On the Theory of Dynamic Damping of Vibrations of Internal-Combustion-Engine Shafts

V. K. NECHAEV
Professor, Doctor of Technical Sciences

Introduction

Modern methods for calculating the torsional vibrations of crankshafts of piston engines make it possible to determine, with a greater or lesser degree of accuracy, those additional twisting stresses that arise in the shaft when the engine operates at any of its critical speeds, under conditions of exact torsional resonance. The criterion for the presence of this resonance is taken to be the coincidence of the angular frequency $\eta_h$ of the resonating harmonics of the torque on the engine shaft with one of the conditional frequencies $\omega_0$ of the natural torsional vibrations of the shafting of the installation under investigation. The ratio

$$ \gamma=\frac{\eta_h}{\omega_0}, $$

hereinafter called the tuning coefficient, assumes, under resonance conditions, the value $\gamma=1$.

The external indication of the resonant character of the oscillatory process is usually an intensive growth of the amplitudes of forced vibrations. Such growth, or development of amplitudes, continues until the input of energy into the oscillating system due to the work of the excitation, performed by the resonating harmonic torques, becomes equal to its expenditure caused by the presence of damping forces in the installation. The maximum resonant amplitude of vibration is determined from the equality of these works for each cycle of vibration. It is precisely this limiting amplitude that is given by the calculation methods for resonant vibrations of crankshafts known at the present time [3, 15, 17, 18].

But such “favorable” (for the development of vibrations) ideal conditions of sufficiently prolonged operation of the installation at an exactly constant value of the tuning coefficient $\gamma=1$ are in fact never realized in practice, since in actual power installations with piston engines both $\eta_h$ and $\omega_0$ are always variable.

The angular frequency $\eta_h$ of the harmonic torque $M_h \sin \eta_h t$ is related to the angular speed of rotation of the shaft $\eta$ and to the order $h$ of this moment by the relation

$$ \eta_h=h\eta. $$

Thus, $\eta_h$ is constant in time only under the condition $\eta=\mathrm{const}$.

In general, in the dynamic investigation of installations with piston engines, one must distinguish:

a) the mean (imaginary or conventional) angular speed $\eta_0$ of rotation of the engine shaft;

b) the instantaneous angular velocity \(\eta_s\) of rotation of the engine shaft, when this shaft, together with the entire shafting of the installation, is regarded as an absolutely rigid body (in torsion);

c) the true instantaneous velocity \(\eta_i\) of rotation of some specified section of the shaft.

The mean angular velocity \(\eta_0\) is determined by the formula

\[ \eta_0=\frac{\pi n}{30}\ \mathrm{sec}^{-1} \]

from the engine speed \(n\) in revolutions per minute, measured, for example, by a totalizing counter. The velocity \(\eta_0\) is usually constant under a constant load on the engine and unchanged regulation of it.

The instantaneous velocity \(\eta_s\) changes continuously as a result of the inequality between the instantaneous value of the variable torque on the engine shaft and the moment of the external load. The specific value of \(\eta_s\) for each instant of time can be found, for example, by means of the well-known Wittenbauer method. Oscillations of this velocity depend on the character of the resultant diagram of tangential forces on the engine shaft and on the law of variation of the moment of the external load with the angle of rotation of the crankshaft.

Obviously, \(\eta_s\) may also be called the instantaneous speed of rotation of the engine shaft, i.e., of all its sections and of all masses rotating with it.

The true instantaneous angular velocity \(\eta_i\), owing to torsional vibrations of the shafting, is not only variable in time but also differs along the length of the shaft of the installation. The velocities \(\eta_i\) are determined analytically only by a detailed calculation of the torsional vibrations of the shafting and by constructing a torsiogram for each shaft section of interest to us separately.

The relative amplitude of oscillations of the velocities \(\eta_s\) and \(\eta_i\) over the course of one cycle of operation of the engine cylinders is characterized, respectively, by the expressions

\[ \delta_s=\frac{\eta_{s\max}-\eta_{s\min}}{\eta_0}, \]

\[ \delta_i=\frac{\eta_{i\max}-\eta_{i\min}}{\eta_0}. \]

The quantity \(\delta_s\)—the “rigid” degree of nonuniformity, or, as it is often called, the degree of flywheel nonuniformity—is calculated by generally known methods, under the assumption of absolute rigidity of the shafting in torsion. Naturally, \(\delta_s\) is the same for all sections of the shaft. The quantity \(\delta_i\), to which we shall give the name of the “elastic” (or true) degree of nonuniformity, is no longer the same for different sections of the shaft. Experimentally, \(\delta_i\) can be found by torsiographing the portion (or, more precisely, the section) of the shaft that interests us.

The “rigid” degree of nonuniformity \(\delta_s\), corresponding to the so-called nodal vibrations of the shaft, enters into \(\delta_i\) through vibrations caused by the principal orders of the torque harmonics that are far from resonance, if by this \(\delta_i\) one understands the full degree of nonuniformity due to the action on the shafting of the entire harmonic spectrum of the engine torque and of the moment of the external load.

All generally accepted methods for calculating vibrations of crankshafts, especially forced vibrations, assume the replacement of all masses of the crank-and-connecting-rod mechanism of each cylinder, which participate in the oscillatory process in a variable manner, by rotating masses with

constant moment of inertia. In fact, however, as is known, the moment of inertia of the masses of the connecting-rod–crank mechanism, reduced to the crank radius, is a periodic function of the angle of rotation \( \alpha \) of the crankshaft. Accordingly, the natural frequency of vibration of the shafting also varies periodically with \( \alpha \), in the interval from some \( \omega_{max} \) to \( \omega_{min} \). Here it is expedient to introduce the concept of the “internal” degree of nonuniformity \( \delta_\theta \) of the oscillating system, defined by the relation

\[ \delta_\theta=\frac{\omega_{max}-\omega_{min}}{\omega_0} \]

and characterizing the range of variation (pulsation) of the natural frequency relative to its mean value \( \omega_0 \). The frequency of these pulsations of the natural frequency depends both on the rotational speed of the engine shaft and on the number of its cylinders and the mutual arrangement of the crankshaft throws.

In contrast to the “internal” degree of nonuniformity \( \delta_\theta \), which is wholly determined by the basic properties of the oscillating system itself, the quantities \( \delta_s \) and \( \delta_i \) may be given the name “external” degrees of nonuniformity (respectively “rigid” and “elastic”), since the latter are, in essence, determined by the external forces acting on the system.

In any installation with piston engines, the continuous oscillations \( \eta \) and \( \omega_0 \) (characterized by the degrees of nonuniformity \( \delta_s \), \( \delta_\theta \), and \( \delta_i \)) at all times deflect the system from that state of a stationary oscillatory process which alone is considered by the ordinary theory of harmonic vibrations. An analogous phenomenon also occurs when the installation operates at any one of its critical speeds.

It appears sufficiently obvious that such a permanent disturbance of the resonant oscillatory process in the shaft of a piston engine, due to external and internal nonuniformity (\( \delta_s \), \( \delta_i \), \( \delta_\theta \)), excludes the possibility of the development of those resonant amplitudes and stresses which are calculated by the usual methods, assuming \( \gamma=\mathrm{const}=1 \). With an increase of \( \delta \) in the oscillating system, an additional, so-called apparent or dynamic, damping seems to increase. At the same time, unlike real damping (for example, internal friction in the shaft, friction in the bearings, etc.), dynamic damping is not connected, at least in the main, with dissipation of energy.

However, the question of dynamic damping of torsional vibrations has scarcely been developed. An analytical accounting for the influence of the degree of nonuniformity (external and internal) on the magnitude of the amplitudes of resonant vibrations of internal-combustion-engine shafts can at present be carried out only with the aid of the well-known Mansey formula [14].

This formula, attractive by virtue of its simplicity and clarity, is cited in many works on the theory of vibrations and is recommended by a number of authors [1, 4, 10, 11, 12] for use in design practice. But there is no unanimity of opinion in the interpretation of Mansey’s formula and in the assessment of its degree of accuracy. Its fundamental premises cannot be considered convincing. Moreover, a critical analysis of Mansey’s formula and of certain views connected with it on the effect of dynamic damping compels one to recognize it as plainly unsatisfactory [5]. Its use may lead to serious errors not only in the quantitative but also in the qualitative assessment of those dynamic processes which take place in the shafting of a power installation with a piston engine.

The purpose of the present work is to carry out a detailed analysis of the influence of nonuniformity of rotation of the engine shaft on the character and development of resonant and nonresonant vibrations in it, and also to establish the true effect of dynamic damping caused by this nonuniformity.

1. General Definitions

In studying the vibrations of internal-combustion-engine shafts, it is necessary to keep in mind the fundamental distinction between the “external” \((\delta_s, \delta_i)\) and “internal” \((\delta_\theta)\) degrees of nonuniformity in installations with piston engines.

“Internal” nonuniformity \((\delta_\theta > 0)\), caused by periodic changes in the moments of inertia of the reduced masses of the engine, is wholly connected with the basic characteristics of the installation. Here the reduced system, at least when \(\delta_\theta\) is not too small, loses the properties of a harmonic resonator, and its oscillations acquire a quasi-harmonic character.

“External” nonuniformity \((\delta_s, \delta_i)\), that is, nonuniformity of shaft rotation, does not introduce such substantial changes into the oscillatory system, which, as in the ideal case \((\delta_s = 0)\), under certain restrictions indicated below, can usually be described by linear differential equations with constant coefficients. Thus, in studying the vibrations of installations with constant reduced masses \((\delta_\theta = 0)\), having \(\delta_s > 0\) (and, to a certain extent, \(\delta_i > 0\)), we retain the possibility of using the entire basic apparatus of the theory of harmonic oscillations, with its spectral approach to the analysis of the effect of complex periodic excitation.

In real oscillatory systems, such as internal-combustion-engine installations, one usually has simultaneously \(\delta_\theta > 0\) and \(\delta_s > 0\), or, more precisely, \(\delta_\theta > 0\) and \(\delta_i > 0\). However, the study of vibrations under these conditions presents exceptional difficulties.

We shall confine ourselves here only to an analysis of the case \(\delta_\theta = 0\), in order to reveal in pure form the effect of the presence of \(\delta_s > 0\) (or \(\delta_i > 0\)) in a linear system, and thereby to establish the concrete significance of shaft-rotation nonuniformity as a factor in the dynamic damping of torsional vibrations in internal-combustion-engine installations.

Each cylinder of a working engine produces on the crankshaft a variable torque \(M_{kr}\), whose magnitude, at any angle of rotation of the crankshaft \(\alpha\), can be determined by the usual methods from the indicator diagram. From the data obtained in this way, the curve \(M_{kr}\) is plotted as a function of \(\alpha\), having, depending on the engine cycle, a period \(2\pi\) or \(4\pi\).

Harmonic analysis of this curve, which as a rule always satisfies the Dirichlet conditions, permits the torque of one cylinder to be represented in the form of a trigonometric series

\[ M_{kr} = M_0 + \sum_{h=1}^{\infty} A_h \sin h\alpha + \sum_{h=1}^{\infty} B_h \cos h\alpha = M_0 + \sum_{h=1}^{\infty} M_h \sin (h\alpha + \varepsilon_h) \tag{1} \]

\[ (h = 1, 2, 3\ldots), \]

for two-stroke engines, and

\[ M_{kr} = M_0 + \sum_{h=\frac12}^{\infty} M_h \sin (h\alpha + \varepsilon_h) \tag{2} \]

\[ (h = \tfrac12, 1, 1\tfrac12, 2\ldots), \]

for four-stroke engines. Here:

\(M_0\)—the constant component of the torque \(M_{kr}\),

\(M_h\)—the amplitude of the harmonic moment (or, more briefly, the amplitude of the harmonic) of order \(h\), determined by the formula:

\[ M_h = \sqrt{A_h^2 + B_h^2}, \]

\(\varepsilon_h\) is the phase angle of this harmonic, calculated by the formula

\[ \varepsilon_h=\operatorname{arctg}\frac{B_h}{A_h} \]

and characterizing the position of the vector \(M_h\) relative to the crank at the instant corresponding to \(\alpha=0\), that is, TDC.

In what follows, for the sake of simplicity in all the derivations, we shall consider only two-stroke engines, that is, take \(h=1, 2, 3\ldots\). To obtain results also suitable for four-stroke machines, it is sufficient in the final formulas to understand by \(h\) not only integral orders, but also half-orders.

The angle of rotation of the crank \(\alpha\), entering into (1) and (2), is related to the time \(t\) by the obvious relation

\[ \alpha=\int_0^t \eta\,dt, \tag{3} \]

where \(\eta\) is the instantaneous angular velocity of rotation of the shaft, or, more precisely, the instantaneous angular velocity of rotation of the crank of the cylinder under consideration. Now from (1):

\[ \begin{aligned} M_{kp} &=M_0+\sum_{h=1}^{\infty} A_h \sin\left(h\int_0^t \eta\,dt\right) +\sum_{h=1}^{\infty} B_h \cos\left(h\int_0^t \eta\,dt\right) \\ &=M_0+\sum_{h=1}^{\infty} M_h \sin\left(h\int_0^t \eta\,dt+\varepsilon_h\right). \end{aligned} \tag{4} \]

In the generally accepted methods of calculating vibrations of crankshafts, direct proportionality between \(\alpha\) and \(t\) is assumed:

\[ \alpha=\eta_0 t, \tag{5} \]

where \(\eta_0\), obviously, represents a certain mean angular velocity of rotation of the shaft over the entire time interval under consideration, that is, at least over the course of one cycle of the working process in the cylinder.

Substitution of (5) into (1) gives the usual spectral representation of the torque \(M_{kp}\) in the form of a sum of elementary moments (harmonics), each of which is regarded as a simple harmonic function of time,

\[ M_{kp}=M_0+\sum_{h=1}^{\infty} A_h\sin h\eta_0 t +\sum_{h=1}^{\infty} B_h\cos h\eta_0 t =M_0+\sum_{h=1}^{\infty} M_h\sin\left(h\eta_0 t+\varepsilon_h\right). \tag{6} \]

Here all \(M_h\), \(A_h\), \(B_h\), \(\varepsilon_h\) are customarily considered constant, independent of time. But this assumption is the less valid the less stable the operating regime of the engine is and, as will be shown below, the greater the nonuniformity of rotation of its shaft.

Under constant load on the engine and with an established course of the thermal processes in the cylinder, the accuracy of series (6) as a characteristic of the true (that is, harmonic in time) spectrum of the torque \(M_{kp}\) is determined entirely by the errors of relation (5), on the basis of which, as the result of a purely geometric (that is, without taking the time factor into account) harmonic analysis of the curve \(M_{kp}=f_0(\alpha)\), time was introduced and series (6) was obtained.

It is obvious that in all real practical problems

\[ \eta \ne \mathrm{const} \]

and, consequently, series (1) passes into (4), where each term of the form:

\[ (M_h)=M_h\sin\left(h\int_0^t \eta\,dt+\varepsilon_h\right) \tag{7} \]

often turns out to be a very complicated function of time \(t\).

Thus, the “geometric” harmonic analysis of the curve \(M_{\mathrm{cr}}=f_0(\alpha)\), the results of which are written in the form of the ordinary series (1), is still far from giving an answer to the question of the concrete characteristics of the harmonic spectrum of the torque \(M_{\mathrm{cr}}\), the components of which, under the conditions of the investigation of forced vibrations of linear systems with constant parameters, must be pure monoharmonic functions of time. To obtain these monoharmonic components of the torque directly, it would be necessary to carry out a harmonic analysis of the curve \(M_{\mathrm{cr}}\), constructed as a function of time \(t\). However, constructing such a curve, at least at the beginning of calculations of shaft vibrations, presents insurmountable difficulties, since the dependence of \(\eta\) on \(t\) is not sufficiently known in advance.

Moreover, the curve \(M_{\mathrm{cr}}=f_0(\alpha)\) also has a quite definite physical meaning, since the principal thermal processes occurring in the engine cylinder are not so much an explicit function of time as a function of the variable volume of the working space of the cylinder, that is, consequently, of the angle of rotation of the crank \(\alpha\). Finally, it simply makes no sense to refrain from using the results of extensive computational work on the harmonic analysis of typical torque diagrams, carried out by a number of investigators [3, 10, 15] and presented either in the form of graphs or tables giving specific values of the harmonic coefficients

\[ \frac{M_h}{Fr} \]

(where \(F\) is the piston area) for each \(h\) and for various values of the mean indicated pressure in the cylinder, \(p_i\). The use of such graphs facilitates the designer’s work, since it frees him from the need to perform laborious computations each time for the harmonic analysis of the curve \(M_{\mathrm{cr}}\).

Thus, it is nevertheless expedient to begin our investigation with series (1), each component of which:

\[ (M_h)=M_h\sin(h\alpha+\varepsilon_h) = M_h\sin\left(h\int_0^t \eta\,dt+\varepsilon_h\right) \tag{8} \]

represents a simple harmonic function of \(\alpha\) and, at the same time, a complicated modulated\(^1\) function of time \(t\).

In accordance with the definitions and terminology accepted in radio engineering [7], we shall distinguish the following principal types of modulation of function (8):

1) Modulation in amplitude, or amplitude modulation. Here:

\[ \eta=\eta_0=\mathrm{const}, \]

\[ \varepsilon_h=\mathrm{const}, \]

\(^1\) By modulation we understand, as usual, the introduction into a purely sinusoidal oscillatory process of certain special, particular periodic deviations.

that is,

\[ \alpha = h\eta_0 t \]

and

\[ (M_h)=M_h\sin(h\eta_0 t+\varepsilon_h), \tag{9} \]

where the amplitude of the moment \(M_h\) is variable in time. The degree of deviation of \((M_h)\) from a purely sinusoidal function of \(t\) is determined by the character of the changes with \(t\) of the amplitude \(M_h\).

2) Modulation in frequency, or frequency modulation. Here:

\[ M_h=\text{const},\qquad \varepsilon_h=\text{const}, \]

while the angular frequency \(\eta\) is variable with \(t\).

3) Modulation in phase, or phase modulation, characterized by the conditions:

\[ \eta=\eta_0=\text{const}, \]

\[ M_h=\text{const}, \]

while the phase angle \(\varepsilon_h\) varies as a function of time. It is easy to show that phase modulation is analogous to frequency modulation. Frequency modulation, through the variable rotational speed of the vector \(M_h\), evidently inevitably leads to changes in \(\varepsilon_h\) and therefore is at the same time a modulation in phase. In view of this, there is no need to consider phase modulation separately.

4) Mixed modulation—when \(M_h\) and \(\eta\) are variable in time.

For the convenience of the further investigation, let us introduce the concepts of gas and inertia harmonics of the torque on the engine shaft. We shall call the harmonic components of the torque on the shaft produced by the pressure of the gases on the piston gas harmonics, and the components of this moment caused by the inertia forces of the reciprocating masses of the connecting-rod-and-crank mechanism inertia harmonics.

Such a differentiation of the components \(M_{\mathrm{cr}}\) is expedient already in the ordinary analysis of vibrations of the shafts of installations operating over a wide range of speeds. It is especially necessary in our investigation, since cyclic oscillations of the angular velocity of rotation of the engine shaft correspondingly change the magnitude of the inertia forces of the reciprocating parts of the connecting-rod-and-crank mechanism and, on the other hand, have practically no effect on the magnitude of the forces of gas pressure on the piston.

For \(\eta=\text{const}\), the inertia harmonics are easily computed analytically, by known formulas, [5]. This allows us to use, for carrying out harmonic analysis (for example, by graphical or tabular methods), the curve of torques constructed on the basis of the indicator diagram without taking inertia forces into account. But, of course, in the final results of this analysis, that is, in all \(M_h\) and \(\varepsilon_h\) entering into series (1), it is subsequently necessary to take due account of the amplitudes and phases of the inertia harmonics.

It is evident that nonuniformity of rotation of the engine shaft leads to frequency modulation of the gas harmonics in expansion (1), and to mixed modulation of the inertia harmonics.

The specific effect of this modulation, undoubtedly different for gas and inertia harmonics, depends on the law of variation of the instantaneous angular velocity of rotation of the crankshaft \(\eta\) with time.

2. Gas Harmonics with Monoharmonic Frequency Modulation

As is known, [6], the actual motion of the crankshaft (or of the entire crankshaft train, if it is regarded as absolutely rigid) is composed of its uniform rotation with mean speed \(\eta_0\) and harmonic oscillations superposed on this rotation:

\[ \alpha=\eta_0 t+\sum_{h=1}^{\infty}\varphi_h \sin(h\eta_0 t-\zeta_h) \tag{10} \]

Here:

\(\varphi_h\) is the amplitude of the forced oscillations of the crankshaft under the action of the harmonic components of order \(h\) of all torques on the shafting;

\(\zeta_h\) is the phase angle of these oscillations.

In the case of a very high stiffness of the engine shaft (as compared with the stiffness of the entire shafting), it may be assumed that its forced oscillations are caused only by the principal harmonics of the torque; moreover, the greatest effect is, of course, produced by the principal harmonics of the lower orders.^1

For the time being taking into account only the strongest of these harmonics, having order \(h_0\), we write the equation of motion of the crankshaft in the form

\[ \alpha=\eta_0 t+\varphi_0 \sin(h_0\eta_0 t-\zeta), \tag{11} \]

or

\[ \alpha=\eta_0 t+\varphi_0 \sin(\nu t-\zeta), \tag{12} \]

where:

\(h_0\) is the order of the harmonic \(M_{h0}\) of the torque on the shaft, which causes nonuniform rotation of the shaft with degree of nonuniformity \(\delta_s\), or, more briefly, the order of cyclic nonuniformity \(\delta_s\);

\(h_0\eta_0=\nu\) is the angular frequency of this harmonic, or the angular frequency of the cyclic nonuniformity \(\delta_s\);

\(\varphi_0\) is the amplitude of the forced oscillations of the crankshaft (shaft) caused by the action of the moment \(M_{h0}\) (for the method of determining it in the case of an absolutely rigid shaft, see the author’s work [6]);

\(\zeta\) is the phase angle of these oscillations, or the phase angle of the cyclic nonuniformity \(\delta_s\).

Differentiation of (11) gives the instantaneous angular velocity of rotation of the crankshaft

\[ \eta=\frac{d\alpha}{dt}=\eta_0+h_0\eta_0\varphi_0\cos(h_0\eta_0 t-\zeta) \]

and its extreme values

\[ \eta^{\max}_{\min}=\eta_0\pm h_0\eta_0\varphi_0. \]

Hence the degree of nonuniformity is

\[ \delta_s=\frac{\eta_{\max}-\eta_{\min}}{\eta_0}=2h_0\varphi_0 \tag{13} \]

and

\[ \varphi_0=\frac{\delta_s}{2h_0}. \tag{14} \]

^1 Here, at the present stage of our investigation, we exclude the possibility of operation of the engine at any critical speed, since only under these conditions:

\[ \delta_s \ll \theta_i . \]

Now

\[ \eta=\eta_0\left[1+\frac{\delta_s}{2}\cos(\nu t-\zeta)\right] \]

and the variable angular frequency of the harmonic (in \(\alpha\)) moment \(M_h\)

\[ h\eta=h\eta_0\left[1+\frac{\delta_s}{2}\cos(\nu t-\zeta)\right] \]

or

\[ h\eta=\eta_h\left[1+\frac{\delta_s}{2}\cos(\nu t-\zeta)\right], \]

where the quantity \(\eta_h\), determined by the expression

\[ \eta_h=h\eta_0, \tag{15} \]

may be regarded as the mean, “carrier” frequency of the moment \(M_h\). The meaning of this term will be explained below.

Using relations (12) and (14), we shall establish the relation between the true angle of rotation of the crankshaft \(\alpha\) and the degree of nonuniformity \(\delta_s\):

\[ \alpha=\eta_0 t+\frac{\delta_s}{2h_0}\sin(\nu t-\zeta) \tag{16} \]

and substitute this value of \(\alpha\) into (8)

\[ (M_h)=M_h\sin\left[h\eta_0t+\frac{\delta_s}{2}\frac{h}{h_0}\sin(\nu t-\zeta)+\varepsilon_h\right] \]

or, on the basis of (15),

\[ (M_h)=M_h\sin\left[\eta_h t+\xi\sin(\nu t-\zeta)+\varepsilon_h\right], \tag{17} \]

where the notation has been introduced

\[ \xi=\frac{h}{h_0}\frac{\delta_s}{2}. \tag{18} \]

Expression (17) now characterizes the actual form of each of the harmonic (in \(\alpha\)) components of series (1), obtained as a result of harmonic analysis of the curve \(M_{kp}=f(\alpha)\).

We shall henceforth call the coefficient \(\xi\), determined by formula (18) or, on the basis of (14), by the expression

\[ \xi=h\varphi_0, \tag{19} \]

the modulation amplitude. It is evident that \(\xi\) represents the absolute amplitude \(\varphi_0\) of oscillation of the crankshaft under consideration, increased \(h\) times, or of the entire crankshaft, if we are interested here only in the effect of the rigid degree of nonuniformity \(\delta_s\).

Introduce the notation

\[ m_h=\frac{(M_h)}{M_h}. \tag{20} \]

Then from (17) we obtain an expression for the “unit” component, harmonic in \(\alpha\) (but, of course, no longer harmonic in \(t\)), of series (1), corresponding to the value \(M_h=1\):

\[ m_h=\sin\left[\eta_h t+\xi\sin(\nu t-\zeta)+\varepsilon_h\right]. \tag{21} \]

Let us proceed to a detailed analysis of this expression. As a result of the analysis we must obtain an answer to the question of the influence of cyclic oscillations of the shaft rotation speed on the characteristics of the truly harmonic (that is, harmonic in time) spectrum of the torque \(M_{\mathrm{cr}}\).

We expand (21):

\[ \begin{aligned} m_h={}&\sin(\eta_h t+\varepsilon_h)\cdot \cos[\xi\sin(\nu t-\zeta)]\\ &+\cos(\eta_h t+\varepsilon_h)\cdot \sin[\xi\sin(\nu t-\zeta)] . \end{aligned} \tag{22} \]

But, as is known, [2, 4]:

\[ \cos(x\sin\varphi)=I_0(x)+2\sum_{q=1}^{\infty} I_{2q}(x)\cos 2q\varphi, \tag{23} \]

\[ \sin(x\sin\varphi)=2\sum_{q=0}^{\infty} I_{2q+1}(x)\sin(2q+1)\varphi, \tag{24} \]

where \(I_q(x)\) is a Bessel function of the first kind, defined by the general expression:

\[ I_q(x)=\left(\frac{x}{2}\right)^q \sum_{p=0}^{\infty} \frac{(-1)^p\cdot\left(\dfrac{x}{2}\right)^{2p}} {p!\Gamma(q+p+1)} . \]

Here \(\Gamma\) is the usual symbol for the gamma function.

Now, by formula (23),

\[ \cos[\xi\sin(\nu t-\zeta)] = I_0(\xi)+2\sum_{q=1}^{\infty} I_{2q}(\xi)\cos 2q(\nu t-\zeta) \]

and, consequently,

\[ \begin{aligned} &\sin(\eta_h t+\varepsilon_h)\cdot \cos[\xi\sin(\nu t-\zeta)]\\ &\qquad = I_0(\xi)\cdot \sin(\eta_h t+\varepsilon_h) +2\sum_{q=1}^{\infty} I_{2q}(\xi)\sin(\eta_h t+\varepsilon_h)\cdot \cos 2q(\nu t-\zeta). \end{aligned} \]

Hence, after elementary transformations, we obtain the final expression for the first term in formula (22):

\[ \begin{aligned} &\sin(\eta_h t+\varepsilon_h)\cdot \cos[\xi\sin(\nu t-\zeta)] = I_0(\xi)\sin(\eta_h t+\varepsilon_h)\\ &\qquad +\sum_{q=1}^{\infty} I_{2q}(\xi)\sin[(\eta_h t+\varepsilon_h)+2q(\nu t-\zeta)]\\ &\qquad +\sum_{q=1}^{\infty} I_{2q}(\xi)\sin[(\eta_h t+\varepsilon_h)-2q(\nu t-\zeta)] . \end{aligned} \tag{25} \]

Further, on the basis of formula (24):

\[ \sin[\xi\sin(\nu t-\zeta)] = 2\sum_{q=0}^{\infty} I_{2q+1}(\xi)\sin(2q+1)(\nu t-\zeta) \]

and, analogously to the preceding,

\[ \cos(\eta_h t+\varepsilon_h)\,\sin[\xi\sin(\nu t-\zeta)] = \sum_{q=0}^{\infty} I_{2q+1}(\xi)\sin[(\eta_h t+\varepsilon_h)+(2q+1)(\nu t-\zeta)] + \]

\[ +\sum_{q=0}^{\infty} I_{2q+1}(\xi)\sin[-(\eta_h t+\varepsilon_h)+(2q+1)(\nu t-\zeta)]. \tag{26} \]

Now substitution of expressions (25) and (26) into (22), after certain transformations, gives

\[ m_h=I_0(\xi)\sin(\eta_h t+\varepsilon_h) +\sum_{q=1}^{\infty} I_q(\xi)\sin[(\eta_h t+\varepsilon_h)+q(\nu t-\zeta)] + \]

\[ +\sum_{q=1}^{\infty}(-1)^q I_q(\xi)\sin[(\eta_h t+\varepsilon_h)-q(\nu t-\zeta)]. \]

From this we obtain the sought final expansion in time of the unit harmonic, with respect to the crank angle \(\alpha\), of the moment \(m_h\), represented by equation (21):

\[ m_h=\sin[\eta_h t+\xi\sin(\nu t-\zeta)+\varepsilon_h] = I_0(\xi)\sin(\eta_h t+\varepsilon_h) + \]

\[ +\sum_{q=1}^{\infty} I_q(\xi)\sin[(\eta_h+q\nu)t+(\varepsilon_h-q\zeta)] + \]

\[ +\sum_{q=1}^{\infty}(-1)^q I_q(\xi)\sin[(\eta_h-q\nu)t+(\varepsilon_h+q\zeta)]. \tag{27} \]

The modulation amplitude \(\xi\) entering here is determined by formulas (18 and 19). It is directly proportional to the cyclic degree of nonuniformity \(\delta_s\) and to the ratio of the order \(h\) of the considered harmonic of the torque to the order \(h_0\) of the harmonic causing this nonuniformity \(\delta_s\).

But, as is known,

\[ I_0(0)=1, \]

\[ I_q(0)=0,\qquad (q\ne0) \]

therefore expression (27), for \(\xi=0\), that is, in the case of absolutely uniform rotation of the shaft, passes into the elementary relation

\[ m_h=\sin(\eta_h t+\varepsilon_h), \]

or

\[ (M_h)=M_h\sin(h\eta_0 t+\varepsilon_h), \]

which underlies all ordinary calculations of vibrations of crankshafts that do not take into account the nonuniformity of rotation of the engine shaft.

Let us consider some particular cases of monoharmonic frequency modulation of the moments \((M_h)\) and \(m_h\), harmonic in \(\alpha\), represented by formulas (17) and (21) and entering as terms into series (1).

1st case. Purely sinusoidal modulation:

\[ \zeta=0;\qquad \varepsilon_h=0. \]

Under these conditions, from (27):

\[ m_h=\sin[\eta_h t+\xi\sin\nu t] = I_0(\xi)\sin\eta_h t + \]

\[ +\sum_{q=1}^{\infty} I_q(\xi)\sin(\eta_h+q\nu)t +\sum_{q=1}^{\infty}(-1)^q\sin(\eta_h-q\nu)t. \tag{28} \]

or in expanded form:

\[ \begin{aligned} m_h&=\sin(\eta_h t+\xi\sin \nu t)=I_0(\xi)\sin\eta_h t+\\ &\quad+I_1(\xi)\sin(\eta_h+\nu)t+I_2(\xi)\sin(\eta_h+2\nu)t+I_3(\xi)\sin(\eta_h+3\nu)t+\cdots\\ &\quad-I_1(\xi)\sin(\eta_h-\nu)t+I_2(\xi)\sin(\eta_h-2\nu)t-I_3(\xi)\sin(\eta_h-3\nu)t+\cdots . \end{aligned} \tag{29} \]

2nd case. Purely cosinusoidal modulation:

\[ \zeta=\pi/2;\qquad \varepsilon_h=0 \]

Here from (27)

\[ \begin{aligned} m_h&=\sin(\eta_h t+\xi\cos\nu t)=I_0(\xi)\sin\eta_h t\\ &\quad+\sum_{q=1}^{\infty} I_q(\xi)\sin\left[(\eta_h+q\nu)t-q\pi/2\right] +\sum_{q=1}^{\infty}(-1)^q I_q(\xi)\sin\left[(\eta_h-q\nu)t+q\pi/2\right] \end{aligned} \tag{30} \]

or, specifically, in expanded form

\[ \begin{aligned} m_h&=\sin(\eta_h t+\xi\cos\nu t)=I_0(\xi)\sin\eta_h t\\ &\quad+I_1(\xi)\cos(\eta_h+\nu)t-I_2(\xi)\sin(\eta_h+2\nu)t-I_3(\xi)\cos(\eta_h+3\nu)t+\cdots,\\ &\quad+I_1(\xi)\cos(\eta_h-\nu)t-I_2(\xi)\sin(\eta_h-2\nu)t-I_3(\xi)\cos(\eta_h-3\nu)t+\cdots . \end{aligned} \tag{31} \]

3rd case.

\[ \zeta=0;\qquad \varepsilon_h=-\pi/2. \]

From (27):

\[ \begin{aligned} m_h&=\cos(\eta_h t+\xi\sin\nu t)=I_0(\xi)\cos\eta_h t+\sum_{q=1}^{\infty}I_q(\xi)\cos(\eta_h+q\nu)t+\\ &\quad+\sum_{q=1}^{\infty}(-1)^q I_q(\xi)\cos(\eta_h-q\nu)t=I_0(\xi)\cos\eta_h t+\\ &\quad+I_1(\xi)\cos(\eta_h+\nu)t+I_2(\xi)\cos(\eta_h+2\nu)t+\cdots\\ &\quad-I_1(\xi)\cos(\eta_h-\nu)t+I_2(\xi)\cos(\eta_h-2\nu)t-\cdots . \end{aligned} \tag{32} \]

4th case.

\[ \zeta=-\pi/2;\qquad \varepsilon_h=0. \]

Here

\[ m_h=\sin(\eta_h t-\xi\cos\nu t). \]

Taking into account that, by the general properties of Bessel functions [2,4], for \(q=0\) and any even \(q\):

\[ I_q(-\xi)=I_q(\xi), \]

and for odd \(q\):

\[ I_q(-\xi)=-I_q(\xi), \]

we may use expansion (31) directly, introducing only the corresponding changes of signs of the individual terms. Then in our case

\[ \begin{aligned} m_h&=\sin(\eta_h t-\xi\cos\nu t)=I_0(\xi)\sin\eta_h t\\ &\quad-I_1(\xi)\cos(\eta_h+\nu)t+I_2(\xi)\sin(\eta_h+2\nu)t+I_3(\xi)\cos(\eta_h+3\nu)t+\cdots\\ &\quad-I_1(\xi)\cos(\eta_h-\nu)t+I_2(\xi)\sin(\eta_h-2\nu)t+I_3(\xi)\cos(\eta_h-3\nu)t-\cdots . \end{aligned} \tag{33} \]

In a completely analogous manner, from (32) one could obtain the expansion in a series of functions

\[ m_h=\cos(\eta_h t-\xi\sin\nu t). \]

The formulas given above, (27)—(33), already provide some material for a number of conclusions about the influence of nonuniformity of rotation of the engine shaft on the usual spectrum—harmonic in \(\alpha\)—of the torque due to gas pressure, obtained as a result of carrying out a harmonic analysis of the curve \(M_{\mathrm{cr}}=f_{0}(\alpha)\). Let us note the most essential of these conclusions.

1. Cyclic (that is, periodic) nonuniformity of rotation of the crankshaft causes frequency modulation of each gas harmonic \(M_h\) entering series (1). As a result of such modulation, the moment \(M_h\) becomes a complex periodic function of time and, according to formulas (27)—(33), is resolved into a whole series of purely harmonic, in time, component moments.

Fig. 1

In Fig. 1, above, the frequency-modulated moment is shown by the solid line

\[ (M_h)=M_h\sin(\eta_h t+\xi\sin \nu t) \]

for one particular case, when

\[ h=4;\qquad \eta_h=4\eta_0;\qquad \nu=\eta_0;\qquad \delta_s=0.2, \]

that is, for the ratio of the frequency of the harmonic under consideration to the frequency of the cyclic oscillations of the shaft rotation speed

\[ \frac{\eta_h}{\nu}=\frac{h}{h_9}=4 \]

and the modulation amplitude

\[ \xi=\frac{h}{h_0}\cdot\frac{\delta_s}{2}=0.4. \]

For comparison, the purely harmonic function of time \(M_h\sin\eta_h t\) is plotted with a dashed line.

In the lower half of Fig. 1, the harmonic spectrum of the moment \(M_4\) is presented in expanded form; in our particular case, according to formula (29), its components are determined by the expressions

\[ M_4 I_0(0.4)\sin 4\eta_0 t;\qquad M_4 I_1(0.4)\sin 5\eta_0 t; \]

\[ - M_4 I_1(0.4)\sin 3\eta_0 t;\qquad M_4 I_2(0.4)\sin 6\eta_0 t; \]

\[ M_4 I_2(0.4)\sin 2\eta_0 t;\ldots\ldots \]

Only the first three components are shown in the diagram; components of higher orders, because of their insignificant magnitude in our particular case, have been discarded.

The diagram in Fig. 1 may, incidentally, serve as an illustration of the statement noted above that frequency modulation is always at the same time also phase modulation, since it leads, for example, to a shift in time of the maxima of the function \((M_h)\) relative to the uniformly distributed-in-time maxima of the function \(M_h\sin\eta_h t\).

1. With monoharmonic frequency modulation, the harmonic-in-\(t\) components of the moment \(M_h\) have the frequencies:

\[ \ldots(\eta_h-2\nu);\quad(\eta_h-\nu);\quad \eta_h;\quad(\eta_h+\nu);\quad(\eta_h+2\nu)\ldots \]

and amplitudes proportional to the corresponding Bessel functions of the first kind of the modulation amplitude, i.e. \(I_q(\xi)\), where \(q=0,1,2\ldots\)

1. Since, in general,

\[ I_0(\xi)<1, \]

it follows that, owing to monoharmonic frequency modulation caused by the nonuniform rotation of the shaft, the amplitude of the moment \(M_h\) is reduced from \(M_h\) to \(M_h I_0(\xi)\). The degree of this reduction depends on the modulation amplitude \(\xi\), determined by formula (18).

1. At the same time, in the harmonic-in-time spectrum of the harmonic-in-\(\alpha\) moment \(M_h\), additional harmonic components, or “satellites,” appear, of the form:

\[ M_h I_q(\xi)\sin(\eta_h\pm q\nu)t, \]

where \(q=1,2,3\ldots\), located in the spectrum symmetrically on both sides of the principal, central “carrier” harmonic

\[ M_h I_0(\xi)\sin\eta_h t, \]

which has the “carrier” frequency \(\eta_h\), determined by formula (15) through the mean angular velocity of shaft rotation \(\eta_0\).

1. The amplitudes of the “carrier” harmonic \(M_h I_0(\xi)\) and of the satellites \(M_h I_q(\xi)\) do not depend on the phase angle \(\varepsilon_h\) or on the phase of the frequency modulation \(\zeta\), and are determined entirely by the modulation amplitude \(\xi\), i.e., according to (18), by the cyclic nonuniformity \(\delta_s\), by the ratio of the carrier frequency \(\eta_h\) to the frequency \(\nu\) of the cyclic oscillations of the shaft-rotation speed, and, of course, by the known laws of variation of the Bessel functions \(I_q(\xi)\) with increasing argument \(\xi\).

2. In the upper part of Fig. 2 and Fig. 3 are shown the harmonic spectra of the moment \(M_h\), determined by formula (29), respectively for two particular cases:

\[ \frac{\eta_h}{\nu}=\frac{h}{h_0}=20;\quad \delta_s=0.1; \]

\[ \frac{\eta_h}{\nu}=\frac{h}{h_0}=4;\quad \delta_s=0.2, \]

and in their lower part is presented the specific relation of the functions \(I_q(\xi)\) to the degree of nonuniformity \(\delta_s\).

Carrier harmonic
\(M_h \eta_0(\xi)\sin \eta_h t\)

Satellites
\(M_h \eta_q(\xi)\sin \eta_h\left(1+q\dfrac{h_0}{h}\right)t\)

Amplitudes \(M_h \eta_q(\xi)\)

Frequencies of the carrier and satellites \(\longrightarrow\)

\(\delta=0\); \(q=0\); \(q=1\); \(q=-1\); \(q=2\); \(q=-2\); \(q=3\); \(q=-3\); \(\eta_h\)

\(\eta_q(\xi)\)

\(\dfrac{h}{h_0}=20\)

Cyclic nonuniformity \(\delta \longrightarrow\)

Curves: \(q=0\), \(q=1\), \(q=2\), \(q=3\)

Fig. 2

These diagrams indicate a definite dependence of the amplitudes of the carrier harmonic and of its satellites on the degree of nonuniformity \(\delta_s\), in espe-

Amplitudes \(M_h \eta_q(\xi)\)

\(\delta=0.2\); \(q=0\); \(q=1\); \(q=2\); \(M_h\); \(\eta_h\)

Frequencies of the carrier and its satellites \(\longrightarrow\)

\(\eta_q(\xi)\)

\(\dfrac{h}{h_0}=4\)

Curves: \(q=0\), \(q=1\), \(q=2\)

Cyclic nonuniformity \(\delta \longrightarrow\)

Fig. 3

…features at a large ratio \(h/h_0\). The dynamic effect \(\delta_s\), that is, the effect of frequency modulation, is the greater the slower this modulation is, the smaller its frequency \(\nu\) in comparison with the carrier frequency \(\eta_h\) of the harmonic, of order \(\alpha\), of the moment \(M_h\) under consideration.

1. At large \(h/h_0\), that is, under slow modulation, even a small nonuniformity in the rotation of the shaft may lead to a considerable decrease in the amplitude \(M_h I_0(\xi)\) of the carrier harmonic and may even cause the latter to disappear completely, when the modulation amplitude \(\xi\) proves to be a root of the Bessel function \(I_0(\xi)\). However, as the amplitude of the carrier harmonic decreases, the amplitudes of the satellites simultaneously increase rapidly.

2. A comparison, for example, of formulas (32) and (33), and an analysis of the general expression (27), show that the phase angle of the frequency modulation \(\zeta\), without changing the amplitudes of the carrier and the satellites, affects the phases of these satellites relative to the phase of the carrier harmonic and the entire shape of the modulated moment \(M_h\).

Fig. 4

The dependence of the form of \(M_h\) on \(\zeta\) is illustrated by the graphs in Figs. 4 and 5. In Fig. 4 the case of synchronous-sinusoidal modulation is presented, when the modulation frequency \(\nu\) is equal to the carrier frequency \(\eta_h\) and, in addition,

\[ \zeta_h=0;\qquad \varepsilon_h=0. \]

Under these conditions formula (27) becomes

\[ m_h=\sin[\eta_h t+\xi\sin\eta_h t]=I_0(\xi)\sin\eta_h t+ \]

\[ +\sum_{q=1}^{\infty} I_q(\xi)\sin(1+q)\eta_h t+ \sum_{q=1}^{\infty}(-1)^q I_q(\xi)\sin(1-q)\eta_h t. \]

Hence

\[ m_h=[I_0(\xi)-I_2(\xi)]\sin\eta_h t+[I_1(\xi)+I_3(\xi)]\sin2\eta_h t \]

\[ +[I_2(\xi)-I_4(\xi)]\sin3\eta_h t+[I_3(\xi)+I_5(\xi)]\sin4\eta_h t+\ldots, \]

or, in general form,

\[ m_h=\sin(\eta_h t+\xi\sin\eta_h t)= \sum_{q=1}^{\infty}\left[I_{q-1}(\xi)+(-1)^q I_{q+1}(\xi)\right]\sin q\eta_h t. \tag{34} \]

In Fig. 5 specific cases are given (for different values of \(\xi\)) of synchronous monoharmonic modulation with phase difference \(\pi/2\), that is

\[ \nu=\eta_h;\quad \zeta=-\pi/2;\quad \varepsilon_h=0. \]

With these data, from (30), after some transformations, we have

\[ m_h=\sin(\eta_h t+\xi\cos\eta_h t)=I_1(\xi)+ \]

\[ +\sum_{q=0}^{\infty}(-1)^q\left[I_{2q}(\xi)+I_{2q+2}(\xi)\right]\sin(2q+1)\eta_h t +\sum_{q=1}^{\infty}(-1)^q\left[I_{2q+1}(\xi)-\right. \]

\[ \left.-I_{2q+3}(\xi)\right]\cos 2q\eta_h t . \tag{35} \]

9. In synchronous modulation \((\eta_h=\nu)\) the symmetry of the arrangement of the satellites with respect to the middle carrier harmonic \(M_h I_0(\xi)\sin\eta_h t\) is completely destroyed. Here all the satellites lie on one side of the carrier harmonic, in the region of higher frequencies.

Fig. 5

10. Finally, formula (35) and Fig. 5 show that, with synchronous modulation with phase \(\zeta=\pi/2\), in the truly harmonic spectrum of the moment \(M_h\) there appears a constant term \(M_h I_1(\xi)\), the magnitude of which increases with increasing modulation amplitude \(\xi\), that is, with increasing \(\delta\) and the ratio \(h/h_0\).

3. Gas Harmonics with a Complex Modulating Frequency

The cases considered above of monoharmonic frequency modulation of the moments \(M_h\), harmonic in \(\alpha\), of course do not yet reflect all the real forms of nonuniform rotation of an engine shaft. Monoharmonic modulation is a consequence of purely sinusoidal oscillations of the instantaneous angular velocity \(\eta\) of rotation of the crankshaft. But in fact, in any engine the velocity \(\eta\) varies according to a complex periodic law.

Assuming, as before, that the crankshaft is very rigid (p. 10), we thereby have grounds to consider that the nonuniformity of its rotation is caused only by the action of harmonic moments of the principal order-

…cycles. Under such conditions formula (10) may be written in the following form, respectively for two-stroke and four-stroke engines:

\[ \alpha=\eta_0 t+\sum_{p=1}^{\infty}\varphi_{pz}\sin(pz\eta_0t-\zeta_{pz}) \tag{36} \]

and

\[ \alpha=\eta_0t+\sum_{p=1}^{\infty}\varphi_{\frac{pz}{2}}\cdot \sin\left(\frac{pz}{2}\eta_0t-\zeta_{\frac{pz}{2}}\right), \]

if \(z\) is the number of cylinders of the engine.

But, analogously to (14),

\[ \varphi_{pz}=\frac{\delta_{pz}}{2pz}, \]

where \(\delta_{pz}\) is the partial degree of nonuniformity of rotation of the crankshaft caused by the action on the shaft of harmonic moments of order \(pz\).

Let us further denote

\[ \nu=z\eta_0, \tag{37} \]

\[ \xi_p=h\varphi_{pz}=\frac{h}{pz}\cdot\frac{\delta_{pz}}{2}. \tag{38} \]

Then from (36)

\[ \alpha=\eta_0t+\sum_{p=1}^{\infty}\frac{\xi_p}{h}\sin(p\nu t-\zeta_{pz}). \]

Substitution of this expression for \(\alpha\) into formula (8) gives the instantaneous value, harmonic in \(\alpha\), of the gas moment \(M_h\) already as a function of time,

\[ (M_h)=M_h\sin\left[\eta_ht+\sum_{p=1}^{\infty}\xi_p\sin(p\nu t-\zeta_{pz})\right]. \tag{39} \]

Here

\(\nu\) — is the angular frequency of the lowest principal harmonic of the torque on the shaft, determined for two-stroke engines by the obvious expression (37),

\(\xi_p\) — is the amplitude of the frequency modulation occurring under the isolated action on the system of harmonics of order \(pz\),

\(\eta_h\) — as usual, is the mean carrier frequency of the moment \(M_h\), that is,

\[ \eta_h=h\eta_0. \]

It is easy to show that under ordinary conditions (except in cases of resonance) the quantities \(\xi_p\) decrease rapidly with increasing index \(p\), especially for large \(z\). Taking these circumstances into account, we shall consider here specifically only the comparatively simple case of modulation of the moment \(M_h\) due to cyclic oscillations of the shaft rotational speed caused by the first two principal harmonics of the torque \(M_{kr}\) (that is, of orders \(z\) and \(2z\) in two-stroke engines and of orders \(z/2\) and \(z\) in four-stroke engines).

We shall confine ourselves in (39) to the values \(p=1\) and \(p=2\), and for simplicity take:

\[ \zeta_z=0;\quad \zeta_{2z}=0. \]

then instead of (39) we shall have

\[ (M_h)=M_h\sin\left[\eta_h t+\xi_1\sin \nu t+\xi_2\sin 2\nu t\right] \tag{40} \]

or, for a unit harmonic \((M_h=1)\):

\[ m_h=\sin\left[\eta_h t+\xi_1\sin \nu t+\xi_2\sin 2\nu t\right]. \tag{41} \]

We expand expression (41):

\[ m_h=\sin(\eta_h t+\xi_1\sin \nu t)\cdot \cos(\xi_2\sin 2\nu t)+ \cos(\eta_h t+\xi_1\sin \nu t)\cdot \sin(\xi_2\sin 2\nu t), \tag{42} \]

but, by formulas (28) and (23),

\[ \sin(\eta_h t+\xi_1\sin \nu t) = I_0(\xi_1)\sin \eta_h t+ \sum_{q=1}^{\infty} I_q(\xi_1)\sin(\eta_h+q\nu)t+ \]

\[ +\sum_{q=1}^{\infty}(-1)^q I_q(\xi_1)\sin(\eta_h-q\nu)t, \]

\[ \cos(\xi_2\sin 2\nu t) = I_0(\xi_2)+2\sum_{q=1}^{\infty} I_{2q}(\xi_2)\cos 4q\nu t. \]

Further, according to (32) and (24),

\[ \cos(\eta_h t+\xi_1\sin \nu t) = I_0(\xi_1)\cos \eta_h t+ \sum_{q=1}^{\infty} I_q(\xi_1)\cos(\eta_h+q\nu)t+ \]

\[ +\sum_{q=1}^{\infty}(-1)^q I_q(\xi_1)\cos(\eta_h-q\nu)t, \]

\[ \sin(\xi_2\sin 2\nu t) = 2\sum_{q=0}^{\infty} I_{2q+1}(\xi_2)\sin(2q+1)2\nu t. \]

Substitution of all these expressions into (42) gives:

\[ \begin{aligned} m_h={}&I_0(\xi_2)\left[ I_0(\xi_1)\sin \eta_h t+ \sum_{q=1}^{\infty} I_q(\xi_1)\sin(\eta_h+q\nu)t+ \sum_{q=1}^{\infty}(-1)^q I_q(\xi_1)\sin(\eta_h-q\nu)t \right] \\ &+\sum_{q=1}^{\infty} I_{2q}(\xi_2)\cos 4q\nu t \left[ I_0(\xi_1)\sin \eta_h t+ \sum_{q=1}^{\infty} I_q(\xi_1)\sin(\eta_h+q\nu)t+ \right.\\ &\left.\qquad\qquad\qquad\qquad +\sum_{q=1}^{\infty}(-1)^q I_q(\xi_1)\sin(\eta_h-q\nu)t \right]+ \\ &+\sum_{q=0}^{\infty} I_{2q+1}(\xi_2)\sin(2q+1)2\nu t \left[ I_0(\xi_1)\cos \eta_h t+ \sum_{q=1}^{\infty} I_q(\xi_1)\cos(\eta_h+q\nu)t+ \right.\\ &\left.\qquad\qquad\qquad\qquad +\sum_{q=1}^{\infty}(-1)^q I_q(\xi)\cos(\eta_h-q\nu)t \right]. \end{aligned} \]

Expanding this expression, after regrouping the terms and elementary trigonometric transformations, we obtain

\[ \begin{aligned} m_h&=\sin(\eta_h t+\xi_1\sin \nu t+\xi_2\sin 2\nu t)=\\ &=I_0(\xi_1)I_0(\xi_2)\sin\eta_h t +I_1(\xi_1)\,[I_0(\xi_2)-I_1(\xi_2)]\sin(\eta_h+\nu)t+\\ &\quad+[I_0(\xi_2)I_2(\xi_1)+I_2(\xi_1)I_2(\xi_2)+I_1(\xi_2)I_0(\xi_1)]\sin(\eta_h+2\nu)t+\ldots\\ &\quad+I_1(\xi_1)[-I_0(\xi_2)-I_1(\xi_2)]\sin(\eta_h-\nu)t +[I_0(\xi_2)I_2(\xi_1)+I_2(\xi_1)I_2(\xi_2)-\\ &\quad-I_1(\xi_2)I_0(\xi_1)]\sin(\eta_h-2\nu)t+ \end{aligned} \tag{43} \]

Here all terms containing factors of the Bessel functions \(I_q(\xi)\) with index \(q \geqslant 3\) have been discarded, in view of their small magnitude for ordinary values of \(\xi\).

Comparison of (43) with the harmonic spectrum (28) of the unit moment

\[ m_h=\sin(\eta_h t+\xi\sin\nu t) \]

shows that, in the case of biharmonic (in the case considered above) nonuniformity of rotation of the engine shaft, the following principal differences arise in the harmonic spectrum \(M_h\) from the corresponding spectrum for the case of monoharmonic nonuniformity:

1. The amplitude of the carrier harmonic is decreased in the ratio

\[ \frac{I_0(\xi_1)I_0(\xi_2)}{I_0(\xi_1)}. \]

1. The amplitudes of the symmetric sidebands, located in the frequency harmonic spectrum at the former distances on both sides of the carrier harmonic, are no longer equal. Thus, for example, the amplitudes of the first sidebands, having frequencies \(\eta_h+\nu\) and \(\eta_h-\nu\), are in the ratio

\[ \frac{I_0(\xi_2)-I_1(\xi_2)} {I_0(\xi_2)+I_1(\xi_2)}, \]

which differs from unity.

1. The amplitudes of all sidebands, in general, increase with increasing \(\xi\) (the specific numerical values of which are limited in practice by the range of \(\delta\) and \(h\) of interest to us), that is, with a slowing of the frequency modulation.

According to formula (37), the amplitude of the frequency modulation \(\xi_p\), for a nonuniformity degree \(\delta_{pz}\) that is always limited (or at least under nonresonant conditions), rapidly decreases with increasing order \(pz\) of the harmonic of the torque on the shaft causing this modulation. Consequently, even in the presence of very complex periodic frequency modulation, taking into account in it only one or, at most, the first two principal harmonic components may be considered sufficiently reliable and justified. For large \(z\) (multicylinder engines), the second of the indicated components already becomes negligibly small.

Thus, when investigating the effect of frequency modulation of the torque in internal-combustion-engine installations, it is usually quite permissible to assume that the angular speed of rotation of the engine shaft (or its crank) oscillates according to a simple sinusoidal law. The errors resulting from this assumption rapidly decrease with increasing number of engine cylinders.

4. Inertial Harmonics of the Torque and Nonuniformity of Shaft Rotation

As is known [3], the torque produced on the engine shaft by the inertia forces of the reciprocating masses of the crank-and-connecting-rod mechanism of one cylinder is represented by the series

\[ M_{\text{in}}=\sum_{h=1}^{\infty} M_h' \sin h\eta_0 t, \tag{44} \]

where

\[ h=1,\ 2,\ 3,\ldots \]

The coefficients \(M_h'\), that is, the amplitudes of the harmonic components of \(M_{\text{in}}\), have the following values:

\[ \left. \begin{aligned} M_1'&=\frac{G_2}{g}r^2\eta_0^2 \left[ \frac{1}{4}\left(\frac{r}{l}\right) +\frac{1}{16}\left(\frac{r}{l}\right)^3 +\frac{15}{512}\left(\frac{r}{l}\right)^5+\ldots \right], \\[4pt] M_2'&=\frac{G_2}{g}r^2\eta_0^2 \left[ -\frac{1}{2} -\frac{1}{32}\left(\frac{r}{l}\right)^4 -\frac{1}{32}\left(\frac{r}{l}\right)^6-\ldots \right], \\[4pt] M_3'&=\frac{G_2}{g}r^2\eta_0^2 \left[ -\frac{1}{4}\left(\frac{r}{l}\right)^2 -\frac{1}{8}\left(\frac{r}{l}\right)^4 -\frac{1}{16}\left(\frac{r}{l}\right)^6-\ldots \right], \\ &\ldots \end{aligned} \right\} \tag{45} \]

Here:

\(G_2\)—the weight of the reciprocating parts of the crank-and-connecting-rod mechanism of one cylinder,

\(r\)—the crank radius,

\(l\)—the length of the connecting rod.

The phase angles of the component harmonic moments in formula (44) are equal to \(0\) or \(\pi\), depending on the sign of the corresponding amplitude \(M_h'\), determined by formulas (45). For the usually small ratio \(r/l\), the decisive role in (44) is played by the harmonic torque of the second order, whose angular frequency is equal to \(2\eta_0\).

Expressions (44) and (45) were obtained analytically, under the assumption of absolutely uniform shaft rotation, that is, \(\eta=\mathrm{const}=\eta_0\). Undoubtedly, rotation of the shaft with a variable, oscillating speed introduces certain changes into the entire harmonic spectrum (44) of the inertial torques on the engine shaft.

It was established above that the cyclic nonuniformity of shaft rotation causes, as it were, a certain transformation of the curve of the “gas” torque \(M_{\text{cr}}\), initially plotted against the crank angle \(\alpha\). Owing to the absence of exact proportionality between \(\alpha\) and \(t\), when \(\delta_s>0\), this curve, when transferred to the \(t\)-axis, acquires a nonuniform stretching along the abscissa axis. Naturally, under such a nonuniform linear deformation, both the shape of the curve \(M_{\text{cr}}\) and all components of its Fourier-series expansion change. However, the curve \(M_{\text{cr}}\) does not undergo any deformation along the ordinate axis, since, as was already noted above, the gas pressure in the cylinder practically does not depend on oscillations of the instantaneous speed of rotation of the crank \(\eta\).

The influence of oscillations of the speed \(\eta\) on the torque due to the inertia forces of the reciprocating masses has a different character. Here these oscillations \(\eta\) not only disturb the proportionality between \(\alpha\) and \(t\), but also lead to corresponding changes in the magnitude of the inertia forces and of their moments on the shaft. Thus, for the curve \(M_{\text{cr}}=f(a)\), constructed according to the usual

…by the formulas of applied mechanics, in reality turn out to be variable scales along both coordinate axes. This, in fact, is the “geometric” effect of mixed modulation.

Let us establish the relation between the harmonic, in time, spectrum of the inertial moment \(M_{\mathrm{in}}\) and the degree of nonuniformity of shaft rotation \(\delta_s\), assuming a purely sinusoidal law of oscillation of the angular velocity of the shaft

\[ \eta=\eta_0\left[1+\frac{\delta_s}{2}\cos(\nu t-\zeta)\right], \tag{46-a} \]

to which, according to (16), there corresponds the following expression for the angle of rotation of the shaft

\[ \alpha=\eta_0 t+\frac{\delta_s}{2h_0}\sin(\nu t-\zeta). \tag{46} \]

Denote

\[ \frac{\delta_s}{2h_0}=\frac{\delta_s}{2}\cdot\frac{\eta_0}{\nu}=\lambda . \tag{47} \]

Then

\[ \begin{gathered} \alpha=\eta_0 t+\lambda\sin(\nu t-\zeta),\\ \eta=\eta_0\left[1+h_0\lambda\cos(\nu t-\zeta)\right], \end{gathered} \tag{47-a} \]

where \(\nu\) is the angular frequency of oscillations of the shaft speed, and \(h_0\) is the order of these oscillations.

We start from the known approximate, but practically sufficiently accurate, expression for the piston travel

\[ x=r(1-\cos\alpha)+\frac{1}{2}\left(\frac{r}{l}\right)r\sin^2\alpha . \tag{48} \]

This equation, obtained from purely geometric considerations, does not directly contain the time \(t\) and, consequently, imposes no restrictions whatever on the form of the relation between \(\alpha\) and \(t\).

From (48), the velocity and acceleration of the piston are

\[ v=\frac{dx}{dt} =r\left(\sin\alpha+\frac{1}{2}\frac{r}{l}\sin2\alpha\right)\cdot\frac{d\alpha}{dt}, \]

\[ j=\frac{d^2x}{dt^2} =r\left(\sin\alpha+\frac{1}{2}\frac{r}{l}\sin2\alpha\right)\cdot\frac{d^2\alpha}{dt^2} + r\left(\cos\alpha+\frac{r}{l}\cos2\alpha\right)\cdot\left(\frac{d\alpha}{dt}\right)^2 . \tag{49} \]

The inertia force of the reciprocating masses of the connecting-rod–crank mechanism

\[ P=\frac{G_2}{g}j \]

creates on the engine shaft a torque \(M_{\mathrm{in}}\), the magnitude of which is determined from the obvious relation:

\[ M_{\mathrm{in}}\frac{d\alpha}{dt}=P\frac{dx}{dt}. \]

or

\[ M_{\mathrm{in}}=\frac{P\,\dfrac{dx}{dt}}{\dfrac{dz}{dt}}. \]

Hence, using expressions (46) and (49), we obtain:

\[ M_{\mathrm{in}}=\frac{G_2}{g}r^2\eta_0^2 \left\{ \left(\cos\alpha+\frac{r}{l}\cos 2\alpha\right) \left[1+\frac{\delta_s}{2}\cos(\nu t-\zeta)\right]^2 -\chi\left(\sin\alpha+\frac{1}{2}\frac{r}{l}\sin 2\alpha\right) \sin(\nu t-\zeta) \right\} \left(\sin\alpha+\frac{1}{2}\frac{r}{l}\sin 2\alpha\right), \tag{50} \]

where the notation has been introduced

\[ \chi=\frac{\delta_s}{2}\frac{\nu}{\eta_0}. \tag{50-a} \]

A detailed investigation of this equation, under the conditions \(\delta_s>0\), presents considerable difficulties, owing to the exceptional complexity of all the calculations. Therefore we shall restrict ourselves here to an analysis of the case \(\dfrac{r}{l}=0\) (a connecting rod of infinitely great length). The qualitative conclusions under such an assumption are hardly likely to change in any essential way.

For \(\dfrac{r}{l}=0\), from (50), after elementary transformations, we have

\[ M_{\mathrm{in}}=\frac{1}{2}\frac{G_2}{g}r^2\eta_0^2 \left\{ \left[1+\frac{\delta_s}{2}\cos(\nu t-\zeta)\right]^2\sin 2\alpha -\chi\sin(\nu t-\zeta)\sin^2\alpha \right\}. \tag{51} \]

For absolutely uniform rotation of the shaft \((\delta_s=0)\), this expression becomes

\[ M_{\mathrm{in}}=\frac{1}{2}\frac{G_2}{g}r^2\eta_0^2\sin 2\alpha, \tag{52} \]

which completely coincides with the data of formulas (45) for \(r/l=0\). Thus, the quantity

\[ \frac{1}{2}\frac{G_2}{g}r^2\eta_0^2 \]

is the amplitude of the inertial harmonic of the second order for \(\delta_s=0\) and \(r/l=0\) (the only harmonic component of \(M_{\mathrm{in}}\) in this case). Making use of this circumstance, let us call in formula (51) the ratio

\[ m_{\mathrm{in}}=\frac{M_{\mathrm{in}}}{\dfrac{1}{2}\dfrac{G_2}{g}r^2\eta_0^2} \tag{53} \]

the unit inertial harmonic moment.

Then from (51):

\[ m_{\mathrm{in}}= \left[1+\frac{\delta_s}{2}\cos(\nu t-\zeta)\right]\sin 2\alpha -\chi\sin(\nu t-\zeta) +\chi\sin(\nu t-\zeta)\cos 2\alpha \tag{54} \]

or, in general form:

\[ M_{\mathrm{in}}=A+B+C+D+E, \tag{55} \]

where:

\[ A=\left(1+\frac{\delta_s^2}{8}\right)\sin 2\alpha, \tag{56} \]

\[ B=\delta_s\sin 2\alpha\cdot \cos(\nu t-\zeta), \tag{57} \]

\[ C=\frac{\delta_s^2}{8}\cos(2\nu t-2\zeta)\sin 2\alpha; \tag{58} \]

\[ D=\chi\sin(\nu t-\zeta)\cos 2\alpha, \tag{59} \]

\[ E=-\chi\sin(\nu t-\zeta). \tag{60} \]

The angle of rotation of the crank \(\alpha\) entering here is determined by the formula (47-a) given above.

Let us consider each of the terms in (55) separately. Thus, from (56) and (47-a):

\[ A=\left(1+\frac{\delta_s^2}{8}\right)\sin\left[2\eta_0 t+2\lambda\sin(\nu t-\zeta)\right] \]

and hence, on the basis of formula (27), putting in it \(\varepsilon_n=0\), we obtain:

\[ \begin{aligned} A={}&\left(1+\frac{\delta_s^2}{8}\right)I_0(2\lambda)\sin 2\eta_0 t +\left(1+\frac{\delta_s^2}{8}\right)\sum_{q=1}^{\infty} I_q(2\lambda)\sin\left[(2\eta_0+q\nu)t-q\zeta\right]+\\ &+\left(1+\frac{\delta_s^2}{8}\right)\sum_{q=1}^{\infty}(-1)^q I_q(2\lambda)\sin\left[(2\eta_0-q\nu)t+q\zeta\right] \end{aligned} \tag{61} \]

Next, analogously from (57):

\[ B=\delta_s\left\{ I_0(2\lambda)\sin 2\eta_0 t\cdot \cos(\nu t-\zeta)+ \right. \]

\[ +\sum_{q=1}^{\infty} I_q(2\lambda)\sin\left[(2\eta_0+q\nu)t-q\zeta\right]\cos(\nu t-\zeta)+ \]

\[ \left. +\sum_{q=1}^{\infty}(-1)^q I_q(2\lambda)\sin\left[(2\eta_0-q\nu)t+q\zeta\right]\cos(\nu t-\zeta) \right\} \]

or, after some transformations,

\[ B=\frac{1}{2}\delta_s\left\{ I_0(2\lambda)\sin\left[(2\eta_0+\nu)t-\zeta\right] +I_0(2\lambda)\sin\left[(2\eta_0-\nu)t+\zeta\right] \right\}+ \]

\[ +\frac{1}{2}\delta_s\sum_{q=1}^{\infty} I_q(2\lambda)\sin\left\{[2\eta_0+(q+1)\nu]t-(q+1)\zeta\right\}+ \]

\[ +\frac{1}{2}\delta_s\sum_{q=1}^{\infty} I_q(2\lambda)\sin\left\{[2\eta_0+(q-1)\nu]t-(q-1)\zeta\right\}+ \]

\[ +\frac{1}{2}\delta_s\sum_{q=1}^{\infty}(-1)^q I_q(2\lambda)\sin\left\{[2\eta_0-(q-1)\nu]t+(q-1)\zeta\right\}+ \]

\[ +\frac{1}{2}\delta_s\sum_{q=1}^{\infty}(-1)^q I_q(2\lambda)\sin\left\{[2\eta_0-(q+1)\nu]t+(q+1)\zeta\right\}. \tag{62} \]

Transforming expression (58) for \(C\) in exactly the same way, we obtain:

\[ \begin{aligned} C=\frac{\delta s^{2}}{16}\Bigg\{& I_{0}(2\lambda)\sin\bigl[(2\eta_{0}+2\nu)t-2\zeta\bigr] +I_{0}(2\lambda)\sin\bigl[(2\eta_{0}-2\nu)t+2\zeta\bigr]+\\ &+\sum_{q=1}^{\infty} I_{q}(2\lambda)\sin\left\{\bigl[2\eta_{0}+(q+2)\nu\bigr]t-(q+2)\zeta\right\}+\\ &+\sum_{q=1}^{\infty} I_{q}(2\lambda)\sin\left\{\bigl[2\eta_{0}+(q-2)\nu\bigr]t-(q-2)\zeta\right\}+\\ &+\sum_{q=1}^{\infty}(-1)^{q}I_{q}(2\lambda)\sin\left\{\bigl[2\eta_{0}-(q-2)\nu\bigr]t+(q-2)\zeta\right\}+\\ &+\sum_{q=1}^{\infty}(-1)^{q}I_{q}(2\lambda)\sin\left\{\bigl[2\eta_{0}-(q+2)\nu\bigr]t+(q+2)\zeta\right\} \Bigg\}. \end{aligned} \tag{63} \]

Let us put in formula (27)

\[ \varepsilon_{h}=\pi/2 \]

and replace in it \(\xi\) and \(\eta_{h}\), respectively, by \(2\lambda\) and \(2\eta_{0}\). Then

\[ \begin{aligned} \cos\bigl[2\eta_{0}t+2\lambda\sin(\nu t-\zeta)\bigr] &=I_{0}(2\lambda)\cos 2\eta_{0}t+\\ &\quad+\sum_{q=1}^{\infty} I_{q}(2\lambda)\cos\bigl[(2\eta_{0}+q\nu)t-q\zeta\bigr]+\\ &\quad+\sum_{q=1}^{\infty}(-1)^{q}I_{q}(2\lambda)\cos\bigl[(2\eta_{0}-q\nu)t+q\zeta\bigr]. \end{aligned} \]

On the basis of this relation we have from (59), after a number of transformations:

\[ \begin{aligned} D=&\frac{\chi}{2}\left\{ I_{0}(2\lambda)\sin\bigl[(2\eta_{0}+\nu)t-\zeta\bigr] -I_{0}(2\lambda)\sin\bigl[(2\eta_{0}-\nu)t+\zeta\bigr] \right\}+\\ &+\frac{\chi}{2}\Bigg\{ \sum_{q=1}^{\infty} I_{q}(2\lambda)\sin\left\{\bigl[2\eta_{0}+(q+1)\nu\bigr]t-(q+1)\zeta\right\}-\\ &\qquad-\sum_{q=1}^{\infty} I_{q}(2\lambda)\sin\left\{\bigl[2\eta_{0}+(q-1)\nu\bigr]t-(q-1)\zeta\right\}+\\ &\qquad+\sum_{q=1}^{\infty}(-1)^{q}I_{q}(2\lambda)\sin\left\{\bigl[2\eta_{0}-(q-1)\nu\bigr]t+(q-1)\zeta\right\}-\\ &\qquad-\sum_{q=1}^{\infty}(-1)^{q}I_{q}(2\lambda)\sin\left\{\bigl[2\eta_{0}-(q+1)\nu\bigr]t+(q+1)\zeta\right\} \Bigg\}. \end{aligned} \tag{64} \]

Now substitution of expressions (61)—(64) into (55), after a corresponding regrouping of terms, gives:

\[ \begin{aligned} M_{\mathrm{in}}={}&-\gamma\sin(\nu t-\zeta)+ \\ &+\left[\left(1+\frac{\delta_s^2}{8}\right)I_0(2\lambda)-\gamma I_1(2\lambda)+\frac{\delta_s^2}{8}I_2(2\lambda)\right]\sin 2\eta_0 t+ \\ &+\sum_{q=1}^{\infty}\left\{\left[\frac{\delta_s^2}{16}I_{q-2}(2\lambda)+\frac{1}{2}(\delta_s+\lambda)I_{q-1}(2\lambda)+\right.\right.\\ &\qquad\left.+\left(1+\frac{\delta_s^2}{8}\right)I_q(2\lambda)+\frac{1}{2}(\delta_s-\lambda)I_{q+1}(2\lambda)+\right.\\ &\qquad\left.\left.+\frac{\delta_s^2}{16}I_{q+2}(2\lambda)\right]\sin\left[(2\eta_0+q\nu)t-q\zeta\right]\right\}+\\ &+\sum_{q=1}^{\infty}\left\{(-1)^q\left[\frac{\delta_s^2}{16}I_{q-2}(2\lambda)-\frac{1}{2}(\delta_s-\lambda)I_{q-1}(2\lambda)+\right.\right.\\ &\qquad\left.+\left(1+\frac{\delta_s^2}{8}\right)I_q(2\lambda)-\frac{1}{2}(\delta_s+\lambda)I_{q+1}(2\lambda)+\right.\\ &\qquad\left.\left.+\frac{\delta_s^2}{16}I_{q+2}(2\lambda)\right]\sin\left[(2\eta_0-q\nu)t+q\zeta\right]\right\}. \end{aligned} \tag{65} \]

Expression (65), obtained by us from (54), is evidently the desired harmonic (in time) spectrum of the unit inertial torque for \(\delta_s>0\) and \(r/l=0\).

Term-by-term multiplication of (65) by

\[ \frac{1}{2}\frac{G_2}{g}\,r^2\eta_0^2 \]

gives the absolute values of the harmonic components of the inertial torque of one cylinder, depending on the degree of nonuniformity of rotation of the engine shaft (or, more precisely, of the crank of the given cylinder).

From series (65) for \(m_{\mathrm{in}}\) it follows that, in the presence of cyclic monoharmonic nonuniformity of shaft rotation, characterized by equation (46-a), in an engine with a very long connecting rod \((r/l\to 0)\), in addition to the “carrier” harmonic of the inertial torque

\[ (m_z)=\left[\left(1+\frac{\delta_s^2}{8}\right)I_0(2\lambda)-\gamma I_1(2\lambda)+\frac{\delta_s^2}{8}I_2(2\lambda)\right]\sin 2\eta_0 t \tag{66} \]

with angular frequency \(2\eta_0\), lateral satellites of the form appear

\[ (m_q)=m_q\sin\left[(2\eta_0\pm q\nu)t\mp q\zeta\right] \tag{67} \]

with frequencies \(2\eta_0\pm q\nu\), located, in the general case, symmetrically with respect to the fundamental, carrier frequency \(2\eta_0\).

Formula (65), in its general features, resembles expression (27), obtained by us above, for the harmonic spectrum of a gas harmonic. However, here, in the case of inertial torques, the satellites (67) no longer

are symmetric in their magnitude (amplitude). In addition, in the spectrum (65), in contrast to the spectrum (27), there appears a new harmonic component

\[ \chi \sin(\nu t-\zeta), \]

synchronous with the frequency modulation and opposite to it in phase.

The amplitudes \(m_q\) of the satellites in the spectrum (65), in general, decrease rapidly as their index \(q\) increases, in accordance with the behavior of the Bessel function \(I_q(2\lambda)\) as its order increases and the argument is small.

The amplitudes \(m_q\) depend strongly on the coefficient \(\lambda\) and, consequently, according to (47), on the degree of nonuniformity \(\delta_s\) and the relative modulation frequency \(\nu/\eta\).

For a more detailed analysis of these dependences, let us consider some principal special cases of frequency modulation of the moment of inertia \(m_{\mathrm{in}}\).

a) Synchronous monoharmonic modulation

Here, for

\[ \nu=2\eta_0 \]

the series (65) may be represented in the form

\[ \begin{aligned} m_{\mathrm{in}}={}&m'_2\sin 2\eta_0 t+m''_2\sin(2\eta_0 t-\zeta)+{}\\ &{}+m'_4\sin(4\eta_0 t-\zeta)+m''_4\sin(4\eta_0 t-2\zeta)+{}\\ &{}+m'_6\sin(6\eta_0 t-2\zeta)+m''_6\sin(6\eta_0 t-4\zeta)+{}\\ &{}+\ldots \end{aligned} \tag{68} \]

On the basis of formulas (47) and (50-a), for \(\nu=2\eta_0\),

\[ \chi=\delta_s;\qquad \lambda=\delta_s/2. \]

Now the coefficients \(m\) entering into the right-hand side of (68), according to (65), are determined by the expressions:

\[ \left. \begin{aligned} m'_2={}&\left(1+\frac{\delta_s^2}{8}\right)I_0(\delta_s)-\delta_s I_1(\delta_s)+\frac{\delta_s^2}{8}I_2(\delta_s),\\[4pt] m''_2={}&-\delta_s,\\[4pt] m'_4={}&\frac{\delta_s^2}{16}I_{-1}(\delta_s)+\delta_s I_0(\delta_s)+\left(1+\frac{\delta_s^2}{8}\right)I_1(\delta_s)\\ &{}+\frac{\delta_s^2}{16}I_3(\delta_s),\\[4pt] m''_4={}&\frac{\delta_s^2}{16}I_0(\delta_s)+\left(1+\frac{\delta_s^2}{8}\right)I_2(\delta_s)-\delta_s I_3(\delta_s)\\ &{}+\frac{\delta_s^2}{16}I_4(\delta_s). \end{aligned} \right\} \tag{69} \]

Of course, here, for the usually small value of \(\delta_s\), one may discard an entire series of terms containing Bessel functions of high orders.

The amplitude and phase angle of the unit harmonic moment of the second order \(m_2\), entering into the spectrum (68), are determined by the geometric summation of \(m'_2\) and \(m''_2\), taking into account their phase angles \(0\) and \(\zeta\). In exactly the same way the amplitudes \(m_4, m_6,\ldots\) of the following components in

(68). The results of similar calculations for various specific values of the degree of nonuniformity are given in Fig. 6.

Analysis of this graph, as well as of formulas (68) and (69), makes it possible to draw the following conclusions concerning the influence of cyclic nonuniformity of shaft rotation on the inertial moment \(M_{in}\) under conditions in which the frequency \(\nu\) of this nonuniformity is equal to twice the mean angular velocity of shaft rotation:

Fig. 6

a) Cyclic nonuniformity of shaft rotation leads to the appearance, in the spectrum of the inertial moment \(M_{in}\), of a whole series of harmonic components with frequencies that are multiples of \(2\eta_0\).

b) As a rule, among all these components the largest magnitude is possessed by the “carrier” harmonic \(m_2\), with frequency \(2\eta_0\). All the remaining harmonics, which are one-sided satellites of the carrier harmonic, have small amplitudes (for moderate \(\delta\)), varying with an increase in their index \(q\).

c) For absolutely uniform shaft rotation, \(\delta_s = 0\), and from (69):

\[ m_2' = 1, \]

\[ m_2'' = m_4' = m_4'' = \ldots = 0, \]

that is, (68) reduces to

\[ m_{in} = 1 \cdot \sin 2\alpha . \]

Consequently, the deviation of \(m_2\) from unity may be regarded as a certain characteristic of the dynamic effect of nonuniform shaft rotation.

d) The amplitude \(m_2\) of the “carrier” harmonic of the inertial moment depends not only on the magnitude \(\delta_s\), but also on the phase angle \(\zeta\) of the cyclic nonuniformity (or frequency modulation). The diagram in Fig. 6 shows that the character of the influence of \(\delta_s\) on the amplitude \(m_2\) is to a large extent determined by the magnitude of \(\zeta\). Here \(m_2\) decreases comparatively rapidly with increasing \(\delta_s\) when \(\zeta = 0\), remains almost unchanged when \(\zeta = \pi/2\), and increases comparatively rapidly when \(\zeta = \pi\).

e) In contrast to \(m_2\), the amplitudes of the following harmonics \(m_4, m_6 \ldots\) practically no longer depend on \(\zeta\) and increase continuously with increasing \(\delta_s\).

\(\beta\)) Monoharmonic modulation with frequency \(\nu = \eta_0\)

Here (68) takes the form:

\[ \left. \begin{aligned} m_{in}={}&m_1'\sin(\eta_0 t+\zeta)+m_1''\sin(\eta_0 t-\zeta)+{}\\ &+m_2'\sin 2\eta_0 t+m_2''\sin(2\eta_0 t-4\zeta)+{}\\ &+m_3'\sin(3\eta_0 t-\zeta)+m_3''\sin(3\eta_0 t-5\zeta)+{}\\ &+\ldots \end{aligned} \right\} \tag{70} \]

The values of all the coefficients \(m'\) and \(m''\) entering here can be found from (69) for

\[ \chi=\delta_s/2;\qquad \lambda=\delta_s/2; \]

while the amplitudes \(m_1\) and \(m_2,\ldots\) of the harmonic components of the inertial moment \(m_{\mathrm{in}}\) are determined by geometrical summation of the corresponding \(m'\) and \(m''\).

The results of similar calculations for various values of \(\delta_s\) are shown in Fig. 7, analysis of which leads to the following conclusions concerning the effect of frequency modulation of the inertial moment \(M_{\mathrm{in}}\) under the conditions \(\nu=\eta_0\):

Fig. 7

a) In contrast to the preceding case \((\nu=2\eta_0)\), here the satellites have frequencies that are multiples of the angular velocity of rotation of the shaft \(\eta_0\).

b) The amplitude of the carrier harmonic \(m_2\) is practically entirely independent of the phase angle \(\zeta\) and decreases only very slowly with increasing \(\delta_s\). Consequently, the dynamic influence of the nonuniformity of shaft rotation on the carrier harmonic of the inertial moment is here quite negligible.

c) The amplitudes of the first satellites \(m_1\) and \(m_3\) increase comparatively rapidly with increasing nonuniformity of shaft rotation.

In an analogous manner one could consider the spectrum (68) for any other values of the ratio \(\nu/\eta_0\) of the frequency of cyclic oscillations of the shaft rotational speed to its mean speed \(\eta_0\). Some results of such investigations are given in Fig. 8, where the dependence is presented of the amplitude \(m_2\) of the “carrier” harmonic of the inertial moment on the engine shaft for various \(\delta_s\) and for two different modulation frequencies (that is, frequencies of oscillation of the angular speed of shaft rotation):

\[ \nu=4\eta_0;\qquad \nu=6\eta_0. \]

This diagram shows that in the first case, for \(\nu=4\eta_0\), there is still a rather sharply expressed dependence of \(m_2\) on \(\zeta\) and \(\delta_s\). Here, just as in the case of synchronous modulation considered above \((\nu=2\eta_0)\),

there prove to be possible cases of an increase in the amplitude of the carrier harmonic \(m_2\), and, consequently, of the forced oscillations of the system caused by it, when the uniformity of shaft rotation deteriorates (the effect of negative dynamic damping). At \(\nu=6\eta_0\), the influence of \(\zeta\) and \(\delta_s\) on \(m_2\) practically disappears. We obtain the same picture also for still larger values of \(\nu\).

Now, summing up all that has been set forth above, it must be recognized that the influence of the nonuniformity of rotation of the engine shaft on the inertial torque

Fig. 8

\(M_{\mathrm{in}}\) is in general insignificant, at least for those moderate values of \(\delta_s\) with which one usually has to deal in practice. This influence is more substantial only in the case of synchronous modulation, when the frequency of oscillations of the shaft speed is equal to the frequency of the second, i.e. the strongest, harmonic of the inertial moment. But even here the dynamic effect of nonuniform shaft rotation is determined mainly by the phase \(\zeta\).

5. Forced oscillations of a linear system under excitation that is monoharmonic with respect to the crank angle \(\alpha\).

In the preceding sections of the present article it was shown that the usual expansion of the curve of the torque on the engine shaft into the trigonometric series (1) with respect to the crank angle \(\alpha\) still does not give the harmonic spectrum that we need in analyzing the forced oscillations of a linear system. In essence, each component \(M_n\) of series (1) is sufficiently complex in its spectral content and contains a whole series of harmonic in time (i.e. truly harmonic) moments in the form of a “carrier” harmonic and lateral satellites of various intensities, the concrete magnitudes of which may be determined from the formulas given above.

The reason for the splitting of each \(M_n\) of series (1) into a carrier harmonic and satellites is, first of all, the nonuniformity of rotation of the crankshaft

of the engine shaft. The physical causes of this splitting are inherent in the basic mechanical properties of linear oscillatory systems with constant masses and stiffnesses [8, 9].

With an increase in the nonuniformity of shaft rotation, as a rule, the amplitudes of the side satellites in the spectrum \(M_h\) increase.

However, the ratio of the amplitudes of the carrier and the satellites and the degree of their increase or decrease with increasing \(\delta\),¹ cannot yet serve as sufficiently good criteria for estimating the influence of the nonuniformity of shaft rotation on its forced vibrations.

The amplitudes of forced vibrations caused in a system by the action of a harmonic (in time) force are determined not only by the amplitude of this force, but also by a whole series of other factors, such as, for example, the ratio of the frequency of the force to the frequency of the natural vibrations of the system, the intensity and character of damping in the elements of the installation, etc. Therefore, in order to estimate the actual influence of the degree of nonuniformity of rotation \(\delta\) on the forced vibrations of a shaft, it is necessary to investigate in detail the effect on a linear oscillatory system of an exciting moment \(M_h\), monoharmonic in \(\alpha\), but complex in its true harmonic spectrum, characterized, for example, by series (27).

Let us consider the forced vibrations of the installation shown in Fig. 9, consisting of a single-cylinder engine having a reduced mass with mean moment of inertia \(\Theta_1\), and a flywheel (propeller) with moment of inertia \(\Theta_2\). Suppose, for simplicity, that the damping moment in the system is proportional to the instantaneous twisting velocity of the shaft during vibrations. The differential equations of motion of both masses here have the form:

Fig. 9

\[ \left. \begin{aligned} \Theta_1 \frac{d^2 \varphi_1}{dt^2} + k\left(\frac{d\varphi_1}{dt}-\frac{d\varphi_2}{dt}\right) + c(\varphi_1-\varphi_2) &= M_1,\\ \Theta_2 \frac{d^2 \varphi_2}{dt^2} - k\left(\frac{d\varphi_1}{dt}-\frac{d\varphi_2}{dt}\right) - c(\varphi_1-\varphi_2) &= M_{11}, \end{aligned} \right\} \tag{71} \]

where \(M_1\) and \(M_{11}\) are the external torques acting, respectively, on the crank of the shaft and on the flywheel; \(k\) is the coefficient of velocity damping.

The moment \(M_{11}\), in the general case, varies with time (or with the angle of rotation of the crank \(\alpha\)):

\[ M_{11} = -M_0 + f_{11}(t), \tag{72} \]

where \(M_0\) is the mean value of the external resisting moment, equal to the mean torque developed by the engine,²

\(M_1\) is the torque on the engine shaft produced by the pressure of the gases in the cylinder and by the inertia forces of the reciprocating masses of the crank mechanism. As indicated above, \(M_1\) is, in the main, an explicit function of the crank rotation angle \(\alpha\).

Harmonic analysis of the curve \(M_1=f_1(\alpha)\) within the limits of one cycle of cylinder operation gives the ordinary trigonometric series

\[ M_1=M_0+\sum_{p=1}^{\infty} M_p \sin (p\alpha+\varepsilon_p'), \tag{73} \]

analogous to (1).

Here

\[ \alpha=\eta_0 t+\varphi_1. \tag{74} \]

Now, instead of (71), one may write:

\[ \Theta_1 \frac{d^2\varphi_1}{dt^2} +k\left(\frac{d\varphi_1}{dt}-\frac{d\varphi_2}{dt}\right) +c(\varphi_1-\varphi_2) = M_0+\sum_{p=1}^{\infty} M_p \sin (p\alpha+\varepsilon_p') \tag{75} \]

\[ \Theta_2 \frac{d^2\varphi_2}{dt^2} -k\left(\frac{d\varphi_1}{dt}-\frac{d\varphi_2}{dt}\right) -c(\varphi_1-\varphi_2) = -M_0+f_{II}(t). \tag{76} \]

Let us represent the right-hand variable part of equation (75) in the form of a sum of three terms

\[ \sum_{p=1}^{\infty} M_p \sin (p\alpha+\varepsilon_p') = f_1(\alpha)+f_2(\alpha)+f_3(\alpha), \tag{77} \]

where

\[ f_1(\alpha)=M_h \sin (h\alpha+\varepsilon_h') \tag{78} \]

is some single harmonic component in \(\alpha\), specifically of interest to us, of the moment \(M_1\) on the engine shaft, causing resonant or near-resonant vibrations in the range of operating speeds of the installation.

Further, here

\[ f_2(\alpha)=M_z \sin (z\alpha+\varepsilon_z') \tag{79} \]

is one of the lower-order harmonic components in \(\alpha\) of the moment \(M_1\), which, even under nonresonant conditions, is the principal cause of nonuniform shaft rotation, that is, above all, determines the existence in the system of a considerable, almost rigid, degree of nonuniformity \(\delta_s\).

In multicylinder engines the order of this harmonic \(z\) is equal to the number of cylinders of the engine, or to half this number, depending on the cycle of the engine.

Finally,

\[ f_3(\alpha)= \sum_{p=1}^{\infty} M_p \sin (p\alpha+\varepsilon_p') -f_1(\alpha)-f_2(\alpha) \tag{80} \]

includes all the remaining components of \(M_1\). In a resonant situation for \(f_1(\alpha)\), all the terms entering into \(f_3(\alpha)\) are outside resonance and therefore do not give especially strong forced oscillations of the shaft. (As will be shown below, the latter assertion is not always sufficiently exact.)

All three terms on the right-hand side of equality (77) are expressed in the form of explicit functions of \(\alpha\), but, of course, ultimately, through equation (74), they turn out to be functions of time \(t\). However, the exact concrete form of the dependence (74) is unknown to us, owing to the absence of any practical possibility of solving the system of equations (75)—(76) in its complete form.

The investigation of these, already nonlinear differential equations, even in the case of a small number of components on the right-hand side of equation (75), presents exceptional difficulties. The complexity of the system of equations (75)—(76) is caused by the dependence of the forced oscillations \(\varphi_1\) of the reduced mass of the engine on the exciting moments acting upon it, and by the simultaneous dependence of the law of variation of these moments with time on the forced oscillations \(\varphi_1\), through the effect of frequency modulation and even of mixed modulation (with respect to the inertial moments).

In view of the nonlinearity of equations (75) and (76), the direct and unconditional application of the principle of superposition is no longer possible here.

Let us denote by \(\varphi_{1,\mathrm{n}}, \varphi_{1,z}, \varphi_{1,3}, \varphi_{1,\mathrm{II}}\) the oscillations of the first mass of our system which would be caused respectively by the moments \(f_1(\alpha)\), \(f_2(\alpha)\), \(f_3(\alpha)\), \(f_{\mathrm{II}}(t)\), in the case of their completely independent, separate action on the system, one at a time, over a sufficiently long interval of time.

With the simultaneous action of all four indicated moments, the total oscillation of the first mass \(\varphi_1\) will no longer be equal to the sum:

\[ \varphi_{1,\mathrm{n}}+\varphi_{1,z}+\varphi_{1,3}+\varphi_{1,\mathrm{II}}, \]

since, owing to mutual modulation, the true (harmonic in \(t\)) spectrum of each of these moments will change, and consequently the effect of their action on the oscillating system will also change. Here the nonuniformity of rotation of the crankshaft and flywheel will be not only different, but also very complex in character.

In order to obtain the possibility of at least an approximate solution of the problem of the forced oscillations of the system in Fig. 9, we shall make the following simplifying assumptions:

a) We shall take the moment of external resistance (the external useful load on the engine) to be constant, i.e. we shall put in formula (72)

\[ f_{\mathrm{II}}(t)=0. \]

b) For the time being, we shall completely exclude from consideration all moments included in \(f_3(\alpha)\), i.e. we shall take in (77)

\[ f_3(\alpha)=0. \]

c) Taking into account the usually large magnitude of the lowest principal harmonic of \(f_2(\alpha)\) and its low frequency, and, consequently, the possibility of only high-frequency modulation of \(f_2(\alpha)\) by the oscillations of the crank caused by the moment \(f_1(\alpha)\), we shall completely neglect the effect of this modulation; that is, in formula (74) we shall replace the angle \(\varphi_1\) by the angle \(\varphi_{1z}\). Then from (79)

\[ f_2(\alpha)=M_z \sin [z(\eta_0 t+\varphi_{1z})+\varepsilon_z]. \tag{81} \]

d) Finally, we shall assume that the nonuniformity of rotation of the shaft caused by the moment \(M_z\) is considerably greater than that “elastic” nonuniformity which the crank receives owing to the forced oscillations of the system under the action of the moment \(M_{\mathrm{n}}\).

The moment \(M_z\), modulated according to (81), causes complex forced oscillations of the entire system and, in particular, of the reduced mass of the engine, i.e. of the crank. The carrier harmonic of the moment \(M_z\) gives oscillations of the crank according to the law:

\[ (\varphi_{1z})=\varphi_{1z}\sin (z\eta_0 t-\zeta) \]

at the “carrier” frequency \(z\eta_0\).

The corresponding additional vibrational velocity of the crank is:

\[ (\varphi_{1z})'=\varphi_{1z}\nu\cos(\nu t-\zeta), \]

where, for the “carrier” frequency of the moment \(M_z\), the notation has been introduced:

\[ \nu=z\eta_0. \]

Now the degree of nonuniformity of rotation is:

\[ \delta = 2\varphi_{1z}\cdot \frac{\nu}{\eta_0}. \]

Hence the total instantaneous angular velocity of rotation of the crankshaft is

\[ \dot{\eta}=\eta_0\left[1+\frac{\delta}{2}\cos(\nu t-\zeta)\right] \tag{82-a} \]

and the instantaneous angle of its rotation is

\[ \alpha=\int_0^t \eta\,dt =\eta_0 t+\eta_0\frac{\delta}{2\nu}\sin(\nu t-\zeta). \]

Substitute this expression for \(\alpha\) into (78):

\[ f(\alpha)=M_h\sin\{h\eta_0 t+\xi\sin(\nu t-\zeta)+\varepsilon_h'\}, \tag{82} \]

where

\[ \xi=\frac{h\eta_0}{\nu}\frac{\delta}{2}=\frac{h\delta}{z\,2}, \tag{83} \]

as before, represents the amplitude of the frequency modulation of the harmonic, with respect to \(\alpha\), of the moment \(f_1(\alpha)\).

Relation (82) already gives the dependence of the harmonic, of interest to us with respect to \(\alpha\), of the moment \(f_1(\alpha)\) on time and takes into account, at least approximately, the effect of the nonuniformity of shaft rotation caused by the moment \(M_z\) on the true time-harmonic spectrum \(f_1(\alpha)\), i.e., \(M_h\).

Now, discarding in (75) and (76) the moments \(M_0\), \(f_3(\alpha)\), and \(f_{II}(\alpha)\), in accordance with the assumptions adopted above, we write the differential equations of motion of both masses in the following form:

\[ \Theta_1\frac{d^2\varphi_1}{dt^2} +k\left(\frac{d\varphi_1}{dt}-\frac{d\varphi_2}{dt}\right) +c(\varphi_1-\varphi_2) = \]

\[ = M_h\sin\{h\eta_0 t+\xi\sin(\nu t-\zeta)+\varepsilon_h'\} + \]

\[ +M_z\sin[z(\eta_0 t+\varphi_{1z})+\varepsilon_z], \tag{84} \]

\[ \Theta_2\frac{d^2\varphi_2}{dt^2} -k\left(\frac{d\varphi_1}{dt}-\frac{d\varphi_2}{dt}\right) -c(\varphi_1-\varphi_2)=0. \tag{85} \]

The first term on the right-hand side of equation (84) does not depend on the vibrations of both masses that it causes, \(\varphi_{1h}\) and \(\varphi_{2h}\); likewise, the second term, \(M_z\), does not depend on \(\varphi_{1h}\) and \(\varphi_{2h}\).

Consequently, equations (84) and (85) are no longer completely nonlinear, and a partial application of the principle of superposition to them is possible. Taking this circumstance into account, we split the system of equations (84)—(85) into two pairs of equations:

\[ \left. \begin{aligned} \Theta_1\frac{d^2\varphi_{1h}}{dt^2} +k\left(\frac{d\varphi_{1h}}{dt}-\frac{d\varphi_{2h}}{dt}\right) +c(\varphi_{1h}-\varphi_{2h}) &= \\ &=M_h\sin[h\eta_0 t+\xi\sin(\nu t-\zeta)+\varepsilon_h'] \\ \Theta_2\frac{d^2\varphi_{2h}}{dt^2} -k\left(\frac{d\varphi_{1h}}{dt}-\frac{d\varphi_{2h}}{dt}\right) -c(\varphi_{1h}-\varphi_{2h}) &=0 \end{aligned} \right\} \tag{86} \]

\[ \left. \begin{aligned} \Theta_1 \frac{d^2 \varphi_{1z}}{dt^2} &+ k\left(\frac{d\varphi_{1z}}{dt}-\frac{d\varphi_{2z}}{dt}\right) + c(\varphi_{1z}-\varphi_{2z}) \\ &= M_z \sin [z(\eta_0 t+\varphi_{1z})+\varepsilon_z'],\\ \Theta_2 \frac{d^2 \varphi_{2z}}{dt^2} &- k\left(\frac{d\varphi_{1z}}{dt}-\frac{d\varphi_{2z}}{dt}\right) - c(\varphi_{1z}-\varphi_{2z})=0. \end{aligned} \right\} \tag{87} \]

Here equations (87) are nonlinear, in contrast to the linear equations (86). Both of these systems of equations are connected with one another by the quantities \(\xi\), \(\nu\), and \(\zeta\), which depend on \(\varphi_{1z}\) and \(z\) and enter into the right-hand side of the first of equations (86). In what follows we shall regard \(\xi\), \(\nu\), and \(\zeta\) as known in advance, determined at least experimentally by torsiographing the engine under investigation. Then the analysis of the forced vibrations of a nonuniformly rotating shaft, caused by the action of a single harmonic, of order \(\alpha\), moment \(M_h\), reduces to the study of two linear differential equations (86).

Dividing the first of equations (86) by \(\Theta_1\), the second by \(\Theta_2\), and subtracting the second equation from the first, we then have

\[ \Theta_1 \frac{d^2 \Psi}{dt^2} + k_1 \frac{d\Psi}{dt} + c_1 \Psi = M_h \sin [h\eta_0 t+\xi \sin(\nu t-\zeta)+\varepsilon_h'], \tag{88} \]

where

\[ \Psi=\varphi_{1h}-\varphi_{2h} \]

is the instantaneous angle of twist of the shaft of our system caused by the action upon it of the moment \(M_h\). Here

\[ k_1=k\frac{\Theta_1+\Theta_2}{\Theta_1\Theta_2}, \]

\[ c_1=c\frac{\Theta_1+\Theta_2}{\Theta_1\Theta_2}. \]

The phase angle \(\varepsilon_h'\), entering into equation (88) and appearing in the harmonic (with respect to \(\alpha\)) analysis of the curve of the engine torque, does not play any appreciable role in the vibratory process under consideration, determining only a shift in time of the forced-vibration curve. Taking this into account, for simplicity we set

\[ \varepsilon_h'=0. \]

Then, instead of (88), we have

\[ \Theta_1 \frac{d^2 \Psi}{dt^2} + k_1 \frac{d\Psi}{dt} + c_1 \Psi = M_h \sin [\eta_h t+\xi \sin(\nu t-\zeta)], \tag{89} \]

where

\[ \eta_h=h\eta_0 \]

is the angular frequency of the harmonic under consideration \(M_h\), of the \(h\)-th order.

Equation (89), like equation (88), is linear; and the principle of superposition is fully applicable to it, according to which the complete forced vibration may be obtained by summing all those individual vibrations that will be produced in the system by each of the harmonic (in time \(t\)) components of the right-hand side of (89), when each acts on the system in isolation.

The exciting moment in the right-hand side of differential equation (89) is a frequency-modulated, monoharmonic

as a function of \(\alpha\), whose harmonic spectrum in time can be determined from the formulas given earlier. Assuming \(h\) to be not very small \((h>2)\), we have sufficient grounds to regard \(M_h\) as a purely gas moment.

Now, using expression (27) and putting \(\varepsilon_h=0\) in it, instead of (89) we obtain

\[ \Theta_1 \frac{d^2\Psi}{dt^2}+k_1\frac{d\Psi}{dt}+c_1\Psi = M_h I_0(\xi)\sin \eta_h t+ \]

\[ +\sum_{q=1}^{\infty} M_h I_q(\xi)\sin \bigl[(\eta_h+q\nu)t-qz\bigr]+ \]

\[ +\sum_{q=1}^{\infty}(-1)^q M_h I_q(\xi)\sin \bigl[(\eta_h-q\nu)t+qz\bigr]. \tag{90} \]

Here the products

\[ M_h I_q(\xi) \]

are the amplitudes of the components, harmonic in \(t\), of the moment \(M_h\). Each of these terms, quite independently of the others, causes harmonic oscillations of the shaft twist angle with a definite amplitude, phase, and frequency.

It is obvious that equation (90), in full accordance with the principle of superposition, has the following steady-state solution:

\[ \Psi=\Phi_0\sin(\eta_h t-\varepsilon_0)+ \]

\[ +\sum_{q=1}^{\infty}\Phi_q\sin\bigl[(\eta_h+q\nu)t-qz-\varepsilon_q\bigr]+ \]

\[ +\sum_{q=1}^{\infty}\Phi_{-q}\sin\bigl[(\eta_h-q\nu)t+qz-\varepsilon_{-q}\bigr]. \tag{91} \]

Here the following notation has been introduced:

\[ \left. \begin{aligned} \Phi_0&=\beta_0\,\frac{M_h I_0(\xi)}{c_1} =\beta_0\Psi_{cm} I_0(\xi),\\[6pt] \Phi_q&=\beta_q\,\frac{M_h I_q(\xi)}{c_1} =\beta_q\Psi_{cm} I_q(\xi),\\[6pt] \Phi_{-q}&=\beta_{-q}\,\frac{M_h I_{-q}(\xi)}{c_1} =\beta_{-q}\Psi_{cm} I_q(\xi), \end{aligned} \right\} \tag{92} \]

where

\[ \Psi_{cm}=\frac{M_h}{c_1}, \]

as usual, is the static angle of twist of the shaft of the system under the action of a constant moment of magnitude \(M_h\).

It is obvious that \(\Phi_0\) is the amplitude of the forced oscillations of the shaft twist angle produced by the “carrier” harmonic component

\[ M_h I_0(\xi)\sin\eta_h t \]

of the moment \(M_h\), complex in time but monoharmonic in \(\alpha\).

Similarly, \(\Phi_q\) and \(\Phi_{-q}\) denote the amplitudes of the oscillations caused by the action, respectively, of each upper

\[ M_h I_q(\xi)\sin\bigl[(\eta_h+q\nu)t-q\xi\bigr] \qquad (q=1,2,3,\ldots) \]

and lower

\[ M_h I_q(\xi)\sin\bigl[(\eta_h-q\nu)t+q\xi\bigr] \qquad (q=1,2,3,\ldots) \]

satellites in the spectrum \(M_h\).

In formulas (92), \(\beta\) is the dynamic coefficient, determined by the formulas:

\[ \left. \begin{aligned} \beta_0&=\frac{1}{\sqrt{(1-\gamma_0^2)^2+4\left(\frac{k_1}{k_c}\right)^2\gamma_0^2}},\\[6pt] \beta_q&=\frac{1}{\sqrt{(1-\gamma_q^2)^2+4\left(\frac{k_1}{k_c}\right)^2\gamma_q^2}},\\[6pt] \beta_{-q}&=\frac{1}{\sqrt{(1-\gamma_{-q}^2)^2+4\left(\frac{k_1}{k_c}\right)^2\gamma_{-q}^2}}. \end{aligned} \right\} \tag{93} \]

respectively for the carrier harmonic and its upper and lower satellites.

Here the tuning coefficient for the carrier harmonic is

\[ \gamma_0=\frac{\eta_h}{\omega_0}, \]

where \(\omega_0\) is the frequency of the natural oscillations of our system. Analogously, for each of the upper and lower satellites:

\[ \gamma_q=\frac{\eta_h+q\nu}{\omega_0}, \]

\[ \gamma_{-q}=\frac{\eta_h-q\nu}{\omega_0}. \]

Finally, \(k_c\), in formulas (93), is the usual critical damping coefficient:

\[ k_c=\sqrt{2\Theta_1 c_1}. \]

Now the phase angles \(\varepsilon\), characterizing the lag of the forced oscillations behind the corresponding exciting moments and entering into equation (81), may be computed from the formulas:

\[ \left. \begin{aligned} \varepsilon_0&=\operatorname{arctg}\,2\frac{k_1}{k_c}\frac{\gamma_0}{1-\gamma_0^2},\\[6pt] \varepsilon_q&=\operatorname{arctg}\,2\frac{k_1}{k_c}\frac{\gamma_q}{1-\gamma_q^2},\\[6pt] \varepsilon_{-q}&=\operatorname{arctg}\,2\frac{k_1}{k_c}\frac{\gamma_{-q}}{1-\gamma_{-q}^2}. \end{aligned} \right\} \tag{93-a} \]

Equation (91) shows that, when a linear system is acted upon by a single exciting moment \(M_h\), sinusoidal with respect to the crank angle \(\alpha\) and therefore frequency-modulated with respect to \(t\), there arises in it a whole complex of harmonic forced oscillations with different amplitudes, frequencies, and phases.

Here the harmonic analyzer of the function \(M_h\) is the linear oscillatory system itself. It is precisely it and its basic physical properties that ensure the physical reality of the analytical expansion of \(M_h\), considered above, into the carrier harmonic and satellites. Each of these harmonic components of \(M_h\) gives rise to a quite definite harmonic oscillatory process, all characteristics of which can be calculated from the formulas given above.

It is easy to see that, in a system under the action of the moment \(M_h\) represented by formula (82), a whole series of resonance cases is possible:

a) the principal resonance with the “carrier” harmonic \(M_h I_0(\xi)\) when

\[ \eta_h=\omega_0; \]

b) a “side” resonance with each of the upper satellites \(M_h I_q(\xi)\) when

\[ \eta_h+q\nu=\omega_0, \]

\[ \eta_h=\omega_0-q\nu; \]

\[ (q=1,\,2,\,3,\ldots); \]

c) a “side” resonance with each of the lower satellites \(M_h I_q(\xi)\) when

\[ \eta_h=\omega_0+q\nu \]

\[ (q=1,\,2,\,3,\ldots). \]

For the values indicated here of the mean angular frequency \(\eta_h\) of the moment \(M_h\), the corresponding dynamic coefficient \(\beta\), determined by one of formulas (93), becomes very large, or even unboundedly large in the absence of damping in the system.

Here we encounter for the first time certain new circumstances, completely unknown in the generally accepted theory of vibrations of shafts of piston engines.

According to this usual theory, each moment \(M_h\), obtained in the harmonic analysis of the curve \(M_{\mathrm{cr}}=f(\alpha)\) and having the mean angular frequency

\[ \eta_h=h\eta_0, \]

gives the installation one definite critical rotational speed of the shaft

\[ \eta_{0\mathrm{cr}}=\frac{\omega_0}{h}. \]

But from what has been said above it is evident that here, in the case of nonuniform rotation of the shaft, even under the isolated action of a single moment \(M_h\), one may expect the appearance of a whole series of critical speeds:

\[ \eta_{q\mathrm{cr}}=\frac{\omega_0\mp q\nu}{h}, \qquad (q=0,\,1,\,2,\ldots), \]

at which conditions arise in the system for the development of strong resonant vibrations.

In exact resonance with the carrier harmonic \(M_h I_0(\xi)\), i.e., at the principal critical rotational speed of the shaft

\[ \eta_{0\mathrm{cr}}=\frac{\omega_0}{h}, \]

all the upper satellites have frequencies \(\eta_h+q\nu\) greater than \(\omega_0\), that is, they are already in the beyond-resonance region, and for them the tuning coefficient \(\gamma_q>1\), while all the lower satellites have \(\gamma_{-q}<1\).

According, for example, to formula (90), the distance between any two components in the harmonic spectrum \(M_h\) is the same and is equal to \(\nu\), that is, to the mean angular velocity of rotation of the shaft multiplied by the order of cyclic nonuniformity. This distance (along the frequency axis), as well as the absolute magnitudes of the frequencies of the carrier harmonic \(\eta_h=h\eta_0\) and of all satellites,

\[ \eta_h \pm q\nu = \eta_0(h \pm qz), \qquad (q=1,2,3\ldots) \]

depend directly on the absolute magnitude of the mean angular velocity of rotation of the shaft and on the order of frequency modulation \(z\). Consequently, when the number of revolutions of the engine changes, the entire harmonic spectrum \(M_h\) (for example, in the upper part of Fig. 2) will shift in the corresponding direction and, at the same time, be compressed or expanded along the abscissa axis in proportion to the change in \(\eta_0\).

To eliminate the difficulties that arise here in the graphical representation of a number of dependencies of interest to us, we shall proceed as follows. We shall construct the harmonic spectrum \(M_h\) (Fig. 10¹, above), plotting along the abscissa axis not the absolute values of the frequencies of the corresponding harmonics, but the ratios of these frequencies to the frequency of the carrier harmonic:

Fig. 10

\[ \rho=\frac{\eta_h \pm q\nu}{\eta_h}=1\pm q\frac{z}{h}, \qquad (q=0,1,2,\ldots). \tag{94} \]

It is obvious that, for small values of \(\eta_0\), only harmonic components of high orders can enter resonance. As \(\eta_0\), that is, the number of revolutions of the engine, increases, harmonics of increasingly lower orders will enter resonance. Thus, the motion along the abscissa axis—

¹ In Figs. 10, 13, 14, 15, 16, and 18 the moment \(M_h\) is denoted by \(\mathfrak{M}_h\).

to the right along the abscissa \(\rho\) there corresponds a decrease of the tuning coefficient of the carrier harmonic (that is, of the basic tuning coefficient) and, conversely, to the value \(\rho=0\) there correspond

\[ \gamma_0=\infty;\qquad \eta_0=\infty. \]

For \(\gamma_0=1\) we have resonance with the carrier harmonic. Resonance with any satellite of number \(\pm q\) will occur when the equality

\[ \frac{\eta_{\mathrm{h}}\pm q\nu}{\omega_0}=1, \tag{95} \]

is satisfied, that is, for the following value of the basic tuning coefficient:

\[ \gamma_0=\frac{1}{1\pm q\frac{\nu}{\eta_{\mathrm{h}}}}=\frac{1}{\rho}. \tag{96} \]

To any arbitrary value of \(\gamma_0\) there corresponds the following value of the tuning coefficient, for example, of the upper satellite, with frequency \(\eta_{\mathrm{h}}+q\nu\):

\[ \gamma_q=\frac{\eta_{\mathrm{h}}+q\nu}{\omega_0} =\frac{\eta_{\mathrm{h}}}{\omega_0}\left(1+\frac{q\nu}{\eta_{\mathrm{h}}}\right) =\rho\gamma_0. \tag{97} \]

Here, of course, for each particular satellite the coefficient \(\rho\) is a completely definite quantity, independent of \(\eta_0\).

Formulas (96) and (97) make it possible to carry out a specific numerical marking of the abscissa axis \(\gamma_0\) in the lower part of Fig. 10. This axis serves as the basis for constructing the resonance curves \(\Phi_0,\Phi_1,\Phi_{-1}\ldots\) by formulas (92) and (93), for any adopted value of the relative damping coefficient \(k/k_c\).

To determine the amplitudes \(\Phi\) of all harmonic oscillations caused by the action of the moment \(M_{\mathrm{h}}\) at any specified number of engine revolutions \(n\), it is now sufficient to calculate the corresponding basic tuning coefficient

\[ \gamma_0=\frac{h\eta_0}{\omega_0}, \]

where

\[ \eta_0=\pi n/30. \]

Then the intersection of the vertical corresponding to this value \(\gamma_0\) with the curves \(\Phi\) (Figs. 11 and 12) ¹ gives the desired amplitudes of the forced harmonic oscillations included in series (91). The phases of these oscillations are easily calculated from formulas (93-a).

Now, by graphical summation of all terms of series (91), we obtain the curve of the complete oscillations of the shaft twist angle under the action of the torque \(M_{\mathrm{h}}\), harmonic in \(\alpha\), in the case of nonuniform rotation of this shaft according to equation (82-a).

The amplitudes \(\Phi_0,\Phi_1,\Phi_{-1}\ldots\) of all the monoharmonic oscillatory processes summed here depend, according to (92) and (93), on a whole series of factors and, above all, on the number of engine revolutions \(n\), that is, on the basic tuning coefficient \(\gamma_0\). At certain specific “critical” values of \(\gamma_0\), determined by formula (96), the forced vibrations

¹ In Figs. 11 and 12 only the curves \(\Phi_0,\Phi_1,\Phi_{-1}\) are shown, and all curves \(\Phi\) for the following satellites are omitted.

Fig. 11

acquire a more or less sharply expressed resonant character due to the occurrence of a resonant situation for one of the components of the harmonic spectrum \(M_h\).

Fig. 12

The degree of intensity of the side resonances depends in many respects on the absolute magnitudes of the amplitudes \(\Phi_0, \Phi_1\ldots\) and on the relations between them. The height of the peaks of the curves \(\Phi\) in Fig. 10 is determined both by the value of the damping coefficient \(k\) in the system and by the amplitudes \(M_h I_q(\xi)\) of the corresponding harmonic components \(M_h\), that is, in particular, by the concrete numerical values of the Bessel functions \(I_q(\xi)\).

But, according to (84):

\[ \xi = \frac{\eta_h}{\nu}\,\frac{\delta}{2}, \]

that is, the amplitudes \(\Phi\) are closely connected with the magnitude of the degree of nonuniformity \(\delta\) and the frequency of this nonuniformity \(\nu\), or, more precisely, with the ratio \(\eta_h/\nu\).

The connection between \(\Phi\) and \(\delta\) is entirely analogous to the dependence of the amplitudes of the harmonic components of the moment \(M_h\) on \(\delta\), considered above and illustrated, for example, by the graphs in Figs. 6–8. As a rule, with an increase in \(\delta\), the amplitudes \(\Phi_q\) and \(\Phi_{-q}\) of the forced vibrations produced by the satellites increase.

The influence of \(\delta\) on the magnitude of the amplitude \(\Phi_0\) of the forced vibrations caused by the carrier harmonic of the moment \(M_h\) may be considerably more complex. An increase in \(\delta\) invariably leads to some decrease in \(\Phi_0\) if \(M_h\) is a gas harmonic, that is, a component of the torque on the shaft produced by the gas pressure in the cylinder. An increase in \(\delta\) may lead to an increase or decrease in \(\Phi_0\), or may even have almost no effect at all on the magnitude of \(\Phi_0\), if \(M_h\) is an inertia harmonic. The effect of the influence of \(\delta\) on \(\Phi_0\) here depends to a strong degree on the phase \(\zeta\), for example, in formula (46).

It follows from (84) that an increase in the ratio \(\eta_h/\nu\), that is, a decrease in the frequency of the cyclic oscillations of the shaft rotational speed, acts on all the amplitudes \(\Phi\) (in the case of gas harmonics) in the same direction as an increase in \(\delta\). The nonuniformity of the shaft rotation has its strongest effect on the composition of the harmonic spectra of moments \(M_h\) of high orders. It is precisely at large \(\eta_h/\nu\) that one should expect the strongest effect of dynamic damping of shaft vibrations due to the nonuniformity of its rotation, manifested in a reduction of the resonant vibrations of the shaft as \(\delta\) increases.

Formula (94) further shows that the position of the satellites in the frequency spectrum of \(M_h\) also depends on the ratio \(\eta_h/\nu\) (Fig. 10, top). With a decrease in \(\nu\), that is, with a slowing of the cyclic oscillations of the shaft rotational speed, the frequency intervals between the satellites are reduced, and all the satellites approach the principal carrier harmonic \(M_h I_0(\xi)\). Simultaneously with this, the amplitudes \(\Phi_q, \Phi_{-q}\) of the vibrations caused by the satellites increase and, for the most part, the amplitudes \(\Phi_0\) of the vibrations caused by the carrier harmonic decrease (Fig. 12).

Very slow oscillations of the angular velocity of shaft rotation, that is, small values of \(\nu\), may considerably reduce the amplitude \(M_h I_0(\xi)\) of the carrier harmonic and even bring it to zero. Thus, as \(\nu\) is decreased, the old, principal resonance with the carrier harmonic gradually weakens, while at the same time the action of the satellites increases.

Especially great here is the role of the first pair of satellites, which give strong forced vibrations with amplitudes \(\Phi_1\) and \(\Phi_{-1}\). The summation of these vibrations, occurring respectively with frequencies \(\eta_h+\nu\) and \(\eta_h-\nu\), which are close to one another at small \(\nu\), gives a complex law of resultant vibrations with beats having the frequency

\[ (\eta_h+\nu)-(\eta_h-\nu)=2\nu . \]

These beats will be expressed the more sharply and clearly, the less the values $\Phi_0$, $\Phi_2$, $\Phi_{-2}$, etc. make themselves felt here. The depth of the beats, in general, increases with increasing $\delta$.

As $\nu \to \infty$, all the curves $\Phi$ merge into one resonance curve $\Phi_0$ of the ordinary type (corresponding to the case $\delta = 0$), and the cyclic oscillations of the shaft rotational speed $\eta$ will here be dynamically equivalent to the corresponding periodic oscillations of the fundamental, carrier frequency $\eta_h$. Let us consider this limiting transition in its most elementary form.

Assume that for all values of $\nu$ the quantities $\delta$, $\eta_0$, and $h$ remain unchanged. Then, by formulas (93a), as $\nu \to 0$:

\[ \gamma_q \to \gamma_0; \qquad \gamma_{-q} \to \gamma_0 \]

\[ \overline{\varepsilon}_q \to \overline{\varepsilon}_0; \qquad \overline{\varepsilon}_{-q} \to \overline{\varepsilon}_0 \]

and by (93):

\[ \beta_q \to \beta_0; \qquad \beta_{-q} \to \beta_0. \]

Consequently, now from (91):

\[ \Psi \to \beta_0 \left[ I_0(\xi) + 2 \sum_{q=1}^{\infty} I_{2q}(\xi) \right] \sin(\eta_h t - \overline{\varepsilon}_0). \]

But, as is known from the general theory of Bessel functions [2], for any $\xi$,

\[ I_0(\xi) + 2 \sum_{q=1}^{\infty} I_{2q}(\xi) = 1, \]

that is, as $\nu \to 0$,

\[ \Psi \to \beta_0 \sin(\eta_h t - \overline{\varepsilon}_0). \]

We have obtained the usual equation of forced oscillations of a linear system under the action of an exciting moment that is monoharmonic in time. Here, however, $\beta_0$ of course varies slowly (for $\nu \ne 0$) in accordance with the periodic oscillations $\eta_0$, $\eta_h$, and $\gamma_0$.

At exact fundamental resonance, that is, when $\eta_h = \omega_0$, and also for small $\xi$ ($\nu \gg 0$) and sufficiently small damping in the system, the effect of the satellites causing oscillations with amplitudes $\Phi_q$ and $\Phi_{-q}$ ($q = 1, 2, 3, \ldots$) may be neglected. Then the dependence of the amplitude of the complete forced oscillations $\Psi$, caused by the moment $M_h$, on the magnitude of the degree of nonuniformity $\delta$ will coincide with the relation between $\Phi_0$ and $\delta$.

But for $\gamma_0 = \mathrm{const}$, according to (92), the amplitudes $\Phi_0$ are proportional to the amplitude of the carrier harmonic $M_h I_0(\xi)$. Consequently, the law, shown above, for example in Figs. 2 and 3, of the variation of $I_0(\xi)$ with increasing $\delta$ is at the same time an approximate characteristic of the degree of decrease of the amplitude $\Psi_{\max}$ of the complete forced resonant vibrations in the system due to dynamic damping caused by cyclic oscillations of the shaft rotational speed.

When the number of revolutions of the engine changes, that is, when $\eta_0$ changes in one direction or another from its critical value $\eta_{0\mathrm{cr}} = \omega_0 / h$, all the satellites in the spectrum $M_h$ shift correspondingly, the state of fundamental resonance is disturbed, and the possibility of side resonances noted above appears. It is obvious that outside the fundamental resonance, when $\gamma_0 \ne 1$, the relation between $\Phi_0$ and $\delta$ no longer characterizes the actual dependence of $\Psi_{\max}$ on $\delta$. Here, in order to clarify the effect of dynamic dampi-

... damping due to $\delta$, it is necessary each time to carry out a specific calculation of the quantity $\Psi_{\max}$ from equation (91) and to compare this $\Psi_{\max}$ with $\Phi_0$ for $\delta=0$.

But, in any case, it is evident that during engine operation under conditions of lateral resonance—for example, with the first satellite—it is in principle quite possible for an effect of negative dynamic damping to appear, when the nonuniformity of shaft rotation leads not to a decrease but to an increase of the total amplitude $\Psi_{\max}$, as compared with the value it would have for $\delta=0$. The probability of occurrence of such negative damping is, of course, the greater the larger the amplitude of the resonating satellite and the smaller the effective damping in the system.

Apparently, negative dynamic damping is also quite possible in nonresonant regions, in the case of large $\delta$ and slow modulation, owing to the large magnitude, for example, of $\Phi_1$ and $\Phi_{-1}$ as compared with $\Phi_0$.

We have considered, in general outline, the phenomenon of dynamic damping of forced vibrations of a nonuniformly rotating shaft acted upon by a single torque, monoharmonic in $\alpha$, $M_h$.

Here we assumed the case of, as it were, external, independent dynamic damping, when the cyclic nonuniformity of shaft rotation was caused by some other components of the torque on the engine shaft, and not by $M_h$. This nonuniformity, through the effect of frequency modulation (primarily), acted on the harmonic spectrum of the moment $M_h$, but, according to the conditional assumptions we adopted, did not depend at all on the moment $M_h$ itself or on the vibrations of the system caused by it.

Thus, the degree of nonuniformity $\delta$ has not yet included that “elastic” nonuniformity of rotation which arises in the system due to the action precisely of this moment $M_h$.

We must now dwell on the possibility of a peculiar “self-damping” of vibrations, due to the fact that the “elastic” nonuniformity of shaft rotation generated by the moment $M_h$ also gives frequency modulation of this moment and thereby, once again, changes the effect of its action on the oscillating system. For an approximate analysis of such phenomena of “self-modulation,” we shall make use of the results obtained above in our study of the effect of frequency modulation, assuming in the appropriate places

\[ \nu=\eta_h . \]

Earlier, in adopting a number of simplifying assumptions given on p. 35, we excluded from consideration the case

\[ h=z \quad \text{or} \quad \nu=\eta_h . \]

Now this restriction is no longer at all necessary for us.

But for the time being, temporarily, let us also assume here that the monoharmonic nonuniformity of shaft rotation, having frequency $\nu=\eta_h$, is caused by action on the system from outside, by some other moments independent of $M_h$.

Then the moment $M_h$ will receive frequency modulation of the same monoharmonic character.

Let us consider the particular case of in-phase modulation

\[ (M_h)=M_h\sin(\eta_h t+\xi\sin\eta_h t), \]

where, by formula (84), for $\eta_h=\nu$,

\[ \xi=\delta/2 . \]

But, according to (34),

\[ M_h \sin\left(\eta_h t+\frac{\delta}{2}\sin \eta_h t\right)= \]

\[ = M_h \sum_{q=1}^{\infty} \left[ I_{q-1}\left(\frac{\delta}{2}\right)+(-1)^q I_{q+1}\left(\frac{\delta}{2}\right) \right]\sin q\eta_h t, \]

that is, here all the satellites, in the harmonic (with respect to time \(t\)) spectrum of the moment \(M_h\), are located on one side of the carrier harmonic and have frequencies that are multiples of \(\eta_h\).

Now, analogously to (89), we obtain the differential equation of forced oscillations of the shaft-twist angle of our two-mass system according to Fig. 9:

\[ \Theta_1\frac{d^2\Psi}{dt^2}+k_1\frac{d\Psi}{dt}+c_1\Psi = \]

\[ = M_h \sum_{q=1}^{\infty} \left[ I_{q-1}\left(\frac{\delta}{2}\right)+(-1)^q I_{q+1}\left(\frac{\delta}{2}\right) \right]\sin q\eta_h t. \]

The stationary part of the solution of this equation has, obviously, the following form:

\[ \Psi=\sum_{q=1}^{\infty}\Phi_{q-1}\sin(q\eta_h t-\varepsilon_q), \tag{98} \]

where

\[ \Phi_{q-1}=\beta_{q-1}\frac{M_h}{c_1} \left[ I_{q-1}\left(\frac{\delta}{2}\right)+(-1)^q I_{q+1}\left(\frac{\delta}{2}\right) \right], \tag{99} \]

\[ \left. \begin{aligned} \beta_{q-1} &= \frac{1}{ \sqrt{(1-\gamma_{q-1}^{\,2})^2+ 4\left(\frac{k_1}{k_c}\right)^2\gamma_{q-1}^{\,2}} },\\[6pt] \gamma_{q-1} &= \frac{q\eta_h}{\omega_0},\\[6pt] \varepsilon_{q-1} &= \operatorname{arctg}\left( 2\frac{k_1}{k_c}\cdot \frac{\gamma_{q-1}}{1-\gamma_{q-1}^{\,2}} \right). \end{aligned} \right\} \tag{100} \]

Expanding the equation of forced oscillations (98), we obtain:

\[ \Psi=\Phi_0\sin(\eta_h t-\varepsilon_0)+ \Phi_1\sin(2\eta_h t-\varepsilon_1)+ \Phi_2\sin(3\eta_h t-\varepsilon_2)+\ldots, \tag{101} \]

Here:

\[ \left. \begin{aligned} \Phi_0 &= \beta_0\frac{M_h}{c_1} \left[ I_0\left(\frac{\delta}{2}\right)- I_2\left(\frac{\delta}{2}\right) \right],\\[6pt] \Phi_1 &= \beta_1\frac{M_h}{c_1} \left[ I_1\left(\frac{\delta}{2}\right)+ I_3\left(\frac{\delta}{2}\right) \right],\\ &\hspace{2em}\cdots\\ &\hspace{2em}\cdots \end{aligned} \right\} \tag{102} \]

where \(\Phi_0, \Phi_1,\ldots\) represent the amplitudes of the forced oscillations caused by the individual components, harmonic in \(t\), of the moment \(M_h\).

Similarly, putting in (27)

\[ \zeta=\pi/2;\qquad \varepsilon_h=0, \]

we consider the moment

\[ (M_h)=M_h\sin(\eta_h t+\xi\cos\eta_h t). \]

For this case, by formula (35), with

\[ \xi=\delta/2 \]

we have

\[ \begin{aligned} M_h\sin\left(\eta_h t+\frac{\delta}{2}\cos\eta_h t\right) = M_h\Bigg\{ I_1\left(\frac{\delta}{2}\right) &+\sum_{q=0}^{\infty}(-1)^q \left[ I_{2q}\left(\frac{\delta}{2}\right) + I_{2q+2}\left(\frac{\delta}{2}\right) \right]\sin(2q+1)\eta_h t \\ &+\sum_{q=0}^{\infty}(-1)^q \left[ I_{2q+1}\left(\frac{\delta}{2}\right) - I_{2q+3}\left(\frac{\delta}{2}\right) \right]\cos 2q\eta_h t \Bigg\}. \end{aligned} \]

Now the equation of forced oscillations, instead of (98), takes the form: ¹)

\[ \Psi= \Phi_0\sin(\eta_h t-\bar{\varepsilon}_0) +\Phi_1\cos(2\eta_h t-\bar{\varepsilon}_1) +\Phi_2\sin(\eta_h t-\bar{\varepsilon}_2)+\ldots, \tag{103} \]

where:

\[ \left. \begin{aligned} \Phi_0&=\beta_0\,\frac{M_h}{c_1} \left[ I_0\left(\frac{\delta}{2}\right)+I_2\left(\frac{\delta}{2}\right) \right],\\ \Phi_1&=\beta_1\,\frac{M_h}{c_1} \left[ I_1\left(\frac{\delta}{2}\right)-I_3\left(\frac{\delta}{2}\right) \right], \end{aligned} \right\} \tag{104} \]

\[ \ldots\ldots\ldots\ldots\ldots \]

The quantities \(\beta\) and \(\varepsilon\) entering here are determined by the preceding formulas (100).

Figures 13 and 14 show the curves of the forced oscillations of the angle of twist \(\Psi\) of the shaft of the two-mass system under consideration, calculated from equations (98) and (103) for the following specific data:

\[ \delta=2,\qquad \frac{k_1}{k_c}=0.1,\qquad \eta_h=\omega_0 \quad\text{(principal resonance).} \]

The value \(\delta=2\) corresponds to such a large nonuniformity of shaft rotation that its angular velocity oscillates within the limits from \(0\) to \(2\eta_0\). Of course, such a case is scarcely possible in practice, even under experimental conditions; nevertheless it has been adopted by us in Figs. 13 and 14 in order to obtain the most illustrative characterization of the influence of \(\delta\).

For similar reasons, a large value of the relative damping \(k_1/k_c\) has also been adopted, since, as damping in the system increases, the effect of the superposition of vibrations caused by satellites upon the vibrations produced by the carrier harmonic under conditions of its resonance is intensified.

The diagrams in Figs. 13 and 14 show that, despite the very strong distortion of the curve of the exciting moment \(M_h\) with time (in comparison—

¹) Here we discard the static effect of the constant term: \(M_h I_1\left(\dfrac{\delta}{2}\right)\).

compared with the pure sinusoidal curve \(M_h\) in a), owing to the large \(\delta\), the curve of the resultant forced oscillations \(\Psi\) differs only slightly from a pure sinusoid. But the nonuniformity of shaft rotation appreciably reduces the amplitude of the forced oscillations \(\Psi_{\max}\) in comparison with the amplitude that would be established in the system under the action of the moment \(M_1\) for \(\delta=0\) (compare curves 2 and 6 in Figs. 13, 14).

Fig. 13

In the figure: “Forced oscillations of the angle \(\Psi\) for
\(M=M_h\sin(\eta_h t+\xi\sin\eta_h t)\), \(\xi=1\), \(k/k_c=0.1\).”

Legend:

1. \(M_h\sin(\eta_h t+\xi\sin\eta_h t)\)
2. \(\Psi=\Psi_0+\Psi_1\)
3. \(\Psi_0=\varphi_0\sin(\eta_h t-\pi/2)\)
4. \(\Psi_1=\varphi_1\sin(2\eta_h t-\varepsilon_1)\)
5. \(M_h\sin\eta_h t\), \(\delta=0\)
6. \(\varphi_0\sin(\eta_h t-\pi/2)\), \(\delta=0\)

As was already noted above, in a system having \(\delta>0\), new critical rotational speeds of the shaft may appear, owing to the presence, in the harmonic spectrum of the moment \(M_h\), of satellites with different frequencies. This situation is illustrated by the diagrams in Figs. 15 and 16, where the resonance curves \(\Psi_{res}/\Psi_{st}\), calculated by us from equations (98) and (103) for various \(\delta\) and \(k_1/k_c\), are shown. Here it is easy to observe that the intensity of the side resonances increases sharply with decreasing damping in the system and with increasing \(\delta\). However, in the majority of practically possible cases \((\delta<0.2)\), the side resonances can hardly have any substantial significance.

Fig. 14

In the figure: “Forced oscillations of the angle \(\Psi\) for
\(M=M_h\sin(\eta_h t+\xi\sin\eta_h t)\), \(\xi=1\), \(k/k_c=0.1\).”

Legend:

1. \(M=M_h\sin(\eta_h t+\xi\sin\eta_h t)\)
2. \(\Psi=\Psi_0+\Psi_1\)
3. \(\Psi_0=\varphi_0\sin(\eta_h t+\pi/2)\)
4. \(\Psi_1=\varphi_1\sin(2\eta_h t-\varepsilon_1)\)
5. \(M_h A_1(\xi)\)
6. \(\varphi_0\sin(\eta_h t-\pi/2)\), \(\delta=0\)

Finally, these diagrams indicate a very insignificant (for \(\delta < 0.2\)) influence of the degree of nonuniformity of shaft rotation on the amplitudes of resonant vibrations (principal resonance) under the conditions:

\[ \eta_h = \nu . \]

Fig. 15.
Inscriptions in the figure: “Forced vibrations \(\psi\)”; “for \(M = M_0 \sin(\eta_h t + \frac{\delta}{2}\sin \eta_b t)\)”; \(k_1/k_c = 0.05\). Axes: \(\psi_{\max}/\psi_{\mathrm{st}}\), \(x_0\). Curve labels include \(\delta = 0.2\), \(\delta = 1\), \(\delta = 2\).

From a number of specific calculations carried out by us in connection with the construction of the curves in Figs. 15 and 16, it follows that, with not too great damping

Fig. 16.
Inscriptions in the figure: “Forced vibrations \(\psi\)”; “for \(M = M_0 \sin(\eta_h t + \frac{\delta}{2}\sin\eta_b t)\)”; \(k_1/k_c = 0.1\). Axes: \(\psi_{\max}/\psi_{\mathrm{st}}\), \(x_0\). Curve labels include \(\delta = 0\), \(\delta = 0.2\), \(\delta = 1\), \(\delta = 2\).

\[ (k_1/k_c \leqslant 0.1), \]

under conditions of resonance with the carrier frequency \((\eta_h = \omega_h)\), the amplitudes \(\Phi_1, \Phi_2, \ldots\) of the vibrations caused by the satellites are very small in comparison with

in comparison with \(\Phi_0\). Therefore, for the case of principal resonance, equations (99) and (104) may be rewritten in the form

\[ \Phi_{0res}=\beta_{0res}\frac{M_h}{c_1}\left[I_0\left(\frac{\delta}{2}\right)-I_2\left(\frac{\delta}{2}\right)\right]. \tag{105} \]

for the exciting moment

\[ M_h\sin\left[\eta_h t+\frac{\delta}{2}\sin\eta_k t\right] \]

and

\[ \Phi_{0res}=\frac{M_h}{c_1}\beta_{0res}\left[I_0\left(\frac{\delta}{2}\right)+I_2\left(\frac{\delta}{2}\right)\right] \tag{106} \]

for the exciting moment

\[ M_h\sin\left[\eta_h t+\frac{\delta}{2}\cos\eta_h t\right]. \]

Comparison of these expressions, as well as Figs. 12 and 13, shows that the phase of the frequency modulation \(\zeta\) in the general formula (27) for the spectrum \(M_h\) has a rather strong influence on the magnitude of the dynamic damping of forced vibrations at large values of \(\delta\).

For \(\delta=0\), expressions (105) and (106) pass into the most ordinary form

\[ \Phi_{res}=\beta_{0res}\frac{M_h}{c_1} \]

and give the amplitude of those resonant vibrations which will be established in the system in the case of uniform rotation of the shaft. The corresponding ratio:

\[ \frac{\Phi_{0res}}{\Phi_{res}}=I_0\left(\frac{\delta}{2}\right)\pm I_2\left(\frac{\delta}{2}\right) \tag{107} \]

is presented as a function of \(\delta\) in Fig. 17.

Fig. 17

Here the shaded region establishes the range of possible variations of the ratio \(\Phi_{0res}/\Phi_{res}\) as a function of the phase angle of the modulation \(\zeta\), for each specific \(\delta\).

It follows from the diagram in Fig. 17 that the dynamic effect of \(\delta\), manifested in a reduction of the amplitude of resonant vibrations under conditions of synchronous modulation \(\eta_h=\nu\), is very small, at least in the range of sufficiently small values of \(\delta\) commonly encountered in practice. On average this effect may be characterized by the ratio

\[ \frac{\Phi_{0res}}{\Phi_{res}}\cong I_0\left(\frac{\delta}{2}\right). \]

For a given nonuniformity \(\delta\) of the shaft rotation imposed from outside, the concrete value of \(\zeta\) depends on the entire complex of conditions under which the oscillatory processes in the installation develop. Taking these conditions into account in the general case, when \(\eta_h \ne \nu\), is very difficult.

The question of the value of \(\zeta\) is solved more simply in the case of “self-damping.”

At the principal resonance, \((\eta_h=\omega_0)\), the carrier harmonic

\[ M_h I_0\left(\frac{\delta}{2}\right)\sin \eta_h t \]

causes forced oscillations

\[ \Phi_{0res}\sin(\eta_h t-\pi/2) \]

with the \(90^\circ\) phase shift usual for resonance.

Now the instantaneous angle of rotation of the crankshaft (neglecting the small \(\Phi_1,\Phi_2,\ldots\)) is

\[ \alpha=\eta_0 t+\Phi_{0res}\sin(\eta_h t-\pi/2). \]

Fig. 18

Consequently, in the case under consideration the harmonic moment \(M_h\sin\alpha\) proves to be modulated according to the law

\[ (M_h)=M_h\sin[\eta_h t+\Phi_0 h\sin(\eta_h t-\pi/2)]. \]

Comparison of this expression with (27) shows that here

\[ \zeta=\pi/2, \]

that is, for estimating the effect of self-damping, in Fig. 17 one should use the curve

\[ \frac{\Phi_{0res}}{\Phi_{res}}=I_0\left(\frac{\delta}{2}\right)+I_2\left(\frac{\delta}{2}\right). \tag{107-a} \]

But under self-damping conditions the value of the “elastic” \(\delta\) can no longer be considered prescribed in advance, in view of the circular relation between \(\delta\) and \(\Phi_{0res}\) through the relation

\[ \delta=2h\Phi_{0res} \tag{108} \]

and the dynamic-damping curve (107-a). This “elastic” \(\delta\) must be determined from the following considerations.

In Fig. 18 the curve according to equation (107-a) has been constructed. It is obvious that the ordinate axis here may also be regarded as the axis of moments, on which the value of the amplitude of the carrier harmonic is laid off:

\[ M_h\left[I_0\left(\frac{\delta}{2}\right)+I_2\left(\frac{\delta}{2}\right)\right] \]

as a function of \(\delta\). Horizontal 1-a gives the amplitude \(M_h\) of this carrier in the case \(\delta=0\).

Let us compute the resonance amplitude \(\Phi_{res}\) for \(\delta = 0\) by the usual formula

\[ \Phi_{res}=\beta_{0res}\frac{M_h}{c_1} \]

and from it determine the corresponding degree of nonuniformity

\[ \delta_0=\beta_{0res}\frac{M_h}{c_1}\,2h. \]

Plotting \(\delta_0\) on the abscissa axis (Fig. 18), we draw through the point found a vertical line up to its intersection with the curve \(\Phi_{0res}/\Phi_{res}\) at point \(a\), and connect \(a\) by a straight line with the origin of coordinates. The intersection of this straight line with the curve \(\frac{\Phi_{0res}}{\Phi_{res}}\) at point \(b\) gives the true “elastic” \(\delta\), and, consequently, through formula (107-a), the true amplitude of the forced vibrations \(\Phi_0\), taking account of the phenomenon of “self-damping.” The relative effect of self-damping is characterized by the ordinate of point \(b\).

Let us summarize some results of the analysis given above of the influence of the magnitude of the degree of nonuniformity of shaft rotation on the amplitudes of its forced vibrations under the action of the moment \(M_h \sin h\alpha\), monoharmonic in the crank angle \(\alpha\).

a) Even a purely monoharmonic nonuniformity of shaft rotation leads to the appearance of distortions and deviations of the curve of forced vibrations from a pure sinusoid in time, owing to the summation of vibrations with different frequencies equal to the frequencies of the harmonic components of the moment \(M_h\).

b) A complex, non-sinusoidal nonuniformity of shaft rotation has almost the same influence on the harmonic spectrum of the moment \(M_h\) as does monoharmonic nonuniformity. Therefore the dynamic influence of complex nonuniformity on the forced vibrations of the shaft differs little from the influence of the principal harmonic component of the curve of the shaft’s angular velocity \(\eta\) as a function of time.

c) Nonuniformity of shaft rotation manifests itself in its forced vibrations through the effect of frequency (sometimes mixed) modulation of the exciting moments acting on the shaft.

d) Frequency modulation, harmonic in \(\alpha\), of the moment \(M_h\) never, except in cases where \(\xi\) is a root of the function \(I_0(\xi)\), eliminates the phenomenon of stationary, i.e. established, resonance at the carrier frequency \(h\eta_0\). The resonance situation is preserved here unchanged, for any values of the degree of nonuniformity \(\delta\). But with increasing \(\delta\) the amplitude \(M_h I_0(\xi)\) of the resonating “carrier” harmonic decreases and, consequently, so does the amplitude of the resonant vibrations caused by it, in accordance with the law of variation of the Bessel function \(I_0(\xi)\).

(This last conclusion, concerning the decrease in the amplitude of the carrier harmonic as \(\delta\) increases, is not always valid with respect to the inertia moments \(M_{in}\).)

Thus, the ordinary nonuniformity of shaft rotation (conditioned by the character of the diagram of tangential forces on the engine shaft) in general does not prevent the development of resonant vibrations during operation of the installation at critical speeds. Periodic oscillations of the angular velocity of the shaft \(\eta\) can lead only to some change (decrease) in the amplitudes of the resonant vibrations and to distortion of their purely sinusoidal (for \(\delta = 0\)) character.

Therefore the generally accepted explanation of the effect of dynamic damping due to \(\delta>0\)—as a permanent disturbance of resonance, by the impossibility of establishing it even for a short interval of time owing to the continuous periodic variation of the frequency \(h\eta\) of each harmonic torque acting on the shaft—cannot be regarded as correct.

Such an explanation is based on a formal and erroneous application of the basic conclusions of the elementary theory of harmonic oscillations. Here, under conditions of nonuniform rotation of the engine shaft, each harmonic \(M_h\) of the torque \(M_{\mathrm{kr}}\) is considered as a “sinusoidal” function of time having a variable frequency.

But, as was shown above, a “sinusoidal” moment with variable frequency, that is, a frequency-modulated moment, differs fundamentally from a truly sinusoidal moment both in its external characteristics and in the effect of its action on a linear oscillatory system. The resonance curve for the frequency-modulated moment \(M_h\) has its own laws and specific features, unknown in the ordinary theory of shaft vibrations.

6. Forced oscillations of a linear system under complex periodic excitation

In the preceding sections of this article it was shown that each component \(M_h\) of the torque \(M_{\mathrm{kr}}\) on the engine shaft, harmonic with respect to the crank angle of rotation \(\alpha\) and entering into the usual expansion of \(M_{\mathrm{kr}}\) in the Fourier series (1), ceases to be a monoharmonic function of time in the case of shaft rotation with variable, oscillating speed. Due to the effect of frequency modulation (and, in relation to the inertial harmonics, also mixed modulation) for \(\delta>0\), the true spectrum, harmonic in \(t\), of each moment \(M_h\), and consequently also of all \(A_h\) and \(B_h\) in formula (1), contains an entire series of monoharmonic components—the carrier harmonic and satellites.

Summation of these components, calculated for all \(A_h\) and \(B_h\), gives the complete frequency spectrum, harmonic in time, of the torque \(M_{\mathrm{kr}}\) on the shaft. It is obvious that now, for \(\delta>0\), the curve \(M_{\mathrm{kr}}=f_0(\alpha)\) differs from the curve \(M_{\mathrm{kr}}=f(t)\) in its form and, possibly, in its character.¹

For a concrete study of the influence of \(\delta\) on the complex torque \(M_{\mathrm{kr}}\) of the engine, let us adopt, as before, the simplest case of oscillations of the angular velocity of crank rotation according to the monoharmonic law (82-a). Under this condition the instantaneous angle of crank rotation is

\[ \alpha=\eta_0 t+\lambda \sin(\nu t-\zeta), \tag{109} \]

where

\[ \lambda=\frac{\eta_0}{\nu}\,\frac{\delta}{2}. \tag{110} \]

Substitution of the values of \(\alpha\) from (109) into series (1) gives

\[ M_{\mathrm{kr}}=M_0+M_{\mathrm{kr}}' + M_{\mathrm{kr}}''. \tag{111} \]

Here

\[ M_{\mathrm{kr}}'=\sum_{h=1}^{\infty} A_h \sin h\left[\eta_0 t+\lambda \sin(\nu t-\zeta)\right], \tag{112} \]

\[ M_{\mathrm{kr}}''=\sum_{h=1}^{\infty} B_h \cos h[\eta_0 t+\lambda \sin(\nu t-\zeta)] . \tag{113} \]

To simplify the further investigation, let us assume that the torque \(M_{\mathrm{kr}}\), represented by expressions (1), (111)—(113), contains only gas harmonics. This is equivalent to the condition that the initial curve \(M_{\mathrm{kr}}=f_0(\alpha)\) was constructed from the indicator diagram without taking into account the inertia forces of the connecting-rod-and-crank mechanism. The inertia moments, however, may be taken into account separately.

The first, constant, term \(M_0\) in (1) and (111) is of no interest. Let us consider the second term, \(M_{\mathrm{kr}}'\) in (111), determined by the series (112).

To expand this expression we shall use formula (27), putting in it

\[ \varepsilon_h=0;\qquad \eta_h=h\eta_0;\qquad \xi=h\lambda . \]

Then from (112)

\[ \begin{aligned} M_{\mathrm{kr}}' &=\sum_{h=1}^{\infty} A_h \left\{ I_0(h\lambda)\sin h\eta_0 t \right.\\ &\quad+\sum_{q=1}^{\infty} I_q(h\lambda)\sin[(h\eta_0+q\nu)t-q\zeta]\\ &\quad\left. +\sum_{q=1}^{\infty}(-1)^q I_q(h\lambda)\sin[(h\eta_0-q\nu)t+q\zeta] \right\}. \end{aligned} \tag{114} \]

Similarly, taking in (27)

\[ \varepsilon_h=\pi/2;\qquad \eta_h=h\eta_0;\qquad \xi=h\lambda, \]

we have from (113):

\[ \begin{aligned} M_{\mathrm{kr}}'' &=\sum_{h=1}^{\infty} B_h \left\{ I_0(h\lambda)\cos h\eta_0 t \right.\\ &\quad+\sum_{q=1}^{\infty} I_q(h\lambda)\cos[(h\eta_0+q\nu)t-q\zeta]\\ &\quad\left. +\sum_{q=1}^{\infty}(-1)^q I_q(h\lambda)\cos[(h\eta_0-q\nu)t+q\zeta] \right\}. \end{aligned} \tag{115} \]

Now, substituting (114) and (115) into (111) and adopting the notation

\[ h_0=\nu/\eta_0, \]

we obtain an already more concrete form of the expression for the torque \(M_{kp}\) as a function of time:

\[ \begin{aligned} M_{kp}=M_0 &+\sum_{h=1}^{\infty} A_h \bigg\{ I_0(h\lambda)\sin h\eta_0 t +\sum_{q=1}^{\infty} I_q(h\lambda)\sin\big[(h+qh_0)\eta_0 t-q\xi\big] \\ &\qquad\qquad +\sum_{q=1}^{\infty}(-1)^q I_q(h\lambda)\sin\big[(h-qh_0)\eta_0 t+q\xi\big] \bigg\} \\ &+\sum_{h=1}^{\infty} B_h \bigg\{ I_0(h\lambda)\cos h\eta_0 t +\sum_{q=1}^{\infty} I_q(h\lambda)\cos\big[(h+qh_0)t-q\xi\big] \\ &\qquad\qquad +\sum_{q=1}^{\infty}(-1)^q I_q(h\lambda)\cos\big[(h-qh_0)t+q\xi\big] \bigg\}. \end{aligned} \tag{116} \]

For each particular value of \(h_0\), and, of course, always an integer one (for two-stroke engines), of the order of the cyclic irregularity of shaft rotation, expression (116), after expanding the sums and the corresponding regrouping of terms, can be brought to the form:

\[ M_{kp}=\overline{M}_0+\sum_{h=1}^{\infty}\overline{A}_h\sin h\eta_0 t+\sum_{h=1}^{\infty}\overline{B}_h\cos h\eta_0 t \tag{117} \]

or

\[ M_{kp}=\overline{M}_0+\sum_{h=1}^{\infty}\overline{M}_h\sin(h\eta_0 t+\varepsilon_h) \tag{118} \]

where

\[ \overline{M}_h=\sqrt{\overline{A}_h^{\,2}+\overline{B}_h^{\,2}}, \tag{119} \]

\[ \varepsilon_h=\operatorname{arctg}\frac{\overline{B}_h}{\overline{A}_h}. \tag{120} \]

The quantities entering here, \(\overline{A}_h\), \(\overline{B}_h\), \(\overline{M}_h\), are the amplitudes of the monoharmonic components, already in time \(t\), of the torque \(M_{kp}\) on the engine shaft¹.

\[ h=\frac{1}{2},\,1,\,1\frac{1}{2},\,2\ldots \]

All \(\overline{A}_h\), \(\overline{B}_h\), and \(\overline{M}_h\) in formulas (117)—(120) are, evidently, complicated functions of all the \(A_h\) and \(B_h\) entering series (1), and also depend on \(h\), \(h_0\), \(\lambda\), and \(\delta\).

A detailed analysis of series (116) in general form presents considerable difficulties and cannot give sufficiently transparent results.

It was shown above that the effect of frequency modulation due to \(\delta\), for each harmonic in \(\alpha\) component \(M_h\) of the torque \(M_{\mathrm{cr}}\) on the engine shaft, increases with an increase in the ratio \(\eta_h/\nu\). In two-stroke engines the smallest possible frequency \(\nu\) of cyclic oscillations of the shaft speed over the course of one revolution is, evidently, equal to the mean angular speed of its rotation, i.e.,

\[ \nu=\eta_0, \]

and, consequently,

\[ h_0=1. \]

Let us consider this case, which is of particular interest to us, when dynamic damping due to the nonuniformity of shaft rotation proves to be the most effective.

Putting \(\nu=\eta_0\) in (114), after a series of not especially complicated, but rather cumbersome, transformations, we obtain:

\[ M_{\mathrm{cr}}' = \sum_{h=1}^{\infty} A_h \sin h[\eta_0 t+\lambda \sin(\eta_0 t-\zeta)] = \overline{A}_0 + \sum_{h=1}^{\infty} \overline{A}_h' \sin h\eta_0 t + \sum_{h=1}^{\infty} \overline{B}_h' \cos h\eta_0 t, \tag{121} \]

where:

\[ \overline{A}_0 = \sum_{p=1}^{\infty} (-1)^p A_p I_p(p\lambda)\sin p\zeta, \tag{122} \]

\[ \overline{A}_h' = \sum_{p=1}^{\infty} A_p \left[ I_{h-p}(p\lambda)\cos(h-p)\zeta - (-1)^{h+p} I_{h+p}(p\lambda)\cos(h+p)\zeta \right], \tag{123} \]

\[ \overline{B}_h' = \sum_{p=1}^{\infty} A_p \left[ - I_{h-p}(p\lambda)\sin(h-p)\zeta + (-1)^{h+p} I_{h+p}(p\lambda)\sin(h+p)\zeta \right]. \tag{124} \]

In analogous fashion, from (115), for \(\nu=\eta_0\):

\[ M_{\mathrm{cr}}' = \sum_{h=1}^{\infty} B_h \cos h[\eta_0 t+\lambda \sin(\eta_0 t-\zeta)] = \overline{B}_0 + \sum_{h=1}^{\infty} \overline{A}_h'' \sin h\eta_0 t + \sum_{h=1}^{\infty} \overline{B}_h'' \cos h\eta_0 t. \tag{125} \]

Here

\[ \bar{B}_0=\sum_{p=1}^{\infty}(-1)^p B_p I_p(p\lambda)\cos p\zeta, \tag{126} \]

\[ \bar{A}_h^{\prime\prime} = \sum_{p=1}^{\infty} B_p \left[ I_{h-p}(p\lambda)\sin (h-p)\zeta + (-1)^{h+p} I_{h+p}(p\lambda)\sin (h+p)\zeta \right], \tag{127} \]

\[ \bar{B}_h^{\prime\prime} = \sum_{p=1}^{\infty} B_p \left[ I_{h-p}(p\lambda)\cos (h-p)\zeta + (-1)^{h+p} I_{h+p}(p\lambda)\cos (h+p)\zeta \right]. \tag{128} \]

Now, substituting all this into the general expression (111), we obtain, as in (117), the true harmonic spectrum of the torque \(M_{\mathrm{cr}}\):

\[ M_{\mathrm{cr}} = M_0+ \sum_{h=1}^{\infty} A_h \sin h\left[\eta_0 t+\lambda \sin(\eta_0 t-\zeta)\right] + \]

\[ + \sum_{h=1}^{\infty} B_h \cos h\left[\eta_0 t+\lambda \sin(\eta_0 t-\zeta)\right], \]

or

\[ M_{\mathrm{cr}} = \bar{M}_0+ \sum_{h=1}^{\infty}\bar{A}_h \sin h\eta_0 t + \sum_{h=1}^{\infty}\bar{B}_h \cos h\eta_0 t, \tag{117} \]

where, specifically:

\[ \bar{M}_0 = M_0+\bar{A}_0+\bar{B}_0 = \]

\[ = M_0+ \sum_{p=1}^{\infty} (-1)^p I_p(p\lambda) \left[ A_p \sin p\zeta+B_p \cos p\zeta \right] \tag{129} \]

\[ \bar{A}_h = \bar{A}_h^{\prime}+\bar{A}_h^{\prime\prime} = \]

\[ = \sum_{p=1}^{\infty} \left\{ \left[ A_p\cos (h-p)\zeta+B_p\sin (h-p)\zeta \right] I_{h-p}(p\lambda) + \right. \]

\[ \left. + \left[ -A_p\cos (h+p)\zeta+B_p\sin (h+p)\zeta \right] (-1)^{h+p} I_{h+p}(p\lambda) \right\} \tag{130} \]

\[ \bar{B}_h = \bar{B}_h^{\prime}+\bar{B}_h^{\prime\prime} = \]

\[ = \sum_{p=1}^{\infty} \left\{ \left[ -A_p\sin (h-p)\zeta+B_p\cos (h-p)\zeta \right] I_{h-p}(p\lambda) + \right. \]

\[ \left. + \left[ A_p\sin (h+p)\zeta+B_p\cos (h+p)\zeta \right] (-1)^{h+p} I_{h+p}(p\lambda) \right\} \]

Finally, these same expressions could also have been written in the more general form (108), with the values \(\overline{M}_h\) and \(\varepsilon_h\) entering here being determined by formulas (119) and (120).

Formulas (129)--(131) are considerably simplified in the particular case when the phase angle of the cyclic nonuniformity is

\[ \zeta = 0. \]

Under this condition, in expression (117):

\[ \overline{M}_0 = M_0 + \sum_{p=1}^{\infty}(-1)^p B_p I_p(p\lambda), \tag{132} \]

\[ \overline{A}_h = \sum_{p=1}^{\infty} \left[ I_{h-p}(p\lambda) - (-1)^{h+p} I_{h+p}(p\lambda) \right] A_p, \tag{133} \]

\[ \overline{B}_h = \sum_{p=1}^{\infty} B_p \left[ I_{h-p}(p\lambda) + (-1)^{h+p} I_{h+p}(p\lambda) \right]. \tag{134} \]

Approximate calculations show that, for

\[ \lambda = \frac{\delta}{2} < 1 \quad \text{and} \quad h > 1 \]

(the first of these conditions is certainly always fulfilled in practice), the second terms under the summation sign on the right-hand side of expressions (130) and (131) may be neglected without appreciable error, and they may be written in the simpler form:

\[ \overline{A}_h = \sum_{p=1}^{\infty} \left[ A_p \cos (h-p)\zeta + B_p \sin (h-p)\zeta \right] I_{h-p}(p\lambda), \tag{135} \]

\[ \overline{B}_h = \sum_{p=1}^{\infty} \left[ - A_p \sin (h-p)\zeta + B_p \cos (h-p)\zeta \right] I_{h-p}(p\lambda), \tag{136} \]

since, for moderate \(\lambda\) and \(h>1\), the Bessel functions of the first kind always satisfy the inequality

\[ I_{h-p}(p\lambda) \gg I_{h+p}(p\lambda). \]

Only for \(h=1\) (the order \(h\) of the harmonic under investigation is equal to the order \(h_{\omega}\) of the frequency modulation), for a large value of \(\delta\), is it necessary to use the exact expressions (130) and (131).

To reveal the concrete relation between the degree of nonuniformity of shaft rotation \(\delta\) and the true harmonic spectrum of the torque \(M_{kr}\), we carried out detailed calculations by formulas (129)--(136) for one single-cylinder two-stroke diesel engine with a mean indicated pressure \(p_i = 7.0\ \mathrm{kg}/\mathrm{cm}^2\).

The amplitudes of the harmonic components (referred to \(1\ \mathrm{cm}^2\) of piston area and \(1\ \mathrm{cm}\) of crank radius) of the torque produced on the shaft of this engine by the gas pressure are given in the table (see below).

\(h\)	\(M_h\), kg·cm	\(A_h\), kg·cm	\(B_h\), kg·cm
1	4.60	4.29	+1.65
2	4.10	4.08	−0.36
3	2.84	2.80	−0.39
4	1.76	1.67	−0.54
5	1.06	0.98	−0.41
6	0.64	0.57	−0.30
7	0.40	0.34	−0.21
8	0.26	0.22	−0.15
9	0.18	0.14	−0.11
10	0.10	0.08	−0.07
11	0.07	0.05	−0.05
12	0.06	0.04	−0.04

Further, as before, we assume that the angular velocity of rotation of the crankshaft varies according to the multiharmonic law (82-a), with

\[ \nu = \eta_0 . \]

Taking concrete numerical values for the degree of nonuniformity \(\delta\) and the phase angle \(\zeta\), by formulas (129)—(136) we calculate the components \(\overline{M}_h\), \(\overline{A}_h\), \(\overline{B}_h\) of the true harmonic, in \(t\), spectrum \(M_{\mathrm{cr}}\), and, by (120), the phase angles \(\bar{\varepsilon}_h\).

Now we compute the ratio

\[ \mu_h = \frac{\overline{M}_h}{M_h}, \]

which characterizes the concrete influence of the nonuniformity of shaft rotation (or, more precisely, of the crankshaft) on each moment \(M_h\), harmonic in \(\alpha\), in the expansion (1). The quantity \(\mu_h\), with a certain degree of convention, may be called the coefficient of dynamic damping due to \(\delta\).

The results of our calculations are presented in Figs. 19—22.

In Fig. 19\(^{1}\) the law of variation of the coefficient of dynamic damping is shown,

\[ \mu_1 = \frac{\overline{M}_1}{M_1}, \]

for the first-order harmonic moment, as a function of \(\delta\) and \(\zeta\).

In Fig. 20 the dependence of \(\mu_1\) on the phase angle \(\zeta\) is presented in greater detail, for certain concrete values of the degree of nonuniformity \(\delta\).

The diagram in Fig. 21 characterizes the relation of the coefficient of dynamic damping \(\mu_h\) to the degree of nonuniformity \(\delta\) for various \(h\).

Here the conclusion made earlier is illustrated: that the effect of dynamic damping is strengthened with an increase in the ratio \(h/h_0\), i.e., with a relative slowing of the oscillations of the rotational speed of the machine shaft.

Finally, Fig. 22 gives the relation between \(\delta\), \(\zeta\), and the true phase angle \(\bar{\varepsilon}_1\) of the harmonic (in time) moment \(M_1 \sin(\eta_0 t + \bar{\varepsilon}_1)\) of the first order.

Throughout, we quite deliberately carried out the calculations even for very large values of \(\delta\), values hardly possible in practice, with the aim of revealing most sharply and distinctly the dynamic effect of nonuniformity of shaft rotation.

\(^{1}\) In Figs. 19, 20, and 21 the coefficient of dynamic damping \(\mu\) is denoted by \(D\).

Analysis of Figs. 19–22 and all the preceding calculations provides sufficient grounds for the following conclusions about the effect of $\delta$.

Fig. 19

In the figure: two-stroke diesel; $p_i = 7\,\mathrm{kg/cm^2}$; $\nu = \eta_0$; $h = 1$; ordinate $D_\nu = \left|\dfrac{K_\nu}{K}\right|$; abscissa $\delta$.

The periodic nonuniformity of shaft rotation, disturbing the exact direct proportionality between the crank angle of rotation $\alpha$ and time-

Fig. 20

In the figure: two-stroke diesel; $p_i = 7\,\mathrm{kg/cm^2}$; $\nu = \eta_0$; $h = 1$; ordinate $D_\nu = \left|\dfrac{K_\nu}{K}\right|$; abscissa $\zeta$. Curves: $\delta = 1.0$, $\delta = 0.2$, $\delta = 0.1$.

Fig. 21

Fig. 22

by \(t\), through the effect of frequency (and, with respect to the inertia harmonics, also mixed) modulation, transforms the entire diagram of torques \(M_{kr}=f_0(\alpha)\) on the engine shaft. This transformation, revealed, for example, when the diagram is reconstructed from the abscissa axis \(\alpha\) to the abscissa axis \(t\), leads to a change in all harmonic components of the torque, both in amplitude and in phase. Thus, for example, the moment

\[ M_h \sin(h\alpha+\varepsilon_h) \]

passes into

\[ \overline{M}_h \sin(h\eta_0 t+\overline{\varepsilon}_h). \]

The influence of the nonuniformity of shaft rotation on its forced vibrations may, to a certain extent, be characterized by the coefficient of dynamic damping

\[ \mu_h=\frac{\overline{M}_h}{M_h}. \]

In a linear system with velocity damping, the amplitude of forced vibrations at resonance is proportional to the amplitude of the resonating harmonic (if, of course, the effect of all the other, nonresonating harmonics is neglected). For \(\delta=0\), a harmonic moment \(M_h\) acts on the system, producing, under resonance conditions, forced vibrations with amplitude \(\Psi_{res}\). In the case of nonuniform shaft rotation (\(\delta>0\)), however, the resonance condition is not disturbed, but the resonating (“carrier”) harmonic already has the amplitude

\[ \overline{M}_h=\mu_h M_h \]

and, consequently, gives forced vibrations of the system with amplitude

\[ \overline{\Psi}_{res}=\mu_h\Psi_{res}. \]

Thus, the coefficient \(\mu_h\) takes into account the dynamic influence of the nonuniformity of shaft rotation on the amplitudes of its resonant vibrations. This justifies the name we have adopted for \(\mu_h\) as the coefficient of dynamic damping due to \(\delta\).

The coefficient \(\mu_h\) is different for each torque harmonic and depends not only on the magnitude of the degree of nonuniformity \(\delta\), but also on a number of other factors, namely on:

a) the ratio \(h/h_0\) of the order of the harmonic moment \(M_h\) under consideration to the order of the cyclic oscillations of the shaft speed (more precisely, to the order of the principal harmonic component of the curve of the instantaneous angular velocity of rotation of the corresponding crank);

b) the form and character of the engine indicator diagram or, more precisely, the character, harmonic in \(\alpha\), of the torque spectrum \(M_{kr}\), the ratio of the amplitudes of the individual components \(M_h\) of this spectrum, their phases, etc.;

c) the phase angle of the cyclic nonuniformity of shaft rotation \(\zeta\).

According to Fig. 22, the nonuniformity of shaft rotation has the strongest effect on harmonic (in \(\alpha\)) moments \(M_h\) of high orders. This circumstance fully agrees with the conclusions drawn earlier concerning the increase in the effectiveness of the influence of \(\delta\) on the structure of the spectrum \(M_{kr}\) as the frequency \(\nu\) decreases in comparison with \(h\eta_0\).

The dependence of the coefficients \(\mu_h\) on the characteristics of the spectrum \(M_{kr}\) in \(\alpha\) is explained by the fact that, when frequency modulation acts, for example, on a single monoharmonic moment in \(\alpha\),

\[ M_h \sin(h\alpha+\varepsilon_h), \tag{a} \]

this moment singles out from itself a whole series of satellites located on both sides of the carrier harmonic. Therefore, in the final result, each moment harmonic in \(t\),

\[ \overline{M}_h \sin (h\eta_0 t-\overline{\varepsilon}_h), \]

in the true harmonic spectrum of \(M_{kp}\) (formula 116) includes not only the carrier harmonic separated out from moment \((a)\), but also all satellites of other components \(M_h\) of the torque that coincide with it in frequency.

Concrete calculations show that a change of the mean indicated pressure \(p_i\) in the engine within moderate limits:

\[ p_i=5 \div 9 \ \text{kg}/\text{cm}^2 \]

leads only to small corrections in the graphs in Figs. 19–22, constructed for \(p_i=7.0\). This fact is not in contradiction with the conclusion given above concerning the relation between the coefficient \(\mu\) and the form of the indicator diagram, since a moderate change in \(p_i\) does not introduce substantial changes into the general pattern of the diagram of tangential forces for the given engine.

According to Fig. 21, the coefficient \(\mu_h\), depending on the phase \(\zeta\), may turn out to be either less than or greater than unity. But the value

\[ \mu_h>1 \]

characterizes an increase in the amplitudes of forced resonant vibrations, simultaneously with a deterioration in the uniformity of rotation of the shaft.

Earlier we established such a possibility of amplification of vibrations due to an increase of \(\delta\) only with respect to inertial harmonics, which for \(\delta>0\) acquired mixed modulation. Here, however, we find the conditions and causes for the occurrence of the phenomenon of negative damping also in a system subjected to the action only of “gas” torques.

As noted above, the amplitude of each moment harmonic in \(t\), \(\overline{M}_h\), includes not only the amplitude of the “carrier” part of the harmonic in \(\alpha\) of the moment \(M_h\), but also the amplitudes of a series of satellites separated, for \(\delta>0\), from the composition of other harmonic in \(\alpha\) components of the torque \(M_{kp}\). Thus, for example, under the conditions of “resonance” of the moment \(M_h\), that is, when

\[ \eta_h=h\eta_0=\omega_0 \]

the resonant vibrations will be produced not only by the carrier part of the “resonating” \(M_h\), but also by the corresponding satellites of other components of the moment \(M_p\), named above. The amplitude of the sum of these synchronous vibrations, of course, depends on the phase relation of all simultaneously resonating harmonics, that is, above all, on the phase \(\zeta\).

It is now evident that the coefficient of dynamic damping \(\mu_h\) may assume values \(>1\), if a whole series of other harmonic in \(\alpha\) components \(M_{kp}\), not resonating in the usual sense of the word, come to the aid of the “resonating”¹ harmonic \(M_h\) through their satellites that are resonating at present.

The phase angle of the frequency modulation \(\zeta\) first appeared for us in the expressions for the true angle of rotation of the crank in the case of simple monoharmonic nonuniformity of its rotation (formula 109).

The approximate determination of the specific value \(\zeta\), in each individual case, presents no especially great difficulties.

It is obvious that the second term in (109) characterizes an oscillatory process superposed on the uniform rotation of the shaft and caused by the action of the moment

\[ \overline{M}_\nu \sin(\nu t+\overline{\varepsilon}_{\mathrm{h}}). \]

For a comparatively small value of the degree of nonuniformity \(\delta\), in the majority of practical cases, in a first approximation one may neglect the phenomenon of automodulation and take:

\[ \overline{M}_\nu = M_\nu, \]

\[ \overline{\varepsilon}_{\mathrm{h}}=\varepsilon_{\mathrm{h}}, \]

where the quantities \(M_\nu\) and \(\varepsilon_{\mathrm{h}}\) are taken from the results of the ordinary harmonic analysis of the curve \(M_{kr}=f_0(\alpha)\).

But, as always, forced oscillations lag behind the exciting force by a certain angle \(\sigma\). Consequently, in our case the crankshafts will oscillate according to the law

\[ \lambda \sin(\nu t+\varepsilon_{\mathrm{h}}-\sigma). \]

Comparison of this expression with the right-hand side of equation (109) gives

\[ \zeta=\sigma-\varepsilon_{\mathrm{h}}. \tag{137} \]

The angle \(\sigma\) entering here, as usual, depends on the magnitudes of the actual velocity damping forces in the system and on the tuning coefficient

\[ \gamma_\nu=\frac{\nu}{\omega_0} \]

and can be calculated from known formulas for each specifically assigned value of \(\gamma_\nu\). Now, with already known \(\zeta\) and \(\delta\), for example from graphs analogous to Fig. 22, the corresponding value of the coefficient of dynamic damping \(\mu_{\mathrm{h}}\) is also determined for all harmonic orders \(h\) of interest to us.

It is obvious that the angle \(\zeta\), depending on a number of specific circumstances, can acquire the most varied values and, accordingly, change not only the magnitude but also the sign of the dynamic damping of the vibrations of the system, caused by the nonuniformity of rotation of the engine shaft. Therefore, contrary to the usual views, we have no grounds whatever for excluding the possibility that certain forced vibrations of the shaft may be amplified with increasing \(\delta\), owing to the effect of negative dynamic damping.

7. Conclusions

1. Periodic oscillations of the rotational speed of the shaft over the course of each of its revolutions affect the forced vibrations of the shaft, essentially only through the “deformation” of the diagram of torques. In this case the harmonic (with respect to \(t\)) moments with amplitudes \(\overline{M}_{\mathrm{h}}\) and phases \(\overline{\varepsilon}_{\mathrm{h}}\) differ from those \(M_{\mathrm{h}}\) and \(\varepsilon_{\mathrm{h}}\) which are determined by harmonic analysis of the ordinary diagram of tangential forces constructed with respect to the crankshaft angle of rotation \(\alpha\), and not with respect to time \(t\), as should in essence be done when properly applying to a linear system the general method of expanding a given force function in a Fourier series.

1. The transition from \(M_h\) and \(\varepsilon_h\) to \(\overline{M}_h\) and \(\overline{\varepsilon}_h\), i.e., the harmonic analysis of the curve \(M_{cr}\) with respect to time, is performed by the oscillating system itself, which responds (under the conditions of its linearity) precisely to the harmonic, with respect to \(t\), components of \(M_{cr}\). The explanation of the modulation effect should, of course, be sought not in any mathematical operations, but in the most fundamental properties of a linear oscillatory system as a harmonic resonator.

2. As a criterion of the intensity of the influence of \(\delta\) on the forced resonant vibrations of a shaft caused, for example, by harmonic moments of the \(h\)-th order, one may take the coefficient of dynamic damping

\[ \mu_h=\frac{\overline{M}_h}{M_h}=\frac{\overline{\Psi}_{res}}{\Psi_{res}}, \]

which characterizes the change in the amplitude of the forced oscillations of the system \(\overline{\Psi}_{res}\) due to \(\delta>0\), in comparison with the amplitude \(\Psi_{res}\) that would be established in the system for \(\delta=0\).

1. Contrary to the thus far only theory for taking into account the influence of \(\delta\), proposed by Ker Wilson and Manley [13, 14], this coefficient of dynamic damping is a very complicated function not only of \(\delta\) and of the ratio \(\frac{h}{h_0}\), but also of a whole series of other factors.

2. Contrary to generally accepted views, it must be recognized that periodic nonuniformity of shaft rotation does not eliminate the possibility of the development of stationary resonant vibrations in the system and, evidently, cannot always lead to a reduction in the amplitudes of forced oscillations. Cases are not excluded in which the nonuniformity of shaft rotation will stimulate the development of forced resonant vibrations.

3. These circumstances, and all our preceding conclusions, show that the theory of resonance suppression in a system having \(\delta>0\), proposed by Ker Wilson and Manley, is completely inconsistent with reality. It does not reflect the essence of dynamic damping or the physical picture of this phenomenon. Its application can lead only to an erroneous assessment of the role and relative significance of dynamic damping of shaft vibrations in internal-combustion-engine installations and, in general, in installations with piston engines.

4. Of course, the theory proposed by us for taking into account the effect of nonuniformity of shaft rotation is also not perfect. It is evident that with an increase in the nonuniformity of shaft rotation, not only is the modulation of the torques intensified, but the actual damping in the system also increases owing to additional energy losses in the shaft, increased impacts in the bearings, etc. But all these losses depend on the individual features of each particular power installation and can hardly be generalized by any theoretical analysis or experiment. Nor is Manley’s formula able to take them into account.

5. The complicated dependence of the coefficient \(\mu_h\) on a whole series of factors apparently excludes the possibility of obtaining any sufficiently reliable, theoretical or experimental universal formula for calculating the dynamic damping of vibrations of shafts of installations having \(\delta>0\).

But, as approximate calculations show, for small, usual values of \(\delta\):

\[ \delta < \frac{1}{20}\div\frac{1}{40} \]

and

\[ \frac{h}{h_0}<6\div10 \]

one may completely neglect the effect of dynamic damping caused by nonuniform rotation of the shaft, since under these conditions the oscillations \(\mu_h\) near unity, depending on \(h\) and \(\zeta\), are very small. Mancy’s formula, however, gives here an extremely exaggerated estimate of the effect \(\delta\).

In the rarer cases encountered in practice, when

\[ \delta > \frac{1}{20} \]

and

\[ h > 6 \div 10, \]

for a detailed accounting of the influence of \(\delta\), it is advisable to carry out a detailed analysis and compute \(\mu_h\) in accordance with the considerations set forth above. A similar analysis is also necessary in experimental investigations of vibrations aimed at clarifying the role of individual types of damping in the overall balance of damping forces in internal-combustion-engine installations.

1. Finally, it should be noted that, for the values of \(\delta\) and \(h/h_0\) usual in practice, the effect of nonuniform shaft rotation may be completely neglected in all calculations of the width of forbidden resonance zones in the range of the engine’s operating speeds.

Only for very large \(\delta\) and \(h/h_0\) is it necessary to shift the boundaries of each of these zones accordingly, proceeding from a detailed accounting of the dynamic effect of the nonuniformity of shaft rotation on the basis of the theory set forth above.

REFERENCES

1. Zhitomirsky B. K.—Diesel Engineering, 1937, Nos. 2, 4.

2. Kuzmin R.—Bessel Functions, ONTI, 1935.

3. Lur’e I. A.—Torsional Vibrations in Diesel Installations, Voenmorizdat, 1940.

4. Whittaker E. and Watson G.—A Course of Modern Analysis, Vol. II, GTTI, 1934.

5. Nechaev V. K.—Dynamic Damping of Vibrations and Mancy’s Formula (manuscript).

6. Nechaev V. K.—Theoretical Torsiograms for ICE Shafts, Izvestiya TII, Vol. 58 (1937), Issue 2.

7. Rytov S.—Modulated Oscillations and Waves. Proceedings of the Physical Institute of the USSR Academy of Sciences, Vol. II (1940), Issue 1.

8. Rytov S. M.—Uspekhi fizicheskikh nauk, Vol. XXIX (1946), Issues 1–2.

9. Papaleksi N. D.—Uspekhi fizicheskikh nauk, Vol. XXXI (1917), Issue 4.

10. Serensen S. V. and Tetelbaum I. M.—Dynamic Strength in Mechanical Engineering, Mashgiz, 1945.

11. Gogin A.—Shipbuilding, 1940, No. 12.

12. Gogin A.—Collection of Reports on the Dynamic Strength of Machine Parts, Academy of Sciences of the USSR, 1946.

13. Kjaer V. A.—Vibrations de torsion dans les vilebrequins des moteurs Diesel, Bulletin Technique du Bureau Veritas, 1931, May.

14. Mancy I.—Théorie des oscillations de torsion des lignes d’arbres de moteurs à combustion interne, Bull. Techn. du Bureau Veritas, 1931, Août.

15. Lewis F.—Trans. Soc. Nav. Arch. and Mar. Eng., 1925.

16. Tuplin W. A.—Torsional Vibrations, New-York, 1934.

17. Ker Wilson W.—Practical Solution of Torsional Vibration Problems, New-York, 1935.

18. Wydler H.—Drehschwingungen in Kolbenmaschinenanlagen, Berlin, 1922.

Noted Misprints

Page	Line	Printed	Should be
7	3 from bottom	result	results
9	2 from top	$\alpha=h\eta_0 t$	$\alpha=\eta_0 t$
10	1 from bottom	$\dot{\delta}_s \cong b_i$	$\dot{\delta}_s \cong \dot{\delta}_i$
19	6 from top	$\displaystyle +\sum_{q=0}^{\infty}(-1)^q J_{2q}(\xi)+$	$\displaystyle +\sum_{q=0}^{\infty}(-1)^q J_{2q}(\xi)+$
19	7 from top	$-J_{2q+3}(\xi)]$	$-J_{2q+3}(\xi)]$
21	4 from bottom	$[J_0(\xi_1)\sin(\eta_h t+$	$[J_0(\xi_1)\sin\eta_h t+$
22	7 from top	$\sin(\eta_h-2\nu)t$	$\sin(\eta_h-2\nu)t+\ldots$
25	3 from bottom	$\displaystyle \left[1+\frac{\dot{\delta}_s}{2}\cos(\nu t-\zeta)\right]$	$\displaystyle \left[1+\frac{\dot{\delta}_s}{2}\cos(\nu t-\zeta)\right]^2$
27	3 from top	$\displaystyle C=-\frac{\dot{\delta}^{s2}}{16}\{$	$\displaystyle C=\frac{\dot{\delta}_s^{\,2}}{16}\{$
30	25 from top	varying	decreasing
31	2 from top	(69)	(65)
31	13 from bottom	$m_9$	$m_3$
34	12 from top	$\displaystyle \sum_{p=1}^{\infty} M_p\sin(pa-\varepsilon_p')$	$\displaystyle \sum_{p=1}^{\infty} M_p\sin(pa+\varepsilon_p')$
38	3 from bottom	terms	term
42	20 from top	$k\ k_c$	$k/k_c$
47	14 from top	$\displaystyle \Psi=\sum_{q=1}^{\infty}\Phi_{q-1}$	$\displaystyle \Psi=\sum_{q=1}^{\infty}\Phi_{q-1}$
48	12 from top	$\displaystyle M_h J_1\left(\frac{\dot{\delta}}{2}\right)$	—
51	5 from top	$\displaystyle \frac{\dot{\delta}}{2}\sin\eta_k t$	$\displaystyle \frac{\dot{\delta}}{2}\sin\eta_h t$
52	23 from top	$\Phi_0$	$\Phi_{0res}$
54	5 from top	accept	recognize
55	17 from top	$\displaystyle \sum_{q=1}^{\infty}(-1)_a$	$\displaystyle \sum_{q=1}^{\infty}(-1)^q$
57	2 from bottom	$M_{kp}'$	$M_{kp}''$
58	3 from bottom	$\overline{B}_h=\overline{B}_h'-\overline{B}_h''$	$\overline{B}_h=\overline{B}_h'+\overline{B}_h''$
64	4 from top	$\overline{M}_h'\sin(h\eta_0 t-\overline{\varepsilon}_h)$	$\overline{M}_h'\sin(h\eta_0 t+\overline{\varepsilon}_h)$
79	4 from bottom	$+1{,}0229$	$+1{,}0223$
87	2 from bottom	$(\varphi_{i\,otn}-\varphi_{i+1,\,otn}^{2})$	$(\varphi_{i\,otn}-\varphi_{i+1,\,otn})^{2}$
90	6 from top	$M_h'=$	$M_r'=$
92	5 from top	$+c_{111}'\varphi_{1\,otn}=$	$+c_{111}'\varphi_{1\,otn}^{2}=$

Izv. TPI, vol. 61, issue 2.

1. We use quotation marks here, taking into account that the moment \(M_h\) has a periodically variable frequency. ↩↩↩↩

2. We assume here a steady operating regime of the engine. ↩