REDUCTION OF AN OSCILLATING SYSTEM

V. K. NECHAEV
Professor, Doctor of Technical Sciences

As is known, the first stage of every calculation of vibrations of the shaft of a power installation with an internal-combustion engine is the determination of the characteristics of the so-called equivalent, or reduced, system. This system, usually consisting of a series of concentrated (“point”) masses connected to one another by sections of a smooth massless shaft, must, in its vibration characteristics, be equivalent to the actual shafting of the installation.

Retaining distributed masses in the equivalent scheme is usually inadvisable, since it leads to a considerable complication of all subsequent calculations. The introduction of such distributed masses may be recommended only in installations with a very long shaft (certain ship installations), and also in calculations of the natural frequencies of vibration of shafts of engines with a large number of cylinders \[4, 6, 9\].

The calculation of the characteristics of the equivalent system is carried out according to known rules and formulas \[1, 5, 8\], obtained both theoretically and experimentally. Experimental data are especially necessary in determining the stiffness of the throw of a crankshaft, since the theoretical analysis of the deformations of a throw arising during its twisting is very complicated and gives results that agree poorly with experience.

As a result of carrying out such calculations, a power installation consisting, for example, of an \(m\)-cylinder in-line engine, a flywheel, and a dynamo, is reduced to a system consisting of \(m+2\) concentrated masses connected to one another by \(m+1\) pieces of an elastic and inertialess, that is, massless, shaft. The moments of inertia of all these masses, as well as the stiffnesses of all shaft sections, are determined by concrete numerical values.

Now, at least in the general case, the analysis of the vibrations of such a system can be reduced to the solution of a system of nonhomogeneous differential equations, the number of which is equal to the number of masses, \(m+2\). The form and character of these equations are determined both by the specific features of the oscillating system under investigation and, to a considerable extent, by the character of the damping forces that oppose the development of vibrations.

But the solution of such a system of equations, even for an installation with a small number of masses, presents exceptional difficulties and can scarcely be carried through to the end in practice, not only in general form but also in a concrete numerical form.

In order to reduce all statements and calculations connected with the investigation of vibrations of the crankshaft of a multicylinder internal-combustion engine, the greatest possible simplification of the complex reduced multimass system is advisable, by reducing it to a system with a smaller number of masses. The character and form of this new, already simplified, reduced system is determined both by the tasks of the subsequent calculations and by the equivalence conditions underlying the reduction.

Thus, the elementary device [7] is widely known, by means of which the calculation of the natural frequencies of a multi-mass system can be approximately reduced to the determination of the natural frequency of a two- or three-mass system. In this way the problem of the laborious determination of the roots of the frequency (secular) equation is reduced to the application of elementary simple formulas for the frequency of a system with two or three masses. This device is based on replacing all the reduced masses of the engine cylinders by one total mass concentrated at the midpoint of the length of the crankshaft.

Another reduction in the number of masses of the system being calculated was applied by us earlier, for example, in the paper “A Graphical Method for Determining the Frequencies of Natural Oscillations” [4]. There it was shown that, in free oscillations of a multi-mass system, the sum of the inertial moments (i.e., the moments of inertia forces) of all the engine masses is equal to the inertial moment of a certain fictitious, imaginary mass having moment of inertia \(\Theta/D_m\) and applied at the position of the last, \(m\)-th mass of the engine. The coefficient \(D_m\) is a variable quantity and, for given moments of inertia \(\Theta\) of all the masses of the installation, as well as for a known stiffness of all sections of the shaft, depends only on the character of the curve of the mode shape of the system’s oscillations and, consequently, on the angular frequency of the oscillations. Numerical values of \(D_m\) for different numbers of engine cylinders and different values of the parameter \(A\), characterizing the oscillation frequency, are given in the author’s paper [3].

Now, for example, the engine—flywheel installation is reduced merely to a two-mass system, whose free-oscillation mode-shape curve coincides with the free-oscillation mode-shape curve of the basic two-mass system on the section between the last mass of the engine and the flywheel (Fig. 1).

Specifically, this means that the inertial moment of all the engine masses is equal to the inertial moment of the reduced mass \(\Theta_r\), if its oscillation amplitude \(\varphi_r\) is equal to the oscillation amplitude \(\varphi_m\) of the last mass of the engine. In free oscillations of the basic system under consideration, engine—flywheel, the ratio of the amplitudes \(\varphi_m\) and \(\varphi_r\) is equal to the ratio of the moments of inertia \(\Theta_1\) and \(\Theta_r\).

Such a reduction, satisfying the condition of equality of the total inertial moment of all engine masses and the inertial moment of the reduced mass \(\Theta_r\), is quite sufficient for calculations of free oscillations, but it can no longer satisfy us in the analysis of forced oscillations, especially in studying questions of the development of, for example, resonant vibrations. The given reduction of masses, ensuring the same frequency of free natural oscillations of both systems (the actual one in Fig. 1-a and the reduced one in Fig. 1-b), does not give their equivalence with respect to a number of other dynamic characteristics.

Suppose that, in free oscillations of both systems \(a\) and \(b\), the oscillation amplitudes of the flywheels \(\varphi_1\) are identical; consequently, the amplitude \(\varphi_m\) of the last engine mass is equal to the amplitude \(\varphi_1\) of the reduced mass with moment of inertia \(\Theta_r\).

At the instant of greatest deflection of the masses, the first system has the potential energy

\[ \Pi=\frac{1}{2}\sum_{i=1}^{m-1} c_{i,i+1}\,(\varphi_i-\varphi_{i+1})^2+\frac{1}{2}c_{m1}(\varphi_m-\varphi_1)^2, \]

and the second, reduced system,

\[ \Pi_1=\frac{1}{2}c_{m1}(\varphi_m-\varphi_1)^2. \]

It is obvious that

\[ \Pi_1 < \Pi . \]

An analogous relation

\[ T_1 < T \]

could also be obtained for the kinetic energies \(T\) and \(T_1\), respectively, of the actual and the reduced systems.

Fig. 1

Fig. 2

Thus, our two systems are inequivalent to one another from the point of view of their kinetic and potential energy at any instant of time. It is clear that the behavior of these systems, under conditions, for example, of receiving the same excitation work during one cycle of oscillation, will be different.

Let us replace the entire left-hand part of the main system, located to the left of the node of oscillation (we consider here only oscillations of the one-node type), by one concentrated mass with moment of inertia \(\Theta_r\) (Fig. 2), situated somewhere to the left of the last, \(m\)-th mass of the engine. Let us denote by \(c_r\) the stiffness of the shaft section between this concentrated mass and the node of oscillation.

Now the left reduced single-mass system (Fig. 2-b) will be fully equivalent to the left part of the main multi-mass system (Fig. 2-a) in all principal dynamic indices, provided only that at any instant of time the following conditions are satisfied:

a) the left halves of both systems (Figs. 2-a and 2-b) must have identical natural frequencies \(\omega\), equal, of course, to the natural frequency \(\omega_0\) of the main multi-mass engine–flywheel system; this is important, for example, in order subsequently to have a completely equivalent assessment of both systems with respect to their remoteness from the resonant state or their proximity to it;

b) both systems must at every instant of time have, respectively, equal magnitudes of kinetic and potential energy; this is necessary for ensuring the equivalence of both systems, for example, under conditions of the development of nonstationary vibrations;

c) for equal amplitudes of the flywheels I in both systems, the nodal sections of the shaft must be loaded by equal twisting moments, i.e., the elastic line on these sections of the shaft in both systems must have the same angle of inclination to the shaft axis; this is necessary in order to ensure the equivalence of both systems from the standpoint of the loads arising in the shaft of the installation during vibrations;

d) under conditions of identical loading of the nodal sections of the shaft, the work of excitation performed, for example, by some harmonic of the total twisting moment in the engine must be the same in both systems;

e) finally, under these same conditions the loss of energy to damping over one cycle of oscillation must be the same in both systems.

The last two conditions, in essence, are entirely independent in character and must be satisfied separately, with the establishment of an appropriate method for reducing the exciting and damping forces throughout the main system. The first three of the conditions named above completely determine the principal characteristics of the reduced two-mass system \(\theta_r, C_r, \varphi_r\) (Fig. 2—b).

Considering the left part of this system, in accordance with the first condition we have

\[ \frac{C_r'}{\Theta_r}=\omega_0^2. \tag{1} \]

The kinetic energy of the left part of the main system is

\[ T=\frac{1}{2}\omega_0^2\sum_{i=1}^{i=m}\Theta_i\varphi_i^2, \]

(we assume here free, monoharmonic oscillations), and the kinetic energy of the left half of the reduced two-mass system is

\[ T_1=\frac{1}{2}\omega_0^2\Theta_r\varphi_r^2. \]

The equivalence condition formulated above in item “b” requires that the equality

\[ T_1=T, \]

i.e.

\[ \frac{1}{2}\omega_0^2\sum_{i=1}^{m}\Theta_i\varphi_i^2 = \frac{1}{2}\omega_0\Theta_r\varphi_r^2, \]

be satisfied, or

\[ \sum_{i=1}^{m}\Theta_i\varphi_i^2=\Theta_r\varphi_r^2 \tag{2} \]

(here \(\varphi\) may be understood as the instantaneous angles of deflection of the corresponding masses).

Similarly, we write the expressions for the potential energy of the left parts of, respectively, the main multimass system and the reduced two-mass system:

\[ \Pi=\frac{1}{2}\sum_{i=1}^{m}c_{i,i+1}(\varphi_i-\varphi_{i+1})^2, \]

where, according to Fig. 2-a,

and

\[ \left. \begin{aligned} \varphi_{m+1}&=0,\\ c_{m,m+1}&=c_{m1}^{\prime},\\ \Pi_1&=\frac{1}{2}\,c_r^{\prime}\cdot\varphi_r^{2}. \end{aligned} \right\} \tag{3} \]

Hence, on the basis of the same condition of item “b,”

\[ \Pi_1=\Pi, \]

or

\[ \sum_{i=1}^{m} c_{i,i+1}(\varphi_i-\varphi_{i+1})^2 = c_r^{\prime}\varphi_r^{2}, \tag{4} \]

where the equalities (3) remain valid.

Further, from the condition of item “b” we have

\[ \varphi_r c_r^{\prime}=\varphi_m c_{m1}^{\prime}. \tag{5} \]

Geometrically, this last relation formulates the natural requirement that there be no kink in the elastic shaft line of the reduced two-mass system at the point where it passes through the vibration node.

Condition \(a\) and formula (1) are not independent and follow entirely from the obvious and necessary equality, under conditions of harmonic undamped oscillations, of the extreme values of \(\Pi\) and \(T\) in each system during each cycle. Indeed, for the left part of the basic system we have the following expressions for the maximum kinetic and potential energies:

\[ T_{\max}=\frac{1}{2}\omega_0^2\sum_{i=1}^{m}\Theta_i\varphi_i^2, \]

\[ \Pi_{\max}=\frac{1}{2}\sum_{i=1}^{m}c_{i,i+1}(\varphi_i-\varphi_{i+1})^2, \]

\[ T_{\max}=\Pi_{\max} = \frac{1}{2}\omega_0^2\sum_{i=1}^{m}\Theta_i\varphi_i^2 = \]

\[ =\frac{1}{2}\sum_{i=1}^{m}c_{i,i+1}(\varphi_i-\varphi_{i+1})^2, \]

if here by \(\varphi\) one understands already the amplitudes of oscillation of the corresponding masses. Hence the angular frequency of the free oscillations of the basic multi-mass system is

\[ \omega_0^2= \frac{ \displaystyle \sum_{i=1}^{m} c_{i,i+1}(\varphi_i-\varphi_{i+1})^2 }{ \displaystyle \sum_{i=1}^{m}\Theta_i\varphi_i^2 } \tag{6} \]

with, of course, the relations (3) taken into account.

Similarly, for the left-hand side of the reduced two-mass system,

\[ \Pi_{1\max}=T_{1\max}=\frac{1}{2}c_r'\varphi_r^2=\frac{1}{2}\Theta_r\varphi_r^2\omega_r^2 \]

and its natural frequency is

\[ \omega_r=\sqrt{\frac{c_r'}{\Theta_r}} \]

or, substituting here the values of \(c_r'\) and \(\Theta_r\) from (4) and (2), we obtain

\[ \omega_r^2=\omega_0^2. \]

Therefore, the independent equations following from conditions “a,” “b,” and “c,” which can be used to determine the basic characteristics \(c_r'\), \(\Theta_r\), \(\varphi_r\) of the reduced system, are only equations (4), (5), and (6).

From equations (4) and (5) we have

\[ \sum_{i=1}^{m} c_{i,i+1}(\varphi_i-\varphi_{i+1})^2=\varphi_m c_{ml}'\varphi_r, \]

and hence the amplitude of vibration of the left-hand, reduced, mass in Fig. 2-b is

\[ \varphi_r= \frac{\displaystyle\sum_{i=1}^{m} c_{i,i+1}(\varphi_i-\varphi_{i+1})^2} {\varphi_m c_{ml}'}, \tag{7} \]

where relations (3) remain valid.

Now, from (2),

\[ \Theta_r= \frac{\displaystyle\sum_{i=1}^{m}\Theta_i\varphi_i^2}{\varphi_r^2}, \]

or, substituting here the value of \(\varphi_r\) from (7),

\[ \Theta_r= \frac{\displaystyle\sum_{i=1}^{m}\Theta_i\varphi_i^2} {\displaystyle\sum_{i=1}^{m} c_{i,i+1}(\varphi_i-\varphi_{i+1})^2} \cdot \frac{(\varphi_m c_{ml}')^2} {\displaystyle\sum_{i=1}^{m} c_{i,i+1}(\varphi_i-\varphi_{i+1})^2}. \]

Hence, using (6), we obtain the expression for the moment of inertia of the left-hand mass of the two-mass reduced system

\[ \Theta_r= \frac{(\varphi_m c_{ml}')^2} {\displaystyle \omega_0^2\sum_{i=1}^{m} c_{i,i+1}(\varphi_i-\varphi_{i+1})^2}. \tag{8} \]

Further, from (1),

\[ c_r'=\Theta_r\omega_0^2, \]

or, according to (8),

\[ c_r'= \frac{\varphi_m^2 c_{ml}'^{\,2}} {\displaystyle\sum_{i=1}^{m} c_{i,i+1}(\varphi_i-\varphi_{i+1})^2}. \tag{9} \]

Thus the stiffness \(c'_r\) of the shaft segment of the reduced system from the left mass to the vibration node is determined (Fig. 2-a). Here, of course,

\[ c_{m,m+1}=c_{m1}', \]

\[ \varphi_m=0. \]

Finally, the stiffness \(c_r\) of the entire shaft of the reduced two-mass system may be computed by the obvious formula

\[ \frac{1}{c_r}=\frac{1}{c'_r}+\frac{1}{c_{10}}, \]

or

\[ c_r=\frac{c'_r\cdot c_{10}}{c'_r+c_{10}}, \tag{10} \]

where \(c_{10}\) is the stiffness of the shaft segment of the original multimass system from the vibration node to the flywheel.

By formulas (7), (8), (10) the principal characteristics of the reduced two-mass system, equivalent to the actual multimass system under conditions of resonant vibrations, are completely determined, when the usual assumption is sufficiently exactly valid that, at resonance or near it, the curve of the shape of the forced vibrations is very close to the shape of the free vibrations of the multimass system without damping [1, 2]. The amplitudes \(\varphi_i\) of the vibrations of the individual masses of the multimass system under consideration that enter into all these formulas may be found from the calculation of the natural vibration frequencies, carried out by tabular methods, or, considerably more rapidly, are computed by the method set forth by the author in his previous works [3, 4]. It is clear that, as numerical values of \(\varphi_i\), the relative amplitudes of the free vibrations may be used here, that is, the amplitudes of vibration of all the masses of the installation, computed on the assumption that the amplitude of vibration of the reduced mass of the first cylinder (the extreme left-hand mass in Fig. 2-a) is equal to unity.

Fig. 3

Now let us turn to the consideration of the items “г” and “д” of the conditions mentioned above on p. 83, which the reduced two-mass system must satisfy in order that it may be used in calculations of forced vibrations of multimass systems.

To ensure complete, not only “geometrical” but also energetic equivalence of the actual and reduced systems with respect to all principal indices, it is necessary also to carry out a reduction of the exciting moments and damping forces in the actual multimass installation under consideration.

Let external exciting harmonic torques \(M_h\sin(\eta_h t+\varepsilon_h)\), with angular frequency \(\eta_h\), act on the reduced masses of the engine cylinders (Fig. 3-a). Of course, it should be assumed that these moments

in the individual cylinders are different in phase, in accordance with the order of alternation of flashes in the cylinders, the engine timing, etc.

As is known [1, 3], the work of all these harmonic moments over one cycle of oscillations under resonance conditions can be determined by the expression

\[ A_{\text{excit}}=\pi M_h \varphi_{1\text{abs}} \overrightarrow{\sum}\varphi_{i\text{rel}}, \tag{11} \]

where, as usual:

\(\varphi_{1\text{abs}}\) is the absolute amplitude of oscillations of the first mass of the system, that is, the reduced mass of the first cylinder of the engine,

\[ \overrightarrow{\sum}\varphi_{i\text{rel}} \]

is the geometric sum of the relative amplitudes of oscillations of the reduced masses of all operating cylinders of the engine, determined from the phase diagram constructed for the exciting moments of the order that interests us.

Let us replace all the moments \(M_h\) acting on the reduced masses of the cylinders of the engine of the actual installation by a single equivalent reduced moment \(M_{h\text{red}}\), applied to the left mass of the reduced two-mass system (Fig. 3-b).

The work of this moment over a cycle of oscillation is:

\[ A'_{\text{excit}}=\pi M_{h\text{red}}\varphi_{r\text{abs}}, \]

where \(\varphi_{r\text{abs}}\) is the absolute amplitude of the forced oscillations of the left mass of the system in Fig. 3-b.

The condition for equivalence of the moments \(M_h\) and \(M_{h\text{red}}\) is

\[ A_{\text{excit}}=A'_{\text{excit}} \]

or:

\[ \pi M_h \varphi_{1\text{abs}}\overrightarrow{\sum}\varphi_{i\text{rel}} = \pi M_{h\text{red}}\varphi_{r\text{abs}}. \]

But, obviously, the following relation can be written between \(\varphi_{r\text{abs}}\), \(\varphi_{1\text{abs}}\), and \(\varphi_{r\text{rel}}\):

\[ \varphi_{r\text{abs}}=\varphi_{1\text{abs}}\cdot\varphi_{r\text{rel}}. \tag{12} \]

Substitution of this value for \(\varphi_{r\text{abs}}\) into the preceding expression for equality of the works of excitation over a cycle of oscillation gives, finally, the following relation for calculating the amplitude of the reduced exciting moment acting on the left mass of the reduced two-mass system:

\[ M_{h\text{red}}=M_h \frac{\overrightarrow{\sum}\varphi_{i\text{rel}}}{\varphi_{r\text{rel}}}, \tag{13} \]

where the \(\varphi_{r\text{rel}}\) entering here is readily determined from formula (7), if only in the latter, on its right-hand side, the corresponding relative amplitudes of oscillations are understood by \(\varphi\), that is,

\[ \varphi_{r\text{rel}}= \frac{ \sum_{i=1}^{m} c_{i,i+1}\left(\varphi_{i\text{rel}}-\varphi_{i+1,\text{rel}}^{2}\right) }{ \varphi_{m\text{rel}}\cdot c_{m1} }, \tag{14} \]

taking account of equality (3).

In what follows, we shall carry out all calculations of forced vibrations for the reduced two-mass system. From such calculations, for example, the absolute resonant amplitude \(\varphi_{r\,abs}\) of the left-hand mass of this system may be found. After this, using formulas (12) and (14), one can already pass to the actual multi-mass original system and determine the absolute amplitude of oscillation of its first mass by the formula:

\[ \varphi_{1abs} = \frac{\varphi_{r\,abs}}{\varphi_{r\,rel}} = \varphi_{r\,abs}\cdot \frac{\varphi_{m\,rel}\cdot c_{m1}'}{ \displaystyle\sum_{i=1}^{m} c_{i,i+1}\left(\varphi_{i,rel}-\varphi_{i+1,rel}\right)^2 }. \tag{15} \]

Further, from \(\varphi_{1abs}\) and the relative amplitudes \(\varphi_{i\,rel}\) known from the calculation carried out earlier, it is easy to compute the absolute amplitudes of oscillation of all masses of the actual installation, the absolute angles of twist of each section of the shaft, and the corresponding additional vibrational stresses arising in this shaft due to vibrations.

Of course, all this is fully valid for the time being only with respect to resonant vibrations, i.e., shaft vibrations arising while the engine is operating at one of the critical speeds. But these calculations too can be carried out concretely only by taking into account the damping forces in the installation.

Let us consider the principal cases of reducing the damping forces. Suppose first that, in the given specific installation, damping forces exist only in the engine and are practically completely subject to the usual velocity law [1], [3]. As is known, the work of velocity damping in the entire engine over one cycle of oscillation can be determined by the expression [3]:

\[ A_{damp} = \pi k_0 F r^2 \omega \varphi_{1abs}^{2} \sum_{i=1}^{m}\varphi_{i\,rel}^{2}, \tag{16} \]

where:

\[ F \text{ — piston area,} \]

\[ r \text{ — crank radius,} \]

\[ \omega \text{ — angular frequency of the forced vibrations,} \]

\[ \sum_{i=1}^{m}\varphi_{i\,rel}^{2} \text{ — the arithmetic sum of the squares of the relative amplitudes} \]

of oscillation of the reduced masses of all cylinders of the engine,

\[ k_0 \text{ — coefficient of velocity damping in the engine.} \]

Let us replace all damping in the engine of the installation in Fig. 3-a by equivalent damping with coefficient \(k_r\), acting on the left-hand mass of the reduced two-mass system in Fig. 3-b. This reduced damping coefficient is determined from the following relations, proceeding from the necessary equality of the energy losses due to damping over one cycle of oscillation in both systems.

Suppose that, in the oscillations of the reduced two-mass system, its left-hand mass oscillates, for example, according to the following harmonic law:

\[ (\varphi_{r\,abs})=\varphi_{r\,abs}\sin\omega t, \]

where by \((\varphi_{r\,abs})\) we denote the instantaneous angle of deflection of the mass, as distinct from the amplitude of oscillation \(\varphi_{r\,abs}\). A damping moment acts on this mass:

\[ M_r=-k_r\frac{d(\varphi_{r\,abs})}{dt}. \]

The corresponding work absorbed by this moment over one cycle of oscillation,

\[ A'_{\text{dempf}}=\int_{0}^{t=2\pi/\omega_{0}} M_r\,d\varphi = \]

\[ =\int_{0}^{t=2\pi/\omega} -k_r \varphi^2 r_{\text{abs}}\omega^2 \cos^2 \omega t\,dt = \pi k_r \omega \varphi^2 r_{\text{abs}} . \tag{17} \]

Equating the right-hand sides of expressions (16) and (17),

\[ \pi k_0 F r^2 \omega \varphi_{1\text{abs}}^2 \sum_{i=1}^{m} \varphi_{i\text{rel}}^2 = \pi k_r \omega \varphi^2 r_{\text{abs}}, \]

and using relation (12), we obtain the final expression for the reduced damping coefficient \(k_r\) in the reduced two-mass system:

\[ k_r = \frac{ k_0 F r^2 \displaystyle\sum_{i=1}^{m} \varphi_{i\text{rel}}^2 }{ \varphi_{r\text{rel}}^2 }. \tag{18} \]

The value \(\varphi_{r\text{rel}}\) entering here is determined by formula (14).

Thus, the system in Fig. 3-a, consisting of a multi-cylinder engine with a flywheel, is now reduced to a two-mass system, where flywheel I remains unchanged, while all the masses of the engine are replaced by a single concentrated mass with moment of inertia \(\Theta_r\) (Fig. 3-b). This new equivalent or reduced system has:

moment of inertia of the left mass \(\Theta_r\), determined by formula (8);

stiffness of the connecting shaft \(C_r\), determined by formula (10);

reduced exciting moment \(M_{h\,red}\), acting on the left mass (formula 13);

reduced coefficient of velocity damping \(k_r\) (formula 18);

relative amplitude of oscillation \(\varphi_r\) of the left mass, expressed in fractions of the oscillation amplitude of the leftmost mass of the actual installation and represented by formula (14).

The monoharmonic moment \(M_{h\,red}\) causes forced oscillations of the reduced system. Of course, \(M_{h\,red}\) will differ in amplitude, phase, and frequency, depending on the order of the harmonic moments \(M_h\) acting on the crankshaft journals of the engine of the actual power installation.

If, in the installation, in addition to velocity damping in the engine, there is also, for example, velocity damping of the propeller (that is, as if flywheel 1 in our preceding diagrams), then the magnitude of the reduced damping coefficient \(k_r\) in the reduced system will change somewhat.

Here we have at our disposal at least two possible choices of the type and character of the equivalent, reduced damping. All damping moments in the engine and the damping moment of the propeller (or dynamo) may be replaced by an equivalent damping moment \(M_r\) applied to the left mass of the reduced two-mass system, or else by a damping moment \(M'_r\) acting in the shaft between the two masses.

In the first case the moment \(M_r\) will depend on the instantaneous angular velocity of the oscillatory motion of the left-hand mass and will be determined by the expression

\[ M_r=-k_r\frac{d(\varphi_{r\,abs})}{dt}. \]

The work of this moment over one cycle of oscillation is:

\[ A_r'=\int_{0}^{t=2\pi/\omega} M_r\,d\varphi =\pi k_r\omega \varphi_{1\,abs}^{2}\cdot \varphi_{r\,otn}^{2}. \tag{19} \]

In the second case, however, the moment \(M_r'\) will depend on the difference of the angular velocities of the oscillatory motion of both masses, that is, it will be determined by the formula

\[ M_r'=-k_r'\frac{d(\varphi_{r\,abs}-\varphi_{1\,abs})}{dt}. \]

The corresponding damping work over one cycle of oscillation is

\[ A_r'=\pi k_r'\omega \varphi_{1\,abs}^{2}(\varphi_{r\,otn}-\varphi_{1\,otn})^{2}. \tag{20} \]

Of course, this latter method of introducing equivalent damping into the reduced system has unquestionable advantages over the first one (formula 19), since here the regularities of vibration of the masses of the installation are taken into account in greater detail.

The work absorbed during one cycle of oscillation by the damping of the propeller can be calculated from the formula:

\[ A_{vinta}=\pi k_{vinta}\omega \varphi_{1\,abs}^{2}. \tag{21} \]

Now the total damping work in the engine, according to (16) and (21),

\[ A_{dempf}=\pi k_0 F r^2\omega \varphi_{1\,abs}^{2} \sum_{i=1}^{m}\varphi_{i\,otn}^{2} \]

\[ +\pi k_{vinta}\omega \varphi_{1\,abs}^{2}. \tag{22} \]

Equating the left-hand sides of (19) and (22), we obtain an expression for the reduced coefficient of velocity damping \(k_r\), determining the damping moment acting on the left-hand mass of the reduced two-mass system:

\[ k_r= \frac{ \pi k_0 F r^2\omega \varphi_{r\,abs}^{2} \sum_{i=1}^{m}\varphi_{i\,otn}^{2} +\pi k_{vinta}\omega \varphi_{1\,abs}^{2} }{ \pi\omega \varphi_{1\,abs}^{2}\varphi_{r\,otn}^{2} } = \]

\[ = \frac{ k_0 F r^2\sum_{i=1}^{m}\varphi_{i\,otn}^{2} +k_{vinta}\varphi_{1\,otn}^{2} }{ \varphi_{r\,otn}^{2} }. \tag{23} \]

Similarly, from comparison of the right-hand sides of (20) and (22) we obtain the value of the reduced damping coefficient

\[ k_r'= \frac{ k_0 F r^2\sum_{i=1}^{m}\varphi_{i\,otn}^{2} +k_{vinta}\varphi_{1\,otn}^{2} }{ (\varphi_{r\,otn}-\varphi_{1\,otn})^{2} }, \tag{24} \]

which makes it possible to determine the reduced damping moment in the shaft of the reduced two-mass system, equivalent from the point of view of losses

energy per cycle of oscillation to the entire velocity damping in the actual power plant.

In the case where, in addition to velocity damping forces, there is also a considerable hysteresis loss of energy in the shaft, it is nevertheless possible—although with a substantially lower degree of accuracy—to introduce into the vibration calculations of the reduced two-mass system the reduced coefficient of velocity damping on the basis of the general relation

\[ k_r=\frac{\sum A_{\text{damp}}}{\pi\omega\varphi_{1\,\text{abs}}^2\varphi_{r\,\text{rel}}}, \tag{25} \]

or

\[ k'_r=\frac{\sum A_{\text{damp}}}{\pi\omega\varphi_{1\,\text{abs}}^2(\varphi_{r\,\text{rel}}-\varphi_{i\,\text{rel}})^2}. \tag{26} \]

Here \(\sum A_{\text{damp}}\) denotes the total loss of energy to damping in the actual plant during one cycle of oscillation. This sum may also include the loss due to hysteresis in the metal of the shaft.

According to the experiments of a number of investigators \([1, 3]\), the loss of energy due to hysteresis of the shaft metal during one cycle of oscillation is characterized by the following general expression:

\[ A_{\text{hist}}=a_z\varphi_{1\,\text{abs}}^{q_z}, \]

where the exponent \(q_z\) is always \(>2\). Thus, the work of hysteresis increases with increasing amplitude of oscillation considerably faster than the work absorbed by velocity damping.

It follows from all the preceding formulas that, in a plant with only velocity damping, the reduced damping coefficient \(k_r\) or \(k'_r\), determined by formulas (23) and (24), proves to be constant, not depending on the absolute magnitude of the amplitudes of oscillation of the masses of the plant. When, however, hysteresis losses are present in the plant, the values \(k_r\) and \(k'_r\) depend to a strong degree on the absolute amplitudes of the shaft twist angles in each section, that is, ultimately, on the amplitudes of oscillation of the masses of the system. Consequently, here one can no longer regard the coefficients \(k_r\) or \(k'_r\) as identical for different critical engine speeds. This, in particular, is the reason for the experimentally observed dependence of \(k_r\), and in general of the damping coefficient, on the number of revolutions and on the operating regime of the engine.

Above, a method was shown for reducing the multi-mass power plant engine—flywheel to a two-mass system equivalent to it (at least in a number of basic characteristics and indices). More complex plants can analogously be approximately reduced to a two-mass system.

Let us consider the power plant in Fig. 4-a, consisting of a multi-cylinder engine, a flywheel with moment of inertia \(\Theta_1\), and a dynamo whose rotor has moment of inertia \(\Theta_{\mathrm{II}}\). Suppose that, in the free oscillations of the given system, the node of oscillation is located between the flywheel and the dynamo. We shall replace this complex multi-mass system by a sufficiently equivalent two-mass system shown in Fig. 4-b. The basis for such a replacement will be the conditions formulated above, on p. 83, imposed by us on the reduction of the engine—flywheel plant according to Fig. 3. From the equal-

of the kinetic energies of the left parts of both systems \(a\) and \(b\) in Fig. 4, analogously to formula (2), we have

\[ \sum_{i=1}^{m} \Theta_i \varphi_{i\,\mathrm{rel}}^{2} + \Theta_I \varphi_{1\,\mathrm{rel}}^{2} = \Theta_r \varphi_{r\,\mathrm{rel}}^{2}. \tag{27} \]

From the necessary equality of potential energies, analogously to (4),

\[ \sum_{i=1}^{m} c_{i,i+1} \left(\varphi_{i\,\mathrm{rel}}-\varphi_{i+1\,\mathrm{rel}}\right)^2 + c'_{I\,II}\varphi_{1\,\mathrm{rel}}^{2} = c'_r \varphi_{r\,\mathrm{rel}}^{2}, \tag{28} \]

where \(c'_r\) is the stiffness of the shaft segment of the reduced two-mass system between the left mass and the node of vibration;

\(c'_{I\,II}\) is the stiffness of the shaft segment of the actual system between the node of vibration and the flywheel.

Finally, from the condition of identical loading by torques of the nodal sections of the shafts of both systems, i.e., from the condition of the absence of fracture of the elastic line of the shaft of the reduced two-mass system, analogously to (5), we write:

Fig. 4

\[ c'_r \varphi_{r\,\mathrm{rel}} = \varphi_{1\,\mathrm{rel}}\,c'_{I\,II}. \tag{29} \]

From these relations it is easy to obtain the following basic characteristics of the reduced system according to Fig. 4-b. From (28) and (29):

\[ \varphi_{r\,\mathrm{rel}} = \frac{ \sum_{i=1}^{m} c_{i,i+1} \left(\varphi_{i\,\mathrm{rel}}-\varphi_{i+1\,\mathrm{rel}}\right)^2 + c'_{I\,II}\varphi_{1\,\mathrm{rel}}^{2} }{ \varphi_{1\,\mathrm{rel}}\,c'_{I\,II} }. \tag{30} \]

From (30) and (27):

\[ \Theta_r = \frac{ \sum_{i=1}^{m} \Theta_i \varphi_{i\,\mathrm{rel}}^{2} + \Theta_I \varphi_{1\,\mathrm{rel}}^{2} }{ \varphi_{r\,\mathrm{rel}}^{2} } = \]

\[ = \frac{ \displaystyle \sum_{i=1}^{m}\Theta_i\varphi_{i\,\mathrm{rel}}^{2}+\Theta_1\varphi_{1\,\mathrm{rel}}^{2} }{ \displaystyle \left[ \sum_{i=1}^{m} c_{i,i+1}\left(\varphi_{i\,\mathrm{rel}}-\varphi_{i+1,\mathrm{rel}}\right)^2 +c'_{\mathrm{III}}\varphi_{1\,\mathrm{rel}}^{2} \right] } \cdot \frac{ \displaystyle \left(\varphi_{1\,\mathrm{rel}}c'_{\mathrm{III}}\right)^2 }{ \displaystyle \left[ \sum_{i=1}^{m} c_{i,i+1}\left(\varphi_{i\,\mathrm{rel}}-\varphi_{i+1,\mathrm{rel}}\right)^2 +c'_{\mathrm{III}}\varphi_{1\,\mathrm{rel}}^{2} \right] }. \]

But it is easy to see that the first factor in the last expression is equal to \(1/\omega_0^2\), where \(\omega_0\) is the natural frequency of the multimass system under consideration; therefore

\[ \Theta_r= \frac{1}{\omega_0^2}\, \frac{ \left(\varphi_{1\,\mathrm{rel}}c'_{\mathrm{III}}\right)^2 }{ \displaystyle \sum_{i=1}^{m} c_{i,i+1}\left(\varphi_{i\,\mathrm{rel}}-\varphi_{i+1,\mathrm{rel}}\right)^2 +c'_{\mathrm{III}}\varphi_{1\,\mathrm{rel}}^{2} }. \tag{31} \]

It is further obvious that, by the fundamental condition of equivalence of the left-hand parts of the two systems,

\[ c'_r=\Theta_r\omega_0^2, \]

and, according to (31),

\[ c'_r= \frac{ \left(\varphi_{1\,\mathrm{rel}}c'_{\mathrm{III}}\right) }{ \displaystyle \sum_{i=1}^{m} c_{i,i+1}\left(\varphi_{i\,\mathrm{rel}}-\varphi_{i+1,\mathrm{rel}}\right)^2 +\left(c'_{\mathrm{III}}\varphi_{1\,\mathrm{rel}}^{2}\right) }. \tag{32} \]

Of course, in all the preceding formulas \(m\) is the number of engine cylinders, and:

\[ c_{m,m+1}=c_{m1}, \]

\[ \varphi_{m+1}=\varphi_1. \]

The relative vibration amplitudes of the masses of the actual installation entering here are determined from the usual calculation of the mode-shape curve of free vibrations.

As before, all exciting moments \(M_h\) of one order may be replaced by a single reduced moment \(M_{h\,red}\), applied to the left-hand mass of the system in Fig. 4-b. The amplitude of this moment is determined by formula (13). Finally, by formulas (18), (23), or (24), the reduced damping coefficient in the reduced system is calculated.

Thus, any multimass system schematically representing a complex power installation with an internal-combustion engine can be approximately reduced to a reduced two-mass system, with all its characteristics concretely known. Of course, the dynamic investigation of such a reduced system is considerably simpler than the investigation of a complex multimass installation, the analysis of which is associated with great and, in many respects still practically insurmountable, mathematical difficulties.

To what extent, then, is the reduction method set forth above universal and reliable? In applying it we had to make use of data,

for example, with respect to the amplitudes of the free, undamped oscillations of the basic multi-mass system. As was already noted earlier, the curve of the shape of the forced resonant vibrations is very close to the curve of the shape of the free undamped oscillations. Consequently, the proposed reduction method is quite reliable and acceptable in the analysis of resonant vibrations.

In other cases, however, when the vibrations of the system occur under off-resonance conditions, the curve of the shape of the forced oscillations proves to be different; to a certain extent the results of the reduction may also turn out to be different. But the use of a reduced two-mass system with characteristics varied as a function, for example, of the excitation frequency complicates our problems so greatly that this must be abandoned. Therefore, also in the investigation of off-resonance vibrations (at least in the region close to resonance), the reduced two-mass system with constant characteristics \(c\), \(\Theta\), and \(k\), independent of the frequency of the exciting moments acting on the crank throws of the engine shaft, is a useful scheme.

LITERATURE

1. Lur’e I. A. — Torsional oscillations in diesel installations. Voenmorizdat, 1940.

2. Natanzon V. Ya. — On forced oscillations of crankshafts at resonance. Tekhnika Vozdushnogo Flota. 1941. No. 2.

3. Nechaev V. K. — Theoretical torsiograms for internal-combustion-engine shafts. Izvestiya TII, vol. 58, 1937, issue 2.

4. Nechaev V. K. — A graphical method for determining the frequencies of natural oscillations. Izvestiya TII, vol. 54, 1934, issue 6. Dieselestroenie, 1934, No. 4.

5. Terskikh V. P. — Torsional oscillations of power installations. Sudpromgiz. 1940.

6. Tetel’baum I. M. — Analysis of torsional oscillations of installations with long shafting. Vestnik Inzhenerov i Tekhnikov. 1935. No. 9.

7. Geiger J. — Mechanische Schwingungen, Berlin, 1927.

8. W. Ker Wilson. — Practical Solution of Torsional Vibration Problems. New-York. 1935.

9. Porter. F. — Trans. A.S.M.E., Vol. 53, 1931.

NOTED MISPRINTS

Izv. TPI, vol. 61, issue 2.