4. Oscillator

4.1 Basic concepts

Measurement of time intervals is based on a generally accepted intuitive idea that the flow of time is even. Accordingly, time-flow measurement is based on the measurement of the parameters of uniform, i.e. evenly moving, or some other even-handed processes. Any device or mechanism that allows or exerts such a thing can be a timer. It has already been pointed out that the Dutch scientist, mathematician and physicist, Christiaan Huygens first proposed, theoretically explained and practically applied the recording and counting of the uniform trace of mechanical oscillations in the measurement of time. Namely, Huygens discovered that free mechanical oscillations are approximately isolated and characterized by very stable own frequencies or periods of oscillation. The mechanism of the timer counts the mechanical oscillations in their uniform trace, it conveniently shows this information and thus measures the flow of time.

In the timer mechanism, two types of oscillators are installed: a physical pendulum, and a balancing wheel (restraint) with a spiral spring. Both types of oscillators perform approximately isolated oscillations under the effect of the restoration force (gravitational in the case of physical pendulum and elastic in the case of a balancing point), as already mentioned, are characterized by a stable, approximately constant, own frequency

4.1.1 Physical pendulum

A massive body acting on the force of gravity, rotating due to its weight around a fixed horizontal axis that does not pass through its center of mass is called a physical pendulum. The physical pendulum is an oscillator that is, as a rule, built into stationary mechanical clocks (wall and large, public - tower) and is shown in Figure 21.

The differential equation describing the free unbroken oscillations of the physical pendulum at the level $𝑂_{𝑥𝑦}$ around the axis is:

$$𝐽\,\frac{d^2𝜑}{dt^2} + mgl \sin 𝜑 = 0\tag{4.1}$$

Which is:

$φ = φ(𝑡)$ - the angle between the vertical fixed axis and the pendulum axis measured at the vertical plane $𝑂_{𝑥𝑦}$; a generalized coordinate that determines the position of the pendulum in the function of time $𝑡$.

$𝐽$ - the moment of inertia of the pendulum relative to the axis of rotation of the pendulum $𝑂_𝑧$
$𝑚$ - pallet mass
$𝑙$ - the distance of the center of mass from the axis of suspension of the pendulum

Figure 21 Physical pendulum

In the case of small oscillations, $\sin φ ≈ φ$, equation 4.1 can be written in the form:
$$\frac{d^2𝜑}{dt^2} + ω_0^2φ = 0\tag{4.2}$$ in which $ω$ is its own angular frequency oscillation of the physical pendulum and is defined by the following expression: $$ω_0 = \sqrt{ \frac{𝑚𝑔𝑙}{𝐽}}\tag{4.3}$$ From equation 4.3 directly follows the formula for the period of small oscillations of the physical pendulum: $$𝑇_0 = 2\pi ω_0 = 2\pi\sqrt{𝐽𝑚𝑔𝑙}\tag{4.4}$$

4.1.2 Balancing wheel

The balance wheel is a massive body, a circular eye around a fixed axis, which passes through its center of mass and performs oscillations under the action of an elastic restitution moment of force. The balance wheel, together with the spiral spring, is installed in mechanical hand and pocket timepieces, stopwatches, chronometers, as well as small stationary clocks and stone alarm clocks. Figure 22 shows a schematic representation of the balance of the wheel and spiral spring assembly together with the overlap, and in Fig. 23 their embodiment in the watch mechanism of the company Omega.

Figure 22 Schematic view of the balance point

Figure 23 Balance spool with the Omega clock

The differential equation describing the free unbroken oscillations of the balance point at the plane $𝑂_{𝑥𝑦}$ around the axis is:

$$𝐽\,\frac{𝑑^2φ}{𝑑𝑡^2} + 𝑘\sin φ = 0 \tag{4.5}$$

In which:

$φ = φ(𝑡)$ is the angle between the fixed axis $𝑂_𝑥$ and the moving axis of the balance point $𝑂_{𝑥_1}$, measured in the plane $𝑂_{𝑥𝑦}$; generalized coordinate that determines the position of the balance point in the function of time 𝑡.

$𝐽$ - the moment of inertia of the pendulum relative to the axis of rotation of the balance point $𝑂_𝑧$

$𝑘$ - coefficient of stiffness of the spiral spring. Assuming that the stiffness coefficient of the spiral spring is constant

$𝑘 = 𝑐𝑜𝑛𝑠𝑡$, equation 4.2 can be written in the form:

$$\frac{𝑑^2φ}{𝑑𝑡^2} + ω_0^2φ = 0 \tag{4.6} $$

in which $ω$ is its own angular frequency of oscillation of the balance point under the influence of the elastic restoring moment of force and is defined by the following expression :

$$ω_0 = \sqrt{𝑘𝐽} \tag{4.7}$$

It follows directly from the equation 4.4 that the formula for the isochronic oscillation period of the balance point is:

$$𝑇_0 = 2\pi\,ω_0 = 2\pi\sqrt{𝐽𝑘} \tag{4.8}$$

It should be noted that the coefficient of stiffness of the spiral spring $𝑘$ is not a constant parameter but that it is a fundamental funk from the angle $φ$, $𝑘 = 𝑘(φ)$. However, as it changes only slightly with an angle $φ$, it can be assumed in the first approximation that it is constant $𝑘≈𝑐𝑜𝑛𝑠𝑡$.

4.2 Nonlinear oscillations of the physical pendulum

4.2.1 Circular error

As in all mechanical systems and in the operation of the oscillator, certain effects on the accuracy of the oscillator operation must be taken into account, due to which the travel of the mechanical timer is never perfectly balanced. In addition to the external effects on the accuracy of the oscillator, there is also a nonlinearity in the pendulum, which depends on the amplitude -circular error.

The differential equation describing the free unbroken oscillations of the physical pendulum in the plane $𝑂_{𝑥𝑦}$ around the axis is:

$$𝜑̈ + 𝜔_0^2\sin 𝜑 = 0\tag{4.9}$$

$ω_0$ is the angular frequency of its own pendulum oscillations, given by the equation 4.7. This is a nonlinear differential equation of the second order, which, in the case of small oscillation amplitudes, is linearized by the approximation $\sin φ ≈ φ$, that is, it turns into a linear differential equation of the second order with constant coefficients. However, the fact is that the free oscillations of the physical pendulum are not harmonic, that the frequency and period of these oscillations are not constant but depend on the oscillation amplitude, that is, the free oscillations of the physical pendulum are not derived. As is known, in the case of arbitrarily large amplitudes, the oscillation period of the physical pendulum is calculated by the formula:

$$𝑇 = 4\,𝐾\big(\sin\frac{θ_0}{2}\big)\sqrt{𝐽𝑚𝑔𝑙} = 4\,𝐾\big(\sin\frac{θ_0}{2}\big) \frac{1}{ω_0}, \tag{4.10}$$

In which $θ_0$ the amplitude is oscillation, and the function $𝐾(𝑘)$ the complete elliptic integral of the first type is defined by the following expression:

$$𝐾(𝑘) = \int_{0}^{\pi/2}\frac{𝑑θ}{\sqrt{1-𝑘^2\sin^2θ}} \tag{4.11}$$

The solution of the complete elliptic integral of the first type can not be expressed through elementary functions but is given in the form of an infinite degree of order:

$$𝑇 = 2\pi\sqrt{𝐽𝑚𝑔𝑙}\big(1 + \frac{1}{4}\sin^2\frac{θ_0}{2} + \frac{9}{64}\sin^4\frac{θ_0}{2} + \frac{225}{2304}\sin^6\frac{θ_0}{2} + ⋯\big) \tag{4.12}$$

If an approximation is introduced in the specified degree

$$\sin\frac{θ_0}{2}≈\frac{θ_0}{2} ⟹\sin^2\frac{θ_0}{2}≈\frac{θ_0^2}{4} \tag{4.13}$$

we get the following approximate expression for the period of oscillation of the pendulum

$$𝑇 = 2\pi\sqrt{𝐽𝑚𝑔𝑙}\big(1 + \frac{θ_0^2}{16}\big) = 2\pi\,ω_0\big(1 + \frac{θ_0^2}{16}\big). \tag{4.14}$$

The relative error of the clock travel can be defined by the following relation

$$𝐻 = \frac{𝑇_0-𝑇}{𝑇_0} = \frac{𝑇_0-(𝑇_0+Δ𝑇)}{𝑇_0} = -\frac{Δ𝑇}{𝑇_0}, \tag{4.15}$$

in which $𝑇_0$ is nominal, and $𝑇$ is changed, that is, the current oscillation period of the pendulum. The pivotal fault of the pendulum is called the relative error of the clockwise induction of the pendulum with the period $𝑇$, the oscillation amplitude $θ_0$, compared to the equivalent ideal isochronic pendulum with period $𝑇_0$ and given by the following relation:

$$𝐻 = \frac{𝑇_0-𝑇}{𝑇_0} = -\frac{θ_0^2}{16} \tag{4.16}$$

Since the amplitude oscillations of the pendulum in modern timers are always less than $3^o$, the member $θ_0^2/16$ for practical purposes sufficiently precisely defines the dependence of the oscillation period of the pendulum by the amplitude. The circular error quantitatively describes the fact that the oscillations of the physical pendulum are not isolated, that is, that the change in the oscillation amplitude of the pendulum also changes its period $𝑇$, which indirectly induces the error of the equilibrium of the clock mechanism and the error of time measurement.

4.2.2 Determination of the circular error by the dual-time perturbation method

The circular error of the pendulum can also be performed using a perturbation account. If, in the differential equation, the oscillation of the physical pendulum (4.9) of the function $\sin φ$ approximates with the first two members of its development in the Maclaurin series expansion:

$$\sin φ ≈ φ − \frac{φ^3}{6}, \tag{4.17}$$ the differential equation (4.9) becomes $$𝜑̈ + ω^2_0φ − \frac{ω^2_0}{6}φ^3 = 0 \tag{4.18}$$ This is a nonlinear differential equation of the second order. It will be solved by the perturbation method of a double time scale, which will determine the period of oscillations, respectively, the circular error of the pendulum. In accordance with the above method, the principles of which are summarized in Chapter 2, the independent variable $𝑡$ is replaced by two variables $𝑡_1$ and $𝑡_2$, $$𝑡_1=𝑡; 𝑡_2=ε𝑡; ε≪1. \tag{4.19}$$ The approximate solution $φ(𝑡_1, 𝑡_2, ε)$ of the differential equation (4.18) is sought in the form of the collection of the initial or zero solution $φ_0(𝑡_1,𝑡_2)$ and first order correction $εφ_1(𝑡1,𝑡2)$, as shown by the expression: $$φ(𝑡_1, 𝑡_2, ξ) ≈ φ_0(𝑡_1, 𝑡_2) + εφ_1(𝑡_1, 𝑡_2) = φ_0 + εφ_1. \tag{4.20}$$ The initial conditions $φ(0) = 𝛷_0, φ̇(0) = 0$ are formulated for both functions $φ_0(𝑡_1, 𝑡_2)$ and $φ_1(𝑡_1, 𝑡_2)$, expressions: $$𝜑_0(0,0)=𝛷_0, 𝜑̇_0(0,0)=0, 𝜑_1(0,0)=0, 𝜑̇_1(0,0)=0\tag{4.21}$$ In accordance with the expressions (4.21), the first and second derivatives of the angular coordinates $φ$ by time are given by the following formulas, $$𝜑̇ = \frac{\partial 𝜑}{\partial t_1}\frac{\partial t_1}{\partial 𝑡}+\frac{\partial 𝜑}{\partial 𝑡_2}\frac{\partial 𝑡_2}{\partial 𝑡} = \frac{\partial 𝜑}{\partial 𝑡_1} + 𝜀\frac{\partial 𝜑}{\partial 𝑡_2},\tag{4.22}$$ $$𝜑̈=\frac{\partial𝜑̇ }{\partial 𝑡_1} + 𝜀\frac{\partial 𝜑̇}{\partial 𝑡_2} = \frac{\partial^2𝜑}{\partial 𝑡_1^2} + 2𝜀\frac{\partial^2𝜑}{\partial 𝑡_1\partial 𝑡_2}+𝜀^2\frac{\partial^2𝜑}{\partial 𝑡_1^2}.\tag{4.23}$$ After neglecting a member with $ε^2$ as a small size of a higher order, the expression (4.22) becomes $$𝜑̈=\frac{\partial^2𝜑}{\partial 𝑡_1^2} + 2𝜀\frac{\partial^2𝜑}{\partial 𝑡_1 \partial 𝑡_2}.\tag{4.24}$$ By substituting expression (4.20) in formula (4.24), a relation is obtained $$φ = \frac{𝜕^2φ_0}{𝜕𝑡_1^2} + ε\frac{𝜕^2φ_1}{𝜕𝑡_1^2} + 2ε\big(\frac{𝜕^2φ_0}{𝜕𝑡_1𝜕𝑡_2} + ε\frac{𝜕^2φ_1}{𝜕𝑡_1𝜕𝑡_2}\big) \tag{4.25}$$ which, after ignoring a member with $ε2$, becomes $$φ = \frac{𝜕^2φ_0}{𝜕𝑡_1^2} + ε\frac{𝜕^2φ_1}{𝜕𝑡_1^2} + 2ε\frac{𝜕^2φ_0}{𝜕𝑡_1𝜕𝑡_2}. \tag{4.26}$$ If the expression (4.20) is replaced in another member of the Maclaurin expansion of the function $\sinφ$, is obtained. $$\frac{1}{6}φ^3 = \frac{1}{6}(φ_0 + εφ_1)^3 = \frac{1}{6}(φ_0^3 + 3φ_0^2εφ_1 + 3φ_1^2ε^2φ_1 + ε^3φ_1^3) \tag{4.27}$$ If in the second member of Maclaurin expansion (4.27) the functions $\sinφ$ neglect the members containing $ε_2$ and $ε_3$, the functions $\sinφ$ become: $$\frac{1}{6}φ^3 = \frac{1}{6}φ_0^3 + \frac{1}{2}φ_0^2εφ_1. \tag{4.28}$$ By substituting the expressions (4.20), (4.26) and (4.28) into the equation (4.18), we obtain the following differential equation: $$𝜑̈ + ω_0^2φ-\frac{ω_0^2}{6}φ^3 = \frac{\partial^2φ_0}{\partial 𝑡_1^2} + ω_0^2φ_0 + ε\bigg(\frac{\partial^2φ_1}{\partial 𝑡_1^2} + 2\frac{\partial^2φ_0}{\partial t_1 \partial t_2} + ω_0^2ω_1 - \frac{ω_0^2}{6ε}ω_0^3 - \frac{ω_0^2}{2}ω_0^2ω_1\bigg) = 0 \tag{4.29}$$ In order for the left side of the equation (4.29) to be equal to zero, and in accordance with the fact that the attenuation factor $ε$ is a small number but different from zero $ε ≠ 0$, it is necessary that the equations: $$\frac{\partial^2φ_0}{\partial 𝑡_1^2} + ω_0^2φ_0 = 0, \bigg(φ_0 = 𝛷_0, \frac{\partial φ_0}{\partial 𝑡_1} = 0, \text{for}\, 𝑡1 = 𝑡2 = 0\bigg) \tag{4.30}$$ $$\frac{𝜕^2φ_1}{𝜕𝑡_1^2} + ω_0^2φ_1 = -2\frac{\partial^2 φ_0}{\partial 𝑡_1 \partial 𝑡_2} + \frac{ω_0^2}{6ε}φ_0^3 - \frac{ω_0^2}{2}φ_0^2φ_1, \bigg(φ_1 = 0, \frac{\partial φ_1}{\partial 𝑡_1} = -\frac{\partial φ_0}{\partial 𝑡_2}, \text{for}\, 𝑡_1 = 𝑡_2 = 0 \bigg) \tag{4.31}$$ with given initial conditions, simultaneously for allowed. The solution of the equation (4.30): $$φ_0 = 𝛷\sin(ω_0𝑡_1 + γ) = 𝛷\sinψ, 𝛷 = 𝑐𝑜𝑛𝑠𝑡, γ = 𝑐𝑜𝑛𝑠𝑡 \tag{4.32}$$ is the harmonic function of the oscillation phase $ψ$.

In order to solve the differential equation (3), the assumption ("Ansatz") is introduced that neither the amplitudes nor the phase difference are of constant size, but that they represent functions of the variable $𝑡_2 (ψ = ψ(𝑡_1, 𝑡_2), 𝛷 = 𝛷 (𝑡_2))$:

$$𝜑_0 = 𝛷(𝑡_2)\sin(𝜔_0𝑡_1 + 𝛾(𝑡_2)) = 𝛷\sin 𝜓. \tag{4.33}$$ In accordance with expression (4.33), the mixed partial derivative of the function $φ_0(𝑡_1, 𝑡_2)$ according to the variable $𝑡_1$ and $𝑡_2$ on the right-hand side of equation (4.31) is: $$\frac{𝜕^2𝜑_0}{𝜕𝑡_1𝜕𝑡_2}=\frac{𝜕𝛷}{𝜕𝑡_2}𝜔_0\cos𝜓−𝛷𝜔_0\frac{𝜕𝛾}{𝜕𝑡_2}\sin𝜓.\tag{4.34}$$ As the members of $φ_0^3$ and $φ_0^2$ appear on the right-hand side of equation (4.31) using additive formulas $$ sin^3𝜓=\frac{3}{4}\sin𝜓−\frac{1}{4}\sin 3𝜓; \sin^2𝜓=\frac{1}{2}−\frac{1}{2}\cos 2𝜓,\tag{4.35}$$

The expression on the right-hand side of the equation of equation (4.31) becomes:

$$-2\frac{𝜕Φ}{𝜕𝑡_2}ω_0\cos ψ + 2Φω_0\frac{𝜕γ}{𝜕𝑡_2}\sin ψ + \frac{ω_0^2}{6ε} Φ^3 \big(\frac{3}{4}\sin ψ - \frac{1}{4}\sin 3ψ\big) + \frac{ω_0^2}{2}φ_1Φ^2 \big(\frac{1}{2}-\frac{1}{2}\cos 2ψ\big)\tag{4.36}$$

In the expression (4.36) all members with the functions $\sin ψ$ and $\cos ψ$ are members of the resonant coercion, due to which secular members appear in the equation solution (4.31).

$$-2\frac{𝜕Φ}{𝜕𝑡_2}ω_0\cos ψ + \big(2Φω_0\frac{𝜕γ}{𝜕𝑡_2} + \frac{ω_0^2}{8ε}Φ^3\big) \sin ψ\tag{4.37}$$

In accordance with the perturbation method of a double time scale, the principles of which have already been summarized in Chapter 2, it is necessary to eliminate the secular members from the solution by annihilating the members of the resonant coercion. An alteration of the member of the resonant coercion with the function $\cos ψ$ was performed by the expression:

$$-2\frac{𝜕Φ}{𝜕𝑡_2}ω_0 = 0,\tag{4.38}$$

from which it follows immediately that the oscillation amplitude is constant $Φ = Φ_0 = 𝑐𝑜𝑛𝑠𝑡$. An alteration of the member of the resonant coercion with the function $\sin ψ$ was performed by the expression:

$$2Φω_0\frac{𝜕γ}{𝜕𝑡_2} + \frac{ω_0^2}{8ε}Φ^3 = 0,\tag{4.39}$$

which represents the differential equation of the first order

$$\frac{𝜕γ}{𝜕𝑡_2} = -\frac{ω_0}{16ε}Φ^2.\tag{4.40}$$

The solution of the differential equation (4.40) reads:

$$γ = \frac{ω_0}{16ε}Φ^2𝑡_2 + γ_0,\tag{4.41}$$

in which $γ_0$ is the integration constant (the phase difference angle) determined from the initial conditions $γ_0 = π/2$. By moving to the permissible $𝑡$ expression $𝑡_2 = ε𝑡$ and in accordance with the solution of the differential equation (4.38), the solution of the differential equation (4.40) becomes:

$$γ = -\frac{ω_0}{16}Φ_0^2𝑡 + \frac{π}{2}\tag{4.42}$$

Finally, the solution of the differential equation (4.9) describing the free unbroken oscillations of the pendulum reads:

$$φ(𝑡) = Φ_0\cos ω_0 \big(1-\frac{Φ_0^2}{16}\big) 𝑡\tag{4.43}$$

The oscillation period is represented by the terms:

$$𝑇 = \frac{2π}{ω_0\big(1-\frac{Φ_0^2}{16}\big)}\tag{4.44}$$

which, by multiplying the name and counters with $\big(1 + \frac{Φ_0^2}{16}\big)$ and ignoring a member of $Φ_0^4$ as a small size of a higher order becomes the expression:

$$𝑇 = \frac{2π}{ω_0} \big(1 + \frac{Φ_0^2}{16}\big).\tag{4.45}$$

The same solution can be obtained using the perturbation method of a double time scale if the differential equation (4.9) is previously transformed into the so-called "Duffing" (Georg Duffing 1861-1944) equation [62]. As in the previously performed procedure, in the differential equation of the oscillation of the physical pendulum (4.9), the function sinφ is approximated with the first two members of its development in the Maclaurin degree line:

$$\sin φ ≈ φ - \frac{φ^3}{6}.\tag{4.46}$$

After this approximation, the differential equation (4.9) becomes:

$$𝜑̈ + ω_0^2φ - ω_0^2\frac{φ^3}{6} = 0\tag{4.47}$$

Since oscillation amplitudes are pendulum small, a member $φ^3$ can be treated as a perturbation, that is, a small disturbance of harmonic oscillations. To explicitly emphasize, a suitable shift of the coordinate φ is introduced:

$$φ = θ√ε, ε≪1,\tag{4.48}$$

which transforms the differential equation (4.9) into a nonlinear so-called. Duffing Differential Equation:

$$𝜃̈ + ω_0^2θ-εω_0^2\frac{θ^3}{6} = 0\tag{4.49}$$

In it, as has already been mentioned, explicitly sees that the cubic term $θ^3$ represents a small disorder of the differential equation of free harmonic oscillations. Nowadays, the Duffing differential equation directly applies the perturbation procedure of the double-time time. The independent variable $𝑡$ is replaced by two variable $𝑡1$ and $𝑡2$,

$$𝑡_1 = 𝑡; 𝑡_2 = ε𝑡,\tag{4.50}$$

A The approximate solution $θ (𝑡1, 𝑡2, ε)$ of the differential equation (4.9) is sought in the form of the collection of the initial or zero solution $θ_0 (𝑡1, 𝑡2)$ and first order correction $εθ_1 (𝑡1, 𝑡2)$, as shown by the expression:

$$θ (𝑡_1, 𝑡_2, ξ) ≈ θ_0 (𝑡_1, 𝑡_2) + εθ_1 (𝑡_1, 𝑡_2) = θ_0 + εθ_1.\tag{4.51}$$

The initial conditions $θ(0) = 𝐴_0, θ̇(0) = 0$ are formulated for both functions $θ_0 (𝑡_1, 𝑡_2)$ and $θ_1 (𝑡_1, 𝑡_2)$, the terms:

$$θ_0(0,0) = 𝐴_0, θ̇_0(0,0) = 0, θ_1(0,0) = 0, θ̇_1(0,0) = 0.\tag{4.52}$$

In accordance with the expressions (4.52), the first and second copies of the angular coordinates $φ$ by time are given by the following formulas, respectively:

$$𝜃̇ = \frac{𝜕θ}{𝜕𝑡_1} + ε\frac{𝜕θ}{𝜕𝑡_2}\tag{4.53}$$ $$𝜃̈ = \frac{𝜕^2θ}{𝜕𝑡_1^2} + 2ε\frac{𝜕^2θ}{𝜕𝑡_1𝜕𝑡_2} + ε^2\frac{𝜕^2θ}{𝜕𝑡_2^2}.\tag{4.54}$$

After neglecting a member with $ε^2$ as a small size of a higher order, the expression (4.54) becomes

$$𝜃̈ = \frac{𝜕^2θ}{𝜕𝑡_1^2} + 2ε\frac{𝜕^2θ}{𝜕𝑡_1𝜕𝑡_2}\tag{4.55}$$

By replacing the expression (4.51) in (4.55), and ignoring a member with $ε^2$ as a small-sized higher-order, we get the following relation:

$$𝜃̈ = \frac{𝜕^2θ_0}{𝜕𝑡_1^2} + ε\frac{𝜕^2θ_1}{𝜕𝑡_2^2} + 2ε\frac{𝜕^2θ_0}{𝜕𝑡_1𝜕𝑡_2}.\tag{4.56}$$

If expression (4.51) is replaced in another member of the Maclaurin development (4.46) of the function $\sin φ$, we obtain:

$$\frac{1}{6}θ^3 = \frac{1}{6}\big(θ_0 + εθ_1\big)^3 = \frac{1}{6}\big(θ_0^3 + 3θ_0^2εθ_1 + 3θ_1^2ε^2θ_1 + ε^3θ_1^3\big).\tag{4.57}$$

If in the second article of the Maclaurin expansion (4.57) of the functions $\sin φ$ neglecting members containing $ε^2$ and $ε^3$, the functions $\sin φ$ become:

$$\frac{1}{6}θ^3 = \frac{1}{6}θ_0^3 + \frac{1}{2}θ_0^2εφ_1.\tag{4.58}$$

By replacing the expressions (4.58), (4.51) and (4.56) in the differential equation (4.9) and neglecting the member containing $ε^2$, the following differential equation is derived:

$$\frac{𝜕^2θ_0}{𝜕𝑡_1^2} + ω_0^2θ_0 + ε \Big(\frac{𝜕^2θ_1}{𝜕𝑡_2^2} + 2\frac{𝜕^2θ_0}{𝜕𝑡_1𝜕𝑡_2} + ω_0^2θ_1 - ω_0^2\frac{θ_0^3}{6}\Big) = 0.\tag{4.59}$$

In order for the left side of equation (4.59) to be equal to zero, and in accordance with the fact that the attenuation factor $ε$ is small but different from zero $ε ≠ 0$, it is necessary that the equations,

$$\frac{𝜕^2θ_0}{𝜕𝑡_1^2} + ω_0^2θ_0 = 0, (θ_0 = 𝐴_0, \frac{𝜕θ_0}{𝜕𝑡_1} = 0, for 𝑡_1 = 𝑡_2 = 0)\tag{4.60}$$ $$\frac{𝜕^2θ_1}{𝜕𝑡_1^2} + ω_0^2θ_1 = -2\frac{𝜕^2θ_0}{𝜕𝑡_1𝜕𝑡_2} + ω_0^2\frac{θ_0^3}{6}, \Big(θ_1 = 0, \frac{𝜕θ_1}{𝜕𝑡_1} = -\frac{𝜕θ_0}{𝜕𝑡_2},\,\,\text{for}\,\, 𝑡_1 = 𝑡_2 = 0\Big).\tag{4.61}$$

for the given initial conditions, simultaneously satisfied. Solving the equation (4.60):

$$θ_0 = 𝐴(𝑡_2) \sin (ω_0𝑡_1 + γ(𝑡_2)) = 𝐴\sin ψ \tag{4.62}$$

it is a harmonic function of the oscillation phase $ψ$. In order to solve the differential equation (3), the assumption ("Ansatz") is introduced that neither the amplitudes nor the phase difference are constant, but that they represent the functions of the variable $𝑡_2$:

$$𝐴 = 𝐴(𝑡_2), ​​ψ = ψ (𝑡_1, 𝑡_2).\tag{4.63}$$

In accordance with the expression (4.62), the mixed partial derivative of the function $θ_0 (𝑡_1, 𝑡_2)$ according to the variable $𝑡_1$ and $𝑡_2$ on the right-hand side of equation (4.61) is:

$$\frac{𝜕^2θ_0}{𝜕𝑡_1𝜕𝑡_2} = \frac{𝜕A}{𝜕𝑡_2}ω_0\cos ψ - 𝐴ω_0\frac{𝜕γ}{𝜕𝑡_2}\sin ψ,\tag{4.64}$$

Since on the right-hand side of the equation (4.61) there are members $θ_0^3$, in accordance with the expression (4.62), using the addition formula,

$$θ_0^3 = 𝐴^3\frac{3}{4}\sin ψ - 𝐴^3\frac{1}{4} \sin 3ψ\tag{4.65}$$

The expression on the right-hand side of the equation of equation (4.61) becomes:

$$-2\frac{𝜕A}{𝜕𝑡_2}ω_0\cos ψ + 2𝐴ω_0\frac{𝜕γ}{𝜕𝑡_2}\sin ψ + \frac{𝐴^3}{8}ω_0^2 \sin ψ - \frac{𝐴^3}{24}ω_0^2\sin 3ψ\tag{4.66}$$

In the expression (4.65) all members with the functions $\sin ψ$ and $\cos ψ$ represent members of resonant coercion, due to which secular members appear in the equation solution (4.61).

In accordance with the perturbation method of a double scale of time, the principles of which are already summarized in Chapter 2, it is necessary to eliminate the secular members from the solution by annihilating the members of the resonant coercion. An alteration of the member of the resonant coercion with the function $\cos ψ$ was performed by the expression:

$$-2\frac{𝜕A}{𝜕𝑡_2}ω_0 = 0,\tag{4.67}$$

from which it follows immediately that the oscillation amplitude is constant $A = A0 = 𝑐𝑜𝑛𝑠𝑡$. An alteration of the member of the resonant coercion with the function $\sin ψ$ was performed by the expression:

$$2𝐴ω_0\frac{𝜕γ}{𝜕𝑡_2} = -\frac{𝐴^3}{8}ω_0^2,\tag{4.68}$$

which represents the differential equation of the first order

$$\frac{𝜕γ}{𝜕𝑡_2} = -\frac{𝐴^2}{16}ω_0.\tag{4.69}$$

The solution of the differential equation (4.69) reads:

$$γ = \frac{ω_0}{16}𝐴^2𝑡_2 + γ_0,\tag{4.70}$$

in which $γ_0$ is the integration constant (the phase difference angle) determined from the initial conditions $γ_0 = π/2$. By moving to the promissory $𝑡$ expression $𝑡_2 = ε𝑡$ and in accordance with the solution of the differential equation (4.67), the differential equation solution (4.69) becomes:

$$γ = -\frac{ω_0}{16}𝐴_0^2ε𝑡 + \frac{π}{2}\tag{4.71}$$

As $φ = \sqrt{ε}θ$, the amplitude $𝐴_0$ of the function $φ(𝑡)$ is $\sqrt{e}$ times greater than the amplitude $𝐵_0, 𝐵_0 = \sqrt{ε}𝐴_0$, directly follows

$$γ = -\frac{𝐴_0^2}{16}ω_0ε𝑡 + \frac{π}{2} = -\frac{𝐵_0^2}{16ε}ω_0ε𝑡 + \frac{π}{2} = -\frac{𝐵_0^2}{16}ω_0𝑡 + \frac{π}{2}.\tag{4.72}$$

In accordance with the above, the solution of the differential equation (4.9) describing the free unbroken oscillations of the pendulum reads:

$$φ(𝑡) = 𝐵_0\cos ω_0 \Big(1-\frac{𝐵_0^2}{16}\Big)𝑡\tag{4.73}$$

The oscillation period is represented by terms

$$𝑇 = \frac{2π}{ω_0 \Big(1-\frac{𝐵_0^2}{16}\Big)}\tag{4.74}$$

which, by multiplying the name and counters with $\big(1 + \frac{𝐵_0^2}{16}\big)$ and ignoring a member of $𝐵_0^4$ as a small size of the higher order becomes the expression:

$$𝑇 = \frac{2π}{ω_0} \Big(1 + \frac{𝐵_0^2}{16}\Big).\tag{4.75}$$

The resulting relation, which contains a member of the piercing circular error, is formally identical to the formula (4.9). In addition to the aforementioned circular error, there are also errors that can be classified as errors that have been caused by an external effect on the oscillator. This section will show the most significant external influences on the oscillator: the influence of temperature (thermal dilation), resistance, density and aerostatic air pressure. All these influences will be shown on the pendulum case in order to show their changes.

4.3 External effects on the oscillator

4.3.1 Thermal dilatations

Changes in the temperature on the oscillation frequency of the pendulum are considered to be the most significant and most harmful [63]. When the temperature changes, the pendulum material is expanded and collected, which according to the formula 4.4 influences the oscillation period. The pendulum length must be precisely defined to maintain precision in the range of 1 second per 100 days. The required precision of 1 sec / 100 days was established in the 1700s with the first mechanical timers and then it was defined that the pendulum length should not change by more than 230 nm [64].

Every material, including the material from which the pendulum is made, changes its dimensions with the change of temperature. According to this principle, the pendulum in the warm environment moves slower than the pendulum in a cooler environment. The formula for linear temperature dilatations is:

$$𝑙 = 𝑙_0 ∙ (1 + α ∙ (𝑡-𝑡_0)) = 𝑙_0 ∙ (1 + α ∙ θ)\tag{4.76}$$

where are they:

$α$ - linear coefficient of thermal (thermal) dilation
$𝑙$ - pendulum length at temperature $𝑡$
$𝑙_0$ - pendulum length at temperature $𝑡_0$

The coefficient $α$ defines a relative change in length per unit depending on the temperature. For some materials the coefficient values ​​α are shown in Table 1

Table 1 Linear coefficients of temperature dilation for some materials

$\underline{\text{Material}}$	$\underline{α\,\,10^{-6} [K^{-1}]}$	$\underline{\text{Material}}$	$\underline{α\,\,10^{-6} [K^{-1}]}$
Invar	1,2	Brass	18-19
Jelly	4	Lead	28
Steel	12	Zinc	39,7

The oscillation period of the physical pendulum, if the circular error is ignored, can be calculated using known formulas:

$$𝑇≈2π\sqrt{\frac{𝐽}{𝑔∙𝑆}} = 2π\sqrt{\frac{\sum 𝑚_𝑖 𝑟_𝑖^2}{𝑔∙\sum 𝑚_𝑖 𝑟_𝑖}} = 2π\sqrt{\frac{𝑙_𝑟}{𝑔}}\tag{4.77}$$

Where are the $𝐽$ and $𝑆$ - the square and static momentum of the inertia, and $𝑔$ - the acceleration of the earth is more difficult. From formula 4.77 it can be clearly seen that:

$$𝐽 = \sum 𝑚_𝑖 𝑟_𝑖^2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,𝑆 = \sum 𝑚_𝑖𝑟_𝑖\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,𝑙_𝑟 = \frac{\sum 𝑚_𝑖 𝑟_𝑖^2}{\sum 𝑚_𝑖 𝑟_𝑖}\tag{4.78}$$

Where are the $𝑚_𝑖$ and $𝑟_𝑖$ - masses and coordinates (relative to the point of hanging) the points of all parts of the pendulum are the so-called. the reduced length of the physical pendulum and, by definition, represents the length of the mathematical pendulum, which has a period of oscillation, as well as the given physical pendulum. The static moment $𝑆$ determines the coordinate of the center of gravity, and the square moment $𝐽$ describes the geometry of the mass distribution around the point of suspension and the center of gravity of the entire pendulum

Since each linear coordinate in the above formulas is subject to temperature dilatations, it can be concluded that the period of its own oscillations of the physical pendulum changes with the change of temperature. Thus, for example, a pendulum with a oscillation period of $T = 2$ seconds, whose carrier is made of ordinary low-carbon steel, loses about 0.5 seconds per day per 1 degree of temperature rise cascade. If the temperature has increased by 10 degrees, a clock with such a pendulum, for 7 days of continuous operation, is delayed for about 36 seconds. Under the same temperature conditions, a watch whose pendulum is geometrically identical but made of white-tree wood will be 12 seconds later in 7 days. Today, most commonly not used pendulum are made of one material, but combinations of materials that allow for the compensation of the change in the length of the pendulum at temperature dilatations. Although it has already been said that wood pendulum is less susceptible to temperature, wood pendants have a problem with moisture [65], thus requiring great protection of wood from moisture before use in the pendulum.

4.3.2. Air Resistance

The force of the air resistance acts on the physical pendulum of the timer, with the exception of the restitution gravity force, which causes the pendulum to lose energy. This energy loss is compensated by the impulse mechanism, which transmits the drive energy to the pendulum by its impulses, and thus ensures the continuity of its oscillations. In the steady-state oscillation mode, the energy balance of the oscillator is achieved: during each oscillation period, the amount of dissipation energy is equal to the amount of energy that the pendulum receives from the average impulse mechanism. The air resistance can in principle be explained by two components: one viscous friction is proportional speed ($~ 𝑣$), and the other resistance of the shape (pressure resistance or dynamic resistance) proportional to the square of the speed ($~ 𝑣^2$). Both components are directed opposite to the velocity direction, so that the total air resistance vector can be represented by the following expression:

$$𝑅⃗(𝑣) = -(𝑏𝑣 + 𝑐𝑣^2) ∙ \frac{𝑣}{𝑣} \tag{4.79}$$

The coefficients $𝑏$ and $𝑐$ are not constants but depend on parameters describing state of gas or air (density, pressure, temperature and humidity) in which the pendulum oscillates. In addition, the parameter $𝑏$ is the function and the viscosity of the air, and $𝑐$ also depends on the geometrical characteristics of the body to which the resistance works. As is known, which of these two components of resistance has the dominant influence on energy dissipation depends on the ratio of inertial and viscous forces in the fluid and is defined by Reynolds number $𝑅𝑒$. For low values ​​of Reynolds number $Re < 10^3$, viscous friction is prevalent, while for its large values ​​$Re > 10^5$, the domino effect has pressure resistance. Since the angular velocity of the pendulum changes according to the harmonic law, and therefore continuously periodically varies from zero to maximum, it is impossible to reliably determine which component of the air resistance predominates during its movement. However, it is justifiable to adopt the assumption that the most significant share of energy dissipation occurs at the same pendulum where Reynolds number is the largest. In accordance with this assumption, in this chapter, a preliminary calculation of the impact of the pressure resistance force, that is, a shape proportional to the square of the velocity will be carried out:

$$𝑅 = 𝑐𝑣^2 = \frac{1}{2}𝐶_𝑥ρ𝑣^2𝑆 \tag{4.80}$$

In the given formula:

- $𝐶_𝑥$ is the resistance coefficient,
- $ρ$ the density of air,
- $𝑣$ the center of gravity of this pendulum and
- $𝑆$ the surface of the characteristic cross-section of this pendulum.

Whether this influence is really dominant and to what extent it ($~𝑣^2$) prevails in relation to the impact of viscous friction ($~𝑣$), it can reliably reveal only the results of the experiment, or possibly computer simulations. Thus, the following calculations refer to the dissipation of the energy of the pendulum under the influence of dynamic resistance, which is proportional to the square of the velocity of the center of gravity of this pendulum. In accordance with this assumption, the pendulum energy loss in the time unit (loss of power) due to air piston air resistance would be determined by the following formula:

$$𝑃 = 𝑅 ∙ 𝑣 = \frac{1}{2}𝐶_𝑥ρ \big| 𝑣^3 \big| 𝑆. \tag{4.81}$$

Then the angular velocity φ̇ (𝑡) is changed according to the law:

$$φ̇ (𝑡) = - φ(𝑡) = φ_0\cos ω𝑡, \tag{4.82}$$

If the angular coordinate φ(𝑡) determines the position of the pendulum as a function of time,

$$ωΦ_0\cos ω𝑡. \tag{4.83}$$ in relations 4.82 and 4.83:

- $ω$ is the angular frequency $ω = 2π/𝑇$, and
- $Φ_0$ amplitude of the pendulum oscillation.

Accordingly, the dissipation of power (energy in a unit of time) would be calculated using the following formula:

$$𝑃 = \frac{1}{2}𝐶_𝑥ρ𝑙^3Φ_0^3ω^3 \big| \cos^3ω𝑡 \big| 𝑆, \tag{4.84}$$ ​​

in which $𝑙$ is the length from the point of suspension to the center of the pendulum. The mean value of the pendulum power dissipation due to dynamic air resistance is defined by the following expression:

$$𝑃_{𝑆𝑅} = \frac{1}{2}𝐶_𝑥ρ𝑙^3Φ_0^3ω^3𝑆\frac{1}{T}\int_{0}^{T} \big| \cos^3ω𝑡 \big| 𝑑𝑡. (4.85)$$

The mean power dissipation value is:

$$𝑃_{𝑆𝑅} = \frac{2}{3π}𝐶_𝑥ρ𝑙^3Φ_0^3ω^3𝑆 = 𝐾ρ𝑙^3Φ_0^3ω^3, 𝐾 = 𝑐𝑜𝑛𝑠𝑡 (1). \tag{4.87}$$

As already pointed out, in the stationary state of oscillation of the pendulum, the energy loss due to the dynamic resistance of the air is equal to the energy that the average impulse mechanism handles the pendulum during each oscillation period:

$$𝑃_{𝑆𝑅}𝑇 = 𝑃_{𝑆𝑅}\frac{2π}{ω} = \frac{4}{3}𝐶_𝑥ρ𝑙^3Φ_0^3ω^2𝑆 = 𝑐𝑜𝑛𝑠𝑡. \tag{4.88}$$

It follows from this fact that, in the steady-state oscillation regime of the pendulum, the average energy of the pendulum is constant. However, if for any reason there is a change in the air density $ρ$, the distance from the point of suspension $𝑙$, the amplitude $Φ_0$ or the angular frequency $ω$, the pendulum will enter a non-stationary oscillation regime with a very complex transition to a new stationary regime that will be distinguished by one the new average energy of the pendulum.

4.3.3 Air density

The change in air density caused by a change in pressure, humidity and / or a change in air temperature, in accordance with formula 4.88, will have a direct impact on the overall energy of the pendulum oscillating, the other energy parameters, and also the timer's travel. It is, in essence, caused by the change of dynamic resistance acting on the pendulum. If 𝑙 = 𝑐𝑜𝑛𝑠𝑡 (no thermal dilatations), from formula 4.88, it follows directly that:

$$ρΦ_0^3ω^2 = 𝑐𝑜𝑛𝑠𝑡. \tag{4.89}$$

As

$$ω^2 = ω_0^2 (1-\frac{Φ_0^2}{8}), \tag{4.90}$$

and in accordance with expression 4.89, follows the relation:

$$ρΦ_0^3 (1-\frac{Φ_0^2}{8}) = 𝑐𝑜𝑛𝑠𝑡. \tag{4.91}$$

The first copy of the relation 4.91 in terms of the air density $ρ$ is given by:

$$Φ_0^3 + 3ρΦ_0^2\frac{𝑑Φ_0}{𝑑ρ}-\frac{Φ_0^5}{8}-5ρ\frac{Φ_0^4}{8}\frac{𝑑Φ_0}{𝑑ρ} = 0, \tag{4.92}$$

The following formula follows:

$$\frac{𝑑Φ_0}{𝑑ρ} = -\frac{Φ_0}{ρ} ∙ \frac{8-Φ_0^2}{24-5Φ_0^2}. \tag{4.93}$$

Formula 4.93 is the first derivative of the oscillation oscillation of the pendulum by the density of air and is a measure of the rate of change of the amplitude $Φ_0$ per unit of air density ρ. Apparently, the change in air density has a direct impact on the change in the oscillation amplitude of the pendulum. And in accordance with the already explained in detail the phenomena that the physical pendulum is not isochronic, but that its oscillation time depends on the amplitude, it follows that the change in the density of the air will change the time of the timer. In order to analyze this effect, it is necessary to first determine the first derivative of the relative clock fault error, defined by the expression (4.16), according to the amplitude of the oscillation of the pendulum:

$$\frac{𝑑𝐻}{𝑑Φ_0} = -\frac{Φ_0}{8}. \tag{4.94}$$

Now, the first derivative of the relative error of the timing of the timer according to the density of the air can now be determined from the following relation:

$$\frac{𝑑𝐻}{𝑑ρ} = \frac{𝑑𝐻}{𝑑Φ_0} ∙ \frac{𝑑Φ_0}{𝑑ρ}. \tag{4.95}$$

By substituting the expressions (4.93) and (4.94) in (4.95), we obtain the formula:

$$\frac{𝑑𝐻}{𝑑ρ} = \frac{Φ_0^2}{8ρ} ∙ \frac{8-Φ_0^2}{24-5Φ_0^2} \tag{4.96},$$

which is a measure of the rate of change of the timing of the timer, ie the relative error of the timer $𝐻$ per unit density of the air $ρ$, resulting from the change in the dynamic resistance of the air acting on the pendulum. Thus, the change in air density changes the force of resistance acting on the pendulum, and thus changes the amount of dissipation of the energy of the pendulum, which results in a change in the amplitude of its oscillations. Since the oscillation period of the pendulum depends on the amplitude of the oscillation over the circular error, the change in the amplitude leads to a change in the oscillation period of the pendulum, that is, of the clockwise travel.

4.3.4. Aerostatic Thrust

The air pendulum oscillating air pistol, with the exception of gravity and air resistance, acts as an aerostatic thrust. In accordance with the Archimedes Act, the aerostatic thrust reduces the weight of the pendulum for the weight of the pendulous air. In this way, the aerostatic thrust reduces the restitution force, the moment in which it relates to the pivoting axis of the pendulum and the cause of its oscillatory movement. The reduction of the weight of the pendulum, due to aerostatic air pressure, is calculated according to the following expression:

$$𝐺 - 𝑃 = 𝑚_𝐾𝑔-𝑚𝑔 = 𝑚_𝐾𝑔\big(1-\frac{𝑚}{𝑚_𝐾}\big) = 𝑚_𝐾𝑔\big(1-\frac{ρ}{ρ_𝐾}\big), \tag{4.97}$$

in which:

- $𝐺$ the weight of the pendulum,
- $𝑃$ aerostatic thrust,
- $𝑚_𝐾$ mass of pendulum
- $𝑚$ mass of pendulous extruded air,
- $ρ_𝐾$ density of piston material,
- $ρ$ density of air.

The differential equation describing the free unbroken oscillations of the physical pendulum in the plane $𝑂𝑥𝑦$ about the axis and the aerostatic air flow is:

$$𝐽\frac{𝑑^2φ}{𝑑𝑡^2} + 𝑚_𝐾𝑔𝑙 \big(1-\frac{ρ}{ρ_𝐾}\big) \sin φ = 0 \tag{4.98}$$

In which

- $φ = φ (𝑡)$ is the angle between the vertical fixed axis and the pivot axis $𝑂𝑥$ measured in the vertical plane $𝑂𝑥𝑦$; a generalized coordinate that determines the position of the pendulum in the function of time $𝑡$.
- $𝐽$ - the moment of inertia of the pendulum relative to the axis of rotation of the pendulum $𝑂𝑧$
- $𝑙$ - the distance of the center of mass from the axis of the pendulum suspension.

In the case of oscillations with small angles of the pendulum deflection, the $\sin φ ≈ φ$, the oscillation period of the physical pendulum on which the aerostatic thrust is acting is determined by the formula:

$$𝑇 = 2π\sqrt{{𝐽}{𝑚_𝐾𝑔𝑙(1-\frac{ρ}{ρ_𝐾})}} = \frac{𝑇_0}{\sqrt{1-\frac{ρ}{ρ_𝐾}}}, \tag{4.99}$$

where $𝑇_0$

$$𝑇_0 = 2π\sqrt{\frac{𝐽}{𝑚_𝐾𝑔𝑙}}. \tag{4.100}$$

oscillation period of the equivalent pendulum oscillating in a vacuum, i.e. Nominal oscillation period. From the formula it follows that the oscillation period of the pendulum in the air is always longer than the period of its oscillation $𝑇_0$ in vacuum. However, if the air density was constant, that fact would have no effect on the timer's travel. The problem of changing the course of the clock mechanism is precisely due to the change in air density, which changes the aerostatic thrust, changes the restoration force, and thus the length of the oscillation period of the pendulum. This impact is the subject of further analysis.

From the fact that the ratio of the density of the air to the density of the material is pendulum small size, much smaller than the unit, $ρ/ρ_𝐾═1$, the formula (4.99) can be developed into the Maclaurin degree order by the variable $ρ$ and approximated with the first two members of this development in the following way:

$$𝑇 = 𝑇_0\sqrt{1-\frac{ρ}{ρ_𝐾}} ≈ 𝑇_0 \big(1 + \frac{ρ}{2ρ_𝐾}\big), \tag{4.101}$$

The relative error of the clock travel $𝐻_𝑃$, defined by the expression (4.92), formed under the influence of aerostatic air pressure, and in accordance with the approximation 4.101 is given by the following expression:

$$𝐻_𝑃 = -\frac{Δ𝑇}{𝑇_0} = -\frac{ρ}{2ρ_𝐾}. \tag{4.102}$$

The first derivative of the relative error of the clock travel, given by the relation (4.102), according to the air density

$$\frac{𝑑𝐻_𝑃}{𝑑ρ} = -\frac{1}{2ρ_𝐾}, \tag{4.103}$$

defines the speed of the change of the timing of the timer, ie, the relative error of the clock stroke $𝐻_𝑃$ per air density unit $ρ$ caused by changing the aerostatic thrust . In order to get an idea of ​​the order of magnitude of the error of the clock travel that was created under the influence of a change in aerostatic thrust, the results of a short budget are stated. If the material is pendulum $(ρ_𝐾 = 11.34 × 103𝑘𝑔 / 𝑚^3)$ and the air density increased by 5% ($Δρ = 0.06𝑘𝑔 / 𝑚^3$), the timer will be delayed for about 0.23 seconds in 24 hours. If the material is pallet steel ($ρ_𝐾 = 7.8 × 103𝑘𝑔 / 𝑚^3$), and the air density increased by 5% ($Δρ = 0.06𝑘𝑔 / 𝑚^3$), the timer will be delayed for about 0.33 seconds in 24 hours. The error is small and the smaller the density of the material is the pendulum.

4.4. Oscillator compensation errors

4.4.1. Compensation of the pendulum heat expansion

The problem of compensation of the thermal expansion of the pendulum (temperature compensation) is over 200 years old and is still an important topic of many research. Compensation methods are numerous and are mostly performed mathematically and experimentally. In its simplest form, the principles of compensating for the effect of stochastic temperature changes on the uniformity of the timing of the timers are based on the choice of the material having the smallest linear coefficient of temperature dilation. As already mentioned in chapter 4.3.1, the firing pallet carrier is less subject to the influence of the temperature than the steel. That is why the first pillars were made of wood and later switched to a combination of materials. Even in the 20th century, with the invention of the "invar" alloy, compensation with only one material can not be completely performed. In order for the compensation to be complete, it is necessary to make the pendulum from two different types of materials.

The compensation based on a combination of steel and brass was first applied by John Harrison (John Harrison 1693-1766) to build his stationary timepieces and a chronometer labeled "N1". It's about the so-called. "Gridiron" - grid, bimetallic pendulum, whose practical solution is illustrated on the left in Fig. 24.

Figure 24 Technical realization of "Gridiron" and cross-section of coaxial steel pendant bearings of steel and zinc pendulum [16]

The pallet carrier is constructed of 4 brass and 5 steel, parallel and symmetrically placed rods, the lengths of which are equal and are connected in such a way steel rods are always wider than, and brass hinges, hanging points.

This method has not yielded satisfactory results and therefore it has been applied to a similar technical solution based on the same principle as the described "gridiron" pendulum, but which uses steel and zinc coaxial tubes instead of steel and brass rods (Figure 24). Since zinc has an extremely large linear coefficient of thermal expansion ($α = 39.7 · 10^{-6} [K-1]$), compensation is achieved only with two steel and one zinc tube.

The most effective solution for compensating heat pallet dilatation's, which is at the same time very simple, is achieved by combining wood and lead. If the pallet carrier is made of white-tree wood, and the lead of the appropriate tube length and weight, the technically almost perfect temperature compensation of the reduced length of the complete pendulum can be achieved. The white fir tree has an extremely small linear coefficient of thermal expansion ($𝑎 = 4 · 10^{-6} [K-1]$), and the lead is large enough ($𝑏 = 28 · 10^{-6} [K-1]$), and their single combination, theoretically , can fully compensate the pendulum dilatation. The preliminary solution of this coupling is shown in Fig. 25. The pendulum carrier is a wooden stick, at the lower end of which a tube of lead is co-axially fixed. The rod extends warmly from the point of hanging downwards, and the lead tip - the opposite.

Figure 25 Temperature compensation of the center of gravity of the pendulum and the origin of the fault residue [16]

The preliminary calculation is based on the determination of the relationship between the length of the rod and the tube in which the thermal shift of the center of gravity of the entire pendulum is neutralized. From the fact that the position of the pendulum center is determined by its static moment $𝑆$ in which the lengths depend on the temperature, then:

$$𝑆 = 𝑚\frac{𝑙_0 (1 + 𝑎ε)}{2} + 𝑀 \big(𝑙_0 (1 + 𝑎ε) -\frac{𝐿_0 (1 + 𝑏ε)}{2}\big); \tag{4.104}$$

In this and the following formulas:

$𝑙_0, 𝐿_0$ - the length of the wooden rod and the lead pipe at the temperature $𝑡_0$;
$m, M$ - mass of wooden stick and lead pipe;
$𝑎, 𝑏$ - linear coefficients of heat dilation of wood and lead,
$ε = 𝑡-𝑡_0$ - temperature difference

To achieve the position of the center of gravity invariant in relation to $ε$, it is enough that the thermal gradient of the static moment is annulled:

$$\frac{𝑑𝑆}{𝑑ε} = \frac{𝑚 ∙ 𝑙_0 ∙ 𝑎}{2} + 𝑀∙ 𝑙_0 ∙ 𝑎 - \frac{𝑀 ∙ 𝐿_0 ∙ 𝑏}{2} \tag{4.105}$$

Since this gradient does not depend on the argument $ε$, it can be concluded that the influence of the temperature on the shift of the pendulum weight will be completely neutralized if the following condition is fulfilled:

$$λ = \frac{𝑙_0}{𝐿_0} = \frac{𝑏}{𝑎}∙\frac{𝑀}{2𝑀 + 𝑚}\tag{4.106}$$

The estimate of this compensation will be made on a case-by-case basis to which is supplied a pendulum described, for which is admitted: $m = 0.4kg$ and $M = 20kg, t = 2 s$. For 10 days of continuous operation, with a temperature rise of $ε = + 100 ℃$, the timer is -2.16 seconds late, which is almost 8 times less than the thermally uncompensated equivalent (-17.28 s). The residual error is due to the fact that the oscillation period of the pendulum is not only a function of the position of the center of gravity but also of the square moment of inertia. Its reduction is the subject of the next, more complex budget.

The origin of the fault of the fault in the course of the timer with a built-in pendulum with a heat-compensated center of gravity is shown in Figure 25. In relation to the fixed pitch of the pendulum, changing its original volume due to a change in temperature, a certain amount of material has changed the square moment of the inertia of the system. This phenomenon must be taken into consideration for better compensations and its neutralization included. It starts from the formula for the relative reduced length of the pendulum, which is a function of the temperature:

$$\frac{𝑙_𝑟}{𝐿_0} = \frac{\frac{1}{12}𝑀(1 + 𝑏ε)^2 + \frac{1}{3}𝑚λ^2(1 + 𝑎ε)^2 + 𝑀 (λ (1 + 𝑎ε) - \frac{1}{2} (1 + 𝑏ε))^2}{ \frac{1}{2}𝑚λ(1 + 𝑎ε) + 𝑀(λ(1 + 𝑎ε) -\frac{1}{2}(1 + 𝑏ε))} \tag{4.107}$$

The first, second and third members of the formula 4.107 are derived from the square mass moments of inertia, namely their own lead lead, wooden stick for the point of suspension and position - lead pipe for the point of suspension. The first and second members of the name of the same formula originate from the static moments of the wooden rod inertia and the lead pipe for the point of suspension. In this analysis, the condition of annihilating the temperature gradient of the relative reduced length of the pendulum is set:

$$\begin{align} \frac{1}{𝐿_0} ∙ \frac{𝑑𝑙_𝑟}{𝑑ε} &= \frac{\frac{2}{3}𝑎λ^2𝑚(1 + 𝑎ε) + \frac{1}{6}𝑀(1 + 𝑏ε) - 2\big(\frac{𝑏}{2}-λ\big) 𝑀 \Big(λ (1 + 𝑎ε) -\frac{1}{2}(1 + 𝑏ε)\Big)}{ \frac{1}{2}λ𝑚 (1 + 𝑎ε) + 𝑀 \Big(λ (1 + 𝑎ε) +\frac{1}{2} (1 + 𝑏ε)\Big)}\\ \tag{4.108} \\ &- \frac{(\frac{𝑎λ}{2}- \big(\frac{𝑏}{2}-𝑎λ\big) 𝑀) ∙ (\frac{1}{3}λ^2𝑚 (1 + 𝑎ε )^2 + \frac{1}{12}𝑀 (1 + 𝑏ε)^2 + 𝑀 (λ (1 + 𝑎ε) -\frac{1}{2} (1 + 𝑏ε)) ^2)} {(\frac{1}{2}λ (1 + 𝑎ε) + 𝑀 (λ (1 + 𝑎ε) -\frac{1}{2}(1+ 𝑏ε)))^2} \end{align}$$

From the fact that the temperature gradient of the reduced length, the pendulum function of the temperature produces that it is not possible, even theoretically, to achieve a complete compensation of the linear temperature dilatation of the pendulum. It is possible to make such a constructive solution that the thermal disorders of their own period of oscillation of the pendulum will be minimized. For such a constructive solution it is most important to determine the value of the parameter $λ = 𝑙_0 / 𝐿_0$ that will annul the gradient $𝑑𝑙_𝑟 / 𝑑ε$ for the average annual temperature of the place where the timer would be placed. The analysis was carried out on the example of a specific timer that is supplied with the pendulum for which the following parameters were adopted: $m = 0.4kg, M = 20kg and T = 2s$. From the condition that ($𝑑𝑙_𝑟 / 𝑑ε = 0$), and for $ε = 0 ℃$, the parameter $λ = 3,0128$ is determined, resulting in full heat compensation of the pendulum at $ε = 0 ℃$. The length of the wooden carrier $𝑙_0$ and the lead pipe $𝐿_0$ are determined from the formula for the period of their own oscillations of the physical pendulum and are $𝑙_0 = 1178.815mm$ and $𝐿_0 = 390.569mm$.

Table 2 shows the error accumulation of three equivalent clock mechanisms, during a 10 day walk, at 7 different temperatures. The first one is supplied with uncompressed pendulum (mark NK), second - pendant with compensated center of gravity (mark KT) according to preliminary calculation, and third - pendant with compensated gradient of reduced length (code KGRD). The results in the table show, above all, the superiority of the compensation of the thermal expansion of the pendulum, which is based on compensating the temperature gradient of the reduced length over the neutralization of the thermal displacement of the center of gravity. In addition, it is noted that a certain level of error always exists, but that in the last column it is almost negligible and for technical application it is more than acceptable.

Table 2 Comparison of compensation quality

$\underline{ε\,\,[℃]}$	$\underline{NK\,\,[𝑠]}$	$\underline{KT\,\,[𝑠]}$	$\underline{KGRD\,\,[𝑠]}$
-30	+54.8416	+6.48685	-0.004220
-20	+34.5607	+4.32520	-0.001876
-10	+17.2802	+2.16292	-0.000469
0	0.0000	0.0000	0.0000
10	-17.2798	-2.16356	-0.000469
20	-34.5593	-4.32775	-0.001876
30	-51.8384	-6.49258	-0.004220

The calculation of the thermal gradient of the relative reduced length of the pendulum, which is explained and implemented in this chapter, can also be applied directly to the combination of the "invar" alloy; steel or brass (teg). In these technical solutions, the issue of unequal thermal conductivity is not of practical significance, since changes in mean annual, even daily air temperatures are extremely slow. Problems of the numerical realization of the budget for ($𝑑𝑙_𝑟 / 𝑑ε = 0$), are almost insurmountable if they were carried out without electronic aids. This may have been the only reason why it was not performed in the 18th and 19th centuries, but the thermal compensation of the pendulum was carried out by trials, gradually shortening it from the lead pipe. All the calculations given in this chapter have been made in the software package "Matlab". The purpose of these calculations is to point to the numerical analysis strategy itself which would shorten or make even more superfluous experiments or laboratory tests with metric adjustments. These experiments and tests are always long-lasting and costly, and with the development of techniques, they become increasingly unnecessary.

Although it leads to a very good compensation of the pendulum heat expansion, the budget presented in the previous chapter has its drawbacks. First of all, it is quite complicated and, because of this complexity, it is not able to cover everything, at least elements of the pendulum.

Instead of a derived budget, one approximate, iterative process can be used with the 3D modeling application (for example, SolidWorks). In this sense, it is necessary to create a 3D model of the pendulum in the mentioned application, and exposes such a model with successive dilatations realized by scaling linear dimensions, with constant checking of the oscillation period. The algorithm of this procedure is shown in Figure 26. The labels in the scheme are identical to the tags used in the previous chapter, except for the codes $τ$ and $τ_0$, which are new. The symbol $τ$ represents the absolute value of the difference between the actual and the nominal pendulum oscillation period $τ = | 𝑇-𝑇_0 |$, and $τ_0$ is the maximum allowed value of this difference. For the adopted initial value of the pendulum $𝐿_0$ and the value of the coefficient $λ = 𝑙_0/𝐿_0$, in which $𝑙_0$ is the length of the lead compensating tube, the 3D modeling application (SolidWorks) calculates the value of the inertia moment of the pendulum relative to the point of suspension, the mass of the rod $𝑚$, the mass Lead tube and position of the center of gravity. A good initial value for the quotient $λ$ can be determined from formula (4.106). Then the period of its own pendulum oscillations is calculated and determines the difference $τ = | 𝑇-𝑇_0 |$. If $τ > τ_0$, it is necessary to correct the length of the pendulum $𝐿_0$, and if $𝑇 > 𝑇_0$, $𝐿_0$ must be abbreviated, and if $𝑇 < 𝑇_0$, $𝐿_0$ must be extended. This procedure is repeated until the condition is satisfied: $τ <τ_0$. Then check the quality of the compensation.

Using the 3D modeling application, scaling all the dimensions of the pendulum, the thermal dilatation of the pendulum is simulated. Scaling is carried out in proportion to the linear coefficients of heat expansion for wood and lead. Now, using the 3D modeling application, the value of the piercing inertia moment in relation to the point of suspension, the mass of the rod $𝑚$, the mass of the lead tube and the position of the center of gravity is again determined. Furthermore, the period of its own pendulum oscillations is calculated and determines the difference $τ = | 𝑇 - 𝑇_0 |$. If $τ > τ_0$, the correction of the quotient $λ = 𝑙_0𝐿_0$ is necessary, so that if $𝑇 > 𝑇_0$, then $λ$ must be increased, and if $𝑇 < 𝑇_0$, $λ$ must be reduced. In accordance with the diagram in figure (26), this procedure is repeated until the condition is satisfied: $τ < τ_0$, which achieves a thermal compensation with the desired desired accuracy. The same algorithm can also be used in the case of compensating for the thermal expansion of the pendulum for the combination of materials invar and steel respectively, invar and brass.

Figure 26 Diagram of iterative procedure compensation of pendulum heat dilatation's

4.4.2 Neutralization and compensation of air density change

The change in air density is due to changes in temperature, humidity and air pressure changes and, in accordance with the assumptions and formulas given in the preceding chapters, affects the equilibrium of the timing of the timer. More than 150 years ago, watchmakers saw the relationship between the change in atmospheric air pressure and the movement of clock movements, so this effect was called the barometric error of the clock. In this paper, this term is not used because it is not in agreement with the very essence of the problem that exactly the density, and not the air pressure, have a direct impact on the oscillation period of the pendulum timer.

The effect of the density change is small and is about the order of magnitude smaller than the effect of thermal dilatations. However, in precision clocks (astronomical clocks, chronometers, high-quality public and tower timers), this error is noticeable and can become inadmissible in the case of long periods of extremely high or low atmospheric pressure.

There is only one way to completely neutralize the effect of changing the air density on the timer stroke. The timer mechanism can, together with the pendulum, be enclosed in a hermetic (glass) vessel, which completely neutralizes the influence of the variation of the atmospheric air density on the change in the air density in which the pendulum oscillates. This method was most commonly applied to astronomical timers. If, for any reason, it is not possible to have a hermetic clock isolation, another, less efficient procedure is applied which can only partially compensate for the change in air density. A mechanical barometer, an aneroid with a small weight, was built on the piano clock. Due to the change in air pressure, the aneroid moves away, changing the momentum of the inertia and the position of the pendulum center, changes the oscillation period, thereby neutralizing the change in the oscillation period of the pendulum due to the change in the aerostatic thrust. This second method, which requires a detailed budget, can only be partially successful. Namely, as the change in air density does not only depend on the pressure, its impact on the timing of the timer and can not be completely neutralized by a process that is exclusively based on pressure measurement.

The simplest compensation procedure, which does not require the use or construction of any additional device, is based on the fact that the change in air density affects the travel of the timer in two different ways. One effect is the consequence of the change in the dynamic resistance acting on the pendulum and is described by formula (4.109), which reads:

$$\frac{𝑑𝐻}{𝑑ρ} = \frac{Φ_0^2}{8ρ} ∙ \frac{8-Φ_0^2}{24-5Φ_0^2}. \tag{4.109}$$

Another factor that influences the change of the timer's movement due to the change in air density is the aerostatic thrust and is defined by the relation (4.110):

$$\frac{𝑑𝐻_𝑃}{𝑑ρ} = -\frac{1}{2ρ𝐾}. \tag{4.110}$$

These two factors are mutually opposed and tend to compensate each other as described by the following formula:

$$\frac{𝑑𝐻_𝑃}{𝑑ρ} + \frac{𝑑𝐻}{𝑑ρ} = 0. \tag{4.111}$$

By substitution of expressions (4.109) and (4.110) in formula (4.111), a relation is obtained:

$$\frac{Φ_0^2}{8ρ} ∙ \frac{8-Φ_0^2}{24-5Φ_0^2}-\frac{1}{2ρ𝐾} = 0, \tag{4.112}$$

which, after compiling, becomes a bicquound equation in the oscillation amplitude $Φ_0$:

$$Φ_0^4- \Big(8 + 20\frac{ρ}{ρ𝐾}\Big) Φ_0^2 + 96\frac{ρ}{ρ𝐾} = 0 \tag{4.113}$$

The solution of the equation (4.113) defines the oscillation amplitude of the pendulum for which the change in the timing of the watch due to changes in dynamic resistance and aerostatic air pressure will be compensated. Two numerical examples illustrate the practical applicability of the above analysis.

If it is a pendulum of lead density $ρ = 11.34 103\,𝑘𝑔/𝑚^3$ and the density of air is $1,225\,𝑘𝑔/𝑚^3$ then the equation (4.113) of the amplitude $Φ_0$ for which the change in the timing of the timers due to the change in dynamic resistance and aerostatic air pressure is compensated by one another:

$$Φ_0^4-8.00216Φ_0^2 + 0.01037 = 0. \tag{4.114}$$

The solutions of the equation (4.114) read:

$$\begin{split}&Φ_0 ≈ ± 0.0360\,𝑟𝑎𝑑 ≈ ± 2.063° \\ &\,\,\,\,\,\text{or} \\ &Φ_0≈ ± 2.8286\,𝑟𝑎𝑑 ≈ ±162,067°.\end{split} \tag{4.115}$$

The solution $Φ_0 ≈ ± 2.063°$ has a physical meaning.

If it is a steel pendulum steel $ρ = 7,85 10^3\,𝑘𝑔/𝑚^3$ and the density of air is $1,225\,𝑘𝑔/𝑚^3$ then according to the equation (4.108) amplitude $Φ_0$ for which the change in the timing of the timers due to the change in dynamic resistance and aerostatic air pressure is compensated

$$Φ_0^4-8.00314Φ_0^2 + 0.01508 = 0 \tag{4.116}$$

The equations solution (4.116) reads:

$$\begin{split}&Φ_0 ≈ ± 0.043413\,𝑟𝑎𝑑 ≈ ± 2.487° \\ &\,\,\,\,\,\text{or} \\ &Φ_0 ≈ ± 2.8286\,𝑟𝑎𝑑 ≈ ± 162.067°\end{split} \tag{4.117}$$

The solution $Φ_0 ≈ ± 2.487°$ has a physical meaning.

In order for this method to be applied, it is necessary to ensure that the pendulum always receives constant amounts of energy from the average impulse mechanism. How this is achieved is the topic of the next chapter. Although the implemented budget is not absolutely correct, because it is based exclusively on the influence of dynamic resistance, or resistance pressure on the pendulum weight, it can be used to estimate the order of magnitude.