Eng. V. K. NECHAEV.

(On the question of torsional vibrations of shafts.)

GRAPHICAL METHOD FOR DETERMINING NATURAL VIBRATION FREQUENCIES

Determining the frequencies of the natural torsional vibrations of a shafting system is the first and most important stage of every calculation of power plants with piston engines for torsional vibrations.

In many practical cases, knowledge of the frequencies of the single-node and two-node vibrations of the reduced system is sufficient for judging the operational reliability of a given power plant from the standpoint of torsional vibrations of its shaft. Such a situation usually occurs in installations operating at a constant number of revolutions.

More detailed calculations, including the determination of the amplitudes of forced vibrations and of the torsional stresses in the shaft arising at various critical speeds, are necessary in installations operating over a broad range of speeds, since here it is not always possible to avoid one or another resonance, even of weak orders.

The general technique for determining the frequencies of the natural (free) torsional vibrations of a shaft having a number of flywheel masses fastened to it, as is known, reduces to solving a system of linear differential equations.

A solution of this system of equations in general form, up to a final ready formula for calculating the natural vibration frequencies of the system, exists only for a shaft with two and three flywheel masses. For systems with a larger number of masses, this approach leads to extremely cumbersome determinants, the expansion of which gives expressions scarcely suitable for practical calculations because of their complexity¹.

At the present time there are a number of more practical methods for determining the frequencies of natural torsional vibrations of multi-mass—

system¹⁾. All of them give the desired solution as the result of a number of trials and successive approximations. A common shortcoming of all these methods must be considered the need to carry out a large amount of computational work. This circumstance hinders their broad introduction into practice.

In most installations with internal-combustion engines there is a series of masses equal in magnitude and uniformly arranged along the shaft (the masses of the connecting-rod-and-crank mechanisms of the engine cylinders). In these cases the solution of the problem can be simplified. For this purpose there are a number of methods for more rapidly determining the natural vibration frequencies of the system²⁾. In most of them all the flywheel masses of all the engine cylinders are replaced by one mass uniformly distributed over the entire length of the crankshaft. This device, being a known assumption, somewhat simplifies the determination of the desired frequencies, but leads to finding the roots of transcendental equations. Only Behrens dispenses with this assumption. But Behrens solved the problem only for the special case when a multi-cylinder engine has on its shaft only one other large flywheel mass (flywheel).

We shall set forth here a more general device, making it possible to determine, by a graphical method and without lengthy trial calculations, the natural vibration frequencies of a shaft with \(n\) equal flywheel masses and two large flywheels (such a case is encountered, for example, in the system of a compressorless diesel engine—flywheel—dynamo, and in a number of screw-propeller ship installations).

The basis of this device is the method of reducing all the masses of the vibrating system to one point, to one section of the shaft.

I.

Let us consider an elastic vibrating system consisting of a series of flywheel masses connected with one another by sections of an elastic shaft (Fig. 1). The shaft lies in absolutely smooth bearings.

Let us denote:

\(\Theta_1, \Theta_2 \ldots \Theta_n\)—the moments of inertia of the flywheel masses of each engine cylinder with respect to the axis of the shaft,

¹⁾ H. Holzer. Die Berechnung der Drehschwingungen. Berlin 1921.
H. Wydler. Drehschwingungen in Kolbenmaschinenanlagen. Berlin 1922.
M. Tolle. Regelung der Kraftmaschinen. Berlin. 1921.
J. Geiger. Mechanische Schwingungen. Berlin 1927.
F. Lewis. Torsional Vibrations in the Diesel Engine. Trans. Soc. Nav. Arch. and Mar. Eng. 1925.
F. Porter. The range and severity of torsional vibration in Diesel engine. Transactions A. S. M. E., Vol. 49—50, 1927—28, and also articles by Gümbel, Dreves, and others in Z. d. V. D. I.; A. Cherevkov. On the computation of torsional vibrations in shafts, Bulletin of Engineers, 1933, No. 7.

²⁾ W. Benz. Automobil—Techn. Zeitschrift, 1930 Heft 27, 1934 Heft 5,
Göller. Z. d. V. D. J., 1930, No. 16,
Behrens. Z. d. V. D. J., 1930, No. 20.
Porter. Transactions A. S. M. E. Vol. 53 (1931); Vol. 51 (1929),
Gorfinkel. Le Génie Civil. 1932, No. 15, 16.

\(\Theta_{\mathrm{I}}, \Theta_{\mathrm{II}}\)—moments of inertia of the masses of flywheels I and II with respect to the same axis,

\(m_1, m_2, \ldots, m_n, m_{\mathrm{I}}, m_{\mathrm{II}}\)—the magnitudes of the corresponding masses, reduced to the crank radius \(r\)¹);

\(c_{12}, c_{23}, \ldots, c_{n-1,n}, c_{n\mathrm{I}}, c_{\mathrm{I}\mathrm{II}}\)—the stiffnesses of the shaft sections between adjacent flywheel masses; the indices at \(c\) denote the numbers of the masses lying at the ends of the given shaft section²);

\(\varphi'_1, \varphi'_2, \ldots, \varphi'_n, \varphi'_{\mathrm{I}}, \varphi'_{\mathrm{II}}\)—respectively, the angles of deviation of the masses \(m_1, m_2, \ldots, m_n, m_{\mathrm{I}}, m_{\mathrm{II}}\) at the given instant of time from their mean, equilibrium position,

\(\varphi_1, \varphi_2, \ldots, \varphi_n, \varphi_{\mathrm{I}}, \varphi_{\mathrm{II}}\)—the angular amplitudes of vibration of the corresponding masses.

For all the masses of the vibrating system in Fig. 1 one may write the following system of differential equations:

\[ \begin{aligned} &\Theta_1 \frac{d^2 \varphi'_1}{dt^2} + c_{12}(\varphi'_1-\varphi'_2)=0,\\ &\Theta_2 \frac{d^2 \varphi'_2}{dt^2} - c_{12}(\varphi'_1-\varphi'_2) + c_{23}(\varphi'_2-\varphi'_3)=0,\\ &\cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\\ &\Theta_{n-1}\frac{d^2 \varphi'_{n-1}}{dt^2} - c_{n-2,n-1}(\varphi'_{n-2}-\varphi'_{n-1}) + c_{n-1,n}(\varphi'_{n-1}-\varphi'_n)=0,\\ &\Theta_n \frac{d^2 \varphi'_n}{dt^2} - c_{n-1,n}(\varphi'_{n-1}-\varphi'_n) + c_{n\mathrm{I}}(\varphi'_n-\varphi'_{\mathrm{I}})=0,\\ &\Theta_{\mathrm{I}}\frac{d^2 \varphi'_{\mathrm{I}}}{dt^2} - c_{n\mathrm{I}}(\varphi'_n-\varphi'_{\mathrm{I}}) + c_{\mathrm{I}\mathrm{II}}(\varphi_{\mathrm{I}}-\varphi'_{\mathrm{II}})=0,\\ &\Theta_{\mathrm{II}}\frac{d^2 \varphi'_{\mathrm{II}}}{dt^2} - c_{\mathrm{I}\mathrm{II}}(\varphi'_{\mathrm{I}}-\varphi'_{\mathrm{II}}) + R=0. \end{aligned} \tag{1} \]

where \(R\) is the so-called residual moment. In forced vibrations of the system it denotes the external exciting moment applied to mass II.

¹ The quantities \(\Theta\) and \(m\) are related to each other by the relation: \(\Theta = mr^2\), thus, for example, \(\Theta_1 = m_1 r^2\), \(\Theta_2 = m_2 r^2\), etc.

² The stiffness of a shaft segment of constant cross-section is in general determined by the formula: \(c=\dfrac{G\cdot J_p}{l}\) kgcm/radian, where \(G\) is the modulus of elasticity of the second kind, \(J_p\) is the polar moment of inertia of the shaft section, and \(l\) is the length of the shaft. For a shaft of variable cross-section, \(c\) may be calculated analytically or graphically.

For free vibrations of the system:

\[ R = 0. \]

As is known, the instantaneous angles of deflection of the vibrating masses can be determined by the following expressions:

\[ \begin{aligned} \varphi'_1 &= \varphi_1 \sin \omega t,\\ \varphi'_2 &= \varphi_2 \sin \omega t,\\ &\cdots\\ \varphi'_{n-1} &= \varphi_{n-1} \sin \omega t,\\ \varphi'_n &= \varphi_n \sin \omega t,\\ &\cdots \end{aligned} \]

where \(\omega\) is the angular speed of vibration.

Substituting these values of \(\omega\) into the system of equations (1), for all \(n\) masses of the engine we obtain:

\[ \left. \begin{aligned} \Theta_1 \omega^2 \varphi_1 - c_{12}(\varphi_1-\varphi_2) &= 0,\\ \Theta_2 \omega^2 \varphi_2 + c_{12}(\varphi_1-\varphi_2) - c_{23}(\varphi_2-\varphi_3) &= 0,\\ &\cdots\\ &\cdots\\ \Theta_{n-1}\omega^2 \varphi_{n-1} + c_{n-2,n-1}(\varphi_{n-2}-\varphi_{n-1}) - c_{n-1,n}(\varphi_{n-1}-\varphi_n) &= 0,\\ \Theta_n \omega^2 \varphi_n + c_{n-1,n}(\varphi_{n-1}-\varphi_n) - c_{n1}(\varphi_n-\varphi_1) &= 0 \end{aligned} \right\} \tag{2} \]

In compressorless diesels, inline automobile and aircraft engines, the reduced moments of inertia of the masses of the connecting-rod-and-crank mechanisms of all cylinders are identical (Fig. 2):

\[ \Theta_1=\Theta_2=\cdots=\Theta_n=\Theta, \]

and the stiffnesses of the crankshaft sections between them are also identical:

\[ c_{12}=c_{23}=\cdots=c_{n-1,n}=c \]

Now from (2), omitting the indices, we obtain:

\[ \left. \begin{aligned} \Theta\omega^2\varphi_1 - c(\varphi_1-\varphi_2) &= 0,\\ \Theta\omega^2\varphi_2 + c(\varphi_1-\varphi_2) - c(\varphi_2-\varphi_3) &= 0,\\ &\cdots\\ \Theta\omega^2\varphi_{n-1} + c(\varphi_{n-2}-\varphi_{n-1}) - c(\varphi_{n-1}-\varphi_n) &= 0,\\ \Theta\omega^2\varphi_n + c(\varphi_{n-1}-\varphi_n) - c_{n1}(\varphi_n-\varphi_1) &= 0 \end{aligned} \right\} \tag{3} \]

whence, after cancellation by \(c\):

\[ \left. \begin{aligned} A\varphi_1-(\varphi_1-\varphi_2)&=0,\\ A\varphi_2+(\varphi_1-\varphi_2)-(\varphi_2-\varphi_3)&=0,\\ &\cdots\\ A\varphi_{n-1}+(\varphi_{n-2}-\varphi_{n-1})-(\varphi_{n-1}-\varphi_n)&=0,\\ A\varphi_n+(\varphi_{n-1}-\varphi_n)-\frac{c_n}{c}(\varphi_n-\varphi_1)&=0 \end{aligned} \right\} \qquad \ldots (4) \]

where:

\[ A=\frac{\Theta\omega^2}{c} \qquad \ldots\ldots\ldots (5) \]

Now the amplitudes of vibration of the engine masses are determined by the following expressions:

\[ \left. \begin{aligned} \varphi_2&=\varphi_1-A\varphi_1,\\ \varphi_3&=\varphi_2-A\varphi_1-A\varphi_2,\\ \varphi_4&=\varphi_3-A\varphi_1-A\varphi_2-A\varphi_3,\\ &\cdots\\ \varphi_n&=\varphi_{n-1}-A\varphi_1-A\varphi_2-\ldots-A\varphi_{n-1} \end{aligned} \right\} \qquad \ldots (6) \]

From the first two expressions of system (6):

\[ \varphi_3=\varphi_1-A\varphi_1-A(\varphi_1-A\varphi_1) =\varphi_1(1-3A+A^2), \qquad \ldots (7) \]

and further, in the same manner:

\[ \varphi_4=\varphi_1(1-6A+5A^2-A^3), \qquad \ldots (8) \]

\[ \varphi_5=\varphi_1(1-10A+15A^2-7A^3+A^4), \qquad \ldots (9) \]

\[ \varphi_6=\varphi_1(1-15A+35A^2-17A^3+8A^4-A^5), \qquad (10) \]

and so on.

Let us successively sum the amplitudes of vibration of the individual engine masses:

\[ \left. \begin{aligned} \varphi_1+\varphi_2&=\varphi_1(2-A),\\ \varphi_1+\varphi_2+\varphi_3&=\varphi_1(3-4A+A^2),\\ \varphi_1+\varphi_2+\varphi_3+\varphi_4&=\varphi_1(4-10A+6A^2-A^3),\\ \varphi_1+\varphi_2+\ldots+\varphi_5&=\varphi_1(5-20A+21A^2-8A^3+A^4),\\ \varphi_1+\varphi_2+\ldots+\varphi_6&=\varphi_1(6-35A+56A^2-25A^3+9A^4-A^5) \end{aligned} \right\} \qquad (11) \]

and so on, or in general form:

\[ \left. \begin{aligned} \sum_{1}^{2}\varphi&=B_2\varphi_1,\\ \sum_{1}^{3}\varphi&=B_3\varphi_1,\\ &\cdots\\ \sum_{1}^{n}\varphi&=B_n\varphi_1, \end{aligned} \right\} \qquad \ldots\ldots\ldots (12) \]

where:

\[ \left. \begin{aligned} B_2 &= 2-A,\\ B_3 &= 3-4A+A^2,\\ B_4 &= 4-10A+6A^2-A^3,\\ B_5 &= 5-20A+21A^2-8A^3+A^4,\\ B_6 &= 6-35A+56A^2-25A^3+9A^4-A^5,\\ &\ldots\ldots\ldots\ldots\ldots \end{aligned} \right\} \qquad ..(13) \]

Thus, the sums of the vibration amplitudes of all the engine masses \(m_1 \ldots m_n\) can be expressed through the amplitude \(\varphi\) of vibration of the first mass and the coefficient \(B\), which depends only on the parameter \(A\). In the general case, for \(n\) engine masses:

\[ \varphi_n=\varphi_1\cdot f(A); \]

\[ \sum_{1}^{n}\varphi=B_n\varphi_1. \]

Let us next compute the ratios:

\[ \left. \begin{aligned} D_2&=\frac{\varphi_2}{\displaystyle\sum_{1}^{2}\varphi} =\frac{1-A}{2-A},\\[1.2em] D_3&=\frac{\varphi_3}{\displaystyle\sum_{1}^{3}\varphi} =\frac{1-3A+A^2}{3-4A+A^2},\\[1.2em] D_4&=\frac{\varphi_4}{\displaystyle\sum_{1}^{4}\varphi} =\frac{1-6A+5A^2-A^3}{4-10A+6A^2-A^3},\\[1.2em] D_5&=\frac{\varphi_5}{\displaystyle\sum_{1}^{5}\varphi} =\frac{1-10A+15A^2-7A^3+A^4}{5-20A+21A^2-8A^3+A^4},\\[1.2em] D_6&=\frac{\varphi_6}{\displaystyle\sum_{1}^{6}\varphi} =\frac{1-15A+35A^2-17A^3+8A^4-A^5}{6-35A+56A^2-25A^3+9A^4-A^5},\\ &\ldots\ldots\ldots\ldots\ldots\\ &\ldots\ldots\ldots\ldots\ldots\\ D_n&=\frac{\varphi_n}{\displaystyle\sum_{1}^{n}\varphi} =\ldots\ldots\ldots\ldots\ldots \end{aligned} \right\} \qquad ..(14) \]

The coefficients \(D\) depend only on \(A\), and therefore vary when \(\omega_1\) changes.

In the section between the last mass \(m_n\) of the engine and the first flywheel, the shaft is twisted by a moment equal to the sum of the inertial moments of all \(n\) masses of the engine¹:

\[ M=\sum_{1}^{n}\Theta \omega^{2}\varphi =\Theta \omega^{2}\sum_{1}^{n}\varphi,\ldots\ldots\ldots\ldots (15) \]

or, since

\[ \sum_{1}^{n}\varphi=\frac{\varphi_n}{D_n}, \]

then

\[ M=\frac{\Theta \omega^{2}\varphi_n}{D_n}.\ldots\ldots\ldots\ldots (16) \]

The inertial moment of a single \(n\)-th (last) mass of the engine is obviously equal to:

\[ M_n=\Theta \omega^{2}\varphi_n\ldots\ldots\ldots\ldots (17) \]

From comparison of (16) and (17):

\[ M=\frac{M_n}{D_n}=\left(\frac{\Theta}{D_n}\right)\omega^{2}\varphi_n. \]

It follows from this that the sum of the inertial moments of all \(n\) equal masses \(m\) of the engine is equal to the inertial moment of a certain fictitious mass having a moment of inertia \(\dfrac{\Theta}{D_n}\) and applied at the place of the last, \(n\)-th mass. All flywheel masses \(m\) of the engine may be replaced by one dynamically equivalent mass \(\dfrac{\Theta}{D_n r^{2}}\).

The coefficient \(D_n\) is a variable quantity and, for given \(\Theta\) and \(c\), depends only on the angular velocity of oscillation (with a given number of engine cylinders as well). The value of \(D_n\) for each number of engine cylinders is determined by formulas (14)².

For free vibrations of the system (Fig. 1), for the two larger masses \(m_{\mathrm{I}}\) and \(m_{\mathrm{II}}\), from (1) we have:

\[ \Theta_{\mathrm{I}}\frac{d^{2}\varphi'_{\mathrm{I}}}{dt^{2}} -c_{n\mathrm{I}}(\varphi'_n-\varphi'_{\mathrm{I}}) +c_{\mathrm{I}\mathrm{II}}(\varphi'_{\mathrm{I}}-\varphi'_{\mathrm{II}})=0, \]

\[ \Theta_{\mathrm{II}}\frac{d^{2}\varphi'_{\mathrm{II}}}{dt^{2}} -c_{\mathrm{I}\mathrm{II}}(\varphi'_{\mathrm{I}}-\varphi'_{\mathrm{II}})=0. \]

By substitution:

\[ \varphi'_I=\varphi_I \sin \omega t, \]

\[ \varphi'_{II}=\varphi_{II}\sin \omega t, \]

these equations are brought to the form:

\[ \Theta_I \omega^2 \varphi_I+c_{n1}(\varphi_n-\varphi_I)-c_{I,II}(\varphi_I-\varphi_{II})=0. \]

\[ \Theta_{II}\omega^2\varphi_{II}+c_{I,II}(\varphi_I-\varphi_{II})=0. \]

Whence:

\[ \varphi_I=\varphi_{II}-\frac{\Theta_{II}\omega^2\varphi_{II}}{c_{I\,II}} \tag{18} \]

\[ \Theta_I\omega^2\varphi_I+\Theta_{II}\omega^2\varphi_{II}+c_{n1}(\varphi_n-\varphi_I)=0, \]

\[ \varphi_n=\varphi_I-\frac{\Theta_I\omega^2\varphi_I+\Theta_{II}\omega^2\varphi_{II}}{c_{n1}} \tag{19} \]

On the section between the last (\(n\)-th) engine mass and the first flywheel, the shaft is twisted by a moment equal to the sum of the inertial moments of the masses \(m_I\) and \(m_{II}\):

\[ M'=\Theta_I\omega^2\varphi_I+\Theta_{II}\omega^2\varphi_{II} \]

Flywheels I and II may be replaced by one new mass having the moment of inertia:

\[ \frac{\Theta_I+\Theta_{II}}{K} \tag{20} \]

and placed at the point of application of the last, \(n\)-th, engine mass. The coefficient \(K\) is determined from the following relation, expressing the necessary equality of the inertial moments of both flywheels I and II and of the replacing mass, having the moment of inertia

\[ \frac{\Theta_I+\Theta_{II}}{K}: \]

\[ \Theta_I\omega^2\varphi_I+\Theta_{II}\omega^2\varphi_{II} = \frac{\Theta_I+\Theta_{II}}{K}\omega^2\varphi_n \tag{21} \]

Whence:

\[ K=\frac{(\Theta_I+\Theta_{II})\varphi_n}{\Theta_I\varphi_I+\Theta_{II}\varphi_{II}} = \frac{(\Theta_I+\Theta_{II})\varphi_n} {\Theta_I\left(\varphi_{II}-\frac{\Theta_{II}\omega^2}{c_{I\,II}}\varphi_{II}\right)+\Theta_{II}\varphi_{II}} \]

But from (18) and (19):

\[ \varphi_n=\varphi_{II}\left( 1-\frac{\Theta_{II}\omega^2}{c_{I\,II}} -\frac{\Theta_I\omega^2}{c_{n1}} -\frac{\Theta_{II}\omega^2}{c_{n1}} +\frac{\Theta_I\Theta_{II}\omega^4}{c_{I\,II}\cdot c_{n1}} \right), \]

therefore:

\[ K=(\Theta_{\mathrm{I}}+\Theta_{\mathrm{II}}) \frac{ \left( 1-\frac{\Theta_{\mathrm{II}}\omega^2}{c_{\mathrm{I,II}}} -\frac{\Theta_{\mathrm{I}}\omega^2}{c_{n\mathrm{I}}} -\frac{\Theta_{\mathrm{II}}\omega^2}{c_{n\mathrm{I}}} +\frac{\Theta_{\mathrm{I}}\Theta_{\mathrm{II}}\omega^4}{c_{\mathrm{I,II}}\cdot c_{n\mathrm{I}}} \right) }{ \Theta_{\mathrm{I}}+\Theta_{\mathrm{II}} -\frac{\Theta_{\mathrm{I}}\cdot\Theta_{\mathrm{II}}\cdot\omega^2}{c_{\mathrm{I,II}}} } \]

Hence, introducing the notation:

\[ \left. \begin{aligned} \frac{\Theta_{\mathrm{I}}}{\Theta}&=\alpha_1; & \frac{\Theta_{\mathrm{II}}}{\Theta}&=\alpha_2,\\ \frac{c_{n\mathrm{I}}}{c}&=\beta_1 & \frac{c_{\mathrm{I,II}}}{c}&=\beta_2 \end{aligned} \right\} \qquad\ldots\ldots\ldots (22) \]

and taking into account the notation (5), we have:

\[ K= \frac{ 1-A\left(\frac{\alpha_2}{\beta_2} +\frac{\alpha_1+\alpha_2}{\beta_1}\right) +A^2\cdot\frac{\alpha_1\alpha_2}{\beta_1\beta_2} }{ 1-\frac{\alpha_1\alpha_2}{\beta_2(\alpha_1+\alpha_2)}A }. \]

Finally, denoting:

\[ \left. \begin{aligned} \frac{\alpha_1\alpha_2}{\beta_2(\alpha_1+\alpha_2)}&=\gamma,\\ \frac{\alpha_2}{\beta_2}+\frac{\alpha_1+\alpha_2}{\beta_1}&=\delta,\\ \frac{\alpha_1\alpha_2}{\beta_1\beta_2}&=\varepsilon \end{aligned} \right\} \qquad\ldots\ldots\ldots (23) \]

we finally obtain:

\[ K=\frac{1-\delta A+\varepsilon A^2}{1-\gamma A} \qquad\ldots\ldots\ldots (24) \]

Such is the relation for the case of two large flywheels I and II. If the installation has only one flywheel (Fig. 3), then:

\[ \left. \begin{aligned} \Theta_{\mathrm{II}}&=0,\\ \alpha_2=0,\quad \gamma=0,\quad \varepsilon&=0,\\ \delta&=\frac{\alpha_1}{\beta_1} \end{aligned} \right\} \qquad\ldots\ldots\ldots (22a) \]

and

\[ K=1-\delta A \qquad\ldots\ldots\ldots (25) \]

In free vibrations of the system there are no external exciting forces, and the inertial moments of the masses at every instant are balanced by the elastic moments of the twisting shaft.

In this case the sum of the inertial moments of all the masses of the system at each instant of time is equal to zero:

\[ \sum_{1}^{n}\Theta \omega^{2}\varphi+\Theta_{\mathrm I}\omega^{2}\varphi_{\mathrm I} +\Theta_{\mathrm {II}}\omega^{2}\varphi_{\mathrm {II}}=0, \]

or:

\[ \sum_{1}^{n}\Theta\varphi+\Theta_{\mathrm I}\varphi_{\mathrm I} +\Theta_{\mathrm {II}}\varphi_{\mathrm {II}}=0 \tag{26} \]

(this is readily obtained by summing all equations (1) for \(R=0\)).

But according to (15), (17), and (20), this expression can also be written in the following form:

\[ \left(\frac{\Theta}{D_n}\right)\omega^{2}\varphi_n+ \frac{\Theta_{\mathrm I}+\Theta_{\mathrm {II}}}{K}\,\omega^{2}\varphi_n=0, \]

or

\[ \frac{\Theta}{D_n}+\frac{\Theta_{\mathrm I}+\Theta_{\mathrm {II}}}{K}=0 \tag{27} \]

Hence it follows that, in the free vibrations of the system, the sum of the moments of inertia of all its masses, reduced to one point, to one section on the shaft, is equal to zero.

Relation (27) can be used as the basis of the following graphical method for determining the natural vibration frequencies of a multi-mass system according to Fig. 2.

Introducing the relative values:

\[ a_1=\frac{\Theta_{\mathrm I}}{\Theta},\qquad a_2=\frac{\Theta_{\mathrm {II}}}{\Theta}, \]

we rewrite expression (27), which is satisfied for the free vibrations of the system (and only for them), in the following form:

\[ \frac{1}{D_n}+\frac{a_1+a_2}{K}=0. \]

Hence:

\[ \frac{1}{D_n}=-\frac{a_1+a_2}{K}, \]

or:

\[ D_n=L, \tag{28} \]

where:

\[ L=-\frac{K}{a_1+a_2}. \tag{29} \]

For the system with two large flywheels I and II according to Fig. 2, in accordance with (24):

\[ L=\frac{1-\delta A+\varepsilon A^2}{(a_1+a_2)(\gamma A-1)}. \tag{30} \]

and for the system with one flywheel (Fig. 3), on the basis of (25):

\[ L=\frac{\delta A-1}{a_1}\ . . . . . . . . . . . . . . . . . . . (31) \]

The quantity \(D_n\), as established above, depends only on the number \(n\) of engine cylinders and the parameter \(A\). For each number of cylinders, the curves \(D_n=f(A)\) can be constructed once and for all in the coordinates \(D_n\) and \(A\), using expression (14).

The quantity \(L\), according to (30) and (31), depends on the parameter \(A\), and also on the relative values of the moments of inertia \(\Theta_{\mathrm{I}}\) and \(\Theta_{\mathrm{II}}\) of both flywheels and on the stiffnesses of the shaft sections between them and the engine.

For each specifically given system, the numerical values of the coefficients \(a_1\), \(a_2\), \(\gamma\), \(\delta\), and \(\varepsilon\), entering into expressions (30) and (31), can be calculated, and \(L\) can be determined as a function of one \(A\):

\[ L=F(A). \]

The quantity \(A\), determined by formula (5),

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

contains, as numerical values known in magnitude, \(\Theta\) and \(c\) for each cylinder of the engine, as well as the as yet unknown numerical value of the angular velocity \(\omega\) of the natural vibrations of the system.

By assigning to \(\omega\) some arbitrary numerical value, one can calculate the quantity \(A\), and from it determine the corresponding numerical values \(D_n\) and \(L\), using formulas (14), (30), or (31).

These found values of \(D_n\) and \(L\) will in general turn out to be unequal to one another, since the equality of \(D_n\) and \(L\), according to (28), can take place only for those values of \(\omega\) which are the angular velocities of the natural vibrations of the system under consideration. It is hardly possible, however, to predict in advance the values of the angular velocities of the natural vibrations with sufficient accuracy.

In order to find those values of \(\omega\) (and consequently also of the parameter \(A\)) for which the relation

\[ D_n=L \]

is satisfied, one should, on the diagram \(D_n=f(A)\) constructed earlier for the given number of engine cylinders, construct point by point the curve \(L=F(A)\), assigning here different numerical values to \(A\) and laying off along the ordinate axis the calculated values of \(L\). The abscissas of the points of intersection of this curve with the curve \(D_n\) will evidently give those values of \(A\) (and consequently also of \(\omega\)) for which the equality \(D_n=L\) is satisfied. These quantities \(A_1, A_2,\ldots\) now determine, on the basis of

…in equation (5) and the corresponding angular velocities of the system’s natural vibrations:

\[ \omega_{\mathrm{I}}=\sqrt{\frac{A_1 c}{\Theta}},\qquad \omega_{\mathrm{II}}=\sqrt{\frac{A_2 c}{\Theta}},\ldots \tag{32} \]

From what has been set forth above it follows that, in order to determine the frequencies of the natural vibrations of a multi-mass system according to Figs. 2 and 3, it is necessary to prepare in advance the curves \(D_n\) for the corresponding number of cylinders. Each such curve is fully suitable for any installation with an engine having the given number of cylinders. The construction of these curves is carried out according to the equations (14) derived above.

At every engine-building plant such curves can be constructed once and for all, for each type of machine produced by it. In this case, to accelerate the calculations still further, the corresponding values of \(\omega\), calculated from relation (5), can also be marked on the abscissa axis alongside the values of \(A\). These curves will be suitable for the rapid determination of the angular velocities of the natural vibrations of any power installation with the given type of engine.

It should be noted that the method described for the graphical determination of the frequencies of the natural vibrations of a system is, in principle, entirely exact, including only the familiar errors inevitable in any graphical construction. With a sufficiently large drawing size these errors can be made negligibly small.

This method can also be applied to installations which, in addition to the engine, have a whole series of flywheel masses with different moments of inertia \(\Theta_{\mathrm{I}}, \Theta_{\mathrm{II}}, \Theta_{\mathrm{III}}\), etc. The author proposes to return to this question in another article.

Let us show the application of the method described by an example. Let us determine the angular velocities of the natural vibrations of the installation in Fig. 4, consisting of a four-cylinder diesel engine, a flywheel, and a dynamo, connected by one common shaft and having the following basic data, obtained after reducing the masses and lengths of the system¹.

The moment of inertia of the flywheel masses of each cylinder of the engine is

\[ \Theta = 3.0\ \text{kg m sec}^2; \]

the moment of inertia of the flywheel mass is \(\Theta_{\mathrm{I}} = 30.0\ \text{kg m sec}^2\);

the moment of inertia of the dynamo mass is \(\Theta_{\mathrm{II}} = 21.0\ \text{kg m sec}^2\);

the stiffnesses of the shaft sections between the masses are:

\[ c = 5.0\cdot 10^6\ \text{kg m/radian}, \]

\[ c_{n\mathrm{I}} = 3.0\cdot 10^6\ \text{kg m/radian}. \]

\[ c_{\mathrm{I}\mathrm{II}} = 2.0\cdot 10^6\ \text{kg m/radian}. \]

In Fig. 5 the curve \(D_4\) for four-cylinder engines has been plotted in advance, according to the corresponding equation (14).

For our installation we calculate the coefficients, in accordance with the notation (22) and (23):

\[ \alpha_1=\frac{\Theta_1}{\Theta}=\frac{30.0}{3.0}=10.0 \]

\[ \alpha_2=\frac{\Theta_{\mathrm{II}}}{\Theta}=\frac{21.0}{3.0}=7.0 \]

\[ \beta_1=\frac{c_{\mathrm{I,II}}}{c}=\frac{3.0\cdot 10^6}{5.0\cdot 10^6}=0.60 \]

\[ \beta_2=\frac{c_{\mathrm{I,II}}}{c}=\frac{2.0\cdot 10^6}{5.0\cdot 10^6}=0.40 \]

\[ \gamma=\frac{\alpha_1\cdot\alpha_2}{\beta_2\cdot(\alpha_1+\alpha_2)} =\frac{10\cdot 7}{0.4\cdot(10.0+7.0)}=10.30 \]

\[ \delta=\frac{\alpha_2}{\beta_2}+\frac{\alpha_2}{\beta_1}+\frac{\alpha_1}{\beta_1} =\frac{7.0}{0.4}+\frac{7.0}{0.6}+\frac{10.0}{0.6}=45.8 \]

\[ \varepsilon=\frac{\alpha_1\cdot\alpha_2}{\beta_1\cdot\beta_2} =\frac{10.0\cdot 7.0}{0.6\cdot 0.4}=291.7 \]

Now, from formula (30),

\[ L=\frac{1-45.8\,A+291.7\,A^2}{17.0\,(10.3\,A-1)}. \]

Assigning here to the parameter \(A\) various values, beginning with \(A=0\), and calculating from the preceding equation the corresponding values of \(L\), we construct, point by point, the curve \(L=F(A)\) on diagram \(D_4\); in Fig. 5 this curve is denoted by I. The intersection of this curve with curve \(D_4\) gives points whose abscissas turned out to be equal to the following quantities:

\[ A_1=0.0720, \]

\[ A_2=0.1512, \]

\[ A_3=0.870, \]

(in Fig. 6 the part of curve \(D_4\) giving the quantities \(A_1\) and \(A_2\), for greater accuracy, is plotted on an enlarged scale).

Now, by formulas (32), we determine the angular velocities of the natural torsional vibrations of the system:

\[ \omega_1=\sqrt{\frac{A_1 c}{\Theta}} =\sqrt{\frac{0.0720\cdot 5\cdot 10^6}{3.0}} =346.0\ \mathrm{sec}^{-1}, \]

\[ \omega_{\mathrm{II}}=\sqrt{\frac{A_2 c}{\Theta}} =\sqrt{\frac{0.1512 \cdot 5 \cdot 10^6}{3.0}} =502\ \mathrm{sec}^{-1}, \]

\[ \omega_{\mathrm{III}}=\sqrt{\frac{A_3 c}{\Theta}} =\sqrt{\frac{0.870 \cdot 5 \cdot 10^6}{3.0}} =1203\ \mathrm{sec}^{-1}, \]

and the corresponding numbers of natural vibrations of the system per minute are

\[ n_{\mathrm{I}}=\frac{30\omega_{\mathrm{I}}}{\pi} =\frac{30 \cdot 346}{3.14} =3305\ \text{vibrations/min.} \]

\[ n_{\mathrm{II}}=\frac{30\omega_{\mathrm{II}}}{\pi} =\frac{30 \cdot 502}{3.14} =4795\ \text{vibrations/min.} \]

\[ n_{\mathrm{III}}=\frac{30\omega_{\mathrm{III}}}{\pi} =\frac{30 \cdot 1203}{3.14} =11500\ \text{vibrations/min.} \]

Let us note that in practice there is no need to construct the entire curve \(L\), as was shown in Fig. 5; it is quite sufficient to confine oneself to constructing only the portions of this curve near the points of its intersection with the curve \(D\).

An analytical calculation of the angular velocities of the natural vibrations of our installation by Tolle’s method \(^{1}\) gave the following values

\[ \omega_{\mathrm{I}}=346.1\ \mathrm{sec}^{-1}, \]

\[ \omega_{\mathrm{II}}=502\ \mathrm{sec}^{-1}, \]

\[ \omega_{\mathrm{III}}=1205\ \mathrm{sec}^{-1}. \]

We give here the tables of these calculations for the following values of \(\omega\) \(^{2}\) (see the Appendix):

Single-node vibrations \(\omega=345;\ 346;\ 347\ \mathrm{sec}^{-1}\),
two-node vibrations \(\omega=501;\ 502;\ 503\ \mathrm{sec}^{-1}\),
three-node vibrations \(\omega=1204;\ 1205\ \mathrm{sec}^{-1}\).

(In calculating Tolle’s tables we used tables of logarithms.)

Drawing attention here to the change in sign of the residual moment \(R\), one can, on the basis of these tables, write the following relations for the angular velocities of the natural vibrations:

\[ 345<\omega_{\mathrm{I}}<347 \]

\[ 501<\omega_{\mathrm{II}}<503 \]

\[ 1204<\omega_{\mathrm{III}}<1205 \]

Hence, by interpolating (using the values of \(R\) indicated in the tables), we obtain the above-stated quantities \(\omega_{\mathrm{I}}, \omega_{\mathrm{II}}, \omega_{\mathrm{III}}\).

\(^{1}\) M. Tolle, Regelung der Kraftmaschinen. 1921.

\(^{2}\) In these tables \(e\) is the elastic moment in the given section of the shaft, \(\varphi\) are the amplitudes of vibration of the corresponding masses, and \(c\) and \(\Theta\) have their former meanings.

The vibration amplitude of the first mass \(\varphi_1\) is taken equal to one radian.

A comparison of the results obtained by the graphical method set forth in the present article with the data obtained by Tolle’s method shows their complete agreement with respect to the angular velocities of the one-node and two-node vibrations. The slight discrepancy in the value of \(\varphi_{\mathrm{III}}\) arose as a consequence of the unavoidable errors of the graphical construction. But this discrepancy is negligible and has practically no significance.

As an illustration, let us now determine the angular velocities of the natural vibrations of the same installation, but without the dynamo \((\Theta_{\mathrm{d}} = 0)\). In this case we arrive at the scheme in Fig. 3. For it, according to formulas (22-a) and (25):

\[ \alpha_1 = \frac{\Theta_1}{\Theta} = 10.0, \]

\[ \beta_1 = \frac{c_{n1}}{c} = 0.6, \]

\[ \delta = \frac{\alpha_1}{\beta_1} = \frac{10}{0.6} = 16.67, \]

and

\[ L = \frac{\delta A - 1}{\alpha_1} = 1.667 A - 0.1. \]

Using this equation, we construct the straight line \(L\) on the diagram \(D_4\) (in Figs. 5 and 6 it is denoted by \(II\)). It intersects the curve \(D_4\) at two points, whose abscissas are:

\[ A_1' = 0.127 \]

\[ A_2' = 0.870 \]

Now the angular velocities of the natural vibrations are:

\[ \omega_{\mathrm{I}} = \sqrt{\frac{A_1' c}{\Theta}} = \sqrt{\frac{0.127 \cdot 5 \cdot 10^6}{3.0}} = 460\ \mathrm{sec}^{-1}, \]

\[ \omega_{\mathrm{II}} = \sqrt{\frac{A_2' c}{\Theta}} = \sqrt{\frac{0.870 \cdot 5 \cdot 10^6}{3.0}} = 1203\ \mathrm{sec}^{-1}, \]

and the corresponding numbers of natural vibrations per minute:

\[ n_1 = 4390\ \text{vibr./min.} \]

\[ n_{\mathrm{II}} = 11500\ \text{vibr./min.} \]

The analytical calculation carried out for this system by Tolle’s method gave the same values:

\[ \omega_{\mathrm{I}} = 460\ \mathrm{sec}^{-1}, \]

\[ \omega_{\mathrm{II}} = 1203\ \mathrm{sec}^{-1}, \]

II.

F. Porter¹ has also shown that, for engines with a sufficiently large number of cylinders, it is quite possible to replace all the flywheel masses of the engine by a single mass uniformly distributed over the entire length of the crankshaft. The error obtained with such a replacement in determining the natural vibration frequencies of installations with multi-cylinder engines is practically negligible, amounting to fractions of a percent and decreasing as the number of cylinders increases.² Thus, for example, for a six-cylinder engine this error is less than 0.5%, and for a ten-cylinder engine it is less than 0.1%.

F. Porter gave a method for the analytical determination of the natural vibration frequencies of such a system, with the flywheel mass of the engine uniformly distributed over the whole crankshaft. But this method is nevertheless rather cumbersome and requires carrying out a number of tabular calculations.

We shall apply here as well the graphical method described above for determining natural vibration frequencies. It will be established below that, for all multi-cylinder engines with a sufficiently large number of cylinders (beginning approximately with six cylinders and more), one and the same curve D is suitable.

Let us denote (Fig. 7):

\(\Theta\) — the moment of inertia of the moving masses of each cylinder of the engine, with respect to the axis of rotation of the shaft,

\(m\) — the mass of the moving parts of each cylinder, reduced to the crank radius \(r\),

\(\Theta_{\mathrm{I}}\) and \(\Theta_{\mathrm{II}}\) — as before, the moments of inertia of the masses of both flywheels with respect to the same axis,

\(l_0\) — the length of the shaft between the axes of neighboring cylinders, reduced to a certain polar moment of inertia \(J_p\) of its cross section,

\(l_{m\mathrm{I}}\) — the length of the shaft between the end of the distributed mass of the engine and flywheel I, reduced to the same \(J_p\),

\(l_{\mathrm{I,II}}\) — the length of the shaft between flywheels I and II, reduced to the same \(J_p\).

With the usual reduction of the masses and lengths of the system, each \(n\)-cylinder engine is divided into \(n\) separate flywheel masses located at a distance \(l_0\) from one another. The total elastic length \(l_1\) of the entire crankshaft is equal to the distance between the two outermost masses of the engine.

In order to be able to consider, approximately, all the flywheel masses of the engine as uniformly distributed over the entire length of its shaft, it is necessary, as shown in Fig. 7, to add \(\frac{l_0}{2}\) to the length \(l_1\) on both sides.

Now I regard the total reduced mass \(\sum_1^n m\) of the connecting-rod-and-crank mechanisms of all cylinders as uniformly distributed along the length \(l = n l_0'\) of the crankshaft. Per unit length of the shaft there is now a mass with moment of inertia:

\[ \frac{\Theta_n}{l}, \]

where

\[ \Theta_n = \sum_1^n \Theta = n\Theta. \]

A segment of the shaft \(l\), of length \(dx\) (Fig. 8), carries a mass \(dm\) with moment of inertia:

\[ d\Theta_n = \frac{\Theta_n}{l}\,dx \]

The stiffness of the shaft \(l\) is evidently equal to:

\[ c_n = \frac{G \cdot I_p}{l} \]

and, correspondingly:

\[ c'_m = \frac{G \cdot I_p}{l'_m}, \]

\[ c_{\mathrm{I,II}} = \frac{G \cdot I_p}{l_{\mathrm{I,II}}}. \]

In the oscillatory motion of the system, the instantaneous angle of rotation \(\varphi\) of each section of the shaft \(l\) depends on the time \((t)\) and on the position of the given section \((x)\) (Fig. 8):

\[ \varphi = f(x,t). \]

At the same time, the inertial moment produced by the elementary mass \(dm\) of the shaft \(l\) is determined by the expression:

\[ dM = -d\Theta_n \cdot \frac{\partial^2 \varphi}{\partial t^2} = -\frac{\Theta_n}{l}\,dx \cdot \frac{\partial^2 \varphi}{\partial t^2}. \]

Let us take in the shaft \(l\) a section at a distance \(x\) from the left-hand end of the system. The sum of the inertial moments of all elementary masses \(dm\) lying to the left of the section \(x\) is equal to the moment twisting the shaft in this section, i.e.

\[ \int_0^x -\frac{\Theta_n}{l}\,dx \cdot \frac{\partial^2 \varphi}{\partial t^2} + G \cdot I_p \cdot \frac{\partial \varphi}{\partial x} = 0 \]

(the second term in this expression denotes the elastic moment of the shaft in the section \(x\); the derivative \(\dfrac{\partial \varphi}{\partial x}\) is evidently a measure of the twisting of the shaft over the segment \(dx\)).

Let us take the partial derivative with respect to \(x\) of both sides of this expression:

\[ \frac{\Theta_n}{l}\frac{\partial^2 \varphi}{\partial t^2} = G \cdot J_p \cdot \frac{\partial^2 \varphi}{\partial x^2}. \]

Denoting:

\[ \frac{G \cdot J_p l}{\Theta_n} = a^2 \tag{34} \]

we thus obtain the so-called differential equation of vibration of a string:

\[ \frac{\partial^2 \varphi}{\partial t^2} = a^2 \frac{\partial^2 \varphi}{\partial x^2}. \]

The general integral of this differential equation in partial derivatives, as is known, may be written in the following form:¹

\[ \varphi = (A_1 \operatorname{Cos} akt + A_2 \operatorname{Sin} akt) \cdot (A_3 \operatorname{Cos} kx + A_4 \operatorname{Sin} kx), \tag{35} \]

where \(A_1, A_2, A_3, A_4\) and \(k\) are constants of integration, determined by the initial and boundary conditions.

Putting:

\[ \begin{aligned} &\text{for } t=0 \text{ and } x=0 \quad \varphi=0,\\ &\text{for } t=0 \text{ and } x=l \quad \varphi=0, \end{aligned} \]

from (35) we obtain:

\[ \varphi = A_2 \operatorname{Sin} akt \cdot (A_3 \operatorname{Cos} kx + A_4 \operatorname{Sin} kx). \tag{36} \]

In free vibrations of the system, the left end of the shaft is free and is not subjected to the action of twisting moments, i.e.:

\[ \text{for } x=0,\quad \frac{\partial \varphi}{\partial x}=0. \]

Under this condition, from (36):

\[ A_4 = 0 \]

and

\[ \varphi = A_2 \operatorname{Sin} akt \cdot A_3 \operatorname{Cos} kx, \]

or:

\[ \varphi = \varphi_0 \operatorname{Sin} akt \cdot \operatorname{Cos} kx, \tag{37} \]

Here \(\varphi_0\) is evidently the amplitude of vibration of the left free end of the shaft \(l\). The product

\[ ak=\omega \]

¹ See, for example, A. N. Krylov, On Certain Differential Equations of Mathematical Physics. 1932.

denotes the angular velocity of vibration of the shaft. Hence

\[ \kappa=\frac{\omega}{a}, \]

and therefore:

\[ \varphi_x=\varphi_0 \operatorname{Sin}\omega t \cdot \operatorname{Cos}\frac{\omega}{a}x \ldots \ldots \ldots (38) \]

We attach the subscript \(x\) to \(\varphi\), since expression (38) gives the magnitude of the instantaneous angle of deflection of the section \(x\) of the shaft \(l\).

The inertial moment of the entire mass \(\sum_1^n m\) of the engine, twisting the shaft over the segment \(l_n\), is equal to:

\[ M=\int_0^l -d\Theta_n\cdot \frac{\partial^2\varphi_x}{\partial t^2} =\int_0^l \frac{\Theta_n}{l}\,dx\cdot \varphi_0\omega^2 \operatorname{Sin}\omega t\cdot \operatorname{Cos}\frac{\omega}{a}x = \]

\[ =\frac{\Theta_n}{l}\varphi_0 a \operatorname{Sin}\omega t\cdot \operatorname{Sin}\frac{\omega l}{a}\ldots \ldots \ldots (39) \]

The entire mass \(\sum_1^n m\) of the engine, uniformly distributed along the shaft \(l\), may be replaced by a certain concentrated mass with moment of inertia \(\frac{\Theta_n}{D}\), applied at the final \((x=l)\) section of the engine shaft. The equivalence of these masses will be achieved if the inertial moments produced by them are equal to each other, i.e., it is necessary that:

\[ \frac{\Theta_n}{l}a\varphi_0\omega \operatorname{Sin}\omega t\cdot \operatorname{Sin}\frac{\omega l}{a} = \frac{\Theta_n}{D}\omega^2\varphi_l \operatorname{Sin}\omega t, \ldots \ldots (40) \]

where \(\varphi_l\) is the amplitude of vibration of the right end of the shaft.

From (38), for \(x=l\):

\[ \varphi_l=\varphi_0 \operatorname{Cos}\frac{\omega l}{a}. \]

Substituting this value of \(\varphi_l\) into equality (40), we find:

\[ D=\frac{\omega l}{a}\cdot \frac{1}{\operatorname{tg}\frac{\omega l}{a}}. \]

But on the basis of (34) and (35), replacing \(\Theta\) in (5) by \(\Theta_n\):

\[ \frac{\omega l}{a} = \frac{\omega l}{\sqrt{\dfrac{G I_p l}{\Theta_n}}} = \sqrt{\frac{\Theta_n \omega^2}{c_n}} = \sqrt{A}. \]

Now finally:

\[ D= \frac{\sqrt{A}}{\operatorname{tg}\sqrt{A}} \qquad \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (41) \]

The coefficient \(D\) here has exactly the same meaning and significance as in the case considered above of the graphical determination of the natural vibration frequencies of a system with a series of identical concentrated masses. The only difference is that in formula (5), for \(A\):

\[ A=\frac{\Theta_n \omega^2}{c_n} \]

one must now understand by \(\Theta_n\) the total moment of inertia of all the flywheel masses of the engine, and by \(c_n\) the stiffness of the entire engine shaft of length \(l\).

The curve \(D\) according to equation (41) is plotted in Fig. 9 in the coordinates \(D\) and \(A\). It is evidently now suitable for any engine with a sufficiently large number of cylinders. Of course, this curve may also be used for engines with a small number of cylinders, but in that case the error in determining the natural vibration frequencies of the system increases somewhat.

For each given installation being calculated, with known masses and reduced lengths or shaft stiffnesses, using relations (22) and (23) we compute the quantity \(L\) from formulas (24) or (25) in the form of an algebraic function of \(A\). Here, by \(\Theta\) and \(c\), in accordance with what was said above, we mean the total moment of inertia of all the flywheel masses of the engine and, respectively, the stiffness of the entire shaft \(l\). The length \(l\) is determined according to the instructions on p. 18.

Now, on the graph \(D=f(A)\), we construct point by point the curve \(L=F(A)\), laying off as ordinates the values of \(L\) calculated for the various assumed values of \(A\). The intersection of this curve with the curve \(D\) gives points whose abscissas \(A\) make it possible to determine the angular velocities \(\omega\) of the natural vibrations of the system from the formula:

\[ \omega=\sqrt{\frac{A c_n}{\Theta_n}}, \]

as before.

For a system with one flywheel, the constructions are considerably simplified, since here the curve \(L\) becomes a straight line determined by equation (25).

Let us apply the method set forth to the installation considered above, on p. 14: a four-cylinder diesel–flywheel. We shall replace this installation by the system in Fig. 7. According to the instructions on p. 18, we calculate for it the following data:

The magnitude of the moment of inertia of the uniformly distributed mass:

\[ \Theta_n = n \Theta = 4 \cdot 3{,}0 = 12{,}0 \ \text{kg m sec}^2 . \]

The stiffness of the shaft of length \(l = n l_0\) (Fig. 7) carrying this mass:

\[ c_n = \frac{c}{n} \]

where \(c\) is the stiffness of the section of shaft between the axes of adjacent cylinders, \(n\) is the number of cylinders; in our case:

\[ c_n = \frac{5 \cdot 10^6}{4} = 1{,}25 \cdot 10^6 . \]

The stiffness \(c'_{n1}\) of the shaft \(l'_{n1}\) between the end of the distributed mass of the engine and the flywheel \(I\) can be found from the expression:

\[ \frac{1}{c'_{n1}} = \frac{1}{c_{n1}} - \frac{1}{2c}, \]

where \(c_{n1}\) is the stiffness of the section of shaft between the axis of the last cylinder and the flywheel \(I\) (see Fig. 4), and \(2c\) is the stiffness of the section of the crankshaft of length

\[ \frac{l_0}{2}, \]

equal to half the distance between the axes of adjacent cylinders. Hence, with (according to the data on p. 14):

\[ c_{n1} = 3 \cdot 10^6 \ \text{kgm/radian}, \]

\[ C = 5 \cdot 10^6 \ \text{kgm/radian}, \]

we obtain:

\[ c'_{n1} = 4{,}28 \cdot 10^6 \ \text{kgm/radian}. \]

In accordance with what was said on p. 22, we calculate the coefficients by formulas (22-a):

\[ \alpha_1 = \frac{\Theta_1}{\Theta_n} = \frac{30{,}0}{12{,}0} = 2{,}5 \]

\[ \beta_1 = \frac{c'_{n1}}{c_n} = \frac{4{,}28 \cdot 10^6}{0{,}25 \cdot 10^6} = 3{,}43 \]

\[ \delta = \frac{\alpha_1}{\beta_1} = \frac{2{,}5}{3{,}43} = 0{,}729 . \]

Now from (25) we obtain the equation for \(L\):

\[ L = \frac{0{,}729\, A - 1}{2{,}5} = 0{,}291\, A - 0{,}4 . \]

Plotting this straight line on diagram \(D\) (Fig. 9), we obtain the following values of the abscissae of the points where it intersects the curve \(D\)¹):

\[ A_1 = 2.07 \]

\[ A_2 = 15.23 \]

Hence the angular velocities of the natural vibrations of the system are:

\[ \omega_{\mathrm{I}} = \sqrt{\frac{A_1 \cdot c_n}{\Theta_n}} = \sqrt{\frac{2.07 \cdot 1.25 \cdot 10^6}{12.0}} = 464.0\ \mathrm{sec}^{-1}, \]

\[ \omega_{\mathrm{II}} = \sqrt{\frac{A_2 \cdot c_n}{\Theta_n}} = \sqrt{\frac{15.13 \cdot 1.25 \cdot 10^6}{12.0}} = 1260\ \mathrm{sec}^{-1}; \]

and the corresponding numbers of vibrations per minute are

\[ n_{\mathrm{I}}=\frac{30\omega_{\mathrm{I}}}{\pi}=4430\ \text{vibrations/min}, \]

\[ n_{\mathrm{II}}=\frac{30\omega_{\mathrm{II}}}{\pi}=12030\ \text{vibrations/min} \,^{2}). \]

From comparison of these values with the data given above, obtained as a result of an analytical calculation by Tolle’s method:

\[ \omega_{\mathrm{I}}=460\ \mathrm{sec}^{-1},\quad \text{and}\quad \omega_{\mathrm{II}}=1205\ \mathrm{sec}^{-1}, \]

one can calculate the magnitude of the error in determining \(\omega\) that arises when all the masses of the engine are replaced by a single uniformly distributed mass. This error proved to be \(0.86\%\) for \(\omega_{\mathrm{I}}\) and \(4.56\%\) for \(\omega_{\mathrm{II}}\).

Such an error in determining the angular velocities of natural vibrations is still permissible in many practical cases, all the more so since, for engines with more than four cylinders, it becomes still smaller.

It should be noted that the use of the method of replacing all the masses of the engine by a single uniformly distributed mass can be justified only when ready-made curves \(D\) are unavailable for the given number of cylinders.

When calculations of torsional vibrations are carried out repeatedly, it is most expedient to plot in advance, on a sufficiently large scale, the curves \(D\) according to equations (14) for all required numbers of cylinders, and to use them in accordance with what was said in Section I of the present article.

¹) In Fig. 9 the straight line \(L\) is denoted by the number 1.

²) In a completely analogous manner one could determine the values \(\omega_{\mathrm{I}}, \omega_{\mathrm{II}}\), etc., also for a system with two large masses according to Fig. (7), by constructing on diagram \(D\) (Fig. 9) the curve \(L\), according to equation (24). We do not give here an example of such a calculation for lack of space.

Appendix

Table 1                                                                                    $\omega = 345\ \mathrm{sec.}^{-1}$

	$e$	$c$	$-\Theta \omega^2$	$\varphi$
	$-3.5707\,10^5$	$=$	$-3.5707\cdot 10^5$	$+1.00000$	1
1,2	$-3.5707$	$:5.10^6$		$-0.07141$
	$-3.3157$	$=$	$-3.5707$	$+0.92859$	2
2,3	$-6.8864$	$:5.10^6$		$-0.13773$
	$-2.8239$	$=$	$-3.5707$	$+0.79086$	3
3,4	$-9.7103$	$:5.10^6$		$-0.19421$
—	$-2.1305$	$=$	$-3.5707$	$+0.59665$	4
4,5	$-11.8408$	$:3.10^6$		$-0.39427$
	$-7.2264$	$=$	$-35.707$	$+0.20238$	5
5,6	$-19.0672$	$:2.10^6$		$-0.95386$
	$+18.7833$	$=$	$-24,995$	$-0.75148$	6
$R=-0.2839\cdot 10^5$

Table 2                                                                                $\omega = 346\ \mathrm{sec.}^{-1}$

	$e$	$c$	$-\Theta \omega^2$	$\varphi$
	$-3.5915\,10^5$	$=$	$-3.5915\cdot 10^5$	$+1.00000$	1
1,2	$-3.5915$	$:5.10^6$		$-0.07183$
	$-3.3335$	$=$	$-3.5915$	$+0.92817$	2
2,3	$-6.9250$	$:5.10^6$		$-0.13850$
	$-2.8361$	$=$	$-3.5915$	$+0.78967$	3
3,4	$-9.7611$	$:5.10^6$		$-0.19522$
	$-2.1315$	$=$	$-3.5915$	$+0.59445$	4
4,5	$-11.8961$	$:3.10^6$		$-0.39614$
	$-7.1223$	$=$	$-35.915$	$+0.19831$	5
5,6	$-19.0184$	$:2.10^6$		$-0.95092$
	$+18.9200$	$=$	$-25.139$	$-0.75261$	6
$R=-0.0384\cdot 10^5$

Table 3

\(\omega = 347\ \mathrm{s}^{-1}\)

	\(e\)	\(c\)	\(-\Theta \omega^2\)	\(\varphi\)
	\(-3.6123 \cdot 10^5\)	\(=\)	\(-3.6123 \cdot 10^5\)	\(+1.00000\)	1
1,2	\(-3.6123\)	\(:\,5 \cdot 10^6\)		\(-0.07225\)
	\(-3.3513\)	\(=\)	\(-3.6123\)	\(+0.92775\)	2
2,3	\(-6.9636\)	\(:\,5 \cdot 10^6\)		\(-0.13927\)
	\(-2.8482\)	\(=\)	\(-3.6123\)	\(+0.78848\)	3
3,4	\(-9.8118\)	\(:\,5 \cdot 10^6\)		\(-0.19624\)
	\(-2.1393\)	\(=\)	\(-3.6123\)	\(+0.59224\)	4
4,5	\(-11.9511\)	\(:\,3 \cdot 10^6\)		\(-0.43170\)
	\(-5.7992\)	\(=\)	\(-3.6123\)	\(+0.16054\)	5
5,6	\(-17.7503\)	\(:\,2 \cdot 10^6\)		\(-0.88751\)
	\(+18.1290\)	\(=\)	\(-25.286\)	\(-0.71697\)	6
\(R = +0.3787 \cdot 10^5\)

Table 4

\(\omega = 501\ \mathrm{s}^{-1}\)

	\(e\)	\(c\)	\(-\Theta \omega^2\)	\(\varphi\)
	\(-7.5300 \cdot 10^5\)	\(=\)	\(-7.530 \cdot 10^5\)	\(+1.00000\)	1
1,2	\(-7.5300\)	\(:\,5 \cdot 10^6\)		\(-0.15060\)
	\(-6.3989\)	\(=\)	\(-7.530\)	\(+0.84940\)	2
2,3	\(-13.9289\)	\(:\,5 \cdot 10^6\)		\(-0.27858\)
	\(-4.2983\)	\(=\)	\(-7.530\)	\(+0.57082\)	3
3,4	\(-18.2272\)	\(:\,5 \cdot 10^6\)		\(-0.36454\)
	\(-1.5453\)	\(=\)	\(-7.530\)	\(+0.20528\)	4
4,5	\(-19.7725\)	\(:\,3 \cdot 10^6\)		\(-0.65842\)
	\(+34.1215\)	\(=\)	\(-75.30\)	\(-0.45314\)	5
5,6	\(+14.3490\)	\(:\,2 \cdot 10^6\)		\(+0.71745\)
	\(-13.9318\)	\(=\)	\(-52.710\)	\(+0.26431\)	6
\(R = +0.4172 \cdot 10^5\)

Table 5

\(\omega = 502\ \mathrm{sec}^{-1}\)

	\(e\)	\(C\)	\(-\Theta\omega^2\)	\(\varphi\)
	\(-7.5601\cdot 10^5\)	\(=\)	\(-7.5601\cdot 10^5\)	\(+1.0000\)	1
1,2	\(-7.5601\)	\(:5\cdot 10^6\)		\(-0.15120\)
	\(-6.4170\)	\(=\)	\(-7.5601\)	\(+0.84880\)	2
2,3	\(-13.9771\)	\(:5\cdot 10^6\)		\(-0.27954\)
	\(-4.3036\)	\(=\)	\(-7.5601\)	\(+0.56926\)	3
3,4	\(-18.2807\)	\(:5\cdot 10^6\)		\(-0.36561\)
	\(-1.5396\)	\(=\)	\(-7.5601\)	\(+0.20365\)	4
4,5	\(-19.8203\)	\(:3\cdot 10^6\)		\(-0.66000\)
	\(+34.5005\)	\(=\)	\(-75.601\)	\(-0.45635\)	5
5,6	\(+14.6802\)	\(:2\cdot 10^6\)		\(+0.73401\)
	\(-11.6940\)	\(=\)	\(-52.921\)	\(+0.27766\)	6
	\(R=-0.0138\cdot 10^5\)

Table 6

\(\omega = 503\ \mathrm{sec}^{-1}\)

	\(e\)	\(C\)	\(-\Theta\omega^2\)	\(\varphi\)
	\(-7.5903\cdot 10^5\)	\(=\)	\(-7.5903\cdot 10^5\)	\(+1.00000\)	1
1,2	\(-7.5903\)	\(:5\cdot 10^6\)		\(-0.15181\)
	\(-6.4380\)	\(=\)	\(-7.5903\)	\(+0.84819\)	2
2,3	\(-14.0283\)	\(:5\cdot 10^6\)		\(-0.28057\)
	\(-4.3084\)	\(=\)	\(-7.5903\)	\(+0.56762\)	3
3,4	\(-18.3367\)	\(:5\cdot 10^6\)		\(-0.36673\)
	\(-1.5248\)	\(=\)	\(-7.5903\)	\(+0.20089\)	4
4,5	\(-19.8615\)	\(:3\cdot 10^6\)		\(-0.66139\)
	\(+34.9530\)	\(=\)	\(-75.903\)	\(-0.46050\)	5
5,6	\(+15.0915\)	\(:2\cdot 10^6\)		\(+0.75457\)
	\(-15.6244\)	\(=\)	\(-53.132\)	\(+0.29407\)	6
	\(R=-0.5329\cdot 10^5\)

Table 7
\(\omega = 1204\ \mathrm{sec.}^{-1}\)

	\(e\)	\(c\)	\(-\Theta\omega^2\)	\(\varphi\)
	\(-43.488 \cdot 10^5\)	\(=\)	\(-43.488 \cdot 10^5\)	\(+1.0000\)	1
1,2	\(-43.488\)	\(:\,5\cdot10^6\)		\(-0.86976\)
	\(-5.664\)	\(=\)	\(-43.488\)	\(+0.13024\)	2
2,3	\(-49.152\)	\(:\,5\cdot10^6\)		\(-0.98304\)
	\(+37.087\)	\(=\)	\(-43.488\)	\(-0.85280\)	3
3,4	\(-12.065\)	\(:\,5\cdot10^6\)		\(-0.24130\)
	\(+47.580\)	\(=\)	\(-43.488\)	\(-1.09410\)	4
4,5	\(+35.515\)	\(:\,3\cdot10^6\)		\(+1.18265\)
	\(-38.509\)	\(=\)	\(-434.88\)	\(+0.08855\)	5
5,6	\(-3.094\)	\(:\,2\cdot10^6\)		\(-0.15470\)
	\(+2.014\)	\(=\)	\(-304.42\)	\(-0.06615\)	6
	\(R=-1.080\cdot10^5\)

Table 8
\(\omega = 1205\ \mathrm{sec.}^{-1}\)

	\(e\)	\(c\)	\(-\Theta\omega^2\)	\(\varphi\)
	\(-43.560 \cdot 10^5\)	\(=\)	\(-43.560 \cdot 10^5\)	\(+1.00000\)	1
1,2	\(-43.560\)	\(:\,5\cdot10^6\)		\(-0.87121\)
	\(-5.610\)	\(=\)	\(-43.560\)	\(+0.12879\)	2
2,3	\(-49.170\)	\(:\,5\cdot10^6\)		\(-0.98342\)
	\(+37.228\)	\(=\)	\(-43.560\)	\(-0.85463\)	3
3,4	\(-11.942\)	\(:\,5\cdot10^6\)		\(-0.23884\)
	\(+47.634\)	\(=\)	\(-43.560\)	\(-1.09347\)	4
4,5	\(+35.692\)	\(:\,3\cdot10^6\)		\(+1.18854\)
	\(-41.414\)	\(=\)	\(-435.60\)	\(+0.09507\)	5
5,6	\(-5.722\)	\(:\,2\cdot10^6\)		\(-0.28610\)
	\(+5.825\)	\(=\)	\(-304.93\)	\(-0.19103\)	6
	\(R=+0.103\cdot10^5\)

SUMMARY.

The determination of frequencies of natural vibrations of a shaft forms the first and most important stage of every calculation of power plants with piston engines.

It is known that a general way for determining the frequencies of natural torsional vibrations of a shaft having several masses fixed on it consists in the solution of a system of linear differential equations. The solution of this system of equations for a shaft with many masses is very cumbersome and does not suit the purposes of practice.

The existing more practical methods for determining the frequencies of natural vibrations give the solution sought as a result of a series of tests and successive approximations, and require a great amount of calculations.

This article describes a graphical method permitting one to determine, without long trial estimates, the frequencies of natural vibrations of a shaft with \(n\) equal masses being an equal distance apart and with one or two large flywheels (this being the case in Diesel flywheel dynamo and in a series of ship screw installations).

This method is based on the following statement proved in this article: the sum of moments of inertia of all the masses of the system reduced to one point and to one section of the shaft, the vibrations of the system being free, is equal to zero.

For the determination of frequencies of natural vibrations by the given method, it is necessary to find the points of intersection of two curves \(D_n = f(A)\) and \(L = F(A)\), constructed in the coordinates \(A\) (the axis of abscissas), and \(D, L\) (the axis of ordinates). The value \(D_n\) depends only on \(n\)—the number of cylinders of the engine—and on the parameter \(A\). A curve \(D_n = f(A)\) may be constructed once and for all for each \(n\). Every such curve is entirely suitable for any installation with an engine having a given number of cylinders.

The value \(L\) depends on the parameter \(A\), and also on the relative value of the moments of inertia of the masses of the system and on the stiffness of the sections of the shaft between them. The parameter \(A\) is a function of the angular speed of natural torsional vibrations.

The method of determining the natural frequencies of vibrations given in the first part of the article proves to be quite precise, including only those mistakes which are unavoidable with any graphical construction.

A comparison of the results of the estimate of one exemplary installation by the given method with those of the estimate by the Tolle method shows their total coincidence.

In the second part of the article the author shows the possibility of a still further simplification of the solution of the problem of determining the frequencies of natural vibrations by replacing all the concentrated masses of the engine by one mass uniformly distributed over its shaft. In this case, for engines with any (but not a too small) number of cylinders, the same curve \(D_n = F(A)\) is suitable.

Fig. 5.

Fig. 6.

Fig. 7.

Fig. 8.

Fig. 9.

1. F. Porter, Transactions A. S. M. E., Vol. 52 (1931). ↩↩↩↩

2. The method proposed by L. Geiger of replacing all the engine masses by a single resultant mass applied at the middle of the length of the crankshaft often leads to very large errors in the determination of \(\omega\), and therefore cannot be recommended. ↩↩