Reports of the Academy of Sciences of the USSR

1. Volume 163, No. 2

Mechanics

Ya. N. Roitenberg

A Corrected Gyrocompass

(Presented by Academician A. Yu. Ishlinskii, 29 XII 1964)

A gyroscopic compass whose sensitive element is mounted on a platform stabilized in the horizontal plane and is brought into the meridian by means of correcting torques applied to the gyroscope \((^1)\) may be called a corrected gyrocompass.

Let us consider a single-rotor corrected gyrocompass, consisting of an astatized gyroscope whose outer gimbal-ring axis is mounted on a platform stabilized in the horizontal plane and, consequently, coincides with the axis \(\xi\) directed along the radius of the terrestrial sphere (Fig. 1). The angles \(\alpha\) and \(\beta\) determine the position of the gyroscope rotor axis \(z\) relative to the reference system \(\xi\eta\zeta\). We shall denote the angular velocity of the gyroscope’s proper rotation by \(\Phi'\).

The projections of the instantaneous angular velocity of the reference system \(\xi\eta\zeta\) onto its axes \(\xi,\eta,\zeta\) will be

\[ u_1=-\frac{v_N}{R},\qquad u_2=U\cos\varphi+\frac{v_E}{R},\qquad u_3=U\sin\varphi+\frac{v_E}{R}\operatorname{tg}\varphi, \tag{1} \]

where \(v_E\) and \(v_N\) are the eastward and northward components of the ship’s velocity relative to the terrestrial sphere, \(R\) is the radius of the terrestrial sphere, \(U\) is the angular velocity of the daily rotation of the terrestrial sphere, and \(\varphi\) is the latitude of the ship’s location.

Restricting ourselves to the study of the precessional motion of the gyrocompass, one may take for its kinetic energy the approximate expression

\[ T\approx {}^{1}\!/\!_{2}Cr^2, \tag{2} \]

where \(C\) is the moment of inertia of the gyroscope rotor relative to its axis \(z\), and \(r\) is the projection of the absolute angular velocity of the gyroscope onto the axis \(z\). It is not hard to see that

\[ r=\alpha'\sin\beta+\Phi' -u_1\sin\alpha\cos\beta+ +u_2\cos\alpha\cos\beta+u_3\sin\beta. \tag{3} \]

Fig. 1

Assuming that the moment of the resistance forces about the gyroscope rotor axis \(z\) is balanced by the active driving torque, we take \(Q_{\Phi}\equiv0\). Since \(\partial T/\partial\Phi'=Cr,\ \partial T/\partial\Phi=0\), we shall have

\[ Cr=H=\text{const}. \tag{4} \]

Taking into account that

\[ \partial T/\partial\alpha'=H\sin\beta,\qquad \partial T/\partial\alpha=-H\left(u_1\cos\alpha\cos\beta+u_2\sin\alpha\cos\beta\right), \]

\[ \partial T/\partial\beta'=0, \tag{5} \]

\[ \partial T/\partial\beta=H\left(\alpha'\cos\beta+u_1\sin\alpha\sin\beta-u_2\cos\alpha\sin\beta+u_3\cos\beta\right), \]

we find the following equations of motion of the gyroscopic compass:

\[ H\beta' \cos\beta + H(u_1\cos\alpha\cos\beta + u_2\sin\alpha\cos\beta)=M_\xi, \]
\[ H(\alpha'\cos\beta + u_1\sin\alpha\sin\beta - u_2\cos\alpha\sin\beta + u_3\cos\beta)=M_{x^*}. \tag{6} \]

Here \(M_\xi\) and \(M_{x^*}\) are the correcting torques applied to the gyrocompass about the axes \(\xi\) and \(x^*\), respectively.

Since the instrument is mounted on a platform stabilized in the horizon, the elevation angle \(\beta\) of the \(z\)-axis of the gyroscope rotor above the plane of the horizon can be measured.

The angle between the direction of the horizontal projection of the \(z\)-axis of the gyroscope and the vector \(\mathbf v\) of the ship’s velocity relative to the terrestrial sphere (Fig. 1) is equal to \(\psi+\alpha\), where \(\psi\) is the ship’s course. Since the angle \(\psi+\alpha\) can be measured, then, if an instrument is available which determines the ship’s velocity relative to the terrestrial sphere, the components of the ship’s velocity \(v\cos(\psi+\alpha)\) and \(v\sin(\psi+\alpha)\) can be determined.

Also assuming the latitude \(\varphi\) of the ship’s position to be known, one can apply to the gyroscope correcting torques formed according to the law

\[ M_\xi=-H\frac{v}{R}\cos(\psi+\alpha)\cos\beta-\mu K\sin\beta, \]
\[ M_{x^*}=-H\frac{v}{R}\sin(\psi+\alpha)\sin\beta+ \]
\[ +H\left[U\sin\varphi+\frac{v}{R}\sin(\psi+\alpha)\operatorname{tg}\varphi\right]\cos\beta+K\sin\beta, \tag{7} \]

where \(\mu\) and \(K\) are certain constant coefficients. Here it is naturally assumed that the latitude of the ship’s position \(\varphi<90^\circ\).

Taking into account that

\[ v\cos(\psi+\alpha)=v_N\cos\alpha-v_E\sin\alpha, \]
\[ v\sin(\psi+\alpha)=v_E\cos\alpha+v_N\sin\alpha, \tag{8} \]

where

\[ v_N=v\cos\psi,\qquad v_E=v\sin\psi, \tag{9} \]

and replacing \(M_\xi\) and \(M_{x^*}\) by their expressions (7), the equations of motion of the gyroscopic compass (6) can be reduced to the form

\[ H\beta'\cos\beta+HU\cos\varphi\sin\alpha\cos\beta+\mu K\sin\beta=0, \]
\[ H\alpha'\cos\beta-HU\cos\varphi\cos\alpha\sin\beta+ \]
\[ +H\frac{v_E}{R}\operatorname{tg}\varphi\cos\beta(1-\cos\alpha) -H\frac{v_N}{R}\operatorname{tg}\varphi\sin\alpha\cos\beta -K\sin\beta=0. \tag{10} \]

It is not difficult to see that the system of differential equations (10) has the particular solution

\[ \alpha=0,\qquad \beta=0. \tag{11} \]

Thus, for any law of motion of the ship \(\mathbf v=\mathbf v(t)\), the direction toward the north is an equilibrium position of the \(z\)-axis of the gyrocompass, i.e. the corrected gyrocompass has no speed deviation.

For satisfactory operation of the gyrocompass, its oscillations relative to the equilibrium position \(\alpha=\beta=0\) must be damped.

The variational equations, which can be obtained from (10) by taking \(\alpha\) and \(\beta\) to be small angles, can be brought to the form

\[ \alpha'-\frac{v_N}{R}\operatorname{tg}\varphi\cdot\alpha- \left(\frac{K}{H}+U\cos\varphi\right)\beta=0, \qquad \beta'+\frac{\mu K}{H}\beta+U\cos\varphi\cdot\alpha=0. \tag{12} \]

Since \(v_N=v_N(t)\), \(\varphi=\varphi(t)\), equations (12) constitute a system of linear differential equations with variable coefficients.

Sufficient conditions for the asymptotic stability of the particular solution (11), which determines the equilibrium position of the \(z\)-axis of the gyroscopic compass, will be found with the aid of a Lyapunov function, which can be constructed by the method proposed in (2).

Denoting

\[ f_1(t)=\frac{v_N}{R}\tg\varphi,\qquad f_2(t)=U\cos\varphi, \tag{13} \]

we reduce equations (12) to the form

\[ \alpha'-\frac{K}{H}\beta=f_1(t)\alpha+f_2(t)\beta,\qquad \beta'+\frac{\mu K}{H}\beta+s\alpha=[s-f_2(t)]\alpha. \tag{14} \]

Here the coefficient \(s\) in the second equation (14) is chosen so that, for the system of equations

\[ \alpha'-\frac{K}{H}\beta=0,\qquad \beta'+\frac{\mu K}{H}\beta+s\alpha=0 \tag{15} \]

the characteristic equation

\[ \gamma^2+\frac{\mu K}{H}\gamma+\frac{sK}{H}=0 \tag{16} \]

has a pair of complex roots

\[ \gamma_1,\gamma_2=\varepsilon\pm\omega i, \tag{17} \]

where

\[ \varepsilon=-\frac{\mu K}{2H},\qquad \omega=\left(\frac{sK}{H}-\varepsilon^2\right)^{1/2}. \tag{18} \]

We now pass to new variables \(x_1\) and \(x_2\), which we introduce by means of the relations

\[ \alpha=x_1,\qquad \beta=\frac{\varepsilon H}{K}x_1+\frac{\omega H}{K}x_2. \tag{19} \]

In accordance with (14), the functions \(x_1\) and \(x_2\) will satisfy the differential equations

\[ x_1'=\left[\varepsilon+f_1(t)+\frac{\varepsilon H}{K}f_2(t)\right]x_1 +\omega\left[1+\frac{H}{K}f_2(t)\right]x_2, \]

\[ x_2'=\varepsilon\left[1-\frac{H}{K}f_2(t)\right]x_2 -\left\{\omega-\frac{K}{\omega H}[s-f_2(t)] +\frac{\varepsilon}{\omega}\left[f_1(t)+\frac{\varepsilon H}{K}f_2(t)\right]\right\}x_1. \tag{20} \]

As a Lyapunov function we take the negative definite function

\[ V=-\frac{1}{2}(x_1^2+x_2^2). \tag{21} \]

Its derivative with respect to time, by virtue of the differential equations (20), will have the form

\[ V'=a_{11}x_1^2+2a_{12}x_1x_2+a_{22}x_2^2, \tag{22} \]

where

\[ a_{11}=-\left[\varepsilon+f_1(t)+\frac{\varepsilon H}{K}f_2(t)\right], \]

\[ a_{12}=\frac{1}{2}\left\{ \frac{\varepsilon}{\omega}f_1(t) -\frac{H}{\omega K}(\omega^2-\varepsilon^2)f_2(t) -\frac{K}{\omega H}[s-f_2(t)] \right\}. \tag{23} \]

\[ a_{22}=-\varepsilon\left[1-\frac{H}{K}f_2(t)\right]. \]

According to Sylvester’s theorem, the quadratic form (22) will be positive definite if, for any instant of time \(t\), the conditions

\[ a_{11}(t)>0,\qquad a_{11}(t)a_{22}(t)-[a_{12}(t)]^2>0 \tag{24} \]

are satisfied.

Conditions (24) are sufficient conditions for the asymptotic stability of the equilibrium position of the gyroscopic compass.

As an example, consider a wide-range compass whose parameters have the values: \(K/H=3.6\ \mathrm{sec}^{-1}\), \(\mu=0.005\). The coefficient \(s\), by which the transformation (14) is determined, and accordingly the parameters of the Lyapunov function, will be taken as \(s=4\cdot10^{-5}\ \mathrm{sec}^{-1}\).

For the indicated parameter values, the sufficient stability conditions (24) are satisfied at any point of the rectangle

\[ -\varphi_m\leqslant \varphi \leqslant \varphi_m,\qquad -v_m\leqslant v_N\leqslant v_m, \tag{25} \]

where \(\varphi_m=85^\circ\), \(v_m=600\ \mathrm{m}\cdot\mathrm{sec}^{-1}\), so that the corrected gyroscopic compass can be used in aviation, as was also noted in work (1).

Let us note that, since \(\varphi'=v_N/R\), the latitude of the vessel’s location is determined by the expression

\[ \varphi(t)=\varphi(0)+\int_0^t \frac{v_N(\tau)}{R}\,d\tau . \tag{26} \]

It follows from what has been set forth that the equilibrium position of the gyroscopic compass preserves stability under any law of variation of the northern component of the vessel’s velocity \(v_N=v_N(t)\), for which \(v_N(t)\) and \(\varphi(t)\) do not leave the region (25).

Moscow State University
named after M. V. Lomonosov

Received
16 XII 1964

CITED LITERATURE

\(^{1}\) P. H. Savet (Ed.), Gyroscopes: Theory and Design, N. Y., 1961, p. 80. \(^{2}\) Ya. N. Roitenberg, Prikl. matem. i mekh., 22, 2 (1958).