Selection of the Material Density of a Small-Sized Gyroscope Rotor from the Standpoint of Highest Accuracy

V. G. Afonin, M. P. Shumskii

(Presented by the scientific-technical seminar of the Department of Gyroscopic Instruments and Devices)

The influence of the material density of a homogeneous rotor on the accuracy of a gyroscope was studied in a paper. Proceeding from the condition that rotors made of materials of different density are geometrically similar, the author came to the conclusion that a homogeneous rotor should not be made heavier by using materials with a greater specific weight. However, as follows from [1], the shape of a homogeneous rotor corresponding to the highest accuracy of a gyroscope depends on the density of the rotor material. In this connection it is natural to pose the question of studying the influence of the density of the rotor material on the accuracy of a gyroscope without using the assumption of geometric similarity of the rotors being compared.

Let us consider a gyroscope in a gimbal suspension. We shall assume that the principal cause of deviation of the gyroscope axis from the specified position is the friction forces in the supports and current leads. Denote:

\(G\) — weight of the gyroscope rotor,
\(G_{\mathrm{B}}\) — weight of the inner ring of the gimbal suspension,
\(G_{\mathrm{C}}\) — weight of the outer ring of the gimbal suspension,
\(V\) — volume of the gyroscope rotor,
\(\rho\) — density of the rotor material,
\(\rho_0\) — density of steel,
\(g\) — acceleration of free fall,
\(\gamma=\rho\cdot g\) — specific weight of the rotor material,
\(M_{\mathrm{B}}, M_{\mathrm{C}}\) — friction moments in the current leads along the inner and outer axes of the suspension, respectively,
\(\lambda\) — coefficient of the moment of friction forces,
\(I_{\mathrm{ed}}\) — moment of inertia of a rotor made of material with density \(\rho=1\),
\(I=\rho\cdot I_{\mathrm{ed}}\) — moment of inertia of the rotor,
\(\Omega\) — angular velocity of the rotor’s own rotation,
\(\omega_1, \omega_2\) — angular velocities of precession with respect to the outer and inner axes of the suspension, respectively.

The friction moments may be the cause of precessional motion with angular velocities

\[ \omega_1= \frac{M_{\mathrm{B}}+\lambda\,[G_{\mathrm{B}}+G]}{I\Omega} = \frac{1}{\rho}\, \frac{M_{\mathrm{B}}+\lambda G_{\mathrm{B}}}{I_{\mathrm{ed}}\Omega} + \frac{\lambda g V}{I_{\mathrm{ed}}\Omega}, \tag{1} \]

\[ \omega_2= \frac{M_{\mathrm{C}}+\lambda\,[G_{\mathrm{B}}+G_{\mathrm{C}}+G]}{I\Omega} = \frac{1}{\rho}\, \frac{M_{\mathrm{C}}+\lambda\,[G_{\mathrm{B}}+G_{\mathrm{C}}]}{I_{\mathrm{ed}}\Omega} + \frac{\lambda g V}{I_{\mathrm{ed}}\Omega}. \tag{2} \]

Analyzing formulas (1) and (2), one can draw the following conclusions:

1. In order for the drift rates not to depend on the density of the rotor, the conditions \(M_B = M_C = G_B = G_C = 0\) must be satisfied, i.e., there must be no friction moments in the current leads, and the gimbal suspension must be weightless.

2. For ordinary gyroscopes that do not satisfy these conditions, the drift rate decreases as the density of the rotor material increases.

3. For given values of \(\rho\), \(\lambda\), \(M_B\), \(M_C\), \(G_B\), \(G_C\), there must exist a quite definite rotor shape with corresponding values of the volume \(V\) and the moment of inertia \(I_{\mathrm{eq}}\), which gives the minimum angular rate \(\omega_1\) or \(\omega_2\). The question of the optimal rotor shape was considered in works [1--3].

Introduce the notation

\[ M_B+\lambda G_B=M_1,\qquad M_C+\lambda [G_B+G_C]=M_2 . \tag{3} \]

In [1] it is shown that, for a rotor whose dimensions are bounded by a sphere of radius \(R\), the optimal shape is that shown in Fig. 1, where the radius \(r\) of the cylindrical recess is determined as follows.

Fig. 1

Let \(V_1\) denote the volume of the rotor shown in Fig. 1 for \(r=0\), and \(I_{1\mathrm{eq}}\) its moment of inertia corresponding to density \(\rho=1\).

Then

\[ V=V_1\left[1-\xi^3\frac{1}{\cos \nu_k}\right], \tag{4} \]

\[ I_{\mathrm{eq}}=I_{1\mathrm{eq}}\left[1-\xi^5 \frac{3}{(3-\sin^2\nu_k)\cos\nu_k}\right], \qquad \xi=\frac{r}{R}. \tag{5} \]

Assume that \(M_i=n_i\lambda g\rho_0 V_1\), where \(n_i\) is a coefficient depending on the specific values of the friction moments in the current leads, the weights of the frames, the coefficient of the friction moment, and the volume \(V_1\). Taking into account (3), (4), (5), we transform formulas (1) and (2) as follows:

\[ \omega_1= \frac{\dfrac{1}{\rho}M_i+\lambda g V}{\Omega I_{\mathrm{eq}}} = \frac{\lambda g V_1}{\Omega I_{1\mathrm{eq}}}\,[3-\sin^2\nu_k]\, \frac{C_i-\xi^3}{C-\xi^5}, \tag{6} \]

where

\[ C=\frac{3-\sin^{2}\nu_{k}}{3}\cos\nu_{k},\qquad C_i=\left[\frac{\rho_0}{\rho}n_i+1\right]\cos\nu_{k},\quad i=1,2. \tag{7} \]

The value of \(\xi\) corresponding to the minimum drift rate is found by setting equal to zero the derivative with respect to \(\xi\) of expression (6). Practical calculations are conveniently carried out by the method of successive approximations using the formula

\[ \xi_n=\frac{r_n}{R}=\sqrt{0.6\,\frac{C}{C_i}+0.4\,\frac{\xi_{n-1}^{5}}{C_i}}. \tag{8} \]

Since, in a certain neighborhood of the optimal value of \(\xi\) (Fig. 2), the drift rate retains a value close to the minimum, it is possible, in designing the instrument, to take values of \(\xi\) somewhat greater than the optimal one.

Fig. 2

Let us consider a problem that makes it possible to obtain a quantitative estimate of the effect achieved by increasing the density of the rotor material. For a steel rotor \([\rho=\rho_0]\), taking \(\nu_k=45^\circ\) and specifying various values of the coefficient \(n_i\), from formulas (6)—(8) we find the corresponding values of the quantity \(k\omega_{i0}\). The results of the calculation are given in Table 1.

Table 1

The results of analogous calculations for a rotor made of a heavy tungsten-based alloy, \(\rho = 2.3\rho_0\), are given in Table 2.

Table 2

The coefficient \(n_i\) has the greater value the smaller the dimensions of the gyroscope are, since, for design and technological reasons, the weight of the gimbals and the friction in the current leads cannot be reduced in proportion to the reduction in rotor volume (the rotor volume decreases in proportion to the third power of its outside radius).

It is seen from the tables that, with an increase in \(n_i\), i.e., with an increase in the fraction of the friction moment due to the presence of current leads and gimbals of the gimbal suspension, the optimum radius \(r=\xi R\) of the cylindrical cavity in the rotor decreases. For a heavy rotor, at the same values of the friction moment due to the current leads and the weight of the gimbals, the optimum hole radius is larger.

Thus, the use of heavy materials creates more favorable conditions for placing the electric motor in small-size gyroscopes. As is seen from Table 2, the drift rate of a gyroscope with a rotor of optimum shape made of a heavy alloy is less than the drift rate of the corresponding steel rotor for any \(n_i\). However, the greatest effect is achieved at large \(n_i\), i.e., in small-size gyromotors.

For the gyrounit of the AGI-1 attitude indicator (rotor diameter 48 mm), the weight of the inner ring is approximately equal to the weight of the rotor, while the weight of the outer ring is half the rotor weight. Thus, the minimum value of the coefficient \(n_i\) (corresponding to the case when friction moments in the current leads are completely absent) in this case is \(n_1=1\) for angular velocity \(\omega_1\) and \(n_2=1.5\) for angular velocity \(\omega_2\). Since both values of the coefficient cannot be used simultaneously, the radius \(r\) of the hole in the rotor should be determined from the mean value of the coefficient \(n=1.25\). From Table 2 we establish that replacing steel by a heavy alloy in this case leads to a 34% decrease in the angular drift rate.

The results obtained apply to a homogeneous rotor. In practice, the great majority of gyroscope rotors have a nonhomogeneous structure. Thus, the average density of the “squirrel cage” of an asynchronous gyromotor is less than the density of steel. This should be taken into account in the practical use of the results of the study of a homogeneous rotor.

Formulas (1), (2) do not take into account the possible displacement of the rotor’s center of gravity relative to the suspension point. The presence of such a displacement leads to the appearance of an additional drift rate

\[ \omega_t=\frac{G\cdot l}{I\Omega}=\frac{g\cdot V\cdot l}{I_{\text{sp}}\cdot \Omega}. \tag{9} \]

As is evident from this formula, the gyroscope drift caused by a displacement \(l\) of the center of gravity cannot be reduced directly by changing the density of the material. At the same time, it is obvious that the ratio \(V/I_{\mathrm{de}}\) is the smaller, the larger the radius \(r\) of the cylindrical cavity of the rotor. As follows from a comparison of Tables 1 and 2, the optimum heavy rotor should have a hole of larger radius than a light one. Consequently, the optimum heavy rotor will be less sensitive to displacement of the center of gravity.

Conclusions

1. The shape of the rotor of the gyroscope being designed must be selected with allowance for the specific values of the friction moments in the current leads, the weight of the frames of the gimbal suspension, and the coefficient of the moment of the friction forces. Design according to the principle of geometric similarity is associated with a loss of accuracy and therefore cannot be recommended.

2. The use of materials with a high specific weight makes it possible, with a rational choice of the rotor shape, to increase the accuracy of the gyroscope substantially and increases the quality factor of the gyromotor. An increase in the quality factor reduces the sensitivity of the gyroscope to displacement of the center of gravity and to the action of accelerations and vibrations.

3. The effect of using heavy materials increases as the dimensions of the gyroscopes being designed are reduced.

Literature

1. M. P. Shumskii. On the question of the rational shape of the rotor of a gyroscope in a gimbal suspension. Proceedings of the Leningrad Institute of Aviation Instrument Engineering, issue 44, L., 1964.

2. V. A. Pavlov. Rational rotor shape determined by the requirements of increased accuracy of gyroscopic instruments. Proceedings of the Leningrad Institute of Aviation Instrument Engineering, issue XIX, L., 1958.

3. V. A. Pavlov. Elements of the synthesis of small-sized gyroscopic instruments. First Interuniversity Scientific and Technical Conference on Problems of Modern Gyroscopy. LITMO, L., 1960.