Full Text
Ya. N. Roitenberg
ON THE MOTION OF A GYROSCOPIC COMPASS UNDER THE ACTION OF RANDOM FORCES
(Presented by Academician A. N. Kolmogorov, 12 IV 1960)
The accuracy of a gyroscopic compass with mercury ballistic vessels under conditions of a ship’s rolling can be ensured, as was shown by Rawlings (¹), by a proper choice of the period of the natural oscillations of the mercury in the ballistic vessels. Rawlings’ theory proceeds from the assumption that the ship’s rolling takes place according to a sinusoidal law. In the present paper the motion of a gyroscopic compass is studied under the assumption that the ship’s rolling is a stationary random process (²) with fractional-rational spectral density.
- The equations of motion of a gyroscopic compass with mercury ballistic vessels during the rolling of a ship may be written in the form (¹)
\[ H\beta' + HU\cos\varphi \cdot \alpha - c_1\vartheta + c\vartheta = Hv_N/R,\qquad H\alpha' + c\vartheta = -HU\sin\varphi, \]
\[ C\vartheta'' + m\vartheta' + \vartheta + \beta = -W_2/g,\qquad A\gamma'' + s\gamma' + lP\gamma = lPW_1/g. \tag{1} \]
Here \(\alpha\) is the angle of rotation of the gyrocompass in azimuth; \(\beta\) is the angle of elevation of the axis of the gyroscope rotor above the horizontal plane; \(\vartheta\) is the angle between the mercury mirror in the ballistic vessels and the axis of the gyroscope rotor; \(\gamma\) is the angle of rotation of the cardan suspension of the gyrocompass about an axis parallel to the axis of the gyroscope rotor; \(H\) is the kinetic moment of the gyroscope; \(U\) is the angular velocity of the diurnal rotation of the terrestrial sphere; \(\varphi\) is the latitude of the place of observation; \(v_N\) is the northern component of the ship’s velocity; \(R\) is the radius of the terrestrial sphere; \(W_1\) and \(W_2\) are the eastern and northern components of the translational acceleration of the support point of the gyrocompass.
Under conditions of rectilinear uniform motion of the ship,
\[ W_1 \simeq -r\theta''\cos\psi,\qquad W_2 \simeq r\theta''\sin\psi, \tag{2} \]
where \(\theta\) is the ship’s roll angle; \(\psi\) is the ship’s course; \(r\) is the distance of the support point of the gyrocompass from the straight line passing through the center of the ship’s rolling and parallel to the ship’s longitudinal axis.
Introduce new variables \(X_1,\ldots,X_4\) by means of the relations
\[ X_1=\alpha-\alpha^*,\qquad X_2=\beta-\beta^*,\qquad X_3=\vartheta-\vartheta^*,\qquad X_4=\gamma-\gamma^*, \tag{3} \]
where \(\alpha^*=v_N/RU\cos\varphi-\lambda\tan\varphi,\ \beta^*=-\vartheta^*=HU\sin\varphi/c,\ \gamma^*=0,\ \lambda=c_1/c.\)
Then equations (1) are transformed to the form
\[ X_1' + \frac{k^2}{U\cos\varphi}X_3 = 0, \]
\[ X_2' + U\cos\varphi\,X_1 - \lambda\frac{k^2}{U\cos\varphi}X_3 - U\sin\varphi\,X_4 + \frac{k^2}{U\cos\varphi}X_3X_4 = 0, \]
\[ X_3'' + \zeta X_3' + n^2X_3 + n^2X_2 = -\frac{r\sin\psi}{g}\,n^2\theta'', \tag{4} \]
\[ X_4'' + \chi X_4' + \rho^2X_4 = -\frac{r\cos\psi}{g}\,\rho^2\theta'', \]
where \(k^2=cU\cos\varphi/H,\ n^2=1/C,\ \rho^2=lP/A,\ \zeta=m/C,\ \chi=s/A.\)
The equations of the first approximation, which are obtained from (4) by discarding terms of second order of smallness, may be represented in the form of the matrix equation
\[ f(D)X^{(0)}=e(D)\theta(t)\qquad (D=d/dt), \tag{5} \]
where
\[ f(D)= \begin{vmatrix} D & 0 & \dfrac{k^2}{U\cos\varphi} & 0\\ U\cos\varphi & D & -\lambda\dfrac{k^2}{U\cos\varphi} & -U\sin\varphi\\ 0 & n^2 & D^2+\zeta D+n^2 & 0\\ 0 & 0 & 0 & D^2+\chi D+\rho^2 \end{vmatrix}, \qquad X^{(0)}= \begin{Vmatrix} X_1^{(0)}\\ X_2^{(0)}\\ X_3^{(0)}\\ X_4^{(0)} \end{Vmatrix}, \qquad e(D)= \begin{Vmatrix} 0\\ 0\\ -\dfrac{r\sin\psi}{g}\,n^2D^2\\ -\dfrac{r\cos\psi}{g}\,\rho^2D^2 \end{Vmatrix}. \tag{6} \]
From equation (5) it follows that
\[ X^{(0)}=Y(D)\theta(t)\qquad \bigl(Y(D)=F(D)e(D)/\Delta(D)\bigr). \tag{7} \]
Here \(F(D)\) is the adjugate matrix for the matrix \(f(D)\), and \(\Delta(D)\) is the determinant of the matrix \(f(D)\),
\[ \Delta(D)=(D^2+\chi D+\rho^2)\Delta_1(D), \tag{8} \]
where \(\Delta_1(D)=D^4+\zeta D^3+n^2D^2+\xi D+k^2n^2,\quad \xi=\lambda n^2k^2/U\cos\varphi.\)
Below we shall need the quantities \(X_3^{(0)}\) and \(X_4^{(0)}\). According to (7), they are determined by the operational expressions
\[ X_3^{(0)}= \left[ -\frac{r\sin\psi}{g}\,\frac{n^2D^4}{\Delta_1(D)} + \frac{r\cos\psi}{g}\, \frac{\rho^2 n^2 U\sin\varphi D^3}{\Delta(D)} \right]\theta(t), \]
\[ X_4^{(0)}= -\frac{r\cos\psi}{g}\, \frac{\rho^2D^2}{D^2+\chi D+\rho^2}\,\theta(t). \tag{9} \]
Let us note that it follows from (9) that
\[ X_3^{(0)}=\Phi(D)X_4^{(0)}, \tag{10} \]
where
\[ \Phi(D)= \frac{n^2\tan\psi}{\rho^2}\, \frac{D^2(D^2+\chi D+\rho^2)}{\Delta_1(D)} - \frac{n^2U\sin\varphi D}{\Delta_1(D)}. \]
Assuming that the ship’s rolling is a stationary random process with spectral density
\[ S(\omega)=L\frac{4\mu\nu^2}{(\omega^2-\nu^2)^2+4\mu^2\omega^2}, \tag{11} \]
we proceed to determining the mathematical expectations of the random processes \(X_1,\ldots,X_4\), which we denote as follows:
\[ x_i=M[X_i]\qquad (i=1,\ldots,4). \tag{12} \]
To determine the functions \(x_i\) \((i=1,\ldots,4)\), in accordance with (4), we shall have the system of differential equations
\[ \dot{x}_1+\frac{k^2}{U\cos\varphi}x_3=0, \]
\[ \dot{x}_2+U\cos\varphi\cdot x_1-\lambda\frac{k^2}{U\cos\varphi}x_3-U\sin\varphi\cdot x_4 =-\frac{k^2}{U\cos\varphi}\,E, \]
\[ \ddot{x}_3+\zeta\dot{x}_3+n^2x_3+n^2x_2=0, \]
\[ \ddot{x}_4+\chi\dot{x}_4+\rho^2x_4=0, \tag{13} \]
where \(E=M[X_3X_4]\).
Restricting ourselves to the first approximation, we shall replace \(E\) in equations (13) by the expression
\[ E_0=M\left[X_3^{(0)}X_4^{(0)}\right]. \tag{14} \]
The integrals of the system of homogeneous equations that is obtained from (13) for \(E=0\) will tend asymptotically to zero as \(t\to\infty\), since the zeros of the characteristic determinant \(\Delta(D)\), according to (8), are located in the left half-plane of the complex variable \(D\). Therefore, during the rolling of the ship, which is a long-duration process, the quantities \(x_1,\ldots,x_4\) will assume their steady-state values
\[ x_1=-\left(k/U\cos\varphi\right)^2 E_0,\qquad x_2^*=0,\qquad x_3^*=0,\qquad x_4^*=0. \tag{15} \]
Here
\[ E_0=\frac{1}{2\pi}\int_{-\infty}^{\infty} S_{43}(\omega)\,d\omega, \tag{16} \]
where \(S_{43}(\omega)\) is the cross-spectral density of the random processes \(X_4^{(0)}\) and \(X_3^{(0)}\),
\[ S_{43}(\omega)=\Phi(i\omega)S_{44}(\omega), \tag{17} \]
and \(S_{44}(\omega)\) is the spectral density of the random process \(X_4^{(0)}\),
\[ S_{44}(\omega)=4\mu\nu^2 L\frac{\rho^4 r^2\cos^2\psi}{g^2}\, \frac{\omega^4}{\left[(\omega^2-\rho^2)^2+\chi^2\omega^2\right]\left[(\omega^2-\nu^2)^2+4\mu^2\omega^2\right]}. \tag{18} \]
Expression (16) can be reduced to the form
\[ E_0=4\mu\nu^2 L(z_1 I+z_2 J), \tag{19} \]
where
\[ z_1=\frac{1}{2g^2}\rho^2 n^2 r^2\sin 2\psi,\qquad z_2=\frac{1}{g^2}\rho^4 n^2 U\sin\varphi\, r^2\cos^2\psi; \tag{20} \]
\[ I=\frac{1}{2\pi}\int_{-\infty}^{\infty} \frac{g_1(i\omega)}{h(i\omega)h(-i\omega)}\,d\omega,\qquad J=\frac{1}{2\pi}\int_{-\infty}^{\infty} \frac{g_2(i\omega)}{h(i\omega)h(-i\omega)}\,d\omega; \tag{21} \]
\[ g_1(i\omega)=\sum_{s=0}^{7} b_s(i\omega)^{14-2s},\qquad g_2(i\omega)=\sum_{s=0}^{7} B_s(i\omega)^{14-2s}, \]
\[ h(i\omega)=\sum_{s=0}^{8} a_s(i\omega)^{8-s}, \tag{22} \]
and the coefficients of the polynomials (22) have the form
\(b_0=0,\ b_1=1,\ b_2=\rho^2+n^2-\chi\xi,\)
\(b_3=(\rho^2+k^2)n^2-\chi\xi,\ b_4=\rho^2 k^2 n^2,\ b_5=b_6=b_7=0,\)
\(B_0=B_1=B_2=0,\ B_3=\zeta,\ B_4=\xi,\ B_5=B_6=B_7=0,\)
\(a_0=1,\ a_1=2\mu+\chi+\zeta,\)
\(a_2=\nu^2+\rho^2+n^2+2\mu(\chi+\zeta)+\chi\zeta,\)
\(a_3=2\mu(\rho^2+n^2+\chi\zeta)+\zeta(\rho^2+\nu^2)+\chi(\nu^2+n^2)+\xi,\)
\(a_4=2\mu(\xi+\chi n^2+\zeta\rho^2)+\chi(\xi+\zeta\nu^2)+n^2(\nu^2+\rho^2+k^2)+\nu^2\rho^2,\)
\(a_5=2\mu(k^2 n^2+\rho^2 n^2+\chi\xi)+\xi(\rho^2+\nu^2)-\chi n^2(\nu^2+k^2)+\zeta\nu^2\rho^2,\)
\(a_6=2\mu(\xi\rho^2+\chi k^2 n^2)+n^2(\nu^2\rho^2+k^2\nu^2+k^2\rho^2)+\chi\xi\nu^2,\)
\(a_7=(2\mu\rho^2+\chi\nu^2)k^2n^2+\xi\nu^2\rho^2,\)
\(a_8=k^2 n^2\rho^2\nu^2\).
The integrals (21), in the case when all zeros of the polynomial \(h(D)\) are located in the left half-plane of the complex variable \(D\), according to Phillips \((^3)\), have the form
\[ I=-M_1/2a_0N,\qquad J=-M_2/2a_0N, \tag{23} \]
where
\[ N= \begin{vmatrix} a_1 & a_0 & 0 & 0 & 0 & 0 & 0 & 0\\ a_3 & a_2 & a_1 & a_0 & 0 & 0 & 0 & 0\\ a_5 & a_4 & a_3 & a_2 & a_1 & a_0 & 0 & 0\\ a_7 & a_6 & a_5 & a_4 & a_3 & a_2 & a_1 & a_0\\ 0 & a_8 & a_7 & a_6 & a_5 & a_4 & a_3 & a_2\\ 0 & 0 & 0 & a_8 & a_7 & a_6 & a_5 & a_4\\ 0 & 0 & 0 & 0 & 0 & a_8 & a_7 & a_6\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & a_8 \end{vmatrix}, \tag{24} \]
where \(M_1\) and \(M_2\) are determinants obtained by replacing the first column in determinant (24) by a column whose elements are \(b_0, b_1, \ldots, b_7\) or \(B_0, B_1, \ldots, B_7\), respectively.
As follows from (15), under ship-rolling conditions the gyroscopic compass has an azimuth deviation. In accordance with (15) and (19), this deviation is determined by the expression
\[ x_1^*=-a I^*\sin 2\psi-bJ^*\cos^2\psi, \tag{25} \]
where
\[ a=2\mu\gamma^2L\rho^2\frac{r^2}{g^2}\left(\frac{k}{U\cos\varphi}\right)^2,\qquad b=4\mu\gamma^2L\rho^4U\sin\varphi\,\frac{r^2}{g^2}\left(\frac{k}{U\cos\varphi}\right)^2, \]
\[ I^*=n^2I,\qquad J^*=n^2J. \tag{26} \]
- As an example, let us determine the deviation of a gyroscopic compass whose parameters have the values \(k=1.24\cdot10^{-3}\ \mathrm{sec}^{-1}\), \(\rho=6\ \mathrm{sec}^{-1}\), \(\lambda=1.2\ \mathrm{sec}^{-1}\), \(\zeta=0.2n\ \mathrm{sec}^{-1}\), \(\xi=10^{-3}n^2\ \mathrm{sec}^{-3}\), \(r=2\ \mathrm{m}\). The latitude of the observation point is \(\varphi=60^\circ\), so that \(U\cos\varphi=3.646\cdot10^{-5}\ \mathrm{sec}^{-1}\). The parameters determining the spectral density of the ship’s rolling are: \(\mu=0.1\ \mathrm{sec}^{-1}\), \(\gamma=0.8\ \mathrm{sec}^{-1}\), \(\sqrt{L}=0.173\simeq10^\circ\).
Fig. 1
For these data, the deviation of the gyrocompass, according to (25), is
\[ x_1^*=-6.72\,I^*\sin 2\psi-3.03\cdot10^{-2}J^*\cos^2\psi. \tag{27} \]
Here, according to (26) and (23), \(I^*=I^*(T)\), \(J^*=J^*(T)\), where \(T=2\pi/n\). The graphs of the functions \(I^*(T)\) and \(J^*(T)\) are given in Fig. 1.
The maximum value of \(J^*\) in the example under consideration is \(0.0037\), so that the second term in expression (27) does not exceed \(0.11\cdot10^{-3}\), or \(0.4\) angular minutes. Neglecting this quantity as small, we obtain from (27) the following expression for the deviation of the gyrocompass:
\[ x_1^*=-6.72\,I^*\sin 2\psi. \tag{28} \]
The deviation \(x_1^*\) is an intercardinal deviation of the gyroscopic compass. It vanishes on the cardinal courses of the ship \((\psi=0;\ 90;\ 180;\ 270^\circ)\) and attains its greatest values on the intercardinal courses \((\psi=45;\ 135;\ 225;\ 315^\circ)\).
To prevent the intercardinal deviation, the parameters of mercury ballistic vessels must be chosen so that the value of the angular frequency \(n\) of the natural oscillations of the mercury mirror is sufficiently close to the value of the argument \(n^*\) at which the function \(I^*(n)\) becomes zero. As can be seen from the graph, the zero of the function \(I^*(n)\) is located near the point \(n=\gamma\). In the example considered, \(n^*=0.843\).
Moscow State University
named after M. V. Lomonosov
Received
6 IV 1960
REFERENCES
- A. L. Rawlings, The Theory of the Gyroscopic Compass and its Deviations, London, 1929.
- A. A. Sveshnikov, Proceedings of the First Interuniversity Conference on Gyroscopy, 1956.
- H. James, N. Nichols, R. Phillips, Theory of Servomechanisms, IL, 1951.