On the Characteristics of Damping Forces in Power Units with Piston Internal Combustion Engines

V. K. Nechaev

A reliable assessment of the degree of danger in the operation of a designed power unit at one or another critical (or near-critical) engine speed is possible only on the basis of calculating the additional vibrational (resonant) stresses that arise in the shaft in this case. But in order to calculate the latter, it is necessary to know the nature and magnitude of the damping forces (resistances) in the unit’s shafting, which limit the amplitudes of resonant vibrations.

As is known, there are grounds to consider that the most important sources of damping of torsional vibrations in a power unit of the engine–flywheel type are resistances having the character of viscous friction (in the crankshaft bearings, etc.) and internal friction (hysteresis) in the material of the unit shaft.

Under these conditions, the energy balance for stationary resonant torsional vibrations of the shaft of a multicylinder engine with a flywheel may be written approximately in the form:

\[ A_{\text{exc}} = A_k + A_2, \tag{1} \]

where

\[ A_{\text{exc}} \]
is the work performed by the exciting (resonating) harmonic components of the torques, created on the shaft by each cylinder of the engine, during one cycle of resonant torsional vibrations of the shaft,

\[ A_k \text{ and } A_2 \]
are the works absorbed during a cycle of vibrations, respectively, by viscous friction in the engine and by internal friction (hysteresis) in the shaft material.

These works are determined by the known expressions \([1]^1\):

\[ A_{\text{exc}} = \pi M_h \Phi_{1\text{abs}} \sum \varphi_i, \tag{2} \]

\[ A_k = \pi F r^2 k \omega \Phi_{1\text{abs}} \sum \varphi_i^2, \tag{3} \]

\[ A_2 = \sum a \Phi_{1\text{abs}}^{\,q} \sum \left[(\varphi_{i+1} - \varphi_i)\frac{c_{i,i+1}}{W_p}\right]^q d^2 l. \tag{4} \]

^1 Here, as everywhere below, it is assumed that all damping forces in the engine (except for hysteretic damping in the crankshaft) obey a linear law, that is, are proportional to the velocity of vibration. Analysis of the degree of validity of this assumption is not included in the tasks of this article.

Formula (4) is given here for the case of a solid (not hollow) shaft. In the general case, the shaft of a power unit consists of several sections made of different steels with different values of \(a\) and \(q\). This is taken into account by the first summation sign on the right-hand side of formula (4).

Here

$M_h$ — the amplitude of the harmonics of the torques, produced on the shaft by each engine cylinder, which cause the resonant vibrations of the shaft under consideration,

$\Phi_{1abs}$ — the absolute amplitude (in radians) of the resonant vibrations of the first mass of the reduced oscillatory system under consideration,

$\varphi_i$ — the relative amplitude of the free vibrations of the $i$-th mass of the installation (with $\varphi_1 = 1$),

$\displaystyle \overrightarrow{\sum} \varphi_i$ — the geometric sum of the relative amplitudes of the free vibrations of the reduced masses of the engine cylinders, determined by means of the phase diagram of the exciting moments,

$\displaystyle \sum \varphi_i^2$ — the algebraic sum of the squares of the relative amplitudes of the reduced masses of the engine cylinders,

$k$ — the coefficient of viscous (linear) damping in the engine,

$a$ — the hysteresis constant, depending on the characteristics of the material of the given shaft section,

$q$ — the hysteresis exponent (also depending on the characteristics of the material of the shaft section),

$\omega$ — the angular frequency of the resonant torsional vibrations of the shaft of the motor installation under consideration (equal to the frequency of its free vibrations),

$c_{i,i+1}$ — the stiffness of the shaft in the section between two adjacent ($i$-th and $i+1$-th) masses of the installation,

$W_p$ — the polar section modulus of the cross section of the shaft of the installation in the given section,

$d$ and $l$ — respectively, the diameter and length of each shaft section,

$F$ and $r$ — respectively, the piston area and the crank radius.

In deriving and applying expressions (2)—(4), it is usually assumed that the curve of the form of the resonant vibrations of the shaft of the motor installation is similar to the curve of the form of the free vibrations of the same shaft. As is known, in systems with small damping such an assumption is quite appropriate and does not introduce any appreciable errors into the subsequent calculations. This assumption is also adopted by us in the further investigation.

Let us write expressions (2) and (4) in a more concise form

\[ A_{ex} = \lambda M_h \Phi_{1abs}, \tag{5} \]

\[ A_k = \mu k \omega \Phi_{1abs}^{2}, \tag{6} \]

where the following designations have been adopted

\[ \lambda = \pi \overrightarrow{\sum} \varphi_i, \tag{7} \]

\[ \mu = \pi F r^2 \sum \varphi_i^2. \tag{8} \]

Let us assume, in order to simplify the further calculations, that the entire shaft of the motor installation is made of one material, with the coefficients $a$ and $q$ constant over the whole length of the shaft.

Then, instead of (3), one may write

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

where

\[ \nu = \sum \left[(\varphi_{i+1}-\varphi_i)\frac{c_{i,i+1}}{W_p}\right]^q d^2 l . \tag{10} \]

According to the assumption adopted above, the quantities \(\lambda\), \(\mu\), and \(\nu\) depend, first of all, on the character of the curve of the mode shape of the free vibrations of the shafting. In addition, the coefficient \(\lambda\) takes into account the number of operating cylinders of the engine, the order \(h\) of the resonating harmonic moments \(M_h\), and, for secondary orders \(h\), the firing order in the engine cylinders. The coefficient \(\mu\) depends on the dimensions (\(F\) and \(r\)) and on the number of cylinders. Finally, the coefficient \(\nu\) depends on the specific dimensions of the shafting \((d,l)\) and on the hysteresis exponent \(q\) (that is, the physical characteristics of the shaft material).

Now, on the basis of (5), (6), and (9), let us write expression (1) of the energy balance at steady resonance in the following form:

\[ \lambda M_h \Phi_{1abc} = \mu k \omega \Phi_{1abc}^{2} + \nu a \Phi_{1abc}^{q}. \tag{11} \]

We have obtained an equation from which one could find (for example, graphically) the resonant amplitude \(\Phi_{1abc}\) of the first mass, which will occur when the engine operates at the given critical speed. From this value \(\Phi_{1abc}\), it is not difficult, by known methods, to compute also the values of the additional vibrational torsional stresses in each section of the shaft that is of interest to us.

But for a concrete solution of this problem it is necessary to know the concrete values of the quantities \(k\), \(a\), and \(q\) for the given (for example, designed) engine installation. However, there is still very little experimental data on the coefficients \(k\), \(a\), and \(q\). The separate experimental determination of the characteristics of the various kinds of damping forces present in engine installations presents great difficulties. The theoretical aspect of the question of damping forces in such installations has likewise been developed very poorly.

For these reasons, equation (11) is still almost not used for practical calculations and is replaced by one of two simpler ones \(^{1)}\)

\[ \lambda M_h \Phi_{1abc} = \mu k_0 \omega \Phi_{1abc}^{2}, \tag{12} \]

\[ \lambda M_h \Phi_{1abc} = \nu a_0 \Phi_{1abc}^{q}. \tag{13} \]

The right-hand sides of these equations are similar, in their structure, to expressions (6) and (9).

Equation (12) is the basis of the well-known linear method for calculating resonant torsional vibrations, proposed at one time by Wydler [3]. Here all the damping present in the engine installation (both linear, viscous—in the engine mechanism—and nonlinear, hysteretic—in the shaft material) is evaluated by a certain generalized coefficient of linear (viscous) damping \(k_0\).

\(^{1)}\) Here and below it is assumed that the exponent \(q\) in the right-hand side of the generalized expression (13) is exactly equal to the actual hysteresis exponent \(q\) for the shaft material of the engine installation under consideration, which enters into the right-hand side of (9) and (10). On the consequences of choosing an arbitrary value for \(q\) in (13), see [6].

The numerical value of this coefficient in any particular engine installation can be found comparatively simply by experiment. For this purpose one should use the relation following from (12),

\[ k_0=\frac{\lambda M_h}{\mu \omega \Phi_{1abc}} \]

and substitute into it the value \(\Phi_{1abc}\), determined directly by torsion-recording the shaft of this installation while it is operating at the critical speed of rotation\(^1\).

In equation (13), all the damping present in the engine installation—linear damping in the engine and nonlinear, hysteretic damping in the shaft—is taken into account (estimated) by means of a certain generalized hysteresis coefficient \(a_0\) and the exponent \(q\).

Equation (13) is based on the well-known proposal of Lewis [4]: to regard the hysteresis losses in the shaft material as the principal source of damping of torsional vibrations in the shaft of an engine installation. The exponent \(q\) entering into (13) is selected in calculations on the basis of experimental data from studies of hysteresis losses in cylindrical steel specimens (Rowett, Dorey, and others), or sometimes, in accordance with the proposal of Ker Wilson [5], is taken equal to three.

From the same experimental data the coefficient \(a\) may also be selected. But with such a choice of the coefficient \(a\), expression (13), which by its very structure does not take into account any types of damping in the engine installation other than hysteretic damping, usually proves to be very inaccurate. This inaccuracy will be the greater, the smaller the relative share of hysteretic damping in the total magnitude of the damping resistances in the given installation. It will therefore be somewhat more correct to substitute into (13) the experimental value of this coefficient \(a_0\), found by processing the results of torsion-recording an engine installation similar to the one being calculated. Such processing can be carried out on the basis of the expression

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

which follows from (13). Naturally, these experimental values of the coefficient \(a_0\) prove to be different from the values of the coefficient \(a\) obtained in laboratory tests on steel specimens. Thus, for example, Porter [1] obtained

\[ a_0=2.84\,a. \]

Some experimental data on the generalized coefficients \(k_0\) and \(a_0\) can be found in the literature on torsional vibrations. However, these data are few in number and, as their comparison and analysis show, extremely contradictory. Attempts to use them in calculations of resonant vibrations in newly designed installations often prove unsuccessful: the results of calculating resonant amplitudes differ greatly from the results of subsequent torsion-recording of the completed installation. Moreover, some experiments indicate the nonconstancy of these coefficients even in one and the same engine installation.

All this deprives the designer of the possibility of choosing the expected values of the generalized coefficients \(k_0\) or \(a_0\) for the installation being designed—

\(^1\) The quantity \(M_h\) entering into this expression for \(k_0\) is found by harmonic analysis of the curve of the torque created on the shaft by one cylinder of the engine; the natural frequency \(\omega\) is determined by calculation and checked by torsion-recording; the coefficients \(\lambda\) and \(\mu\) can be computed from formulas (7) and (8) given above.

... sufficiently substantiated and reliable. Of course, the designer’s position will improve somewhat as further experimental data on the coefficients \(k_0\) and \(a_0\) are accumulated. However, even the availability of a large amount of experimental material on these coefficients still cannot substantially increase the reliability of their selection in calculations in the design department.

It is shown below that the generalized coefficients \(a_0\) and \(k_0\) generally cannot be regarded as at all satisfactory characteristics of damping in a power unit. These coefficients may change greatly even when the engine operating regime changes and, all the more, when the design characteristics of the power unit change.^1

Let us consider the question of the generalized coefficients \(k_0\) and \(a_0\) as applied to units in which the entire shaft is made of one grade of steel.

Introduce the ratio

\[ \alpha = \frac{A_z}{A_k}. \tag{14} \]

Now, instead of (1), one may write

\[ A_{\mathrm{exc}} = A_k(1+\alpha) \]

or

\[ A_{\mathrm{exc}} = A_z\left(1+\frac{1}{\alpha}\right) \]

and instead of (11):

\[ \lambda \Phi_{1\mathrm{abs}} M_h = \mu k \omega \Phi_{1\mathrm{abs}}^2(1+\alpha) \tag{15} \]

or

\[ \lambda \Phi_{1\mathrm{abs}} M_h = \nu a \Phi_{1\mathrm{abs}}^q \left(1+\frac{1}{\alpha}\right). \tag{16} \]

Comparison of (15) and (16), respectively, with (12) and (13) gives

\[ k_0 = k(1+\alpha), \tag{17} \]

\[ a_0 = a\left(1+\frac{1}{\alpha}\right). \tag{18} \]

Here, as before, the coefficients \(k\) and \(a\) separately characterize the actual damping (respectively, viscous damping in the engine and hysteretic damping in the shaft), while the coefficients \(k_0\) and \(a_0\) are generalized characteristics of the entire damping in the power unit.

It follows from (17) and (18) that the values of the generalized coefficients \(k_0\) and \(a_0\), for any constant values of the coefficients \(k\) and \(a\) (determined by the design and condition of the engine, as well as by the material of the unit shaft), depend strongly on the ratio \(\alpha\) (Fig. 1).

We shall assume here and throughout below that the laws of viscous (linear) damping in the engine mechanism and hysteretic (nonlinear) damping in the shaft of the power unit, as well as their numerical characteristics \(k\), \(a\), and \(q\), do not change when the engine load or speed regimes change.

^1 Similar conclusions are also valid for the coefficients that account for damping in the method for calculating resonant vibrations proposed by K. Wilson.

Fig. 1

Then the laws governing the variation of the generalized coefficients \(k_0\) and \(a_0\) will depend only on the laws to which the coefficient \(\alpha\) is subject. But, according to (6), (7), and (14),

\[ \alpha=\frac{\gamma a}{\mu k\omega}\Phi_{1\mathrm{abs}}^{q-2} \tag{19} \]

or

\[ \alpha=b_1\Phi_{1\mathrm{abs}}^{q-2}, \tag{20} \]

where it is denoted that

\[ b_1=\frac{\gamma a}{\mu k\omega}. \tag{21} \]

Now, instead of (17) and (18), one may write

\[ k_0=k\left(1+b_1\Phi_{1\mathrm{abs}}^{q-2}\right), \tag{22} \]

\[ a_0=a\left(1+\frac{1}{\Phi_{1\mathrm{abs}}^{q-2}}\right). \tag{23} \]

These dependences \(k_0\) and \(a_0\), for \(b_1=\mathrm{const}\) and different values of \(q\), are shown in Fig. 2.

For further investigation it is expedient to relate the amplitude \(\Phi_{1\mathrm{abs}}\) of the resonant oscillations of the shaft, entering into (22), (23), to the amplitude \(M_h\) of the exciting harmonic moments that cause these oscillations. From (12),

\[ \Phi_{1\mathrm{abs}}=\frac{\lambda M_h}{\mu k_0\omega} \]

or, taking into account dependence (17),

\[ \Phi_{1\mathrm{abs}}=\frac{\lambda M_h}{\mu k(1+\alpha)\omega}. \]

Substituting this expression for \(\Phi_{1\mathrm{obs}}\) into (19), we obtain

\[ \alpha(1+\alpha)^{q-2} = \frac{\nu a \lambda^{q-2}}{(\mu k \omega)^{q-1}}\,M_h^{q-2}. \tag{24} \]

Solving this equation with respect to \(\alpha\) in general form (for arbitrary \(q\)) does not appear possible. The problem is simplified only in the special case when

\[ q=3 \]

Fig. 2

(which coincides, for example, with the proposal of K. Wilson). Here, instead of (24), we have

\[ \alpha=\frac{b_2}{1+\alpha}\,M_h, \]

where the notation adopted is

\[ b_2=\frac{\lambda \nu a}{\mu^2 k^2 \omega^2}. \tag{25} \]

Hence

\[ \alpha=\sqrt{\frac{1}{4}+b_2 M_h}-\frac{1}{2} \]

and

\[ k_0=k\left(\frac{1}{2}+\sqrt{\frac{1}{4}+b_2 M_h}\right), \tag{26} \]

\[ a_0 = a\left(1+\frac{1}{\sqrt{\frac{1}{4}+b_2 M_h}-\frac{1}{2}}\right). \tag{27} \]

We have obtained analytical dependences of the generalized coefficients \(k_0\) and \(a_0\) on the amplitude \(M_h\) of the resonating harmonic moments (for \(q=3\)), which, for \(b_2=\mathrm{const}\), have the form shown in Fig. 3.

Fig. 3

We shall use the expressions obtained above, (17)—(27), and the graphs (Figs. 1—3), primarily to clarify the influence of the load and speed regimes of the engine on the values of the generalized coefficients \(k_0\) and \(a_0\) (for constant \(k\) and \(a\)). In doing so, we shall confine ourselves only to the analysis of resonant oscillations of the shaft, that is, to the investigation of cases of engine operation at critical rotational speeds.

Influence of the engine load

Suppose that the engine operates at various loads (from idling to maximum), but at a constant rotational speed \(n\), equal to one of the critical rotational speeds of the shaft of the power unit,

\[ n=n_{\mathrm{cr}}=\frac{N}{h}, \]

where \(h\) is the order of this critical number of revolutions, and

\[ N=\frac{30}{\pi}\,\omega . \]

It is easy to see that under this condition \((n=n_{kr}=\mathrm{const})\) the coefficients \(\lambda\), \(\mu\), and \(\nu\), entering into the expressions for \(a\), \(b_1\), and \(b_2\), remain unchanged for any engine load. Therefore, when the coefficients \(k\), \(a\), and \(q\) do not depend on the engine load (as was stipulated by us above), the coefficients \(b_1\) and \(b_2\) also remain unchanged.

Now from expressions (17), (18), and (24) it follows that the generalized damping coefficients \(k_0\) and \(a_0\) inevitably change when the amplitude of the exciting moments \(M_h\) changes. The specific character of these changes can be established by substituting into (17) and (18) the values of the coefficient \(a\) calculated from (24) for a series of selected values of the amplitude of the moment \(M_h\). In particular, in a sufficiently characteristic case, for \(q=3\), the dependence of the generalized coefficients \(k_0\) and \(a_0\) on \(M_h\) is represented by the graphs given above in Fig. 3.

But the amplitudes of the exciting moments \(M_h\) of each order \(h\), as a rule, increase with an increase in the mean indicated pressure \(p_i\) in the cylinder, that is, with an increase in engine load.

Approximately, it may be assumed that the amplitude \(M_h\) (for \(h=\mathrm{const}\)) is directly proportional to \(p_i\) [2].

Consequently, when the engine load changes (while it is operating at some definite critical number of revolutions), the generalized damping coefficients \(k_0\) and \(a_0\) do not remain constant: the coefficient \(k_0\) increases, and the coefficient \(a_0\) decreases with increasing \(p_i\) (Fig. 3).

This influence of \(p_i\) on the coefficient \(k_0\) will be the stronger, and on the coefficient \(a_0\) the weaker, the larger, for example, the coefficient \(b_2\), determined by formula (25), is; that is, the lower the natural frequency of oscillations of the installation shaft \(\omega\), the larger \(a\), \(q\), and the ratio \(\lambda/\mu^2\) are. But it is easy to see that this latter ratio usually has its maximum value for the principal orders \(h\), when the geometric sum of the relative amplitudes \(\xi_i\) in expression (7) becomes an algebraic sum (or an arithmetic sum, if the node of oscillations is located outside the engine).

Thus, the influence of a change in engine load on the generalized coefficient \(k_0\) will be strongest (and on the coefficient \(a_0\) weakest) in installations with multicylinder engines having large masses and a long shaft, when they operate at the principal critical numbers of revolutions.

It follows already from this, in particular, that the Porter correction indicated is entirely accidental in its absolute magnitude and does not and cannot have any universal significance.

Influence of the engine speed regime and of the order \(h\) of the critical numbers of revolutions

When the speed regime of the engine changes, the latter (in accordance with our condition on p. 10) as if passes at once from one critical number of revolutions \(n_{kr}\) to another \(n'_{kr}\), and then operates for a sufficiently long time at this new critical number of revolutions. During this transition the order of the resonating harmonic moments will change from

\[ h=\frac{N}{n_{kr}} \]

to

\[ h'=\frac{N}{n'_{kp}} . \]

To different orders \(h\) and \(h'\) there correspond, naturally, different values of the amplitudes \(M_h\) and \(M_{h'}\) of the resonant moments. Consequently, the transition of the engine from one critical speed \(n_{kp}\) to another \(n'_{kp}\) is to a certain degree analogous to the cases considered above of a change in engine load. According to (17) and (18), the difference between the values of \(M_h\) and \(M_{h'}\) leads to the appearance of a difference between the values of the damping coefficients \(k_0\) (or \(a_0\)) in these two speed regimes.

But upon transition to another \(n_{kp}\), the coefficients \(b_1\) and \(b_2\) may also change. When the engine operates at different critical speeds corresponding to one and the same, for example, single-node form of shaft vibration, the coefficients \(\mu\) and \(\nu\), determined by formulas (8) and (10) and entering into (21) and (25), do not depend on \(h\).

However, the coefficient \(\lambda\) is closely connected with \(h\). For a given unchanged mode shape of the free vibrations of the shafting, its value depends on the form of the phase diagram of the exciting moments constructed for the given \(h\). For all principal critical speeds of the given installation (with, for example, one single-node form of shaft vibration), the phase diagrams are identical, that is \(\lambda=\mathrm{const}\). Therefore here

\[ b_1=\mathrm{const};\quad b_2=\mathrm{const}. \]

Thus, for the principal orders \(h\), the influence of the number \(h\) of these orders on the generalized coefficients \(k_0\) and \(a_0\) is exerted only through the effect of the change in the amplitude \(M_h\), already considered above. Here the graphs in Fig. 2 remain valid, and for \(q=3\) also the graphs in Fig. 3; they retain their validity for the entire discrete series of values of \(h\) that are principal orders for the motor installation under consideration.

Analogous graphs can also be constructed for series of successive secondary orders \(h\). But, of course, in each such series one should include only those orders \(h\) to which one and the same value of \(\lambda\) corresponds, that is, the same form (scheme) of the phase diagram of the exciting moments.

The coefficient \(\lambda\) for secondary orders \(h\) usually has\(^1\) a smaller value than for principal orders. Correspondingly, the coefficients \(b_1\) and \(b_2\) also have smaller values for secondary orders. Therefore the dependence of the generalized coefficient \(k_0\) on \(h\) proves to be weaker at secondary critical speeds (and that of the coefficient \(a_0\) stronger) than is the case at principal critical speeds.

Figures 4 and 5 give, as an example, graphs of the dependences of the generalized coefficients \(k_0\) and \(a_0\) on the amplitude of the exciting moments \(M_h\) for various \(h\), as applied to one four-cylinder four-stroke in-line engine.

For the same engine a graph is given in Fig. 6.\(^2\) Here, in quadrant I the dependence according to Fig. 4 is repeated; in quadrant IV the horizontal—

\(^1\) Except for cases when the vibration node is located near the middle of the engine.

\(^2\) This graph was calculated and constructed on the assumption that the ratio of linear and hysteretic resistances in the sample motor installation under consideration is such that, when the engine operates with nominal load at one of the principal critical speeds,

\[ \alpha=0.5, \]

that is,

\[ A_k=A_2=0.5\,A_{\mathrm{exc}} . \]

For convenience of notation on the graph, \(k=1\) has been adopted.

4-cylinder, 4-stroke engine.
Firing order: 1–2–4–3

Vertical axis: $\dfrac{K_{\mathrm{в}}}{K}$

Curve labels:

• $h = 2, 4, 6, \ldots$
• $h = \frac{1}{2}, 1\frac{1}{2}, 2\frac{1}{2}, \ldots$
• $h = 1, 3, 5, \ldots$

Horizontal axis: $M_k$

Nominal load condition

Fig. 4

Vertical axis, left: $\dfrac{a_0}{a}$
Vertical axis, right: $\dfrac{a_0}{a}$

Curve labels:

• $h = \frac{1}{2}, 1, 1\frac{1}{2}, \ldots$
• $h = 2, 4, 6, \ldots$
• $h = 1, 3, 5, \ldots$

Horizontal axis: $M_k$

Nominal load condition

Fig. 5

The amplitudes of the exciting moments \(M_h\) for different orders are shown by the lower segments

\[ \left(h=2,\,2 \frac{1}{2},\,3,\ldots\right). \]

From these data, in the manner clear from the drawing, in quadrant II the dependence of the generalized coefficient \(k_0\) on the order \(h\) of the critical speeds has been obtained (for single-node vibrations). An analogous graph could be constructed for the generalized coefficient \(a_0\), using Fig. 5 as the basis for this.

Fig. 6

It is easy to see that the large “scatter” of points present here follows from the essence of the phenomena under consideration. It is explained by the different relative weights of the terms on the right-hand side of expression (11) for different \(h\), owing to the nonlinear character of damping in the engine installation.

Obviously, the experimental determination of the generalized damping coefficients \(k_0\) and \(a_0\) from the results of torsiographing one and the same engine at different critical speeds must inevitably lead to a considerable “scatter” of experimental points. This scatter cannot be smoothed out or averaged (as is often done) on the basis of the usual references to inevitable experimental errors, inaccu-

of the measurements, etc. (although such errors always exist and must be taken into account).

In one and the same engine installation, the generalized coefficient \(k_0\) (or \(a_0\)) necessarily has different values at different critical engine speeds. The fluctuations (scatter) of these values of the coefficient \(k_0\), depending on \(h\), will be strongest (and those of the coefficient \(a_0\) weakest) at high engine loads, in installations with a low frequency \(\omega\), and with a shaft made of a material with large hysteresis (that is, with large \(a\) and \(q\)).

Influence of the Number of Nodes of Shaft Vibration

Here it must be borne in mind that, in passing, for example, from single-node to two-node vibrations, in the expressions given above for the coefficients \(\alpha\), \(b_1\), and \(b_2\) in any particular engine installation, not only the frequency \(\omega\) and the coefficients \(\lambda\) and \(\mu\) change, but also the coefficient \(\nu\), which depends directly on the relative angles of twist of all sections of the installation shaft.

A detailed comparison of the values of \(k_0\) or \(a_0\) for single-node and two-node vibrations can be carried out only on the basis of specific numerical calculations for a particular engine installation. Therefore we shall confine ourselves here to the following remarks.

Let us compare two cases of resonant vibrations of the shaft of some engine installation: 1) in a single-node mode, with a certain amplitude \(\Phi_{1abs}\), and 2) in a two-node mode, with the same amplitude \(\Phi_{1abs}\). According to (19), these two cases will correspond to different values of the coefficients \(\alpha\):

\[ \alpha_1=\frac{\nu_1 a\Phi_{1abs}^{2}}{\mu_1 k\omega_1};\qquad \alpha_2=\frac{\nu_2 a\Phi_{1abs}^{2}}{\mu_2 k\omega_2}. \]

Here the indices 1 and 2 at \(\mu\), \(\nu\), and \(\omega\) indicate the number of nodes of the corresponding vibration mode. Hence

\[ \frac{\alpha_2}{\alpha_1} = \frac{\dfrac{\nu_2}{\mu_2}\,\omega_1}{\dfrac{\nu_1}{\mu_1}\,\omega_2}. \]

If, approximately, one assumes

\[ \frac{\nu_2}{\mu_2}=\frac{\nu_1}{\mu_1}, \tag{28} \]

then

\[ \alpha_2=\alpha_1\frac{\omega_1}{\omega_2}, \tag{29} \]

that is, for two-node vibrations (with frequency \(\omega_2\)) the coefficient \(\alpha\) will have a smaller value than for single-node vibrations (with frequency \(\omega_1\)). Accordingly, by (17) and (18), the generalized damping coefficients \(k_0\) and \(a_0\) will be different.

If condition (28) is sufficiently valid, then, for two-node vibrations of the shaft of some installation with a certain amplitude \(\Phi_{1abs}\), the coefficient \(k_0\) will be smaller, and the coefficient \(a_0\) larger, than the values of these

coefficients in single-node vibrations of the same installation, with the same amplitude \(\Phi_{1abc}\).

But usually

\[ \frac{\nu_2}{\mu_2} > \frac{\nu_1}{\mu_1} \]

therefore the dependence of the coefficients \(k_0\) and \(a_0\) on the number of vibration nodes just noted may in fact prove to be less strong than follows from (29). Here a concrete investigation is needed of the change in the direction of variation of the ratio

\[ \frac{\nu}{\mu\omega}\,\Phi_{1abc}^{q-2} \]

in the transition, for example, from single-node to two-node vibrations of the shaft of the motor installation under consideration.

It was shown above that the generalized damping coefficients \(k_0\) and \(a_0\) depend substantially on the operating regime of the engine. Some concrete conclusions were obtained concerning the directions of their change when the engine load and speed change.

It may be noted that, in all these conclusions, the directions of change of the coefficients \(k_0\) and \(a_0\) turned out, as a rule, to be opposite. This was a consequence of the relations of the coefficients \(k_0\) and \(a_0\) with the coefficient \(\alpha\), represented by expressions (17) and (18).

But these expressions cannot be regarded as universal. They were obtained by comparing the general equation (11) of the energy balance of vibrations in a motor installation with the “generalized” equations (12) and (13).

However, equation (11) is only approximate and simplified, taking into account only two, usually the most important, terms of the energy-loss balance in resonant torsional vibrations (linear damping and hysteresis). With a different structure of the right-hand side of equation (11), it is, of course, possible to obtain somewhat different dependences between the generalized and actual coefficients than those represented by formulas (17) and (18).

Above, as one of the essential assumptions, it was accepted that the entire shaft of the motor installation is made of one and the same material, with the hysteresis constant \(a\) and hysteresis exponent \(q\) identical for all sections of the shaft. Only under this condition is formula (9) for the work \(A_z\) valid.

But in most practical cases, different sections of the shaft of an installation are made of different steels, with different \(a\) and \(q\).

Let us consider the case where the crankshaft and the shaft outside the engine are made of two different steels with, respectively, hysteresis characteristics \(a_1, q_1\) and \(a_2, q_2\).

Then, instead of (9), one should write

\[ A_z = \nu_1 a_1 \Phi_{1abc}^{q_1} + \nu_2 a_2 \Phi_{1abc}^{q_2} \tag{30} \]

and instead of (11)

\[ \lambda M_h \Phi_{1abc} = \mu k \omega \Phi_{1abc}^{2} + \nu_1 a_1 \Phi_{1abc}^{q_1} + \nu_2 a_2 \Phi_{1abc}^{q_2}, \tag{31} \]

where obviously

\[ \nu_1 + \nu_2 = \nu, \]

where \(\nu_1\) is determined by a formula similar to (10).

Comparison of expression (31) with the generalized equations of the energy balance

\[ \lambda M_h \Phi_{1abs}=\mu k_0 \omega \Phi_{1abs}^{2} \]

and

\[ \lambda M_h \Phi_{1abs}=\nu a_0 \Phi_{1abs}^{q_a} \]

leads to new dependences for the generalized damping coefficients \(k_0\) and \(a_0\):

\[ \left. \begin{aligned} k_0&=k\left[1+\frac{\nu_1 a_1}{\mu k\omega}\Phi_{1abs}^{q_1-2} +\frac{\nu_2 a_2}{\mu k\omega}\Phi_{1abs}^{q_2-2}\right],\\[4pt] a_0&=a\left[\frac{\nu_1}{\nu} +\frac{\mu k\omega}{\nu a_1\Phi_{1abs}^{q_1-2}} +\frac{\nu_2 a_2}{\nu a_1}\cdot \frac{1}{\Phi_{1abs}^{q_1-q_2}}\right]. \end{aligned} \right\} \tag{32} \]

Let us denote, analogously to (19),

\[ \alpha_1=\frac{\nu_1 a_1}{\mu k\omega}\Phi_{1abs}^{q_1-2}. \tag{33} \]

Then, instead of (32), one may write

\[ k_0=k\left[1+\alpha_1+\frac{\nu_2 a_2}{\mu k\omega}\Phi_{1abs}^{q_2-2}\right], \]

\[ a_0=a\left[\frac{\nu_1}{\nu}+\frac{1}{\alpha_1} +\frac{\nu_2 a_2}{\nu_1 a_1}\Phi_{1abs}^{q_2-q_1}\right]. \]

But

\[ \frac{\nu_2 a_2}{\mu k\omega}\Phi_{1abs}^{q_2-2} =\alpha_1\frac{\nu_2 a_2}{\nu_1 a_1}\Phi_{1abs}^{q_2-q_1}. \]

Therefore, finally,

\[ k_0=k\left[1+\alpha_1+\alpha_1\frac{\nu_2 a_2}{\nu_1 a_1}\Phi_{1abs}^{q_2-q_1}\right], \tag{34} \]

\[ a_0=a\left[\frac{\nu_1}{\nu}+\frac{1}{\alpha_1} +\frac{\nu_2 a_2}{\nu_1 a_1}\Phi_{1abs}^{q_2-q_1}\right]. \tag{35} \]

Let us note that the first two terms in these expressions for \(k_0\) and \(a_0\) almost coincide with the terms in brackets on the right-hand sides of expressions (17) and (18), considered earlier, obtained for the case when the entire shaft of the installation was made of one material.

From (33) it can be seen that the laws governing the variation of the coefficient \(\alpha_1\) entering into (34) and (35), when, for example, the engine load regime changes, have the same character as the laws considered earlier for the variation of the coefficient \(\alpha\) entering into (17) and (18).

But the presence, in the brackets on the right-hand sides of expressions (34) and (35), of third terms containing \(\Phi_{1abs}\) leads to some change in the laws that were obtained above on the basis of an analysis of expressions (17) and (18).

For \(q_1=q_2=q\), expressions (34) and (35) reduce completely to (17) and (18), since here

\[ \alpha_1=\alpha;\qquad \nu_1=\nu . \]

For \(q_2>q_1\), the dependence of the generalized coefficient \(k_0\), for example on \(\Phi_{1abs}\), proves to be stronger, and the dependence of the coefficient \(a_0\) weaker, than we established earlier. With a sufficiently large difference \(q_2-q_1>0\), the coefficient \(a_0\), in contrast to the conclusions drawn earlier, will also increase with increasing \(\Phi_{1abs}\). It is evident that now, as the amplitude of the exciting moments \(M_h\) (or \(P_i\)) increases, both generalized coefficients \(k_0\) and \(a_0\) will increase.

Thus, in power units there may be cases in which the generalized damping coefficients \(k_0\) and \(a_0\) change in opposite directions, as well as in the same direction, when the engine load or speed regimes are changed.

In the present article we did not set ourselves the goal of a detailed study of the question of the generalized damping characteristics of torsional vibrations in power units with piston engines. Such a study can hardly be successfully carried out in general form without drawing on extensive experimental material and without a cumbersome numerical analysis of a number of typical concrete cases.

The purpose of the article is to show that the generalized damping coefficients \(k_0\) and \(a_0\), used in calculations according to the methods of Wydler and Lewis (as well as K. Wilson), are not constants even for one and the same installation. They change already when the speed or load regimes of the engine are changed, when the engine passes from single-node vibrations to two-node vibrations, or from the principal critical speeds to secondary ones.^1

These coefficients are too inconstant to have universal significance and to characterize damping in power units with any reliability. It is precisely for this reason that calculations of resonant torsional vibrations by the methods of Wydler, Lewis, and K. Wilson, based on the use of “experimental” values of the generalized coefficients \(k_0\) and \(a_0\), obtained by torsiographing installations “analogous” or “similar” to the one being calculated, yield satisfactory results only in rare cases.

The accumulation of a large amount of experimental data by torsiographing even a large number of diverse power units will not ensure an increase in the accuracy of our calculations, because of the inevitable (and necessarily large) “scatter” of these data.

It will be considerably more useful to carry out experimental and theoretical studies of linear damping in power units and to accumulate experimental data separately for the actual coefficients of viscous (linear) and hysteresis damping, \(k\), \(a\), and \(q\). Only this path can lead to a noticeable increase in the accuracy of calculating resonant torsional vibrations in comparison with the accuracy attainable at the present time.

^1 In the two following articles published in the present collection, an equally strong dependence of these generalized coefficients on the design parameters of the power unit is also shown.

References

1. Nechaev, V. K. Theoretical torsiograms for the shaft of an internal combustion engine. Proceedings of the Tomsk Industrial Institute, vol. 58, issue 2, 1937.

2. Melkumov, T. M. and Neiman, I. Sh. (eds.). Aircraft Piston Engines. Oborongiz, 1950.

3. Wydler, H. Drehschwingungen in Kolbenmaschinenanlagen. Berlin, 1922.

4. Lewis, F. Trans. Soc. Nav. Arch. and Mar. Eng., 1925.

5. Ker Wilson, W. Practical Solution of Torsional Vibration Problems. London, 1935.

6. Nechaev, V. K. On the relationships between the damping characteristics of torsional vibrations and certain parameters of an engine installation. Proceedings of the Tomsk Polytechnic Institute, vol. 85, 1957.