STUDY OF OSCILLATIONS OF THE MOVABLE SYSTEM OF A BALANCING MACHINE WITHOUT RIGID CONNECTIONS TO THE ENVIRONMENT

P. T. MAL’TSEV

(Presented by the scientific seminar of the Departments of Gyroscopic Instruments and Devices and Mechanics)

This article presents the results of a study of a balancing machine whose oscillatory system has six degrees of freedom. Two varieties of balancing machines of this type developed by us have been put into operation at one of the enterprises of the electrical industry and provide balancing of rotors of electric machines at operating rotational speeds [1, 2].

Unbalanced masses distributed along the length of the rotor, when it rotates, create centrifugal forces of inertia. It is known that the entire set of unbalanced centrifugal forces of inertia can be replaced by a force \(C\) passing through point \(O\), and by a moment \(M_{\partial}\) with respect to this point (Fig. 1). In the general case, the vector of the moment \(M_{\partial}\) and the vector of the force \(C\) form some angle \(\gamma\).

\[ C = m \varepsilon_c \omega^2, \tag{1} \]

\[ M_{\partial} = m \varepsilon_{\partial} \omega^2 l, \tag{2} \]

where \(m=\dfrac{G}{g}\) is the mass of the rotor;

\(\varepsilon_c\) is the displacement of the rotor’s center of gravity from the axis of rotation (the rotor eccentricity);

\(\varepsilon_{\partial}\) is the conventional eccentricity of the rotor under dynamic unbalance.

The unbalanced centrifugal force \(C\) and the dynamic moment \(M_{\partial}\) can be resolved into horizontal and vertical components (Fig. 1b).

\[ \begin{aligned} C_y &= C \cdot \sin \omega t,\\ C_z &= C \cdot \cos \omega t, \end{aligned} \tag{3} \]

\[ \begin{aligned} M_{\partial y} &= M_{\partial}\cos(\omega t+\gamma),\\ M_{\partial z} &= M_{\partial}\sin(\omega t+\gamma). \end{aligned} \tag{4} \]

Under the action of the vertical component of the force \(C_z\), the movable system performs oscillations along the \(z\)-axis, and the vertical component of the moment \(M_{\partial z}\) causes torsional oscillations about the \(z\)-axis. The horizontal component of the force \(C_y\) causes transverse oscillations of the system along the \(y\)-axis and torsional oscillations about the \(x\)-axis. Under the acti-

Fig. 1. Diagram of the oscillatory system of a balancing machine.

of the horizontal component of the moment \(M_{dy}\), the system undergoes torsional oscillations about the \(y\) axis.

Acceleration of the rotor to operating speed on the balancing machine under consideration is carried out in a technological stator mounted on a platform suspended on springs. The distribution of the masses of the movable system is chosen so that its center of gravity \(O_m\) and the center of stiffness \(O_1\) of the elastic elements lie on the same vertical, which is at the same time perpendicular to the axis of rotation of the rotor. The mass and moments of inertia of the foundation are made 10–12 times greater than the mass and moments of inertia of the movable system.

When the rotor rotates, the system will be acted upon by forces caused by unbalance, inertia forces, the elastic forces of the springs, and resistance forces. In the general case, under the action of the indicated forces, the movable system, having six degrees of freedom, performs small oscillations along the \(x, y, z\) axes and torsional oscillations about these axes. In studying the oscillations of this system, it is assumed that there are no clearances in the rotor bearings and that the movable system (rotor—stator—platform) oscillates as a rigid body.

The most convenient and simple method for solving problems of the dynamics of a non-free system is the method of Lagrange’s equations; in this case the motion of the system is studied in a generalized coordinate system, that is, in mutually independent parameters whose variation determines the motion of the system [3]. The number of these parameters is equal to the number of degrees of freedom; therefore, according to Lagrange, the number of differential equations can be made equal to the number of degrees of freedom. The solution of these equations gives the coordinates determining the position of the system at each instant of time.

To derive the differential equations of oscillation of the system under the action of the rotor unbalance, we shall take as generalized coordinates the coordinates \(x, y, z\) of the translational displacement of the center of gravity and the angles of rotation of the system \(\varphi, \psi, \zeta\) about the \(x, y, z\) axes passing through the center of gravity \(O_m\). To compose the differential equations of oscillation of the movable system of the balancing device, we apply Lagrange’s equation of the second kind:

\[ \frac{d}{dt}\left(\frac{\partial T}{\partial \dot q_i}\right) -\frac{\partial T}{\partial q_i} +\frac{\partial \Pi}{\partial q_i} +\frac{\partial \Phi}{\partial \dot q_i} = Q_i, \tag{5} \]

where \(T\) is the kinetic energy of the system;

\(\Pi\) is the potential energy of the system;

\(\Phi\) is the dissipative function (caused by viscous resistance, depends on the velocity \(\dot q_i\));

\(q_i\) is a generalized coordinate;

\(Q_i\) is the generalized disturbing force corresponding to the generalized coordinate.

The kinetic energy \(T\), potential energy \(\Pi\), and dissipative function \(\Phi\) of the oscillatory system in generalized coordinates have the following expressions:

\[ T = \frac{1}{2}M\dot x^{2} +\frac{1}{2}M\dot y^{2} +\frac{1}{2}M\dot z^{2} +\frac{1}{2}I_x\dot\varphi^{2} +\frac{1}{2}I_y\dot\psi^{2} +\frac{1}{2}I_z\dot\zeta^{2}; \tag{6} \]

\[ \Pi = \frac{1}{2}\kappa_z z^{2} +\frac{1}{2}\kappa_z(B_y\varphi)^{2} +\frac{1}{2}\kappa_z(B_x\psi)^{2} +\frac{1}{2}\kappa_y(y-h_0\varphi)^{2} + \]

\[ +\frac{1}{2}\kappa_x(x-h_0\psi)^{2} +\kappa_y(B_x\zeta)^{2} +\frac{1}{2}\kappa_x(B_y\zeta)^{2}; \tag{7} \]

\[ \Phi=\frac{1}{2}\nu_x\dot{x}^{2}+\frac{1}{2}\nu_y\dot{y}^{2}+\frac{1}{2}\nu_z\dot{z}^{2} +\frac{1}{2}\nu_\varphi\dot{\varphi}^{2} +\frac{1}{2}\nu_\psi\dot{\psi}^{2} +\frac{1}{2}\nu_\zeta\dot{\zeta}^{2}, \tag{8} \]

where \(M\) is the mass of the entire movable system (rotor—stator—platform);

\(I_x, I_y, I_z\) are the moments of inertia of the movable system with respect to the axes \(x, y, z\);

\(\kappa_x, \kappa_y, \kappa_z\) are the total stiffnesses of the elastic elements in the directions of the axes \(x, y, z\);

\(B_y, B_x\) are the distances from the center of gravity to the axes of the elastic elements in the directions of the axes \(y, x\), respectively;

\(h_0\) is the distance between the center of gravity \(O_m\) and the supporting surface of the platform.

The generalized forces for each coordinate have the following expressions:

\[ \left. \begin{aligned} Q_z &= C_z = C\cdot \cos \omega t,\\ Q_y &= C_y = C \sin \omega t,\\ Q_x &= C_x = 0,\\ Q_\zeta &= M_{\mathrm{d}z}=M_{\mathrm{d}}\sin(\omega t+\gamma),\\ Q_\varphi &= C_y(h-h_0)=C(h-h_0)\sin\omega t,\\ Q_\psi &= M_{\mathrm{d}y}=M_{\mathrm{d}}\cos(\omega t+\gamma), \end{aligned} \right\} \tag{9} \]

where \(h\) is the distance from the axis of rotation to the center of stiffness of the shock absorbers.

After substituting into equation (5) the values \(T, \Pi, \Phi, Q_i\) for each coordinate and carrying out the simplest transformations, we obtain the following system of differential equations:

\[ M\ddot{z}+\nu_z\dot{z}+\kappa_z z=C\cos\omega t, \tag{10а} \]

\[ I_z\ddot{\zeta}+\nu_\zeta\dot{\zeta}+(\kappa_yB_x^2+\kappa_xB_y^2)\zeta =M_{\mathrm{d}}\sin(\omega t+\gamma), \tag{10б} \]

\[ M\ddot{x}+\nu_x\dot{x}+\kappa_x(x-h_0\psi)=0, \tag{10в} \]

\[ I_y\ddot{\psi}+\nu_\psi\dot{\psi}+(\kappa_zB_x^2+\kappa_xh_0^2)\psi-\kappa_xh_0x =M_{\mathrm{d}}\cos(\omega t+\gamma), \tag{10г} \]

\[ M\ddot{y}+\nu_y\dot{y}+\kappa_y(y-h_0\varphi)=C\sin\omega t, \tag{10д} \]

\[ I_x\ddot{\varphi}+\nu_\varphi\dot{\varphi}+(\kappa_zB_y^2+\kappa_yh_0^2)\varphi-\kappa_yh_0y =C(h-h_0)\sin\omega t. \tag{10е} \]

As a result of solving the system of differential equations, the following expressions were obtained for the amplitudes and phase shifts of the oscillations of the movable system in each coordinate.

The amplitude and phase shift of the oscillation along the \(z\)-axis are

\[ z_0=\frac{m}{M}\,\varepsilon_c\cdot\sigma_z, \tag{11} \]

\[ \delta_z=\operatorname{arctg}\frac{2\beta_z\omega}{\lambda_z^2-\omega^2}. \tag{12} \]

The amplitude and phase shift of the torsional oscillations about the \(z\)-axis are

\[ \zeta_0=\frac{m}{M}\,\frac{\varepsilon_{\mathrm{d}}\,l}{\rho_y^2}\,\sigma_\zeta; \tag{13} \]

\[ \delta_\zeta=\operatorname{arctg}\frac{2\beta_\zeta\omega}{\lambda^2-\omega^2}. \]

Amplitudes and phase shifts of the oscillations \((x_0,\delta_x)\) along the \(x\) axis and of the angular oscillations \((\psi_0,\delta_\psi)\) about the \(y\) axis.

\[ x_0=\frac{m}{M}\frac{\varepsilon_d l h_0}{\rho_y^2}\sigma_x; \tag{15} \]

\[ \psi_0=\frac{m}{M}\frac{\varepsilon_d l}{\rho_y^2}\sigma_\psi; \tag{16} \]

\[ \delta_x=\delta_\psi=\operatorname{arctg}\left\{ -\frac{ 2\left[ \dfrac{(\beta_x+\beta_\psi)\omega}{\lambda_x^2\lambda_\psi^2} -\dfrac{\beta_x\omega}{\lambda_x^2} -\dfrac{\beta_\psi\omega}{\lambda_\psi^2} \right] }{ \dfrac{\omega^4}{\lambda_x^2\lambda_\psi^2} -\dfrac{\omega^2}{\lambda_x^2} -\dfrac{\omega^2}{\lambda_\psi^2} -\dfrac{4\beta_x\beta_\psi\omega^2}{\lambda_x^2\lambda_\psi^2} -\dfrac{\lambda_x^2 h_0^2}{\lambda_\psi^2\rho_y^2} +1 } \right\}. \tag{17} \]

Amplitudes and phase shifts of the oscillations \((y_0,\delta_y)\) along the \(y\) axis and of the angular oscillations \((\varphi_0,\delta_\varphi)\) about the \(x\) axis.

\[ y_0=\frac{m}{M}\varepsilon_c\sigma_y; \tag{18} \]

\[ \varphi_0=\frac{m}{M}\varepsilon_c\sigma_\varphi; \tag{19} \]

\[ \delta_y=\delta_\varphi=\operatorname{arctg}\left\{ -\frac{ \left[ \dfrac{(\beta_y+\beta_\varphi)\omega^3}{\lambda_y^2\lambda_\varphi^2} -\dfrac{\beta_y\omega}{\lambda_y^2} -\dfrac{\beta_\varphi\omega}{\lambda_\varphi^2} \right] }{ \dfrac{\omega^4}{\lambda_y^2\lambda_\varphi^2} -\dfrac{\omega^2}{\lambda_y^2} -\dfrac{\omega^2}{\lambda_\varphi^2} -\dfrac{4\beta_y\beta_\varphi\omega^2}{\lambda_y^2\lambda_\varphi^2} -\dfrac{\lambda_y^2 h_0^2}{\lambda_\varphi^2\rho_\varphi^2} +1 } \right\}. \tag{20} \]

\(\sigma_z,\delta_\zeta,\sigma_x,\sigma_\psi,\sigma_y,\sigma_\varphi\) are the dynamic coefficients for the corresponding coordinates, equal to

\[ \sigma_z= \frac{\dfrac{\omega^2}{\lambda_z^2}} { \sqrt{ \left(1-\dfrac{\omega^2}{\lambda_z^2}\right)^2 -\dfrac{4\beta_z^2\omega^2}{\lambda_z^4} } }; \tag{21} \]

\[ \delta_\zeta= \frac{\omega^2}{\lambda_\zeta^2} \frac{1}{ \sqrt{ \left(1-\dfrac{\omega^2}{\lambda_\zeta^2}\right)^2 +\dfrac{4\beta_z^2\omega^2}{\lambda_\zeta^2} } }; \tag{22} \]

\[ \sigma_x= \frac{\dfrac{\omega^2}{\lambda_\psi^2}} { \sqrt{ \left[ \dfrac{\omega^4}{\lambda_x^2\lambda_\psi^2} -\dfrac{\omega^2}{\lambda_x^2} -\dfrac{\omega^2}{\lambda_\psi^2} -\dfrac{4\beta_x\beta_\psi\omega^2}{\lambda_x^2\lambda_\psi^2} -\dfrac{\lambda_x^2 h_0^2}{\lambda_\psi^2\rho_y^2} +1 \right]^2 + 4\left[ \dfrac{(\beta_x+\beta_\psi)\omega^3}{\lambda_x^2\lambda_\psi^2} -\dfrac{\beta_x\omega}{\lambda_x^2} -\dfrac{\beta_\psi\omega}{\lambda_\psi^2} \right]^2 } }; \tag{23} \]

\[ \sigma_\psi= \frac{\omega^2}{\lambda_\psi^2} \sqrt{ \frac{ \left(1-\dfrac{\omega^2}{\lambda_x^2}\right)^2 +\dfrac{4\beta_x^2\omega^2}{\lambda_x^4} }{ \left[ \dfrac{\omega^4}{\lambda_x^2\lambda_\psi^2} -\dfrac{\omega^2}{\lambda_x^2} -\dfrac{\omega^2}{\lambda_\psi^2} -\dfrac{4\beta_x\beta_\psi\omega^2}{\lambda_x^2\lambda_\psi^2} -\dfrac{\lambda_x^2 h_0^2}{\lambda_\psi^2\rho_y^2} +1 \right]^2 + \frac{ \left(1-\dfrac{\omega^2}{\lambda_x^2}\right)^2 +\dfrac{4\beta_x^2\omega^2}{\lambda_x^4} }{ 4\left[ \dfrac{(\beta_x+\beta_\psi)\omega^3}{\lambda_x^2\lambda_\psi^2} -\dfrac{\beta_x\omega}{\lambda_x^2} -\dfrac{\beta_\psi\omega}{\lambda_\psi^2} \right]^2 } } }; \tag{24} \]

\[ \sigma_y=\frac{\omega^2}{\lambda_y^2} \sqrt{ \frac{ \left(1-\frac{\omega^2}{\lambda_x^2}\right)^2+\frac{4\beta_x^2\omega^2}{\lambda_x^2} }{ \left[ \frac{\omega^4}{\lambda_y^2\lambda_\varphi^2} -\frac{\omega^2}{\lambda_y^2} -\frac{\omega^2}{\lambda_\varphi^2} -\frac{4\beta_y\beta_\varphi\omega^2}{\lambda_y^2\lambda_\varphi^2} -\frac{\lambda_y^2h_0^2}{\lambda_\varphi^2\rho_x^2} +1 \right]^2 + \frac{ \left(1-\frac{\omega^2}{\lambda_x^2}\right)^2+\frac{4\beta_x^2\omega^2}{\lambda_x^2} }{ +4\left[ \frac{(\beta_y+\beta_\varphi)\omega^3}{\lambda_y^2\lambda_\varphi^2} -\frac{\beta_y\omega}{\lambda_y^2} -\frac{\beta_\varphi\omega}{\lambda_\varphi^2} \right]^2 } } \tag{25} \]

\[ \sigma_\varphi=\frac{\omega^2}{\lambda_\varphi^2} \sqrt{ \frac{ \left[ \frac{\omega^2}{\lambda_y^2}\left(1-\frac{h}{h_0}\right)+\frac{h}{h_0} \right]^2+\frac{4\beta_\varphi^2\omega^2}{\lambda_\varphi^4} }{ \left[ \frac{\omega^4}{\lambda_y^2\lambda_\varphi^2} -\frac{\omega^2}{\lambda_y^2} -\frac{\omega^2}{\lambda_\varphi^2} -\frac{4\beta_y\beta_\varphi\omega^2}{\lambda_\varphi^2\lambda_y^2} -\frac{\lambda_y^2h_0^2}{\lambda_\varphi^2\rho_x^2} +1 \right]^2 + \frac{ \left[ \frac{\omega^2}{\lambda_y^2}\left(1-\frac{h}{h_0}\right)+\frac{h}{h_0} \right]^2+\frac{4\beta_\varphi^2\omega^2}{\lambda_\varphi^4} }{ +4\left[ \frac{(\beta_y+\beta_\varphi)\omega^3}{\lambda_y^2\lambda_\varphi^2} -\frac{\beta_y\omega}{\lambda_y^2} -\frac{\beta_\varphi\omega}{\lambda_\varphi^2} \right]^2 } } \tag{26} \]

\[ \beta_z=\frac{\nu_z}{2M},\qquad \beta_\xi=\frac{\nu_\xi}{2I_z},\qquad \beta_x=\frac{\nu_x}{2M},\qquad \beta_\psi=\frac{\nu_\psi}{2I_y},\qquad \beta_y=\frac{\nu_y}{2M},\qquad \beta_\varphi=\frac{\nu_\varphi}{2I_x} \]

— the damping coefficients for the corresponding coordinates;

\[ \lambda_z=\sqrt{\frac{\kappa_z}{M}};\qquad \lambda_x=\sqrt{\frac{\kappa_x}{M}};\qquad \lambda_y=\sqrt{\frac{\kappa_y}{M}};\qquad \lambda_\xi=\sqrt{\frac{\kappa_y B_x^2+\kappa_x B_y^2}{I_z}} \]

\[ \lambda_\psi=\sqrt{\frac{\kappa_z B_x^2+\kappa_x h_0^2}{I_y}};\qquad \lambda_\varphi=\sqrt{\frac{\kappa_z B_y^2+\kappa_y h_0^2}{I_x}} \]

— the natural frequencies of oscillation of the movable system along the axes \(x, y\), and about these axes, respectively.

The total vertical displacement of a point with coordinates \(B_y, h_0, B_x\) of the movable system (Fig. 1, b) will be equal to

\[ Z=z_0+B_x\psi_0+B_y\varphi_0, \]

where \(z_0\) is the independent vertical displacement of the given point;

\(B_x\psi_0\) is the vertical displacement of the point caused by rotation of the movable system through the angle \(\psi_0\) about the \(y\)-axis;

\(B_y\varphi_0\) is the vertical displacement of the point caused by rotation of the movable system through the angle \(\varphi_0\) about the \(x\)-axis.

The total displacement of a point on the line of intersection of the vertical plane passing through the rotor axis of rotation and the center of gravity of the system with the supporting surface of the platform \((B_y=0)\) will be equal to

\[ Z=z_0+B_x\psi_0. \]

On this line there must be some point whose vertical displacements \(z_0\) and \(B_z\psi_0\), under the action of the force and moment of unbalance \(m_2r_2\) in plane II, will be equal but opposite in sign [4], that is, \(z_0=-B_z\psi_0\), whence \(B_z=-\frac{z_0}{\psi_0}\).

Annotations in the figure:

\[ \lambda_\varphi = 3\ \text{cm},\quad \lambda_1 = 42\ \text{cm},\quad \lambda' = 50\ \text{cm} \]

\[ f,\ \text{cm} \]

1, 2, 3 — for \(\beta_{[[unclear]]} = \varphi_{[[unclear]]} = 0\) and, respectively,
\(\dfrac{h_0}{\rho_y}=0;\ \dfrac{h_0}{\rho_y}=0.5;\ \dfrac{h_0}{\rho_y}=1.0\)

4, 5, 6 — for \(\beta_{[[unclear]]}=0.1\lambda_\varphi,\ \beta_{[[unclear]]}=0.1\lambda_\varphi\), and, respectively,
\(\dfrac{h_0}{\rho_y}=0;\ \dfrac{h_0}{\rho_y}=0.5;\ \dfrac{h_0}{\rho_y}=1.0\)

7, 8, 9 — for \(\beta_{[[unclear]]}=0.3\lambda_\varphi,\ \beta_{[[unclear]]}=0.3\lambda_\varphi\), and, respectively,
\(\dfrac{h_0}{\rho_y}=0;\ \dfrac{h_0}{\rho_y}=0.5;\ \dfrac{h_0}{\rho_y}=1.0\)

Fig. 2. Dependence of the distance to the center of oscillations on the angular velocity of rotation of the rotor being balanced.

Substituting into this expression the values of \(z_0\) and \(\psi_0\) from formulas (11) and (16), we obtain:

\[ B_2=-\frac{\rho_y^2\sigma_z}{l_2\delta_\psi}. \tag{27} \]

The quantity \(B_2\) determines the position of the point whose oscillation does not depend on the unbalance in plane II. Similarly, the distance \(B_1\) is found, which determines the position of another point whose oscillations do not depend on the unbalance in plane I,

\[ B_1=\frac{\rho_y^2\sigma_z}{l_1\sigma_\psi}. \tag{28} \]

Figures 2 and 3 show the dependences \(B_2=F(f)\) and the phase characteristics of the system oscillations.

Fig. 3. Phase shift of oscillations as a function of the angular velocity of rotor rotation.

From the graph in Fig. 2 it is seen that, other conditions being equal, the value of \(B_2\) changes with a change in the rotational frequency. In the beyond-resonance region, with increasing \(f=\dfrac{\omega}{2\pi}\), the value \(B_2\) asymptotically approaches the value determined from the relation \(B_2=\dfrac{\rho_y^2}{l_2}\). At the same time, at a rotational frequency in the region between the values \(\lambda'_z\) and \(\lambda'_x\), the distance \(B_2\) depends to a considerable degree both on the value of the ratio \(\dfrac{h_0}{\rho_y}\) and on the damping coefficient.

From the graphs in Fig. 3 it follows that the phase shift of the independent vertical oscillations (curve 1) at a rotational frequency above \(\lambda'_z\) approaches \(180^\circ\), whereas the phase shift of oscillations about the \(y\)-axis (curves 2, 3, 4) approaches \(180^\circ\) only at a rotor rotational frequency above the value \(\lambda'_x\). At a rotor rotational frequency \(f<2\lambda'_x\), the phase difference \(\delta=\delta_z-\delta_\psi\) may be quite significant and, thus, in the balancing process for different ratios of static-

large errors will occur in determining the angular coordinate of the unbalance from the static and dynamic components of the unbalance. Only when \(f > 4\lambda'_x\) will the phase difference \(\delta = \delta_z - \delta_\psi\) lie within \(3^\circ\). The graphs also show that if \(\omega\) exceeds by a factor of 3 the largest of the natural-oscillation frequencies \((\lambda'_\psi\) or \(\lambda'_x)\) of the movable system, then changing the ratio \(\dfrac{h_0}{\rho_y}\) from 0 to 1.0 has practically no effect on the magnitude of the phase shift \(\delta_\psi\) of the rotational oscillations.

Studies carried out for various ratios of the values \(\lambda'_x\), \(\lambda'_z\), and \(\lambda'_\psi\) showed the same result.

Conclusions

1. A movable system without rigid connections to the environment makes it possible to select two points on the platform (for installing sensors), the oscillations of each of which are determined by the unbalance only in one balancing plane and do not depend on the unbalance in the other plane. This eliminates the need for a device included in the measuring circuit to remove the influence of unbalance in one plane when balancing in the other.

2. Operation of the balancing machine at rotor rotational speeds \(\omega\) in the intervals between \(\lambda_x - \lambda_z\), \(\lambda_x - \lambda_\psi\), or \(\lambda_\psi - \lambda_z\) is not recommended, since in this case large errors are possible in determining the angular coordinate of the unbalance.

3. When designing the mechanical system of a balancing machine, one should strive to make the values \(\lambda_x\), \(\lambda_z\), and \(\lambda_\psi\) approximately equal to one another.

4. Any of the natural-oscillation frequencies of the movable system should be 4–5 times lower than the expected frequency of the forced oscillations caused by rotor unbalance.

References

1. P. T. Mal’tsev. Method and device for dynamic balancing of rotors of electrical machines, taking account of their flexibility. Izv. TPI, vol. 117, TGU Press, 1963.

2. P. T. Mal’tsev, Yu. V. Gorst. Device for dynamic balancing of rotors of electrical machines. Topic No. 28—64—1272/58. Advanced scientific-technical and production experience, GOSINTI, 1964.

3. F. M. Dimentberg, K. T. Shatalov, A. A. Gusarov. Oscillations of Machines. Mashinostroenie, 1964.

4. M. P. Kovalev, S. P. Morzhakov, K. S. Terekhova. Dynamic balancing of rotors of gyroscopic systems. Oborongiz, 1962.