On Relationships Between the Damping Characteristics of Torsional Vibrations and Certain Parameters of a Powerplant

V. K. Nechaev

In the author’s preceding article [1], a strong dependence was established of the generalized damping coefficients of torsional vibrations of the shaft of a powerplant with a piston engine on the speed and load operating conditions of the engine.

It can also be shown that these generalized coefficients \(k_0\) and \(a_0\) vary when the elastic \((c)\) and inertial \((\theta)\) characteristics of the powerplant change.\(^1\)

For this purpose it is necessary to carry out a detailed analysis of the influence of the values of the stiffnesses \(c_{i,i+1}\) of shaft sections and the moments of inertia \(\theta_i\) of the powerplant masses on the coefficient \(a\) entering into the expressions

\[ k_0 = k(1+a), \tag{1} \]

\[ a_0 = a\left(1+\frac{1}{a}\right) \tag{2} \]

for the generalized coefficient of linear damping \(k_0\) and the generalized hysteretic coefficient \(a_0\).

Here \(k\) is the actual coefficient of viscous (linear) damping in the mechanism of each engine cylinder;
\(a\) is the hysteresis constant for the material of the installation shaft.

For an installation whose entire shaft is made of one grade of steel, the coefficient \(a\) is determined from the approximate equation [1]:

\[ a(1+\alpha)^{q-2} = \frac{\nu\lambda^{q-2}}{(\mu k\omega)^{q-1}}\,M_h^{q-2}. \tag{3} \]

Thus, the coefficient \(a\) depends on a whole series of quantities: \(\lambda,\mu,\nu,\omega,k,\alpha,q,M_h\), of which the first four are closely connected with the values of the moments of inertia \(\theta_i\) and stiffnesses \(c_{i,i+1}\) of the reduced powerplant.\(^2\)

Here we shall confine ourselves to a brief investigation of questions concerning the dependence of the generalized coefficients \(k_0\) and \(a_0\) on the position of the node of shaft vibrations, on the frequency of free vibrations of the shaft \(\omega\), and on the absolute dimensions of the powerplant.

\(^1\) An experimental investigation of the dependence of the generalized damping coefficients on the magnitude of the moment of inertia of one of the installation masses is the subject of A. T. Bolgov’s article [4], published in this collection.

\(^2\) We retain here all the notation adopted earlier [1]. The necessary additional explanations are given below.

Dependence of the coefficients \(k_0\) and \(a_0\) on \(\varphi_{cp}\) and on the position of the shaft-oscillation node

Let us first consider the simplest case, when in a multi-cylinder engine only one of its cylinders is operating; in this cylinder a resonating¹ exciting harmonic moment \(M_h\) is produced, while all the remaining cylinders operate at idle and in them \(M_h = 0\).

Here, instead of the general expression for the coefficient \(\lambda\) [1]:

\[ \lambda = \pi \sum^{\to} \varphi_i \tag{4} \]

we have

\[ \lambda = \pi \varphi_x, \]

where \(\varphi_x\) is the relative amplitude of the free oscillations of the reduced mass of the operating cylinder.

Now, instead of (3), one may write

\[ \alpha(1+\alpha)^{q-2} = \frac{\nu a}{(\mu k\omega)^{q-1}} (\pi\varphi_x)^{q-2} M_h^{q-2}. \tag{5} \]

Whence, in the particular case \(q = 3\):

\[ \alpha = \sqrt{ \frac{1}{4} + \frac{\nu a}{\mu k\omega}\,\pi\varphi_x M_h } - \frac{1}{2}. \tag{6} \]

The coefficients \(\mu, \nu, a\) entering here, as well as the frequency \(\omega\), are characteristics of the free oscillations of the shaft of the installation and do not depend on the number of the operating cylinder or on the number of cylinders brought into operation².

When any other cylinder is brought into operation instead of the given cylinder, in expressions (5) and (6) only the quantity \(\varphi_x\) changes; it depends, obviously, on the distance of the operating cylinder from the node of the shaft oscillations (Fig. 1a).

Consequently, in the case considered by us of the operation of one cylinder, the values of the generalized damping coefficients \(k_0\) and \(a_0\) of the engine installation depend on the amplitude \(\varphi_x\), that is, on the number of the operating cylinder or on the distance of the point of application of the exciting moment \(M_h\) from the node of shaft oscillations.

The coefficient \(k_0\) decreases, and the coefficient \(a_0\) increases as \(\varphi_x\) decreases. In this case the dependence of the coefficient \(k_0\) on \(\varphi_x\) proves to be entirely analogous to the dependence of \(k_0\) on the cylinder load or on the amplitude \(M_h\), considered by us earlier [1].

Fig. 1

It is obvious that in installations with multicylinder engines analogous relationships must exist also when all cylinders are operating. Here, as a characteristic of the mean distance of the points of application of the exciting moments \(M_h\) from the vibration node, one may use the mean relative amplitude of the free vibrations of the reduced masses of the engine cylinders

\[ \varphi_{cp}=\frac{1}{m}\sum_{1}^{m}\varphi_i, \]

where \(\sum \varphi_i\) is the algebraic sum of all these amplitudes,
\(m\) is the number of engine cylinders.

Now, for the principal orders \(h\), instead of (4) we have

\[ \lambda=\pi m \varphi_{cp} \]

and instead of (5) and (6)

\[ \alpha(1+\alpha)^{q-2} = \frac{(\pi m\varphi_{cp})^{q-2}}{\mu^{q-1}} \cdot \frac{\nu a}{(k\omega)^{q-1}}M_h^{q-2}, \tag{7} \]

\[ \alpha= \sqrt{ \frac{1}{4} + \frac{\nu a}{\mu k\omega}\,\pi m\varphi_{cp}M_h } -\frac{1}{2}. \tag{8} \]

In the case where the vibration node is located far outside the engine, the relative amplitudes \(\varphi_i\) of all its cylinders are practically identical (Fig. 1b) and equal to unity. Therefore here

\[ \lambda \cong \pi m; \qquad \frac{(m\varphi_{cp})^{q-2}}{\mu^{q-1}}\cong 1 \]

and the coefficient \(\alpha\) is determined from the equation

\[ \alpha(1+\alpha)^{q-2} = \frac{\pi^{q-2}\nu a}{(k\omega)^{q-1}}M_h^{q-2} \]

or, for \(q=3\), from

\[ \alpha= \sqrt{ \frac{1}{4} + \frac{\pi \nu a}{k\omega}M_h } -\frac{1}{2}. \]

If, however, the vibration node is located in the middle of the engine (Fig. 1c), then

\[ \lambda \cong 0; \qquad \varphi_{cp}\cong 0 \]

and

\[ \alpha=0. \]

It is obvious that already in these two limiting cases the coefficients \(k_0\) and \(a_0\) will have different values.

A change in the quantity \(\varphi_{cp}\) in a powerplant may arise, for example, as the result of a displacement of the vibration node (and of the corresponding change in the curve of the shaft-vibration mode) when replacing the flywheel by another one with a different moment of inertia, when installing or removing counterweights on the crankshaft, when changing the length or diameter of the intermediate shaft between the engine and the flywheel, etc.

It is obvious that all such “structural” changes in the powerplant entail definite changes in the magnitude of the generalized damping coefficients \(k_0\) and \(a_0\). But the regularities of the change

these coefficients as functions of $\varphi_{cp}$ will be more complicated than was established above in analyzing the case of switching the entire engine load from one cylinder to another.

When the stiffness of some section of the shaft, or the moment of inertia of one of the installation masses, is changed, the character of the mode-shape curve of the free vibrations of the shaft will change, the node of vibration will shift, and the natural frequency of the shaft $\omega$ will become different. These changes in the mode-shape curve of the free vibrations and in the position of the node will lead not only to a change in the mean amplitude $\varphi_{cp}$ of the free vibrations of the engine masses, but also to new numerical values of the coefficients $\mu$ and $\nu$.

Thus, now in the right-hand sides of expressions (5) and (6) only the coefficients $k, a, q$ and, perhaps, the amplitude of the exciting moments $M_h$ will remain constant (if the change in $\omega$ is small and we are still considering only the operation of the engine at the critical number of revolutions).

A detailed investigation of the dependences of the coefficients $\lambda, \mu, \nu$, of the frequency $\omega$, and of the coefficient $\alpha$, which is of special interest to us, on the values of the stiffnesses $c_{i,i+1}$ of the shaft sections and the moments of inertia $\Theta_i$ of the masses of the powerplant is scarcely possible in general form. Cumbersome calculations are unavoidable here.

We shall confine ourselves here to a single example. In Figs. 2 and 3 a number of curves are presented for one powerplant consisting of a four-cylinder in-line engine and a flywheel (or generator). The curves were calculated for the case in which, in this installation, the value of the moment of inertia $\Theta_M$ of the flywheel (generator armature) is varied while the stiffnesses of all shaft sections and the moments of inertia $\Theta$ of the reduced masses of each engine cylinder remain unchanged. The computations were carried out on the basis of the formulas given above, and also in paper [1]. The initial parameters adopted for the calculation are indicated on the graphs. In the calculations it was assumed that the sources of damping in the installation under consideration are only the engine cylinders (linear damping, characterized by the coefficient $k$) and the shaft of the powerplant (hysteretic damping, characterized by the hysteresis constant $a$ and the exponent $q$). Generator damping was not taken into account. The amplitude $M_h$ of the exciting harmonic moments was taken as constant, independent of $\omega$.

In Fig. 2, as functions of the ratio $\dfrac{\Theta_M}{\Theta}$, there are shown the values of the coefficients $\mu, \nu$, as well as the mean relative amplitude $\varphi_{cp}$ of the vibrations of the engine masses, the frequency $\omega$ of the free vibrations of the shaft, and the distance $X$ from the vibration node to the middle of the engine.

Here we note the comparatively rapid growth of the coefficient $\nu$ as the ratio $\dfrac{\Theta_M}{\Theta}$ decreases. According to [1], this coefficient enters into the expression for the work absorbed by hysteretic resistances during a cycle of resonant vibrations,

\[ A_z = \nu a \Phi_{1abc}^{q}. \tag{9} \]

The growth of the coefficient $\nu$ is caused by an increase in the relative angles of twisting of the shaft and indicates an increase in the specific weight of hysteretic losses as the ratio $\dfrac{\Theta_M}{\Theta}$ decreases (excluding the case of very small values of this ratio).

In Fig. 3, for the same engine, there are shown the dependences on the ratio $\dfrac{\Theta_M}{\Theta}$ of the absolute vibration amplitudes $\Phi_{1abc}$ (of the first engine mass),

Fig. 2

Fig. 3

and \(\Phi_{5abc}\) (flywheel), the coefficient \(a\), and the generalized damping coefficients \(k_0\) and \(a_0\).

These graphs indicate that, in engine installations, there is a rather strong and complex dependence of the generalized coefficients \(k_0\) and \(a_0\) on such apparently purely design parameters as the mass moments of inertia of the installation.

Similar graphs could also be constructed for the case, for example, of varying the length or diameter of the intermediate shaft between the engine and the flywheel.

The above-mentioned relationship between the generalized linear damping coefficient \(k_0\) and the mean amplitude \(\varphi_{\mathrm{av}}\) of the free oscillations of the reduced engine masses was first noted by Shannon [6] in his experiments with a four-cylinder automobile engine driven in rotation by an electric motor. On the basis of these experiments Shannon gives a graph of the dependence of the quantity

\[ \frac{\theta\omega}{k_0} \]

on \(\varphi_{\mathrm{av}}\) (Fig. 4a).

For comparison, on the basis of Figs. 2 and 3 we have constructed a graph (Fig. 4b) for the four-cylinder engine with flywheel considered above as an example.

Fig. 4

As can be seen from our graph, the dependence of the coefficient \(k_0\) on \(\varphi_{\mathrm{av}}\) may have a considerably more complex character,

than Shannon established. The reason for this is the complex simultaneous change, together with \(\varphi_{cp}\), of the quantities \(\mu,\nu,\omega\) entering into (7) and (8).

In different engine installations, with identical values in them of \(\Theta\), \(\varphi_{cp}\), and \(\omega\), the values of the coefficients \(\mu\) and \(\nu\) may turn out to be different. The characteristics of the hysteretic properties of the shaft material may also differ—the constant \(a\) and the exponent \(q\). In attempts to carry out calculations of resonant vibrations of different installations by means of Shannon’s graph cited above, all these concrete circumstances will remain unaccounted for. The results of the calculations will be of low reliability, all the more so because not only the graph but also the entire method proposed by Shannon for calculating \(k_0\) through \(\varphi_{cp}\) takes absolutely no account of the influence on the coefficient \(k_0\) of the magnitude of the absolute amplitude of resonant oscillations \(\Phi_{1abs}\), or of the amplitude of the exciting moments \(M_h\).

All this gives grounds for regarding the dependence proposed by Shannon (Fig. 4a) only as an empirical expression of particular experimental data obtained on one experimental installation, with certain specific and not exactly accounted-for values of the coefficients \(\mu,\nu,a,q\), and for definite values of \(M_h\) that occurred in the experiments. It is clear that dependences of this kind cannot have any universal significance.

It is hardly possible that much more can be provided by the newer method, developed in England, for taking damping into account in engine installations, described in Dorey’s article [8]. Here it is proposed to choose the amplification coefficient at resonance \(\beta\) (in calculation according to K. Wilson) as a function of the magnitude of the equilibrium amplitude \(\varphi_0\) of the free end of the installation shaft, using for this purpose a universal graph relating \(\beta\) and \(\varphi_0\). On the whole, this method, constructed on fairly extensive experimental material for ship installations, deserves attention and detailed analysis and verification. However, one of its obvious shortcomings is the absence of consideration of the concrete hysteretic characteristics of the shaft material and of the possibly different specific weight of linear damping in the engine.

It may be thought that more reliable data on the laws governing the variation of damping in engine installations should be provided by the calculation and study of graphs similar to Figs. 2 and 3, computed and constructed for installations of the type of interest to us, and by a detailed comparison of these graphs with the results of experimental torsiographing.

On the choice of the exponent \(q\) in determining the generalized coefficient of hysteretic damping \(a_0\)

Above, in considering the question of the influence of the design parameters of an engine installation on the generalized damping coefficients \(k_0\) and \(a_0\), and also in article [1], it was assumed that the exponent \(q\) in the generalized expression for the entire damping work in an engine installation over a cycle of resonant oscillations

\[ A_{\mathrm{exc}} = A_k + A_h = \nu a_0 \Phi_{1abs}^{q} \tag{10} \]

was exactly equal to the actual hysteretic exponent \(q\) for the shaft material of the specific installation under consideration.

Only under this condition could one write, instead of (10),

\[ \mu k \omega \Phi_{1abs}^{2} + \nu a \Phi_{1abs}^{q} = \nu a_0 \Phi_{1abs}^{q} \tag{11} \]

and obtain from this the expression

\[ a_0=a\left(1+\frac{1}{\alpha}\right), \]

which we used above and in paper [1] in analyzing the regularities in the variation of the experimental values of the coefficient \(a_0\).

However, equality of the exponents \(q\) in both parts of expression (11) can be ensured only in rare cases.

The question of choosing the exponent \(q\) must inevitably arise for the experimenter as soon as he sets himself the task of experimentally determining the coefficient \(a_0\) in the powerplant under investigation from the results of torsion-graphing. This experimental value of the coefficient \(a_0\) is determined by the expression [1]

\[ a_0=-\frac{\lambda M_h}{\nu \Phi_{1abc}^{\,q-1}} . \tag{12} \]

From this it is easy to see that the particular magnitude of the coefficient \(a_0\), for known from experiment values of \(\Phi_{1abc}\) and \(M_h\), depends strongly on the value of \(q\) adopted by the experimenter for his calculations according to (12), which enters into (12) not only as the exponent on \(\Phi_{1abc}\), but also in the expression for the coefficient \(\nu\) ^1):

\[ \nu=\sum \left[(\varphi_{i-1}-\varphi_i)\frac{c_{i,i+1}}{w_p}\right]^q d^2l . \tag{13} \]

But the experimenter rarely has exact data regarding the value of \(q\) for the material of the shaft of his experimental powerplant. The literature data on the hysteresis characteristics of steels are extremely limited. The experimental determination of the coefficients \(a\) and \(q\) is rather complicated and requires special experiments.

Therefore, when calculating experimental values of the coefficient \(a_0\) by (12), for the hysteresis exponent \(q\) one often adopts some approximate (orientational) value, equal to 2.3 or 3, in accordance with the recommendations of Lewis and K. Wilson, given by them as applied to the usual calculations of the amplitudes of resonant torsional vibrations of powerplant shafts.

We shall show that with such an “arbitrary” choice of the magnitude of the hysteresis exponent \(q\) in the right-hand sides of expressions (10) and (12), we may obtain a greatly distorted picture of the regularities to which the generalized coefficient \(a_0\) is subject.

Let us consider the general case when in the right-hand sides of expressions (10) and (12) some exponent \(q_0\) is adopted, different from the actual exponent \(q\) for the shaft material of the powerplant of interest to us. Then, for

\[ q_0\ne q \]

instead of (11) one should write

\[ \mu k\omega\Phi_{1abc}^2+\nu a\Phi_{1abc}^q=\nu_0 a_0 \Phi_{1abc}^{q_0}, \]

where the coefficients \(\nu\) and \(\nu_0\) are calculated by (13) for the exponents \(q\) and \(q_0\), respectively.

^1) Here and below it is assumed that the entire shaft of the powerplant is made of one material, with hysteresis characteristics \(a\) and \(q\) constant over the entire length of the shaft.

Hence we obtain an expression for the generalized coefficient

\[ a_0=\frac{\nu a}{\nu_0}\cdot \frac{1}{\Phi_{1\mathrm{abs}}^{q_0-q}} +\frac{\mu k\omega}{\nu_0 a}\cdot \frac{1}{\Phi_{1\mathrm{abs}}^{q_0-2}} \]

or

\[ a_0=a\,\frac{\nu}{\nu_0}\cdot \frac{1}{\Phi_{1\mathrm{abs}}^{q_0-q}} \left[1+\frac{\mu k\omega}{\nu a}\cdot \frac{1}{\Phi_{1\mathrm{abs}}^{q-2}}\right]. \tag{14} \]

On the other hand, introducing, as before [1], the concept of the generalized coefficient of linear damping \(k_0\), instead of (11) we have

\[ \mu k\omega \Phi_{1\mathrm{abs}}^2+\nu a\Phi_{1\mathrm{abs}}^q =\mu k_0\omega \Phi_{1\mathrm{abs}}^2, \]

whence

\[ k_0=k(1+\alpha), \]

where

\[ \alpha=\frac{A_2}{A_k}=\frac{\nu a}{\mu k\omega}\,\Phi_{1\mathrm{abs}}^{q-2}. \]

Now, instead of (14), one may write

\[ a_0=a\,\frac{\nu}{\nu_0}\cdot \frac{1}{\Phi_{1\mathrm{abs}}^{q_0-q}} \left(1+\frac{1}{\alpha}\right). \tag{15} \]

We have obtained for the coefficient \(a_0\) an expression more complicated than formula (2), given at the beginning of the present article.

It is evident from (15) that the value \(a_0\) depends strongly on the correctness of the choice of the calculated exponent \(q_0\). Only when

\[ q_0=q \]

and the resulting equality

\[ \nu_0=\nu \]

holds does expression (15) become expression (2), which we investigated earlier in article [1].

Formula (2) also served as the basis for calculating and constructing the specific dependence of the generalized coefficient \(a_0\) on the moment of inertia \(\Theta_M\) of the flywheel for a four-cylinder-engine—flywheel installation (Fig. 2). In this installation the coefficient \(\alpha\) decreases with increasing \(\Theta_M\), which is a consequence of the change in the character of the curve of the mode shape of the free vibrations of the shaft. This decrease in \(\alpha\), according to (2), leads to an increase in \(a_0\) with increasing \(\Theta_M\), as was shown in Fig. 3. But it also follows from Fig. 3 that, with increasing \(\Theta_M\), the amplitude \(\Phi_{1\mathrm{abs}}\) of the resonant vibrations of the left (free) end of the installation shaft increases continuously (at \(M_h=\mathrm{const}\)).

It is obvious that, when choosing \(q_0>q\), according to (15), we obtain a different character of variation of the coefficient \(a_0\) as a function of the magnitude \(\Theta_M\) of the flywheel moment of inertia, compared with that shown in Fig. 3: 1) the generalized coefficient \(a_0\) will now decrease with increasing \(\Theta_M\), and this decrease will be the more rapid, the greater the difference \(q_0-q\).¹

This indicates the need to exercise particular caution when analyzing experimental data for the coefficient \(a_0\) and when processing the results of experiments. An incorrect choice of the exponent (when \(q_0 \ne q\)) may lead to a distorted picture of the regularities in the variation of the generalized coefficient \(a_0\) and to completely random “experimental” values of this coefficient¹.

Consequently, there is every reason to believe that an incorrect choice of the exponent \(q_0\) in processing the results of torsiographing is one of the substantial sources of the wide scatter, reported in the literature, of experimental values of the coefficient \(a_0\), which the designer encounters when attempting to select a suitable value of this coefficient for the installation being calculated.

Dependence of the coefficients \(k_0\) and \(a_0\) on the frequency \(\omega\) of the free vibrations of the installation shaft

It follows from expression (3) that, for \(M_h=\mathrm{const}\), a decrease in the frequency \(\omega\) of the free vibrations of the shaft leads to a decrease in the coefficient \(\alpha\), provided only that the decrease in \(\omega^{q-1}\) is not offset by an increase in the ratio

\[ \frac{\nu \lambda^{q-2}}{\mu^{q-1}}. \]

A decrease in \(\omega\) can be achieved, for example, by increasing the moment of inertia \(\Theta_M\) of the flywheel, by installing counterweights on the crankshaft, by reducing the stiffness of the intermediate shaft, etc. In all these cases, of course, the curve of the mode shape of the free vibrations of the shaft will change and, as a consequence, the coefficients \(\lambda\), \(\mu\), and \(\nu\) will change. As a result, the dependence of the coefficient \(\alpha\) (and hence also of the coefficients \(k_0\) and \(a_0\)) on the frequency \(\omega\) may prove complex and may differ both in magnitude and in direction. Its specific character can be established only on the basis of detailed calculations for each type of motor installation of interest to us.

Figure 5 shows, as an example, the dependences of the generalized coefficients \(k_0\) and \(a_0\) on \(\omega\) for the four-cylinder engine considered above, in which the decrease in the frequency \(\omega\) is achieved by increasing the moment of inertia \(\Theta_M\) of the flywheel.

Dependence of the coefficients \(k_0\) and \(a_0\) on the absolute dimensions of the motor installation

Consider any two geometrically similar installations of the type multicylinder in-line engine—flywheel². In doing so, for simplicity, we shall assume that the moments of inertia \(\Theta\) of the reduced masses of all cylinders in each of the engines are equal to one another; the stiffnesses \(c\) of the shaft sections between each two adjacent cylinders are also equal. The shafts of both installations are made from the same grade of steel.

Let all linear dimensions of the second installation be \(p\) times greater than those of the first. Then

\[ \begin{gathered} c_2=p^3 c_1; \qquad c_{m2}=p^3 c_{m1} \\ \Theta_2=p^5 \Theta_1 : \qquad \Theta_{M2}=p^5 \Theta_{M1}, \end{gathered} \tag{16} \]

where \(c\) is the stiffness of the shaft section between adjacent cylinders,

\(c_m\) is the stiffness of the intermediate shaft between the last cylinder and the flywheel,

\(1\) and \(2\) are indices referring, respectively, to the first (smaller) and second (larger) installation.

Fig. 5

It can be shown [2] that, for given constant ratios \(\dfrac{c_m}{c}\) and \(\dfrac{\Theta_m}{\Theta}\), the character of the curve of the mode shape of the free vibrations of the installation shaft depends only on the dimensionless parameter

\[ A=\frac{\Theta \omega^2}{c}. \]

In free, for example single-node, vibrations of the shaft, this parameter acquires a certain definite numerical value \(A_0\), depending only on the number of engine cylinders and on the ratios \(\dfrac{c_m}{c}\) and \(\dfrac{\Theta_m}{\Theta}\).

Obviously, in the two geometrically similar installations under consideration, with engines having the same number of cylinders,

\[ \frac{c_{m1}}{c_1}=\frac{c_{m2}}{c_2}, \]

\[ \frac{\Theta_{M1}}{\Theta_1}=\frac{\Theta_{M1}}{\Theta_2}, \]

and therefore

\[ A_{01}=A_{02}, \]

that is,

\[ \frac{\Theta_1\omega_1^2}{c_1}=\frac{\Theta_2\omega_2^2}{c_2}. \]

Here \(\omega_1\) and \(\omega_2\) are the frequencies of free oscillations of the shafts of the first and second installations, respectively. Hence

\[ \omega_2^2=\frac{\Theta_1}{\Theta_2}\,\frac{c_2}{c_1}\,\omega_1^2 \]

or, taking (16) into account,

\[ \omega_2=\frac{\omega_1}{p}. \tag{17} \]

Thus, with an increase in the absolute dimensions of the engine installation, the frequency \(\omega\) of free oscillations of its shaft decreases in inverse proportion to the scale factor \(p\).

But for similar curves of the shape of free oscillations of the shaft in both installations under consideration,

\[ \lambda_2=\lambda_1 \]

and

\[ \nu_2=p^3\nu_1, \]

\[ \mu_2=p^4\mu_1, \]

which follows from (13) and the expression [1]:

\[ \mu=\pi Fr^2 \sum \varphi_i^2. \]

Now, on the basis of (3), one may write

\[ \alpha_1(1+\alpha_1)^{q-2} = \frac{\nu_1 a\lambda_1^{q-2}}{(\mu_1 k_1\omega_1)^{q-1}} \,M_{h1}^{q-2}, \]

\[ \alpha_2(1+\alpha_2)^{q-2} = \frac{\nu_2 a\lambda_2^{q-2}}{(\mu_2 k_2\omega_2)^{q-2}} \,M_{h2}^{q-2}. \]

Hence, assuming that in both installations the actual coefficients of linear damping are identical,

\[ k_1=k_2=k, \tag{18} \]

and also the hysteresis characteristics \(a\) and \(q\) of the shaft material are identical, we have

\[ \frac{\alpha_2(1+\alpha_2)^{q-2}}{\alpha_1(1+\alpha_1)^{q-2}} = \frac{\nu_2}{\nu_1} \left(\frac{\mu_1\omega_1}{\mu_2\omega_2}\right)^{q-1} \left(\frac{M_{h2}}{M_{h1}}\right)^{q-2} \]

or, on the basis of (16) and (17),

\[ \frac{\alpha_2(1+\alpha_2)^{q-2}}{\alpha_1(1+\alpha_1)^{q-2}} = \frac{1}{p^{3(q-2)}} \left(\frac{M_{h2}}{M_{h1}}\right)^{q-2}. \tag{19} \]

Let us assume that the working processes in the cylinders of both installations being compared are completely identical. Then one may write

\[ M_{h2}=p^3M_{h1}. \]

Now, from (19),

\[ \frac{\alpha_2(1+\alpha_2)^{q-2}}{\alpha_1(1+\alpha_1)^{q-2}}=1, \]

that is,

\[ \alpha_2=\alpha_1. \]

Hence, according to (2), it follows that the generalized damping coefficients \(k_0\) and \(a_0\) do not depend on the absolute dimensions of the powerplant. Of course, this conclusion is valid only when condition (18) is fulfilled, that is, if the actual coefficients of linear damping in the cylinders of geometrically similar installations are identical and do not depend on the absolute dimensions of the cylinders.

Attempts to relate the damping coefficient \(k\) in the cylinder mechanism to the design parameters of the cylinder were made already by Holzer [5] and Shannon [6], and later by Draminsky [7].

Writing the expression for the work of damping in the mechanism of one cylinder over a cycle of resonant oscillations in the form (in our notation):

\[ A_c=\pi k_h\omega \Phi_{\mathrm{abs}}^2, \tag{20} \]

Holzer gave for the damping coefficient in the cylinder the empirical expression

\[ k_h=0.04\,\Theta\omega, \tag{21} \]

obtained by him on the basis of processing the results of torsiographing a number of four-cylinder diesels.

Comparing (20) with the expression previously adopted by us [1], in accordance with Wydler, for the work of damping in one cylinder,

\[ A_c=\pi kFr^2\omega \Phi_{\mathrm{abs}}^2 \tag{22} \]

and taking (21) into account, we obtain an expression for the coefficient \(k\) of interest to us, from Holzer’s data,

\[ k=\frac{k_h}{Fr^2}=0.04\,\frac{\Theta\omega}{Fr^2}. \tag{23} \]

Shannon [6] established a relationship between the coefficient \(k\), the mean relative amplitude of the free oscillations of the reduced cylinder masses

and the cylinder parameters in the form of the graph given above (Fig. 4a). From this graph [3]

\[ k=\frac{1}{55-45\varphi_{\mathrm{av}}}\cdot\frac{\Theta\omega}{Fr^2}. \]

Hence, taking into account the possible limits of variation of the mean amplitude

\[ \varphi_{\mathrm{av}}=0\div 1 \]

we have

\[ k=(0,018\div 0,100)\,\frac{\Theta\omega}{Fr^2}, \tag{24} \]

which on the average approximately coincides with Holzer’s data¹.

Draminskii [7], assuming that the principal source of linear damping in the cylinder is the energy loss in the oil film in the bearings, obtained the expression for the damping work in one main bearing (per cycle of oscillations)

\[ A=\pi m_a r\Phi_{abc}^{2}\omega^2\,\frac{eh}{\sin\delta} \]

or, for the entire cylinder (also taking into account the damping in the connecting-rod bearing),

\[ A_c \cong 1,5\pi m_a r\Phi_{abc}^{2}\omega^2\,\frac{eh}{\sin\delta}. \tag{25} \]

Here, in addition to the notation adopted by us earlier,

\[ \begin{aligned} m_a&\text{ is the unbalanced rotating mass of the connecting-rod-and-crank mechanism of the cylinder, reduced to the crank radius;}\\ e&\text{ is the radial clearance in the bearing,}\\ \delta&\text{ is a certain angle (usually about }15^\circ\text{).} \end{aligned} \]

Comparing (25) and (22), we obtain an expression for the coefficient of linear damping \(k\) of one cylinder according to Draminskii’s data

\[ k=1,5\,\frac{m_a\omega eh}{Fr^2\sin\delta}. \tag{26} \]

But in two geometrically similar powerplants, in addition to (16) and (17), the relations

\[ r_2=pr_1;\qquad F_2=p^2F_1; \]

\[ m_{a2}=p^3m_{a1}. \]

also hold. It may also be assumed that

\[ e_2=pe_1. \]

Now from (23), (24), or (26) it is easy to obtain

\[ k_1=k_2, \]

that is, that the coefficients of linear damping in the engine cylinder, determined according to Holzer, Shannon, or Draminsky, do not depend on the absolute dimensions of the powerplant. Consequently, in geometrically similar installations the generalized coefficients \(k_0\) and \(a_0\) must be identical.

However, the relations of Holzer, Shannon, and Draminsky cited above cannot be considered sufficiently reliable. The question of damping in engine cylinders still requires further study.

Experimental values of the generalized coefficient \(k_0\), obtained for large diesel engines, usually prove to be several times greater than the values of \(k_0\) found in torsiographing automobile and aircraft engines. One of the reasons for this difference should evidently be regarded as the fact that ordinary carbon steel, having larger values of \(a\) and \(q\), is used for manufacturing the shafts of large installations, rather than the high-alloy steels used for the shafts of high-speed light-type engines. As was indicated earlier [1], an increase in \(a\) and \(q\), other conditions being equal, leads to an increase in \(\alpha\), that is, to an increase in the generalized coefficient of linear damping \(k_0\).

REFERENCES

1. Nechaev V. K. On the question of the characteristics of damping forces in powerplants with piston internal-combustion engines. Izvestiya Tomskogo politekhnicheskogo instituta, vol. 85, 1957.

2. Nechaev V. K. A graphical method for determining the natural frequencies of oscillations. Izvestiya Tomskogo industrial'nogo instituta, vol. 54, issue 6, 1935.

3. Nechaev V. K. Theoretical torsiograms for the shaft of an internal-combustion engine. Izvestiya Tomskogo industrial'nogo instituta, vol. 58, issue 2, 1937.

4. Bolgov A. T. On the question of the dependence of the characteristics of damping forces in powerplants on the moment of inertia of the flywheel. Izvestiya Tomskogo politekhnicheskogo instituta, vol. 85, 1957.

5. Holzer, H. Die Berechnung der Drehschwingungen. Berlin, 1921.

6. Shannon J. Damping Influences in Torsional Oscillation, Engineering, Dec. 20, 27, 1935.

7. Draminsky, Crankshaft Damping, J.M.E. Proc., Applied Mechanics. Vol. 159, (War Emergency Issue No. 46), 1948.

8. Dorey S. F. Government Rules, The Engineer, Centenary Number, p. 248, 1956.

1. The numerical coefficients in formulas (23 and 24) were obtained on the basis of processing the results of torsiographing powerplants. Consequently, it would be more correct to consider that these formulas give values not of the actual \((k)\), but of generalized \((k_0)\) damping coefficients, referred to one cylinder and in some way including also hysteresis damping in the shaft of the powerplant. But at present we are interested only in the structure of these formulas; therefore the replacement in them of the coefficient \(k_0\) by the coefficient \(k\) may be considered appropriate here. ↩↩↩↩

2. By geometric similarity we here mean the similarity of all linear dimensions (lengths and diameters) in both installations. ↩↩