $φ̇ (ψ)$, so the error of the mean pulse mechanism does not exist $𝑅 = 0$.

The restoration moment of the spiral spring force, the moment of the viscous friction force and the forced moment of force of the average impulse mechanism, operates on the timer oscillator. In the steady-state oscillation mode, according to (7.19), the restoration moment of the spiral spring force can be described by the following expression:

$$𝑆=−𝑘∙𝜑=−𝑘∙𝛷\sin 𝜓,\tag{7.30}$$

the moment of the viscous friction force by the formula

$$𝐷=−𝑐∙𝜑̇=−𝑐∙𝛷𝜔_0\cos 𝜓,\tag{7.31}$$

in accordance with the expressions (7.21) and (7.27), the forced moment of force by the relation:

$$𝑀(𝜓)=𝐽∙𝜇_0\cos(𝜓∓𝛿).\tag{7.32}$$

For the physical explanation of the phenomenon of errors of the average impulse mechanisms, it is of particular importance that the force of force described by the relation (7.32) can be represented as a sum of two components:

$$𝑀(𝜓)=𝐽∙𝜇_0\cos(𝜓∓𝛿) = 𝐽∙𝜇_0\cos 𝛿∙\cos 𝜓 ± 𝐽∙𝜇_0\sin 𝛿∙\sin 𝜓\tag{7.33}$$

The first component $𝐽 ∙ μ_0\cos δ ∙ \cos ψ$ is always in the contraction with the moment of the viscous friction force $-𝑐 ∙ Φω_0\cos ψ$. These two moments algebraically sum up and, in the quasi-oscillatory oscillation regime, algebraically annihilate:

$$+𝐽∙𝜇_0\cos 𝛿∙\cos 𝜓 − 𝑐∙𝜔_0𝛷\cos 𝜓=0.\tag{7.34}$$

From here it follows immediately that the forced force component $𝐽∙𝜇_0\cos 𝛿∙\cos 𝜓$ describes the pulse function of the pulse mechanism, the component that compensates for the energy losses of the clock oscillator.

In the case where the phase difference is positive $δ > 0$, the second component of the forced force of the force has a positive sign: $+𝐽 ∙ μ_0\sin δ ∙ \sin ψ$ and algebraically collects with the restoring moment of the force of the spiral spring $-𝑘∙Φ\sin ψ$. In accordance with the fact that, in this case, the second component of the forced force of the force in the contraction with the restoration moment of the spiral spring force, that second component reduces the intensity of the restitution torque of the force

$$+𝐽∙𝜇_0\sin 𝛿∙\sin 𝜓 − 𝑘∙𝛷\sin 𝜓,\tag{7.35}$$

creating an effect that is equivalent to lessening the stiffness of the spiral oscillator spring. As a consequence, this effect induces the prolongation of the oscillation period, and thus the slowing down of the clock, is termed bradyron. The diagrams of the formulas (7.19), (7.20), (7.30), (7.31), (7.32) as well as the components $𝐽 ∙ μ_0\cos δ ∙ \cos ψ$ and $𝐽 ∙ μ_0\sin δ ∙ \sin ψ$ describing the bradychronic effect of the error of the mean impulse mechanism are shown in Fig. 70.

Figure 70 Diagrams of the formulas that describe the bradycron effect of the error of a mean pulse mechanism

The tachychroni effect of the error of the mean pulse mechanism is shown in Figure 71. In this case, in which the phase difference is negative $δ < 0$, the second component of the forced moment of the force has a negative sign: $-𝐽 ∙ μ_0\sin δ ∙ \sin ψ$ and algebraically collects with the restoring moment of the spiral spring force $-𝑘 ∙ Φ\sin ψ$. In accordance with the fact that, in this case, the second component of the forced force of the force is a phase-in phase with the restoration moment of the spiral spring force, that second component increases the intensity of the restitution torque of the force

$$−𝐽∙𝜇_0\sin 𝛿∙\sin 𝜓−𝑘∙𝛷\sin 𝜓,\tag{7.36}$$

creating an effect that is equivalent to increasing the stiffness of the spiral oscillator spring. As a consequence, this effect induces the shortening of the oscillation period, and thus the acceleration of the clock speed, is called tachychron. The diagrams of the formulas (7.19), (7.20), (7.30), (7.31) and (7.32) as well as the components $𝐽 ∙ μ_0\cos δ ∙ \cos ψ$ and $-𝐽 ∙ μ_0\sin δ ∙ \sin ψ$ describing the tachyronous effect of the error of the average impulse mechanism are shown in the figure (71).

Figure 71 Diagrams of formulas that describe the tachyroni effect of the error of a mean impulse mechanism

In the case when the force of force is the average pulse mechanism in the phase (δ = 0) with the angular oscillation oscillator speed,

$$𝑀(ψ) = 𝐽 ∙ μ_0\cos ψ,\tag{7.37}$$ he is in contradiction with the moment of viscous friction force $−𝑐∙𝛷𝜔_0\cos 𝜓$, with it algebraically collects and, in the quasi-oscillatory oscillation regime, algebraic anulas: $$+ 𝐽 ∙ μ_0\cos ψ - 𝑐 ∙ ω_0Φ\cos ψ = ​​0.\tag{7.38}$$

As there is no component of the forced force of the force in this case which affects the restoration moment of the spiral spring force, there is no change in the angular frequency of the oscillator. In other words, when the phase difference is zero, $δ = 0$, the error of the mean pulse mechanism does not exist, and the timer stroke remains unchanged. This isochronic effect of the error of a mean impulse mechanism. The diagrams of the formulas (7.19), (7.20), (7.30), (7.31) and (7.37) describing this effect are shown in Figure 72.

Figure 72 Diagrams of formulas that describe the isochronous effect of an error of a mean impulse mechanism

Before the end, it is necessary to emphasize the general conclusion of this analysis which explained the physical meaning of the phenomenon of changing the angular frequency oscillation of the clock oscillator induced by the force of force of the average impulse mechanism, that is, the error phenomenon of the average impulse mechanisms. It is shown that the phase difference between the angular velocity of the oscillator and the forced force of the force is directly responsible for the change in the frequency of the oscillator. If the phase difference is positive, the forced force momentarily is delayed by the angular velocity of the oscillator, the error of the average impulse mechanism is negative and leads to the bradyron effect of prolonging the oscillation period and slowing down the clock speed. If the phase difference is negative, the forced force momentum is in phase relative to the angular velocity of the oscillator, the error of the average impulse mechanism is then positive and leads to the tachyron effect of shortening the period of oscillations and slowing down the clock speed. Only in a special case, when the mentioned phase difference does not exist, there is no error in the average impulse mechanism, and therefore neither the change in the oscillation period nor the change in the clock stroke. A detailed analysis of the oscillator oscillation oscillation oscillator oscillator also shows the physical cause of the error of the average impulse mechanisms. Namely, if the phase difference between the angular velocity of the oscillation and the forced moment of force exists, only one component of the moment of the force of the average impulse mechanism compensates for the energy losses of the oscillator. The energy of the other component is either surrendered or subtracted from the energy of the spiral springs and thus apparently changes its stiffness, which directly affects the frequency oscillation frequency. In the case of a bradyron effect, stiffness is reduced, and in the case of a tachyron effect, stiffness increases. The described phenomena are not the result of the linear theory of oscillations, but precisely the nonlinear dynamics of the clock mechanisms, which in this dissertation are mathematically treated by the methods of the perturbation account.