ON SOME FEATURES OF RESONANT TORSIONAL VIBRATIONS OF SHAFTS IN MOTOR INSTALLATIONS

V. K. NECHAEV

The modern practical methods for calculating resonant torsional vibrations of shafts in motor installations with piston engines are based, along with certain others, on the following propositions:

1) the elastic curve of the shaft (the so-called curve of the form of its vibrations) at resonance is similar to the elastic curve (that is, the mode-shape curve) of the free vibrations of this shaft in the absence of damping;

2) when a single resonating harmonic exciting moment acts on a linear oscillatory system, the forced vibrations of the point at which this moment is applied lag behind the moment by an angle of \(90^\circ\).

As a rule, the methodology for experimentally determining damping coefficients in motor installations from the results of torsiographing is also based on these same propositions.

Proposition 1, sometimes called Widler’s rule, is sufficiently substantiated by a number of sample calculations (Widler) and by theoretical analysis [1]. It is very valuable for calculation practice, since it makes it possible to determine the resonant stresses in the shaft, and also to compute experimental values of the damping coefficients in an installation from the results of torsiographing one end of the shaft.

Proposition 2 is borrowed from the elementary theory of resonant vibrations in a linear system with one degree of freedom and is usually considered entirely natural and valid for multimass systems.

However, in applying these propositions it is usually not taken into account that in the shaft of a motor installation there are not only elastic but also rigid vibrations. The rigid vibrations are often very small in comparison with the elastic vibrations, and neglecting them does not lead to noticeable errors.

But K. Wilson [4] already noted that the rigid vibrations of the shaft of an installation may be the cause of substantial distortions of torsiograms and, as a consequence, lead to large errors in determining shaft twist angles from the results of torsiographing.

In our article [2] it was shown that the ratio between the rigid and elastic components of the forced vibrations of any shaft section of interest to us depends strongly on the distance \(x\) from the point of application of the exciting moment (which causes these vibrations) to the node of the shaft’s free vibrations. As \(x\) decreases, the relative weight of the rigid vibrations increases rapidly.

Such a dependence is already found in an ideal linear oscillatory system without damping [2]. It sometimes becomes still more pronounced and strong in systems with damping. It must be taken into account in the experimental investigation of questions of vibration damping in shafts of motor installations, as well as in the experimental determi-

in these installations of vibrational stresses by means of seismic torsiographs.

We shall show this using as an example a simple two-mass system (Fig. 1). At a certain intermediate point \(o\) of the shaft of this system there acts an external exciting harmonic moment \(M\sin \eta t\).

Consider three typical cases of linear damping in such a system:

a) on the left mass 1 there acts a damping moment (Fig. 1a)

\[ M_1=-k_1\frac{d\varphi_1}{dt}; \]

at points \(o\) and 2 and in the shaft there are no damping forces,

b) at point \(o\), in addition to the exciting moment \(M\sin \eta t\), there acts a damping moment (Fig. 1b)

\[ M_0=-k_0\frac{d\varphi_0}{dt}; \]

there are no other sources of damping in the system,

c) on masses 1 and 2 there act damping moments equal in magnitude and always opposite in direction (Fig. 1c)

\[ M_1=-M_2=-k_v\left(\frac{d\varphi_1}{dt}-\frac{d\varphi_2}{dt}\right); \]

their magnitude is proportional to the rate of twisting of the shaft during vibrations; there are no other damping moments in the system.

Fig. 1

Case c may be regarded as a linearized representation of hysteretic resistances in the shaft: the moments \(M_1\) and \(M_2\) (unlike the moments \(M_1\) and \(M_0\) in cases a and b) are absent during rigid-body oscillations of the shaft.

Case a. Damping—only on the left mass (Fig. 1a)

Here the works of excitation and damping over a cycle of resonant oscillations are related by the equality

\[ \pi M\Phi_{0\mathrm{res}}\sin\delta=\pi k_1\omega\Phi_{1\mathrm{res}}^2, \tag{1} \]

where \(\Phi_{0\mathrm{res}}\) and \(\Phi_{1\mathrm{res}}\) are the absolute amplitudes of the resonant oscillations, respectively, of the left mass 1 and of the point \(o\) at which the exciting moment \(M\) is applied; \(\delta\) is the angle of lag of the oscillations of point \(o\) behind the moment \(M\).

Hence

\[ k_1=\frac{M\Phi_{0\mathrm{res}}\sin\delta}{\omega\Phi_{1\mathrm{res}}^2}. \tag{2} \]

The same dependence for \(k_1\) could also be obtained analytically (see below) by integrating the system of differential equations for the given elastic system.

But for complex installations (with many masses) this approach proves to be practically unsuitable. Therefore energy dependences of the form (1) are widely used in the calculation and analysis of resonant torsional vibrations of shafts in motor installations. However, as shown below, the formal application of dependences (1), for example in the experimental determination of damping coefficients in motor installations, can lead to serious errors.

Thus, for example, expression (2) could be used to calculate the experimental value of the coefficient \(k_1\) in the installation of Fig. 1a from the results of torsion-graphing the shaft in section 1, when the installation operates at the critical speed. But for this it is necessary to find experimentally, in addition to \(\Phi_{1\mathrm{res}}\), also \(\Phi_{0\mathrm{res}}\) and the angle \(\delta\).

However, the experimental determination of the angle \(\delta\) in torsion-graphing requires special devices if we wish to ensure sufficient accuracy of this determination. Torsion-graphing intermediate sections of the shaft, for example section 0, is usually inconvenient.^1

Therefore, in most cases one has to confine oneself to torsion-graphing one of the end sections of the shaft, 1 or 2, and thereby determine \(\Phi_{1\mathrm{res}}\) or \(\Phi_{2\mathrm{res}}\).

Accordingly, for these two cases expression (2) may be rewritten in the form

\[ k_1 = \frac{M\sin\delta} {\omega \Phi_{1\mathrm{res}}\left(\dfrac{\Phi_{1\mathrm{res}}}{\Phi_{0\mathrm{res}}}\right)} \tag{3} \]

or

\[ k_1 = \frac{M\sin\delta} {\omega \Phi_{2\mathrm{res}} \left(\dfrac{\Phi_{1\mathrm{res}}}{\Phi_{2\mathrm{res}}}\right) \left(\dfrac{\Phi_{1\mathrm{res}}}{\Phi_{0\mathrm{res}}}\right)} . \tag{4} \]

Hence, using the previously noted propositions 1 and 2, one can obtain the following calculation formulas for computing the experimental values of the damping coefficient \(k_1\) from the results of torsion-graphing, respectively, sections 1 and 2 of the shaft of the installation in Fig. 1a:

\[ k'_{1\mathrm{e}} = \frac{M} {\omega \Phi_{1\mathrm{res}} \left(\dfrac{\Phi_{1\mathrm{rel}}}{\Phi_{0\mathrm{rel}}}\right)} , \tag{5} \]

\[ k''_{1\mathrm{e}} = \frac{M} {\omega \Phi_{2\mathrm{res}} \left(\dfrac{\Phi_{1\mathrm{rel}}}{\Phi_{2\mathrm{rel}}}\right) \left(\dfrac{\Phi_{1\mathrm{rel}}}{\Phi_{0\mathrm{rel}}}\right)} . \tag{6} \]

The ratios of relative amplitudes \(\Phi_{\mathrm{rel}}\) entering here are easily calculated on the basis of the curve of the mode shape of the free vibrations of the shaft, obtained by the usual methods in calculating the free vibrations of the elastic system under consideration.

But in fact, at resonance,

\[ \delta \ne 90^\circ;\quad \sin\delta < 1 \]

^1 Here, and everywhere below, it is assumed that torsion-graphing is carried out with torsion graphs of the seismic type.

due to the presence of forced vibrations of the point \(O\) at which the exciting torque \(M\) is applied. For the same reason, Widder’s rule is also violated. In fact,

\[ \left. \begin{aligned} \frac{\Phi_{1\mathrm{res}}}{\Phi_{0\mathrm{res}}} &= \varepsilon_{10}\, \frac{\Phi_{1\mathrm{rel}}}{\Phi_{0\mathrm{rel}}},\\[6pt] \frac{\Phi_{1\mathrm{res}}}{\Phi_{2\mathrm{res}}} &= \varepsilon_{12}\, \frac{\Phi_{1\mathrm{rel}}}{\Phi_{0\mathrm{rel}}} \end{aligned} \right\}, \tag{7} \]

where \(\varepsilon_{10}\) and \(\varepsilon_{12}\) are certain coefficients different from unity. Substituting (7) into (3) and (4),

\[ k_1 = \frac{M}{ \omega \Phi_{1\mathrm{res}} \left( \frac{\Phi_{1\mathrm{rel}}}{\Phi_{0\mathrm{rel}}} \right)} \cdot \frac{\sin \delta}{\varepsilon_{10}} \tag{8} \]

or

\[ k_1 = \frac{M}{ \omega \Phi_{2\mathrm{res}} \left( \frac{\Phi_{1\mathrm{rel}}}{\Phi_{2\mathrm{rel}}} \right) \left( \frac{\Phi_{1\mathrm{rel}}}{\Phi_{0\mathrm{rel}}} \right)} \cdot \frac{\sin \delta}{\varepsilon_{10}\varepsilon_{12}} . \tag{9} \]

Comparison of (8) and (9) with, respectively, (5) and (6) gives

\[ \left. \begin{aligned} k_1 &= \sigma' k'_{1e},\\ k_1 &= \sigma'' k''_{1e} \end{aligned} \right\}, \tag{10} \]

where the notation is

\[ \left. \begin{aligned} \sigma' &= \frac{\sin \delta}{\varepsilon_{10}},\\[6pt] \sigma'' &= \frac{\sin \delta}{\varepsilon_{10}\varepsilon_{12}} . \end{aligned} \right\}. \tag{11} \]

The quantities \(\sigma'\) and \(\sigma''\) should be regarded as correction coefficients by which it is necessary to multiply the experimental values of the coefficients \(k'_{1e}\) and \(k''_{1e}\), calculated by the usual simplified formulas (5 and 6), in order to obtain the true value of the damping coefficient \(k_1\) in the installation under investigation.

To determine the magnitudes of the coefficients \(\sigma'\) and \(\sigma''\), we shall find the coefficients \(k'_{1e}\), \(k''_{1e}\) and the angle \(\delta\) analytically.

For the installations in Figs. 1a and 1b one may write the following system of differential equations\(^1\)

\[ \left. \begin{aligned} \Theta_1 \frac{d^2\varphi_1}{dt^2} + k_1 \frac{d\varphi_1}{dt} + c_1(\varphi_1-\varphi_0) &= 0,\\[6pt] \Theta_2 \frac{d^2\varphi_2}{dt^2} - c_2(\varphi_0-\varphi_2) &= 0,\\[6pt] c_1(\varphi_0-\varphi_1) + c_2(\varphi_0-\varphi_2) + k_0 \frac{d\varphi_0}{dt} &= M \sin \eta t \end{aligned} \right\}. \tag{12} \]

\(^1\) As applied to the installation in Fig. 1a. Here \(k_0=0\).

The steady-state part of the solutions of these equations has the usual form

\[ \varphi_0=A_0\sin\eta t+B_0\cos\eta t=\Phi_0\sin(\eta t-\delta), \]

\[ \varphi_1=A_1\sin\eta t+B_1\cos\eta t=\Phi_1\sin(\eta t-\delta_1), \]

\[ \varphi_2=A_2\sin\eta t+B_2\cos\eta t=\Phi_2\sin(\eta t-\delta_2), \]

where

\[ A_0=a_2A_2;\quad B_0=a_3B_2 \]

\[ A_1=a_2\,\frac{\alpha_1A_2+\Psi_1B_2}{\alpha_1^2+\Psi_1^2}; \quad B_1=a_2\,\frac{\alpha_1B_2-\Psi_1A_2}{\alpha_1^2+\Psi_1^2}, \]

\[ A_2=-\frac{M}{\eta}\,\frac{(\alpha_1^2+\Psi_1^2)}{Y^2+Z^2}\,Y, \]

\[ B_2=-\frac{M}{\eta}\,\frac{(\alpha_1^2+\Psi_1^2)}{Y^2+Z^2}\,Z. \]

Here

\[ Y=\alpha_1a_2\eta\Theta_1+(\alpha_1^2+\Psi_1^2)\eta\Theta_2-a_2\Psi_1k_1, \]

\[ Z=a_2(\alpha_1^2+\Psi_1^2)k_0+\alpha_1a_2k_1+a_2\Psi_1\eta\Theta_1, \]

\[ \alpha_1=1-\gamma^2(1-a_1), \]

\[ \alpha_2=1-\gamma^2(1+a_2), \]

\[ \Psi_1=\eta-\frac{k_1}{c_1};\quad \gamma=\frac{\eta}{\omega}, \]

\[ a_1=\frac{x}{L_1};\quad a_2=\frac{x}{L_2}, \]

where \(\omega\) is the angular frequency of the free vibrations of the system under consideration,

\[ \omega=\sqrt{\frac{c(\Theta_1+\Theta_2)}{\Theta_1\Theta_2}}; \]

\(c_1\) is the stiffness of the shaft segment between mass 1 and the point \(o\) of application of the moment \(M\);

\(L_1\) and \(L_2\) are the distances from the node of free vibrations \(z\) to masses 1 and 2.

For the case of resonant vibrations, when

\[ \gamma=1;\quad k_0=0 \]

from the preceding expressions, after a number of elementary transformations, we obtain

\[ \Phi_{1\,\mathrm{res}}=\beta_{\mathrm{res}}\frac{Ma_1}{\omega^2\Theta_1} =\beta_{\mathrm{res}}a_1\Phi_{\mathrm{st}}\frac{\Theta_1+\Theta_2}{\Theta_1}, \tag{13} \]

\[ \Phi_{2\,\mathrm{res}}=\xi\Phi_{\mathrm{st}}\frac{\Theta_1+\Theta_2}{\Theta_2} =\xi\Phi_{\mathrm{st}}, \tag{14} \]

\[ \Phi_{0\mathrm{res}}=\xi a_1\Phi_{\mathrm{zh}}\frac{\Theta_1+\Theta_2}{\Theta_1}, \tag{15} \]

\[ B_{2\mathrm{res}}=-\frac{M}{\omega k_1}\frac{\Theta_1}{\Theta_2}a_1, \tag{16} \]

where

\[ \xi=\sqrt{(1-a_1)^2+\beta_{\mathrm{res}}^{\,2}a_1^{\,2}}, \tag{17} \]

\[ \beta_{\mathrm{res}}=\frac{\Phi_{2\mathrm{res}}}{\Phi_{2\mathrm{zh}}}=\frac{\Theta_1\omega}{k_1}. \tag{18} \]

Here

\[ \Phi_{\mathrm{zh}}=\frac{M}{(\Theta_1+\Theta_2)\omega^2} \tag{19} \]

is the amplitude of the rigid oscillations of the shaft, and

\[ \Phi_{2\mathrm{zh}}=\Phi_{\mathrm{zh}}\frac{\Theta_1+\Theta_2}{\Theta_2} \]

is the amplitude of the rigid oscillations that would occur in the installation under consideration for \(\Theta_1=0\).

Now from (14) and (16)

\[ \sin\delta=\frac{B_{0\mathrm{res}}}{\Phi_{0\mathrm{res}}} =\frac{B_{2\mathrm{res}}}{\Phi_{2\mathrm{res}}} =\frac{\Theta_1\omega a_1}{\xi k_1} \]

or, taking (18) into account,

\[ \sin\delta=\frac{\beta_{\mathrm{res}}a_1}{\xi}. \tag{20} \]

From (13), (15), and (20) it follows that, for \(M=\mathrm{const}\) and \(\gamma=1\), the amplitudes \(\Phi_{0\mathrm{res}}\), \(\Phi_{2\mathrm{res}}\), and \(\sin\delta\) decrease with decreasing \(x\) according to nonlinear laws (Fig. 2).

From (7)

\[ \varepsilon_{10}=\frac{\Phi_{1\mathrm{res}}}{\Phi_{0\mathrm{res}}}\cdot \frac{\Phi_{0\mathrm{otn}}}{\Phi_{1\mathrm{otn}}}. \]

or, on the basis of (13) and (15), and taking into account that in the two-mass system of Fig. 1

\[ \frac{\Phi_{0\mathrm{otn}}}{\Phi_{1\mathrm{otn}}}=\frac{x}{L_1}=a_1, \tag{21} \]

we have

\[ \varepsilon_{10}=\frac{\beta_{\mathrm{res}}a_1}{\xi}. \]

Similarly, from (7), (13), (14), and taking into account that

\[ \frac{\Phi_{2\mathrm{otn}}}{\Phi_{1\mathrm{otn}}}=\frac{\Theta_1}{\Theta_2}, \]

we obtain

\[ \varepsilon_{12}=\frac{\beta_{\mathrm{res}}a_1}{\xi}. \]

Now, according to (11)

\[ \left. \begin{aligned} \sigma'&=1,\\ \sigma''&=\frac{\xi}{\beta_{\mathrm{res}}a_1}, \end{aligned} \right\} \tag{22} \]

where \(a_1\) and \(\xi\) depend on \(x\). A plot of \(\sigma'\) and \(\sigma''\) as functions of \(a_1\) for the case \(\beta_{\mathrm{res}}=5\) is shown in Fig. 3.

Fig. 2

The obtained dependences give grounds for the following conclusions.

1. According to (22), \(\sigma'=1\) for any \(a_1\). Consequently, when torsion-graphing the left end of the shaft of the installation in Fig. 1a (where linear damping with coefficient \(k_1\) acts) and when processing the results of this torsion-graphing by formula (5), we shall always obtain the correct value of the damping coefficient

\[ k'_{13}=k_1 \]

independently of the magnitude of \(x\) ^1).

1. As is seen from Fig. 3, the coefficient \(\sigma''\) is always greater than unity. Thus, when torsion-graphing the right end of the shaft of the installation according to Fig. 1a and processing the torsion-graphing results by the usual energy method, we shall always obtain an underestimated value of the experimental coefficient \(k''_{13}\) as compared with its true value \(k_1\). Thus, for \(\beta_{\mathrm{res}}=5\) and \(a_1=0.1\) we have \(\sigma''=2.05\), i.e., an error in determining \(k_1\) by a factor of two. With further decrease of \(a_1\), this error increases still more.

2. From (13)—(15), for \(x=0\),

\[ \Phi_{1\mathrm{res}}=\Phi_{0\mathrm{res}}=0, \]

\[ \Phi_{2\mathrm{res}}=\Phi_{2ж}=\frac{M}{\Theta_2\omega^2} \]

and the elastic curve of the resonant vibrations of the shaft assumes a peculiar form (Fig. 4). When the exciting moment \(M\) acts exactly at the node of the free vibrations (\(x=0\)) and damping is present only on the left mass (\(k_1>0\)), the amplitude of the right mass at \(\gamma=1\) proves to be completely independent of the magnitude of \(k_1\). Here the right mass acts as an ideal dynamic damper for the left half of the system. For a sufficiently large ratio \(\Theta_1/\Theta_2\) (that is, for sufficiently small \(\Theta_2\)) the magni—

^1) Consideration of the question of possible errors of torsion-graphing associated with the design and parameters of the seismic torsionograph is not included in the tasks of the present article.

the value \(\Phi_{2\mathrm{zh}}\) may turn out to be considerably greater than the amplitude of rigid vibrations \(\Phi_{\mathrm{zh}}\), which is determined by the usual formula (19)\(^{1)}\).

Fig. 3

It is not difficult to show that the dependence of \(\Phi_{2\mathrm{res}}\) on \(\gamma\) at \(x=0\) has an explicitly resonant character. But this resonant peak, which of course is easily recorded on the torsiogram for \(\Phi_2\), cannot be any characteristic of damping at the left-hand mass. Obviously, when \(x\) is close to zero, torsiographing the right-hand end of the installation according to Fig. 1a is meaningless.

Fig. 4

\(^{1)}\) Of course, this conclusion is fully valid only in the absence of damping in the right-hand part of the system.

Case 6. Damping—only at the point of application of the moment \(M\) \((k_1=0;\ k_0>0)\)

Here, for the resonance case, one may write

\[ \pi M\Phi_{0\text{res}}\sin\delta=\pi k_0\Phi_{0\text{res}}^{2}\omega . \]

Hence

\[ k_0=\frac{M\sin\delta}{\Phi_{0\text{res}}\omega}. \]

Let us take

\[ \frac{\Phi_{0\text{res}}}{\Phi_{1\text{res}}} = \varepsilon_{10} \frac{\Phi_{0\text{rel}}}{\Phi_{1\text{rel}}}, \]

\[ \frac{\Phi_{0\text{res}}}{\Phi_{2\text{res}}} = \varepsilon_{02} \frac{\Phi_{0\text{rel}}}{\Phi_{2\text{rel}}}, \]

then

\[ k_0= \frac{M\sin\delta} {\omega\Phi_{1\text{res}}\varepsilon_{01} \left(\dfrac{\Phi_{0\text{rel}}}{\Phi_{1\text{rel}}}\right)} \]

or

\[ k_0= \frac{M\sin\delta} {\omega\Phi_{2\text{res}}\varepsilon_{02} \left(\dfrac{\Phi_{0\text{rel}}}{\Phi_{2\text{rel}}}\right)}. \]

Hence, taking according to provisions 1 and 2 (p. 36)

\[ \delta=90^\circ, \]

\[ \varepsilon_{01}=\varepsilon_{02}=1, \]

we obtain the calculation expressions for determining the experimental values of the coefficient \(k_0\) from the results of torsiographing, respectively, the left and right ends of the shaft,

\[ k'_{0\text{e}}= \frac{M} {\omega\Phi_{1\text{res}} \left(\dfrac{\Phi_{0\text{rel}}}{\Phi_{1\text{rel}}}\right)}, \]

\[ k''_{0\text{e}}= \frac{M} {\omega\Phi_{2\text{res}} \left(\dfrac{\Phi_{0\text{rel}}}{\Phi_{2\text{rel}}}\right)}. \]

Comparison of these with (23) gives

\[ k_0=\sigma' k'_{0\text{e}}, \]

\[ k_0=\sigma'' k''_{0\text{e}}, \]

where it is denoted

\[ \sigma'=\frac{\sin\delta}{\varepsilon_{01}}, \]

\[ \sigma''=\frac{\sin\delta}{\varepsilon_{02}}. \tag{24} \]

To obtain the necessary dependences for \(\sigma\), we shall use the results, given above, of the general solution of the system of differential equations (12). For \(k_1=0\) and \(\gamma=1\),

\[ A_1=A_2=A_0=0, \]

\[ \Phi_{1\mathrm{res}}=\frac{M}{k_0 a_1 \omega}, \]

\[ \Phi_{2\mathrm{res}}=\frac{M}{k_0 a_2 \omega}, \]

\[ \Phi_{0\mathrm{res}}=\frac{M}{k_0\omega}. \]

Hence

\[ \sin \delta=\frac{B_{0\mathrm{res}}}{\Phi_{0\mathrm{res}}}=1, \]

that is, \(\delta=90^\circ\), independently of \(\Phi_{zh}\) and \(x\).

Further, on the basis of (21) one can obtain

\[ \varepsilon_{01}=1. \]

Now from (24)

\[ \sigma''=\sigma'=1 \]

and

\[ k_{0e}''=k_{0e}'=k_0. \]

Thus, in the case where damping is present only at the point \(o\) of application of the exciting moment \(M\), the usual processing of the torsiogram \(\Phi_1\) or \(\Phi_2\) on the basis of rules 1 and 2 gives the correct result independently of \(x\) and of the magnitude \(\Phi_{zh}\).

At resonance the amplitudes \(\Phi_{1\mathrm{res}}\) and \(\Phi_{2\mathrm{res}}\) increase without bound as \(x\) decreases, despite the presence of damping at the point \(o\), where the exciting moment \(M\) is applied. At the same time, the amplitude \(\Phi_{0\mathrm{res}}\) of point \(o\) remains unchanged for any \(x\). This amplitude satisfies the condition

\[ M=\Phi_{0\mathrm{res}}k_0\omega, \]

where the right-hand side is the amplitude of the damping moment. Here (for \(\gamma=1\)) the inertial resistances of both masses mutually balance each other, and the moment \(M\) overcomes only the damping at point \(o\).

Case c. Internal damping in the shaft (Fig. 1c)

Here the energy balance at resonance can be written in the form

\[ \pi M\Phi_{0\mathrm{res}}=\pi k_{ve}\omega\left(\Phi_{2\mathrm{res}}-\Phi_{1\mathrm{res}}\right)^2 . \tag{25} \]

Hence, on the basis of Widler’s rule, a computational expression is obtained for determining the experimental values of the damping coefficient \(k_v\) from the results of torsiographing, respectively, the right and left ends of the shaft,

\[ k_{ve}''= \frac{\Phi M_{0\mathrm{res}}} {\Phi_{2\mathrm{res}}^{2}\left(1-\dfrac{\Phi_{1\mathrm{rel}}}{\Phi_{2\mathrm{rel}}}\right)^{2}\omega} = \frac{M\Theta_1\Theta_2 a_1} {(\Theta_1+\Theta_2)^2\omega\Phi_{2\mathrm{res}}}, \tag{26} \]

\[ k_{v\mathrm{e}}'=\frac{M\Theta_1\Theta_2 a_2}{(\Theta_1+\Theta_2)^2\omega\Phi_{1\mathrm{rez}}}. \]

The right-hand side of expression (25) is written for the case of a linearized estimate of hysteresis losses in the shaft through the linear coefficient \(k_v\), or, more precisely, through its experimental value \(k_{v\mathrm{e}}\). If such linearization is undesirable, instead of (25) one could write

\[ \pi M\Phi_{0\mathrm{rez}}=\pi a_2(\Phi_{2\mathrm{rez}}-\Phi_{1\mathrm{rez}})^q \]

and from this determine the hysteresis coefficient

\[ a_2= \frac{ M\left(\dfrac{\Phi_{0\mathrm{otk}}}{\Phi_{2\mathrm{otk}}}\right) }{ \Phi_{2\mathrm{rez}}^{\,q-1} \left(1-\dfrac{\Phi_{1\mathrm{otk}}}{\Phi_{2\mathrm{otk}}}\right)^q }, \]

if the exponent \(q\) is known.

Let us compose, for the installation in Fig. 1v, the system of differential equations

\[ \Theta_1\frac{d^2\varphi_1}{dt^2} +k_v\left(\frac{d\varphi_1}{dt}-\frac{d\varphi_2}{dt}\right) +c_1(\varphi_1-\varphi_0)=0, \]

\[ \Theta_2\frac{d^2\varphi_2}{dt^2} -k_v\left(\frac{d\varphi_1}{dt}-\frac{d\varphi_2}{dt}\right) -c_2(\varphi_0-\varphi_2)=0, \]

\[ c_1(\varphi_0-\varphi_1)+c_2(\varphi_0-\varphi_2)=M\sin\eta t. \]

Its solution gives the following expressions for the amplitudes of the forced oscillations of sections 1, 2, and \(o\) of the shaft at resonance \((\gamma=1)\)

\[ \Phi_{1\mathrm{rez}}=\Phi_{\mathrm{zh}}\sqrt{\frac{a_2^2}{\Psi^2}+1}, \]

\[ \Phi_{2\mathrm{rez}}=\Phi_{\mathrm{zh}}\sqrt{\frac{a_1^2}{\Psi^2}+1}, \tag{27} \]

\[ \Phi_{0\mathrm{rez}}=\Phi_{\mathrm{zh}} \sqrt{\left[1-(1-a_1)(1+a_2)\right]^2+\left(\frac{a_1a_2}{\Psi}\right)^2}, \]

where

\[ \Psi=\frac{k\omega}{c}, \]

and for the quantities \(\Phi_{\mathrm{zh}}\), \(a_1\), and \(a_2\) the expressions given on pp. 40--41 are valid.

It follows from this that, for \(M=\mathrm{const}\) and \(\Phi_{\mathrm{zh}}=\mathrm{const}\), and as \(x\to 0\),

\[ \Phi_{2\mathrm{rez}}\to\Phi_{\mathrm{zh}}, \]

\[ \Phi_{1\mathrm{rez}}\to\Phi_{\mathrm{zh}}, \]

that is, the resonance peak on the torsiogram disappears, despite the presence of formal resonance \((\eta=\omega)\).

Let us determine the true value of the damping coefficient \(k_v\) from the results of the analytical solution of the problem under study. For example, from (27), after elementary transformations, we obtain

\[ k_v=\frac{M\Theta_1\Theta_2 a_1}{(\Theta_1+\Theta_2)^2\omega\Phi_{2\,rez}} \cdot \frac{1}{\sqrt{1-\dfrac{\Phi_{zh}^{2}}{\Phi_{2\,rez}^{2}}}} . \]

Comparison of this expression with (26) gives

\[ k_v=\sigma'' k_{v e}'', \]

where

\[ \sigma''=\frac{1}{\sqrt{1-\dfrac{\Phi_{zh}^{2}}{\Phi_{2\,rez}^{2}}}} \]

or, taking (27) into account,

\[ \sigma''=\sqrt{1+\frac{\Psi^2}{a_1^2}} . \tag{28} \]

The graph of the dependence of the coefficient \(\sigma''\) on \(a_1\) for \(\Psi\)-const is shown in Fig. 5. Thus, for example, at

\[ \Psi=0.1;\qquad a_1=0.1 \]

from (28) we have

\[ \sigma''=1.44, \]

therefore the experimental value \(k_{v e}''\), calculated by (26) on the basis of the results of torsiographing section 2, will be 30% less than the actual value of the coefficient \(k_v\).

It follows from (28) that the experimentally found value of the damping coefficient \(k_{v e}\) will be the smaller in comparison with the true value \(k_v\), the smaller \(x\) is and the larger \(\Psi\) is, that is, the larger \(k_v\), \(\omega\), and the smaller \(c\) are. Torsiographing both ends of the shaft and the usual method of processing torsiograms (based on the application of propositions 1 and 2) give identical and correct results only when \(\Phi_{zh}=0\).

Thus, in all the cases considered above (Fig. 1), we have obtained regularities of one and the same character.

Fig. 5

When the distance \(x\) is decreased, the specific weight of rigid oscillations at the point \(o\) of application of the exciting moment \(M\) increases, which leads to a deviation of the phase angle \(\delta\) from its theoretical value of \(90^\circ\) at resonance and to a corresponding decrease in the work of excitation; as a result, the amplitudes of the resonant oscillations of the system masses decrease. Finally, as \(x\to 0\), manifestations of partial resonance are possible, when one part of the system proves to be a dynamic damper for the other part of it.

As a result, at small \(x\) the basic design assumptions 1, 2 are violated, though they are usually sufficiently accurate at large \(x\).

A consequence of this may be, for example, a substantial difference in the experimental values of the damping coefficient in two identical installations differing from one another only in the distance \(x\) between the point of application of the exciting moment and the node of the shaft’s free vibrations.

This leads, finally, to a “scatter” of experimental values of the damping coefficients calculated from the results of torsiographing different sections of the installation shaft. The correct choice of the torsiographing location in installations with small \(x\) largely predetermines the reliability and accuracy of the subsequent results of processing torsiograms. As was shown above, the influence of rigid vibrations disappears here only if the torsiogram is taken from that section (portion) of the shaft where the damping moment to be determined acts. Consequently, the generally known recommendation—to take torsiograms, whenever possible, from those shaft sections of the installation that have the largest amplitudes of resonant vibrations—cannot be regarded as universal.

LITERATURE

1. Natanzon V. Ya. On forced vibrations of crankshafts at resonance. Tekhnika vozdushnogo flota, No. 2, 1941.

2. Nechaev V. K. On the “rigid” and “elastic” nonuniformity of rotation of a crankshaft. Izvestiya Tomskogo politekhnicheskogo instituta, vol. 75, 1954.

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

4. Ker Wilson W. Torsional Vibration Amplitudes at Non-Resonant Speeds, Inst. Mech. Eng. Proc. Vol. 153, No. 3, 1945.