Assoc. Prof. V. K. Nechaev.

Shift of Critical Speeds

(On the question of torsional vibrations of shafts).

Every power installation with piston engines usually has a whole series of so-called critical speeds, or critical numbers of revolutions. These critical speeds may be the cause of very serious disturbances in the normal operation of the installation.

Operation of an engine at critical numbers of revolutions often leads to fracture of the shaft, breakage of spokes in the flywheel or generator, fracture of the transmission mechanism, etc., or, in the best case, to increased wear of the machine parts, loud noise, and shocks during its operation¹.

The critical speeds of an installation are all those speeds at which one of the natural frequencies of vibration of the shafting coincides with the frequency of one of the harmonics of the torque on the engine shaft. At these speeds the corresponding harmonic of the torque enters into resonance with the natural torsional vibrations of the shaft and causes strong forced vibrations of large amplitude.

As is known, harmonic analysis of the torque curve on the shaft of a piston engine gives a series of elementary harmonic moments, with periods equal to

\[ \frac{1}{1},\quad \frac{1}{2},\quad \frac{1}{3} \]

and so on, of the duration of one complete cycle of the working process in the cylinder. For example, analysis of the torque diagram of a four-stroke internal-combustion engine gives elementary harmonic moments of the following frequencies (by frequency we here mean the number of complete cycles of variation of the harmonic per minute):

\[ \frac{1}{2}n,\quad n,\quad 1\frac{1}{2}n,\quad 2n,\quad 2\frac{1}{2}n \text{, etc.} \]

where \(n\) is the number of revolutions of the engine shaft per minute. The first of these harmonics is customarily called the harmonic of order \(1/2\), the second the harmonic of the 1st order, the third the harmonic of order \(1\frac{1}{2}\), etc. In general, the order of a given torque harmonic determines the number of complete cycles of its variation during one revolution of the engine shaft. The amplitudes of the elementary harmonic moments decrease as their order increases.

The decomposition of the torque curve into a whole series of harmonics and the presence, in every multi-mass system, of several natural frequencies of torsional vibration lead to the result that any installation, even one not especially complex, has very many critical speeds.

The order of each critical speed (critical number of revolutions) is named in accordance with the order of that torque harmonic which, at these revolutions, resonates with the natural vibrations of the system. Thus, for example, for the installation: a four-cylinder four-stroke engine—flywheel (a system with five masses), having four²

for the frequencies of natural vibrations \(n_{\mathrm{I}}, n_{\mathrm{II}}, n_{\mathrm{III}}, n_{\mathrm{IV}}\), there exist the following values of critical speeds:

\[ \text{Single-node vibrations}\ldots n_{\mathrm{I}\,\mathrm{cr}}=\frac{n_{\mathrm{I}}}{i}\ \text{rpm.} \]

\[ \text{Two-node vibrations}\quad\ldots n_{\mathrm{II}\,\mathrm{cr}}=\frac{n_{\mathrm{II}}}{i}\ \text{rpm.} \]

\[ \text{Three-node vibrations}\quad\ldots n_{\mathrm{III}\,\mathrm{cr}}=\frac{n_{\mathrm{III}}}{i}\ \text{rpm.} \]

\[ \text{Four-node vibrations}\quad\ldots n_{\mathrm{IV}\,\mathrm{cr}}=\frac{n_{\mathrm{IV}}}{i}\ \text{rpm.} \]

where \(i\) assumes each of the following values:

\[ i=\tfrac{1}{2},\ 1,\ 1\tfrac{1}{2},\ 2,\ 2\tfrac{1}{2},\ 3\ \text{etc.} \]

The value of the coefficient \(i\) determines the order of the given critical speeds\(^{3}\). For practical purposes, only the critical speeds corresponding to single-node and two-node vibrations are of importance. Critical speeds for vibrations with three or more nodes are entirely insignificant, since they lie far beyond the operating speeds of engines.

All critical speeds of an installation may, according to F. Lewis, be divided into two principal classes\(^{4}\):

1) large or principal critical speeds, whose order is a multiple of the number of firings in the engine per one revolution of its shaft, and

2) small or secondary critical speeds—all the rest.

The amplitudes of forced vibrations at the principal critical speeds are determined by the magnitude of the resonating harmonic of the torque, the number of cylinders, the magnitude and arrangement of the masses and elasticities of the system, and the magnitude of the damping forces, and cannot be changed by altering the order of succession of firings in the engine cylinders.

The “strength” of the small (secondary) critical speeds depends to a considerable degree on the order of succession of firings in the engine.

For the installation indicated above with a four-cylinder engine, the following will evidently be the principal critical speeds:

\[ \frac{n_{\mathrm{I}}}{2};\quad \frac{n_{\mathrm{I}}}{4};\quad \frac{n_{\mathrm{I}}}{6};\quad \frac{n_{\mathrm{I}}}{8};\quad \ldots\ \text{rpm, etc.} \]

\[ \frac{n_{\mathrm{II}}}{2};\quad \frac{n_{\mathrm{II}}}{4};\quad \frac{n_{\mathrm{II}}}{6};\quad \frac{n_{\mathrm{II}}}{8};\quad \ldots\ \text{rpm, etc.,} \]

since in this engine we have two firings per one revolution of its shaft; all other critical speeds for single-node and two-node vibrations will be secondary (critical speeds corresponding to \(n_{\mathrm{III}}\) and \(n_{\mathrm{IV}}\), according to what was said above, are neglected).

\(^{3}\) Obviously, in power installations with two-stroke engines there can be only critical speeds of integer orders.

\(^{4}\) F. Lewis. Torsional Vibrations in the Diesel Engines. Trans. Soc. Nav. Arch. 1925; see also N. Urvantsev. Critical Speeds in Diesel Installations, 1931.

The most dangerous for the operation of the installation are the principal critical numbers of revolutions for single-node oscillations. Operation of the engine at these, or close to these, speeds must under no circumstances be permitted, since in that case, as a result of strong forced oscillations, inadmissibly high stresses arise in the shaft. These stresses increase as the order of the given principal critical speeds decreases. The principal critical speeds may exclude a considerable part of the range of operating speeds of the engine that had previously been planned by the designer in designing the shipboard installation.

In the same way, operation of the engine is also impermissible at critical (or near-critical) speeds of low orders for two-node oscillations.

Prolonged operation of the installation at speeds close to the principal critical numbers of revolutions of comparatively high orders for two-node oscillations, or at secondary critical numbers of revolutions, may be permitted only after a detailed investigation of the magnitude of the amplitudes of the forced oscillations and of the additional stresses arising thereby in the shaft.

Determination of the natural torsional-vibration frequencies of the shafting rather often leads to the discovery of a complete, or sufficiently close, coincidence of the specified operating number of revolutions of the engine with one or another of its critical speeds. To eliminate the dangers of resonance arising in this case, it is necessary to make certain definite changes in the installation, so as to shift the given critical speeds beyond the range of the engine’s operating speeds. These necessary changes may sometimes prove to be very substantial. For these reasons it is inadvisable to begin calculating torsional oscillations only after the final design of the entire installation has been completed. Still more impermissible is it to check the installation for torsional oscillations only after all working drawings have been released for production.

The question of completely eliminating critical speeds from the engine’s operating speed range is solved very simply only for installations with a constant rotational speed (stationary installations). In shipboard installations, whose number of revolutions varies over wide limits, it is not always possible to avoid operation at critical speeds completely. But even here, at the very least, it is necessary to place the unavoidable (and only secondary) critical speeds in the region of the least-used engine speeds. The principal critical speeds must lie outside the range of operating revolutions of the installation.

To eliminate the dangers of resonance at one or another engine speed, the following methods are available³:

1. Changing (shifting) the operating numbers of revolutions of the engine away from the critical speeds close to them.
2. Changing the order of succession of the firing impulses in the engine cylinders.
3. Changing the magnitude of the masses or the elastic properties of the system.
4. Introducing a damper.

Shifting the operating speeds of the engine is the simplest means of eliminating resonance in stationary installations. But this change is practically permissible only in rare cases, owing to a number of operational considerations.

By appropriate use of the firing order, it is possible to reduce substantially the effect of certain small (secondary) critical speeds. The corresponding choice of the arrangement likewise has an effect—

of the shaft knees. These possibilities of weakening some (and only secondary) critical speeds were first pointed out by F. Lewis^6).

Changing the magnitude of the masses or the elastic properties of the system changes the numbers of its natural vibrations and thus leads to a shift of all the critical speeds of the installation in one direction or the other. Changing the masses of the system is far less effective than changing the elastic properties of the shafting. The flywheel or generator is usually located near a node of vibration on the shaft; the amplitude of its vibration is small. Therefore, even for a comparatively small change in the number of natural vibrations of the system, it is necessary to change the mass of the flywheel considerably, which is not always possible under the conditions of the required uniformity of engine operation.

Changing the elastic properties of the shafting by altering the length of the shaft, the dimensions of its cross-section, or by inserting an elastic coupling should be regarded as the most suitable means for shifting the critical speeds of an installation in the desired direction, at the designer’s discretion.

The installation of a damper is the last measure, advisable only when it is impossible to use the above-mentioned methods for eliminating resonance. By damping strong forced torsional vibrations of the shaft, the damper permits operation of the engine at critical (or near-critical) speeds. Installing a damper requires space, increases the weight of the engine, and somewhat lowers its mechanical efficiency; existing damper designs are suitable only for high-speed engines of not very great power (automobile and aircraft engines).

Here we shall confine ourselves to considering the method of shifting critical speeds by changing the magnitude of the masses and the elastic properties of the oscillating system. A shift of the critical speeds is equivalent to changing the frequencies of the natural vibrations of the shafting in the same direction.

An increase in the frequencies of the natural vibrations of the system can be achieved:

a) by increasing the stiffness of the shafting by thickening its individual sections, or by reducing their length (it is especially convenient to increase the diameter or reduce the length of the connecting shaft between the engine and the flywheel or generator);

b) by reducing the masses of the system, for example by choosing a flywheel or generator with a smaller flywheel moment.

A decrease in the stiffness of the shafting (decreasing diameters and increasing lengths), as well as an increase in the flywheel masses of the system, lowers the frequency of natural vibrations.

A decrease in the shaft diameter does not always entail an increase in the stresses arising in it, since the resulting removal of the range of operating speeds from resonance often fully compensates for the reduction in the resisting moment of its section.

To determine the magnitude of the necessary changes in the masses and elastic characteristics of the system, one usually has to use the method of Tolle^7), who developed this question in the greatest detail. But calculations by this method are rather complicated and lengthy, since for each trial calculation it is necessary to carry out a series of tabular computations (Tolle’s Vorwärts-Rückwärts calculation).

The graphical method for determining the frequencies of natural vibrations, set forth by the author in his previous article^8), makes it possible to solve the problem

^6) See his work cited above; see also I. Morris, Crankshaft, Vibration, Automobile Engineering, February, 1935.

^7) Tolle. Regelung der Kraftmaschinen, 1922, 2 Aufl.

^8) V. K. Nechaev. “A graphical method for determining the frequencies of natural vibrations,” Izvestiya TII, vol. 54, issue 6, 1935, and also Dizelestroenie, No. 4, 1934.

finding the magnitude of the necessary changes in the system required to obtain the desired number of natural vibrations, is comparatively simple and does not involve lengthy calculations.

This determination of the necessary changes proves especially simple and clear for a system with one flywheel, i.e. for the case most frequent in practice.

I. System with one flywheel.

In the author’s article cited, it was established that the frequencies of the natural vibrations of the engine–flywheel system (Fig. 1) are determined by constructing the straight line \(L\), defined by the equation:

Fig. 1.

\[ L = -\frac{A}{\beta_1}\,\frac{1}{x_1}. \tag{1} \]

on the diagram \(D_n\), drawn beforehand for the given number of engine cylinders. Here:

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

\[ \beta_1=\frac{c_{n1}}{c}, \tag{3} \]

\[ x_1=\frac{\Theta_1}{\Theta}. \tag{4} \]

where:

\(\Theta\) — the moment of inertia of the flywheel masses of each engine cylinder,

\(\Theta_1\) — the moment of inertia of the flywheel,

\(c\) — the stiffness of the section of the crankshaft between the axes of adjacent cylinders,

\(c_{n1}\) — the stiffness of the shaft between the last engine cylinder and the flywheel,

\(\omega\) — the angular velocity of vibration.

The value \(D_n\) depends on \(A\) and on the number of engine cylinders. The abscissas of the points of intersection of the straight line \(L\) with the curve \(D_n\) give the values \(A_1, A_2 \ldots\), from which the angular velocities of the natural vibrations of the system are determined by the formula: ³)

\[ \omega_{\mathrm{I}}=\sqrt{\frac{A_1c}{\Theta}} \]

\[ \omega_{\mathrm{II}}=\sqrt{\frac{A_2c}{\Theta}}\quad \text{and so on.} \tag{5} \]

To shift the obtained natural frequencies of vibration of the shafting in a certain direction and by a certain amount, it is necessary, in the proper manner, to change the stiffness \(c_{n1}\) of the shaft between the last engine cylinder and the flywheel, or the moment of inertia \(\Theta_1\) of the flywheel, or both simultaneously. Of course, the fundamental possibility is also not excluded—

³) See the author’s work cited above.

changes in the stiffness of the crankshaft and in the flywheel masses of each cylinder. But in practice this is usually not feasible. Therefore we exclude such a case from our consideration.

It follows from formula (1) that the magnitude of the angle of inclination of the straight line \(L\) to the axis of abscissas depends on the ratio \(c_n\) and \(c\).

The straight line \(L\) rises more steeply when \(c_n\) is decreased; in this case the points \(M\) of its intersection with the curve \(D_n\) move to the left (Fig. 2). The value of \(A\), and consequently the frequencies of the natural vibrations, decrease.

The magnitude of the segment \(OB=-{}^{1}/\alpha_1\), cut off by the straight line \(L\) on the axis of ordinates, is entirely determined by the ratio of the moments of inertia \(\Theta_1\) and \(\Theta\). When \(\Theta_1\) is decreased, the segment \(OB\) increases and the points of intersection of the straight line \(L\) with the curve \(D_n\) move to the right, which corresponds to an increase in the frequencies of natural vibrations.

From these considerations there follows the following method for determining the magnitudes of the masses and the elastic properties of the shafting that are necessary in order to obtain the specified, required frequencies of the natural vibrations of the system.

Fig. 2.

a) Change in the stiffness \(c_n\) of the shaft between the last cylinder of the engine and the flywheel (with the masses of the system unchanged).

For a specified frequency of natural vibrations \(n_1\) or \(n_{\mathrm{II}}\) we compute the corresponding angular velocity of vibration

\[ \omega=\frac{\pi n}{30} \]

and from it, using formula (2), we calculate the value \(A\), which determines the abscissa of the required point of intersection of the straight line \(L\) with the curve \(D_n\). On this abscissa we find the corresponding point \(M_1\) on the curve \(D_n\) (Fig. 2). Through the point \(B\) on the axis of ordinates, determined by the value

\[ OB=-{}^{1}/\alpha_1 \]

and the point \(M_1\), we draw the straight line \(M_1B\). The tangent of the angle of inclination \(\gamma\) of this straight line to the axis of abscissas (taken, in view of the possible difference of scales along the two coordinate axes, not from the drawing with a protractor, but computed as the ratio of any ordinate of this straight line to the corresponding abscissa) gives the value \({}^{1}/\beta_1\), whence \(\beta_1\) is determined, and after it the new, required value of the stiffness \(c_{n1}\):

\[ c_{n1}=\beta_1 c \]

This new, necessary value of \(c_n\) can be computed directly from the formula. From the drawing:

\[ \operatorname{tg}\gamma=\frac{1}{\beta_1}=\frac{D_{M_1}+{}^{1}/\alpha_1}{A} \]

or

\[ \beta_1=\frac{A}{D_{M_1}+1/\alpha_1}, \]

and

\[ c_{n1}=c\frac{A}{D_{M_1}+1/\alpha_1} . \tag{6} \]

Here \(D_{M_1}\) is the ordinate of the point \(M_1\), lying on the curve \(D_n\) and corresponding to the abscissa \(A_1\), calculated for the given \(\omega\).

b) Changing \(\Theta_1\) of the flywheel (with the elastic properties of the system unchanged).

Through the point \(M_1\), found on the curve \(D_n\) in the manner indicated above, we draw the straight line \(M_1B_1\), parallel to the initial straight line \(L\) (Fig. 3).

This straight line cuts off on the ordinate axis the segment \(OB_1\). The required value of the moment of inertia \(\Theta_1\) of the flywheel is now determined as follows:

\[ \frac{\Theta_1}{\Theta}=\alpha_1=\frac{1}{OB_1}, \]

or

\[ \Theta_1=\frac{\Theta}{OB_1}, \tag{7} \]

where \(\Theta\), as before, denotes the moment of inertia of the flywheel masses of each cylinder of the engine.

The abscissa of the point \(M_2\) of intersection of the straight line \(M_1B_1\) with the second branch of the curve \(D_n\) determines the value of the second natural frequency of the system.

c) Simultaneous change of \(\Theta_1\) and \(c_n\).

The change of \(c_n\) or \(\Theta_1\) considered in items a) and b) makes it possible to obtain a prescribed value of one of the natural frequencies of the system. This prescribed frequency value is determined from the conditions for a sufficient displacement of dangerous critical speeds away from the range of operating speeds of the installation. Most often, in this case, one has to deal with the critical speeds of one-node vibrations. But a change in the masses or in the elastic properties of the system leads to a change in all the natural frequencies and, consequently, to a displacement of all critical speeds. When the dangerous critical speeds of one-node vibrations are shifted, a new superposition of strong critical speeds of two-node vibrations on the operating speeds of the engine may appear. Therefore, when determining the necessary changes in the installation, it is advisable to choose the desired new values of the frequencies of both one-node and two-node vibrations. To realize these frequencies it is necessary simultaneously to change \(\Theta_1\) and \(c_n\). The method set forth here makes it easy to determine the magnitude of these changes.

For the selected \(\omega_1\) and \(\omega_{11}\), we calculate the corresponding values \(A_1'\) and \(A_2'\):

\[ A_1'=\frac{\Theta\omega_1^2}{c};\qquad A_2'=\frac{\Theta\omega_{11}^2}{c}, \]

representing the abscissas of the required points of intersection of the new straight line \(L'\) with the curve \(D_n\). From these abscissas we find the sought points \(M_1\) and \(M_2\) of the curve \(D\) (Fig. 4). Through \(M_1\) and \(M_2\) we draw the straight line \(B_2 M_1 M_2\). It cuts off on the ordinate axis the segment \(\overline{OB_2}\). Now the new required value \(\Theta_1\) is determined from the expression:

\[ \Theta_1=\frac{\Theta}{\overline{OB_2}}. \]

The magnitude of the corresponding required stiffness \(c_{\text{п}}\) between the last cylinder of the engine and the flywheel is computed by the formula:

\[ c_{\text{п}}=\beta_1 c \]

Fig. 4.

The coefficient \(\beta_1\) entering here is determined from the obvious relations:

\[ \frac{1}{\beta_1}=\frac{D_{M_1}+\overline{OB_2}}{A_1} \]

or

\[ \frac{1}{\beta_1}=\frac{D_{M_2}+\overline{OB_2}}{A_2} \tag{8} \]

Here \(D_{M_1}\) and \(D_{M_2}\) are respectively the ordinates of the points \(M_1\) and \(M_2\). The segment \(\overline{OB_2}\) is taken in absolute value.

From the preceding it is clear that by changing \(\Theta_1\) and \(c_{\text{п}}\) it is not always possible to obtain any two prescribed natural frequencies of the system. The point \(B_2\) (Fig. 4) of intersection of the straight line drawn through \(M_1\) and \(M_2\) with the ordinate axis cannot, in essence, lie above the origin \(O\), since a positive value of the segment \(\overline{OB_2}\) corresponds to a negative value of \(\Theta_1\), which is impossible. This circumstance must be taken into account when choosing the desirable values of the natural frequencies.

II. System with two flywheels \(\Theta_{\mathrm{I}}\) and \(\Theta_{\mathrm{II}}\).

To determine the natural frequencies of the system according to Fig. 6, it is necessary, on the diagram \(D_n=f(A)\) for the given number of engine cylinders, to construct the curve \(L\) by the equation\(^{10)}\):

\[ L=\frac{1-\delta A+\varepsilon A^2}{(\alpha_1+\alpha_2)(\gamma A-1)} \tag{9} \]

The abscissas of the points of intersection of this curve with the curve \(D_n\) give the values \(A_1, A_2\), etc., corresponding to the angular velocities:

\[ \omega_{\mathrm{I}}=\sqrt{\frac{A_1 c}{\Theta}};\quad \omega_{\mathrm{II}}=\sqrt{\frac{A_2 c}{\Theta}} \quad \text{and so on.} \]

\(^{10)}\) See the author’s work cited above.

free vibrations. Now the frequencies of the natural vibrations are:

\[ n_1=\frac{30\omega_1}{\pi}\quad \text{and so on.} \]

In formula (9), for \(L\):

\[ \gamma=\frac{\alpha_1\alpha_2}{\beta_2(\alpha_1+\alpha_2)}, \]

\[ \delta=\frac{\alpha_1+\alpha_2}{\beta_1}+\frac{\alpha_2}{\beta_2}, \]

\[ \varepsilon=\frac{\alpha_1\alpha_2}{\beta_1\beta_2}, \]

where

\[ \alpha_1=\frac{\Theta_1}{\Theta};\qquad \alpha_2=\frac{\Theta_{\mathrm{II}}}{\Theta} \]

\[ \beta_1=\frac{c_{\mathrm{II}}}{c};\qquad \beta_2=\frac{c_{\mathrm{I,II}}}{c}. \]

Let us introduce the indicated values for \(\gamma,\delta\), and \(\varepsilon\) into formula (9):

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

This equation of the curve \(L\), obtained by the author in the preceding article, for the convenience of further investigations, may be rewritten in the following form:

\[ L= \frac{ 1-\frac{\alpha_2}{\beta_2}A }{ \frac{\alpha_1\alpha_2}{\beta_2}-(\alpha_1+\alpha_2) } +\frac{A}{\beta_1}, \]

or

\[ L= \frac{\beta_2-A\alpha_2}{\alpha_1\alpha_2A-\beta_2(\alpha_1+\alpha_2)} +\frac{A}{\beta_1}\ . . . . . . . . (10) \]

The last expression makes it possible to determine the magnitude of those changes in the masses and elastic characteristics of the shaft line which are necessary in order to obtain the prescribed, desired frequencies of the system’s natural torsional vibrations.

For this it is necessary, as before, from the prescribed angular velocity of the natural vibrations \(\omega_{\mathrm{I}}\) or \(\omega_{\mathrm{II}}\), to compute the corresponding value \(A\), using formula (2), and from it to find on the curve \(D_n\) the point \(M\) (Fig. 5). The curve \(L\) for the modified system must pass through the point \(M\), i.e. the equality must be satisfied:

\[ L=D_M, \]

or, according to (10):

\[ D_M= \frac{\beta_2-\alpha_2A}{\alpha_1\alpha_2A-\beta_2(\alpha_1+\alpha_2)} +\frac{A}{\beta_1}\ . . . . . . . (11) \]

where \(A\) is the abscissa of the necessary (required or desired) point \(M\) of intersection of the curve \(L\) with the curve \(D_n\), corresponding to the required frequency of the natural vibrations; \(D_M\) is the ordinate of this point.

The latter equality (11) is in general satisfied by a whole series of combinations of the values of the coefficients \(\alpha_1,\alpha_2,\beta_1,\beta_2\), characterizing the relative

Fig. 5.

magnitudes of the masses and of the elastic characteristics of the system. Considering the value of three of these coefficients as known, from this one can determine the value of the fourth coefficient, by which the required magnitude of the corresponding mass or of the stiffness of a shaft segment is then easily calculated.

It must be borne in mind that the most effective will be a change in the stiffness of the shaft on that segment on which, for the given \(\omega\), there will be found

node of vibration. In a system with two flywheels (Fig. 6), it is most expedient to change the shaft stiffness in the section between both flywheels (when shifting the critical speeds corresponding to single-node vibrations).

Fig. 6.

To obtain the basic calculation formulas, we solve equation (11) with respect to each of the four coefficients entering into it.

a) Change in the stiffness \(c_{n}\) of the shaft between the last cylinder of the engine and the flywheel.

The required new value \(c_{n}\) is found from the formula:

\[ c_{n}=\beta_{1}c \]

where \(\beta_{1}\) is determined from the basic equation (11):

\[ \beta_{1}= \frac{A}{ D_{M}-\dfrac{\beta_{2}-\alpha_{2}A}{\alpha_{1}\alpha_{2}A-(\alpha_{1}+\alpha_{2})\beta_{2}} } \ldots\ldots\ldots (12) \]

b) Change in the stiffness \(c_{\mathrm{I\,II}}\) of the shaft between flywheels I and II.

The required new value of the stiffness is:

\[ c_{\mathrm{I\,II}}=\beta_{2}c. \]

The coefficient \(\beta_{2}\) is calculated by the formula:

\[ \beta_{2}=\alpha_{2}A- \frac{\alpha_{1}(A-\beta_{1}D_{M})-\beta_{1}} {(\alpha_{1}+\alpha_{2})(A-\beta_{1}D_{M})-\beta_{1}} \ldots\ldots\ldots (13) \]

which is easily obtained from equality (11).

c) Change in the moment of inertia of flywheel I.

The new value \(\Theta_{1}\), giving the specified frequency of the natural vibrations of the system:

\[ \Theta_{1}=\alpha_{1}\Theta \]

The coefficient \(\alpha_{1}\) is determined from the basic equation (11),

\[ \alpha_{1}= \frac{\alpha_{2}\beta_{2}}{A\alpha_{2}-\beta_{2}} + \frac{\beta_{1}}{A-D_{M}\beta_{1}} \ldots\ldots\ldots (14) \]

d) Change in the moment of inertia of flywheel II.

The required new value of the moment of inertia of the second flywheel:

\[ \Theta_{\mathrm{II}}=\alpha_{2}\Theta \]

The coefficient \(\alpha_{2}\) is calculated by the formula:

\[ \alpha_{2}=\beta_{2}- \frac{1+\alpha_{2}(D_{M}-A/\beta_{1})} {A+(A\alpha_{1}-\beta_{2})(D_{M}-A/\beta_{1})} \ldots\ldots\ldots (15) \]

obtained from equation (11).

e) Simultaneous variation of the magnitudes of the moments of inertia of the masses and of the elastic characteristics of the system.

Calculation according to items a), b), c), or d) makes it possible to determine the magnitude of those changes in one of the masses or stiffnesses of the system which are necessary in order to obtain one prescribed natural frequency.

The calculation method proposed by the author also makes it possible to satisfy the requirement of realizing two prescribed frequencies (single-node and two-node) of the shafting’s natural torsional vibrations.

For the prescribed (required) values of the angular velocities $\omega_I$ and $\omega_{II}$ of the single-node and two-node natural vibrations of the system, we compute the corresponding values $A_1$ and $A_2$ (the abscissae of the necessary points of intersection of the curves $L$ and $D$) (Fig. 5). From these abscissae we find the points $M_1$ and $M_2$ on the curve $D_n$ and measure their ordinates $D_{M1}$ and $D_{M2}$.

Now we compose expressions (11):

\[ D_{M1}=-\frac{\beta_2-\alpha_2 A_2}{\alpha_1\alpha_2 A_1-(\alpha_1+\alpha_2)\beta_2}+\frac{A_1}{\beta_1}\ldots\ldots (16) \]

\[ D_{M2}=-\frac{\beta_2-\alpha_2 A_2}{\alpha_1\alpha_2 A_2-(\alpha_1+\alpha_2)\beta_2}+\frac{A_2}{\beta_1}\ldots\ldots (17) \]

Hence, for example, by jointly solving these two equations one can determine the required values of $\beta_1$ and $\beta_2$, if for all the remaining coefficients $(\alpha_1,\alpha_2,A_1,A_2)$ their numerical values are substituted.

In general, using these equations, one can determine the necessary magnitude of two of the four coefficients $\alpha_1,\alpha_2,\beta_1,\beta_2$, ensuring the obtaining of two prescribed frequencies of the shafting’s natural torsional vibrations.

A joint solution of equations (16) and (17) in general form is inexpedient, in view of the cumbersomeness of the expressions thereby obtained.

We shall show the application of the method set forth on a particular example.

In the example given in the author’s preceding article, for a power plant consisting of a four-cylinder engine, a flywheel, and a dynamo, and having:

\[ \begin{aligned} \Theta &= 3.0\ \text{kgm}\cdot\text{sec}^2, & c &= 5.0\cdot 10^6\ \text{kgm/radian},\\ \Theta_1 &= 30.0\ \text{''}, & c_{nI} &= 3.0\cdot 10^6\ \text{''},\\ \Theta_{II} &= 21.0\ \text{''}, & c_{II} &= 2.0\cdot 10^6\ \text{''}, \end{aligned} \]

\[ \begin{aligned} \alpha_1 &= 10.0,\\ \alpha_2 &= 7.0,\\ \beta_1 &= 0.60,\\ \beta_2 &= 0.40, \end{aligned} \]

the following values of the natural-vibration frequencies were obtained:

\[ \begin{aligned} n_I &= 3305\ \text{vibrations/min},\\ n_{II} &= 4795\ \text{''} \end{aligned} \]

and, correspondingly:

\[ \begin{aligned} \omega_I &= 346.0\ \text{sec}^{-1},\\ \omega_{II} &= 502\ \text{sec}^{-1}. \end{aligned} \]

The curves \(L\) corresponding to these characteristics of the oscillating system are denoted in Fig. 5 by I.

Let us determine the magnitude of the necessary changes in the masses and elastic properties of the plant shafting which ensure, for example, an increase in the frequency of the system’s natural single-node oscillations to the following value:

\[ n_1 = 3600 \text{ oscill./min.} \]

In the general case, the prescribed increase (or decrease) can be achieved by changing one of the quantities \(\Theta_{\mathrm{I}}, \Theta_{\mathrm{II}}, c_{n\mathrm{I}}, c_{\mathrm{III}}\).

1. Increasing \(n_1\) by changing \(\beta_1\), i.e., the rigidity \(c_{n\mathrm{I}}\) of the shaft between the last engine cylinder and the flywheel.

From equation (10):

\[ L=\frac{\beta_2-A\alpha_2}{\alpha_1\alpha_2A-(\alpha_1+\alpha_2)\beta_2}+\frac{A}{\beta_1} \]

it follows that, with an increase of \(\beta_1\), the curve \(L\), in its first branch, goes ever more gently and more gently, starting from the same point on the ordinate axis, determined by the segment

\[ -\frac{1}{\alpha_1+\alpha_2}. \]

In this case the point of intersection of the curve \(L\) with the curve \(D_n\) moves to the right; thereby the coefficient \(A\) increases and the angular velocity of single-node oscillations \(\omega_1\) increases.

The prescribed \(n_1=3600\) corresponds to

\[ \omega_1=\frac{3.14\cdot 3600}{30}=377\ \text{sec.}^{-1}, \]

\[ A_1=\frac{\theta\omega^2}{c}=\frac{3.0\cdot 377}{5\cdot 10^6}=0.0852 \]

For this value of \(A\), from the graph of the curve \(D_4\) we find \({}^{11)}\)

\[ D_{m1}=0.164 \]

Now, by formula (12), we calculate the new required value of \(\beta_1\):

\[ \beta_1= \frac{A_1}{ D_{m1}- \dfrac{\beta_2-\alpha_2A_1}{\alpha_1\alpha_2A_1-(\alpha_1+\alpha_2)\beta_2} } = \frac{0.0852}{ 0.164- \dfrac{0.4-7.0\cdot 0.0852}{70\cdot 0.0852-17\cdot 0.4} } =-1.204 \]

The obtained negative value of \(\beta_1\) shows that no change in \(c_{n\mathrm{I}}\) can produce the required \(\omega_1\) in the given plant.

To determine the limiting, highest \(\omega_1\) attainable by changing \(c_{n\mathrm{I}}\), we construct on the diagram \(D_4\) (Fig. 5) the curve \(L\) for \(\beta_1=\infty\), using the same equation (10):

\[ L=\frac{\beta_2-A\alpha_2}{\alpha_1\alpha_2A-(\alpha_1+\alpha_2)\beta_2} = \frac{0.4-7A}{70A-17\cdot 0.4} \]

This curve is denoted in Fig. 5 by II. The intersection of its first branch with the curve \(D_4\) gives the limiting value of \(A\):

\[ A_{\max}=0.0825 \]

\[ {}^{11)}\ \text{This value for } D_4 \text{ can also be calculated directly by the formula:} \]

\[ D_{m1}= \frac{1-6A_1+5A_1^2-A_1^3} {4-10A_1+6A_1^2-A_1^3} \]

Whence the limiting angular velocity:

\[ \omega_1=\sqrt{\frac{A_{\max}\cdot c}{\Theta}} =\sqrt{\frac{0.0825\cdot 5\cdot 10^6}{3.0}} =371\ \mathrm{sec}^{-1}, \]

and

\[ n_{1\max}=\frac{30}{3.14}\cdot 371=3545\ \text{rev/min}. \]

2. Increasing \(n_1\) by changing \(\beta_2\), i.e., the stiffness of the shaft between the flywheel and the dynamo.

Changing \(c_{II}\) is more effective, since in single-node oscillations of the system under consideration the node of oscillation lies on this section.

By formula (13), for \(D_{M1}=0.164\),

\[ A_1=0.0852, \]

we have:

\[ \beta_2=\alpha_2 A_1- \frac{\alpha_1(A_1-\beta_1D_{M1})-\beta_1} {(\alpha_1+\alpha_2)(A_1-\beta_1D_{M1})-\beta_1} = \]

\[ =0.0852\cdot 7\cdot \frac{10\cdot(0.0852-0.0984)-0.6} {17\cdot(0.0852-0.0984)-0.6} =0.530 \]

Thus, in order to obtain \(n_1=3600\), it is necessary to make

\[ c_{II}=\beta_2\cdot c=0.530\cdot 5.0\cdot 10^6 =2.65\cdot 10^6\ \text{kgm/radian}. \]

On the section between the flywheel and the dynamo it is necessary to increase the shaft diameter, relative to the initial one, in the ratio:

\[ \sqrt[4]{\frac{2.65\cdot 10^6}{2.0\cdot 10^6}}=1.072 \]

(assuming the shaft to have a round solid cross-section), or by 7.2%, which is practically easy to accomplish.

3. Increasing \(n_1\) by changing \(\alpha_1\), i.e., the moment of inertia \(\Theta_1\) of the flywheel.

From formula (14), for \(A=0.0852\) and \(D_{M1}=0.164\):

\[ \alpha_1= \frac{\alpha_2\beta_2}{A\alpha_2-\beta_2} + \frac{\beta_1}{A-D_{M1}\beta_1} = \frac{7\cdot 0.4}{0.0852\cdot 7-0.4} + \frac{0.6}{0.0852-0.164\cdot 0.6} =-44.0 \]

The negative value of \(\alpha_1\) shows that by decreasing \(\Theta_1\) it is impossible to attain the prescribed \(n_1=3600\).

Let us determine the limiting \(n_1\) obtained when the flywheel is reduced to \(\Theta_1=0\). From formula (10), for \(\alpha_1=0\),

\[ L=\frac{A}{\beta_1}+\frac{A}{\beta_2}-\frac{1}{\alpha_2} =A\left(\frac{1}{\beta_1}+\frac{1}{\beta_2}\right)-\frac{1}{\alpha_2}, \]

i.e., the curve \(L\) becomes a straight line cutting off on the ordinate axis the segment

\[ -\frac{1}{\alpha_2}=-\frac{1}{7}=-0.143. \]

This straight line, denoted in Fig. 5 by III, gives the limiting value:

\[ A^{III}=0.0765, \]

which corresponds to:

\[ \omega_{1\,max}=\sqrt{\frac{A^{\mathrm{III}}\cdot c}{\Theta}} =\sqrt{\frac{0.0765\cdot 5\cdot 10^{6}}{3}} =356\ \mathrm{sec}^{-1}, \]

\[ n_{1\,max}=\frac{30\cdot 356}{3.14}\simeq 3400\ \text{oscillations/min}. \]

4. Increasing \(n_1\) by changing \(\alpha_2\), i.e., the moment of inertia \(\Theta_{\mathrm{II}}\) of the dynamo rotor.

From formula (15), for \(A=0.0852\) and \(D_{M1}=0.164\):

\[ \alpha_2=\beta_2\cdot \frac{1+\alpha_1(D_{M1}-A/\beta_1)} {A+(A\alpha_1-\beta_2)(D_{M1}-A/\beta_1)} = \]

\[ =0.4\cdot \left[ \frac{ 1+10\left(0.164-\frac{0.0852}{0.6}\right) }{ 0.0852+(10\cdot 0.0852-0.4) \left(0.164-\frac{0.0852}{0.6}\right) } \right] =5.13, \]

instead of the initial \(\alpha_2=7.0\), adopted in the principal calculation. Such a value of \(\Theta_{\mathrm{II}}\) of the dynamo can sometimes, in design, be realized by selecting a dynamo with a smaller flywheel moment.

From the calculations given above it follows that the required increase in the frequencies of single-node oscillations to \(n_1=3600\) is feasible in the present installation only by reducing the moment of inertia \(\Theta_{\mathrm{II}}\) of the dynamo, or by increasing the stiffness \(c_{\mathrm{II}}\) of the shaft between the dynamo and the flywheel.

But changing each of these quantities generally leads to a change in the frequency not only of single-node, but also of two-node oscillations. The resulting shift of the critical speeds corresponding to two-node oscillations may sometimes prove undesirable, since the new values of the indicated critical speeds may coincide with the operating speeds of the engine.

It was established above that, in order to increase the frequency of single-node oscillations of the installation under consideration to \(n_1=3600\) oscillations/min, it is sufficient to make the stiffness of the shaft between the flywheel and the dynamo equal to \(c_{\mathrm{II}}=2.65\cdot 10^6\) kgm/radian, while keeping unchanged all the other characteristics of the system \(\alpha_1,\alpha_2,\beta_1,c,\Theta\).

The new value of \(c_{\mathrm{II}}\) corresponds to:

\[ \beta_2=0.530. \]

With these data, by formula (10) we obtain the following equation of the curve \(L\):

\[ L= \frac{\beta_2-A\alpha_2} {\alpha_1\alpha_2 A-\beta_2(\alpha_1+\alpha_2)} +\frac{A}{\beta_1} = \frac{0.530-7A}{70A-0.530\cdot 17} +\frac{A}{0.6}. \]

This curve in Fig. 5 is denoted by IV. The point of intersection of its second branch with the curve \(D_4\) has abscissa:

\[ A^{\mathrm{IV}}=0.1628 \]

which corresponds to:

\[ \omega_{\mathrm{II}}= \sqrt{\frac{A^{\mathrm{IV}}\cdot c}{\Theta}} = \sqrt{\frac{0.1628\cdot 5\cdot 10^{6}}{3.0}} =521\ \mathrm{sec}^{-1}, \]

and the frequency of its own two-node vibrations:

\[ n_{\mathrm{II}}=\frac{30}{3.14}\,521=4970\ \text{vibr./min.}, \]

whereas the original elastic system had:

\[ n_{\mathrm{II}}=4795\ \text{vibr./min.} \]

To preserve the former value \(n_{\mathrm{II}}=4795\) vibr./min. at the new value \(n_1=3600\) vibr./min., certain changes must be made simultaneously in two quantities, \(\alpha_2\) and \(\beta_2\). The magnitude of these changes may be found as follows.

The required angular velocities of vibration are:

\[ \omega_1=\frac{\pi\cdot 3600}{30}=377\ \text{sec}^{-1}, \]

\[ \omega_{\mathrm{II}}=\frac{\pi\cdot 4795}{30}=502\ \text{sec}^{-1}, \]

which correspond to:

\[ A_1=\frac{3\cdot 377^2}{5\cdot 10^6}=0.0852, \]

\[ A_2=\frac{3\cdot 502^2}{5\cdot 10^6}=0.1512, \]

\[ D_{M1}=0.164, \]

\[ D_{M2}=\frac{1-6A_2+5A_2^2-A_2^3}{4-10A_2+6A_2^2-A_2^3}=0.0777^{12)} \]

Now let us use, for example, equation (15) and substitute into it first the values \(A_1\) and \(D_{M1}\), and then the values \(A_2\) and \(D_{M2}\). Substituting the values \(A_1\) and \(D_{M1}\):

\[ \alpha_2=\beta_2\, \frac{1+\alpha_1\left(D_{M1}-A_2/\beta_1\right)} {A_1+\left(A_1\alpha_1-\beta_2\right)\left(D_{M1}-\dfrac{A_1}{\beta_1}\right)} = \]

\[ =\beta_2\, \frac{1+10\left(0.164-\dfrac{0.0852}{0.60}\right)} {0.0852+\left(0.0852\cdot 10-\beta_2\right)\left(0.164-\dfrac{0.0582}{0.60}\right)} = \beta_2\frac{1.22}{0.1039-0.022\beta_2}\ .\ .\ .\ (a) \]

Substituting into (15) the values \(A_2\) and \(D_{M2}\):

\[ \alpha_2=\beta_2\, \frac{1+\alpha_1\left(D_{M2}-A_2/\beta_1\right)} {A_2+\left(A_2\alpha_1-\beta_2\right)\left(D_{M2}-A_2/\beta_1\right)} = \]

\[ =\beta_2\, \frac{1+10\left(0.0777-\dfrac{0.1512}{0.60}\right)} {0.1512+\left(1.512-\beta_2\right)\left(0.0777-\dfrac{0.1512}{0.60}\right)} = \beta_2\frac{0.743}{0.113-0.174\beta_2}\ .\ .\ .\ (b) \]

\(^{12)}\) This value \(D_{M2}\), representing the ordinate of the second required point of intersection of the curve \(L\) with the curve \(D_4\), may also be taken directly from the diagram for \(D_4\), from the known abscissa of this point \(A_2=0.1512\).

Equating the right-hand sides of expressions (a) and (b),

\[ \frac{1.22}{0.1039 - 0.022\beta_2} = \frac{0.743}{0.113 - 0.174\beta_2}. \]

Hence:

\[ \beta_2 = 0.309, \]

and now, for example, from (b)

\[ \alpha_2 = 0.309\, \frac{0.743}{0.113 - 0.74 \cdot 0.309} = 3.88. \]

Thus, in order to raise the frequency of the single-node vibrations of the system to \(n_1 = 3600\), while preserving the former frequency of the two-node vibrations \(n_{\mathrm{II}} = 4795\), it is necessary to reduce the moment of inertia of the dynamo rotor to

\[ \Theta_{\mathrm{II}} = \alpha_2 \Theta = 5.13 \cdot 3.0 = 15.39\ \text{kgm sec}^2, \]

and to reduce the stiffness of the shaft between the dynamo and the flywheel to the following value:

\[ c_{\mathrm{II}} = \beta_2 c = 0.530 \cdot 5 \cdot 10^6 = 2.65 \cdot 10^6\ \text{kgm/rad}, \]

while keeping \(\Theta\), \(\Theta_1\), \(c_{\mathrm{II}}\), and \(c\) as before.

In determining the new values of \(\beta_2\) and \(\alpha_2\), instead of equation (15) one could equally well have used equation (12).

In an entirely analogous way, one could find the necessary new values of \(\beta_2\) and \(\alpha_2\) for any other (practically attainable) value of \(n_{\mathrm{II}}\), with the value of \(n_1\) likewise specified.

None of the methods of calculating torsional vibrations known to the author, either in the Russian or in the foreign literature, provides a similar possibility of determining the necessary measures for shifting two natural frequencies in the desired direction and by the required amount.

1. See, for example, ZdVDl, 1929, p. 1383. Engineering, 1930, p. 551 ff. ↩

2. The subscript at \(n\) denotes the number of vibration nodes. ↩

3. See, for example, L. Geiger, Mechanische Schwingungen, Berlin, 1927. ↩