Drifts of a Single-Axis Floated Gyrostabilizer on an Oscillating Base
Abstract Generated abstract
This paper analyzes the drift of a single-axis floated gyrostabilizer mounted on a base undergoing small angular oscillations, a situation relevant to gyrostabilized platforms using such devices as reference sensors. Starting from truncated equations of motion with interframe correction, it derives an integral equation for inertial drift rate and obtains first-approximation expressions for harmonic base motions and disturbing moments, including conditions under which systematic drift is absent or arises from equal-frequency components. The analysis is extended to dry friction in the outer gimbal and to stationary random disturbances, yielding approximate formulas that relate systematic drift to phase relations, correction-circuit gain, friction, and correlation functions between base angular velocities and disturbing moments.
Full Text
DRIFTS OF A SINGLE-AXIS FLOATED GYROSTABILIZER ON AN OSCILLATING BASE
A. M. MALYSHENKO
(Presented by the scientific seminar of the Department of Automation and Telemechanics)
In the present article, the drifts of a single-axis floated gyrostabilizer mounted on a base executing oscillations with small amplitudes of the angles $\alpha_i$, for which
\[ \sin \alpha_i \simeq \alpha_i;\qquad \cos \alpha_i \simeq 1. \]
are analyzed.
Such a situation occurs, in particular, in gyrostabilized platforms with tracking systems that use single-axis gyrostabilizers as reference position sensors.
If a right orthogonal trihedron of axes $xyz$ is attached to the base on which the gyrostabilizer is mounted, and the gyrostabilizer is oriented as shown in Fig. 1, then the motion of the gyrostabilizer may be described by the following set of truncated differential equations:
\[ H\dot{\varepsilon}-H\Omega_x\theta=-H\Omega_y+M_{\mathrm{MK}}-M_{ob}; \tag{1} \]
\[ J\ddot{\theta}+\mu\dot{\varepsilon}+H\Omega_x\varepsilon=-H\Omega_z+M_y+M_{\varepsilon b}. \tag{2} \]
In equations (1) and (2) the following notation is adopted:
$\theta$ — angle of rotation of the outer ring (housing) of the gyrostabilizer relative to the base;
$\varepsilon$ — angle of rotation of the outer ring of the gyrostabilizer relative to the inner ring (float);
$H$ — intrinsic kinetic moment of the gyroscope;
$\mu$ — coefficient of viscous friction between the float and the gyroscope housing;
$\Omega_x,\Omega_y,\Omega_z$ — projections of the absolute angular velocity of the base, respectively, on the axes $x,y,z$;
$M_{ob}, M_{\varepsilon b}$ — moments of disturbing forces about the outer and inner axes of suspension of the gyroscope;
$M_y$ — control moment;
$M_{\mathrm{MK}}$ — moment of interframe correction of the gyrostabilizer.
In what follows we assume that
\[ M_y=0; \]
\[ M_{\mathrm{MK}}=-k\varepsilon, \]
Fig. 1
where \(k\) is the transmission coefficient of the inter-frame correction circuit.
Under these conditions, equations (1) and (2) are written in the form:
\[ H\dot{\varepsilon}+k\varepsilon-H\Omega_x\theta=-H\Omega_y-M_{\psi}; \tag{3} \]
\[ H\dot{\theta}+\mu\varepsilon+H\Omega_x\varepsilon=-H\Omega_z+M_{\varepsilon b}. \tag{4} \]
To determine the drift rate of the gyrostabilizer in inertial space, \(\Omega_z+\dot{\theta}\), we pass from the system of equations (3) and (4) to the integral equation
\[ \dot{\theta}(t)=-\Omega_z(t)+\frac{1}{H}M_{\varepsilon b}(t)+\frac{\mu}{H^2}M_{\theta b}(t)+\frac{\mu}{H}\left[\Omega_y(t)- \right. \]
\[ \left. -\Omega_x(t)\int_0^t \dot{\theta}(t_1)\,dt_1\right] -\left[\frac{\mu k}{H^2}-\Omega_x(t)\right]\left\{\varepsilon_0 e^{-t/T}- \right. \]
\[ \left. -\int_0^t e^{(t_1-t)/T}\left[\Omega_y(t_1)+\frac{1}{H}M_{\theta b}(t_1)+\Omega_x(t_1)\times \right.\right. \]
\[ \left.\left. \times \int_0^{t_1}\dot{\theta}(t_2)\,dt_2\right]dt_1\right\}. \tag{5} \]
Here the time constant is
\[ T=\frac{H}{k}. \tag{6} \]
Equation (5) makes it possible, by the method of successive approximations, to analyze the motions of the gyrostabilizer for arbitrary angular oscillations of the base with frequencies lower than the natural frequency of the gyrostabilizer.
If it is assumed that
\[ \Omega_i(t)=\Omega_{im}\sin(\omega_i t+\varphi_i),\quad i=x,\ y,\ z; \tag{7} \]
\[ M_\varepsilon=M_{\varepsilon 0}+M_{\varepsilon m}\sin(\omega_\varepsilon t+\varphi_\varepsilon); \tag{8} \]
\[ M_\theta=M_{\theta 0}+M_{\theta m}\sin(\omega_\theta t+\varphi_\theta), \tag{9} \]
then, in the steady-state operating regime, the drift rate of the gyrostabilizer according to (5) will, in the first approximation, be equal to
\[ \Omega(t)=\Omega_z(t)+\dot{\theta}(t)= \frac{M_{\varepsilon 0}}{H} +\frac{M_{\varepsilon m}}{H}\sin(\omega_\varepsilon t+\varphi_\varepsilon)+ \]
\[ +\frac{\mu}{k}M_{\theta m}\sin(\omega_\theta t+\varphi_\theta) +\frac{M_{\varepsilon 0}}{H}\Omega_{xm}\sin(\omega_x t+\varphi_x)+ \]
\[ +\frac{\mu}{H}\Omega_{ym}\sin(\omega_y t+\varphi_y) +\left[\frac{\mu k}{H^2}-\Omega_{xm}\sin(\omega_x t+\varphi_x)\right]+ \]
\[ +\frac{M_{\theta m}}{\sqrt{1+T^2\omega_\theta^2}} \sin(\omega_\theta t+\varphi_\theta+\psi_\theta) +\frac{\Omega_{ym}T}{\sqrt{1+T^2\omega_y^2}} \sin(\omega_y t+\varphi_y+\psi_y), \tag{10} \]
where
\[ \psi_\theta=\operatorname{arctg}(-\omega_\theta T); \tag{11} \]
\[ \psi_y=\operatorname{arctg}(-\omega_y T). \tag{12} \]
On the basis of (10) it may be concluded that, in the absence of constant moments \(M_{\varepsilon 0}\) about the precession axis and when the circular frequencies of variation of the disturbing moments and of the projections of the angular velocity of the base on the axes \(x,\ y,\ z\) are unequal, the systematic drift of the gyrostabilizer is absent.
For \(M_{\varepsilon 0}=0\), systematic drift of the gyrostabilizer can arise only in the presence of equal-period components of the harmonics of oscillations \(\Omega_x\) and of one of the variables \(\Omega_y\), \(M_\theta\), or of all three simultaneously. The rate of systematic drift in the latter case will be approximately equal to:
\[ \Omega_c(t)=- \frac{T}{\sqrt{1+T^2\omega^2}}\, \Omega_{xm}\Omega_{ym}\cos(\varphi_x-\varphi_y-\psi)- \]
\[ -\frac{\Omega_{xm}M_{\theta m}}{2k\sqrt{1+T^2\omega^2}} \cos(\varphi_x-\varphi_M-\psi), \tag{13} \]
where
\[ \psi=\operatorname{arctg}(-\omega T), \tag{14} \]
\(\omega\) is the circular frequency of oscillations.
If it is necessary to refine the value of the drift rate of the gyrostabilizer, then, using the method of successive substitutions, one should substitute the drift-rate value found in the first approximation into the original equation (5). In practice, when estimating the drift rate of the gyrostabilizer, one may restrict oneself to solution (10), since the subsequent approximations \(\Omega_i(t)\), as shown by the solution of the problem on an analog model, differ little from the value \(\Omega_1(t)\).
In the moment of the external forces acting on the outer gimbal of the gyroscope, with careful balancing of the latter, the moment of dry-friction forces predominates. The influence of dry friction on the systematic drift of a single-axis gyrostabilizer is analyzed below for the case in which the gyrostabilizer is mounted on a platform in a three-degree-of-freedom Cardan suspension and the rings of the latter rotate through small angles \(\alpha, \beta, \gamma\).
When the dry-friction characteristic about the stabilization axis is approximated by a function of the form
\[ M=-M_T \operatorname{sign}\dot{\theta} \tag{15} \]
and under the assumption that
\[ \varepsilon_0=0;\qquad M_{\varepsilon 0}=0;\qquad M_{\mathrm{MK}}=-k\varepsilon, \]
the truncated equations of motion of the gyrostabilizer take the form:
\[ H\Omega_x\theta+H\dot{\varepsilon}-k\varepsilon = H\Omega_y-M_T\operatorname{sign}\dot{\theta}; \tag{16} \]
\[ H\dot{\theta}+\mu\dot{\varepsilon}+H\Omega_x\varepsilon = -H\Omega_z. \tag{17} \]
We shall restrict ourselves to the special case of motion of the base when
\[ \alpha=\alpha_m\sin(\omega t+\varphi_\alpha); \tag{18} \]
\[ \beta=\beta_m\sin(\omega t+\varphi_\beta). \tag{19} \]
Using the previously obtained equation (5) for this special case, we obtain the following expression for \(\dot{\theta}\), without taking into account terms of the second and higher orders of smallness:
\[ \dot{\theta}(t) = \alpha_m\beta_m\omega\cos(\omega t+\varphi_\alpha)\cdot \sin(\omega t+\varphi_\beta) - \frac{\mu}{H^2}M_T\operatorname{sign}\dot{\theta} + \]
\[ + \frac{\mu}{H}\beta_m\omega\cos(\omega t+\varphi_\beta) + \left[ \frac{\mu k}{H}-\alpha_m\omega\cos(\omega t+\varphi_\alpha) \right] \times \]
\[ \times \left[ \frac{1}{H}\int_0^t e^{\frac{t_1-t}{T}} M_T\operatorname{sign}\dot{\theta}\,dt - \int_0^t e^{\frac{t_1-t}{T}}\beta_m\omega\cos(\omega t+\varphi_\beta)\,dt_1 \right]. \tag{20} \]
On the basis of (17), one may assume that, under the adopted conditions,
\[ \operatorname{sign}\dot{\theta}\simeq -\operatorname{sign}\dot{\varepsilon}, \]
since practically
\[ \dot{\varepsilon}\gg \frac{H}{\mu}\Omega_x\varepsilon. \]
Under the assumption that
\[ \frac{M_\theta}{H}\ll \Omega_{ym}, \]
the established value of the precession rate \(\dot{\varepsilon}\) of the gyrostabilizer is equal to
\[ \dot{\varepsilon}\simeq - \frac{\omega^2T}{\sqrt{1+T^2\omega^2}}\, \beta_m\sin(\omega t+\varphi_\beta+\psi). \tag{21} \]
Here, as before,
\[ \psi=\operatorname{arctg}(-\omega T). \]
Thus, for the adopted mode of oscillation of the base,
\[ \operatorname{sign}\dot{\theta} = \operatorname{sign}\,[\sin(\omega t+\varphi_\beta+\psi)] . \tag{22} \]
If this periodic function is expanded in a trigonometric series,
\[ \operatorname{sign}[\sin(\omega t+\varphi_\beta+\psi)] = \frac{4}{\pi} \left[ \sin(\omega t+\varphi_\beta+\psi) + \frac{\sin(3\omega t+\varphi_\beta+\psi)}{3} +\ldots \right], \]
and we confine ourselves only to its first component, then we shall have
\[ \operatorname{sign}\dot{\theta} \simeq \frac{4}{\pi}\sin(\omega t+\varphi_\beta+\psi). \tag{23} \]
After substituting (23) into (20) and integrating, we obtain the steady-state value of the drift velocity of the gyrostabilizer
\[ \dot{\theta}(t) = \alpha_m\beta_m\cos(\omega t+\varphi_\alpha)\cdot\sin(\omega t+\varphi_\beta) - \frac{4\psi}{\pi H^2}M_T\sin(\omega t+\varphi_\beta+\psi) + \]
\[ + \frac{\mu}{H}\beta_m\omega\cos(\omega t+\varphi_\beta) + \left[ \frac{\mu k}{H^2} - \alpha_m\omega\cos(\omega t+\varphi_\alpha) \right] \times \tag{24} \]
\[ \times \left[ \frac{4M_T}{\pi k\sqrt{1+T^2\omega^2}} \sin(\omega t+\varphi_\beta-2\psi) - \frac{\omega T\beta_m}{\sqrt{1+T^2\omega^2}} \cos(\omega t+\varphi_\beta+\psi) \right]. \]
The constant component of the velocity of relative motion of the gyrostabilizer is then equal to
\[ \dot{\theta}_c = -\frac{1}{2}\alpha_m\beta_m\omega\sin(\varphi_\alpha-\varphi_\beta) + \frac{\omega^2T}{2\sqrt{1-T^2\omega^2}} \alpha_m\beta_m\cos(\varphi_\alpha-\varphi_\beta-\psi) + \]
\[ + \frac{2\alpha_m\omega M_T}{\pi k\sqrt{1+T^2\omega^2}} \sin(\varphi_\alpha-\varphi_\beta-2\psi), \tag{25} \]
and the drift velocity in inertial space is
\[ \Omega_c = \Omega_z+\dot{\theta} = \frac{\omega^2T}{2\sqrt{1+T^2\omega^2}} \alpha_m\beta_m\cos(\varphi_\alpha-\varphi_\beta-\psi) + \]
\[ + \frac{2M_T\omega\alpha_m}{\pi k\sqrt{1+T^2\omega^2}} \sin(\varphi_\alpha-\varphi_\beta-2\psi). \tag{26} \]
With simultaneous oscillations of the platform about the three axes of the gimbal suspension with approximately identical ranges of variation of the oscillation angles \(\alpha\), \(\beta\), \(\gamma\), and with a sufficiently high gain of the interframe correction circuit of the gyrostabilizer,
\[ \operatorname{sign}\dot{\theta} \simeq -\operatorname{sign}\dot{\gamma}. \]
If, in this case, the angles \(\alpha\) and \(\beta\) vary according to (18) and (19), and
\[ \gamma=\gamma_m\sin(\omega t+\varphi_\gamma), \]
then the additional component of the systematic drift velocity of the gyrostabilizer, caused by dry friction about the axis of the outer frame, is approximately equal to the quantity
\[ \Omega_M=-\frac{2a_m M_T}{\pi k\sqrt{1+T^2\omega^2}}\cos(\varphi_a-\varphi_\gamma-\psi). \tag{27} \]
When the oscillations of the gyrostabilizer base and the moments of the disturbing forces about its suspension axes are random in character, the mathematical expectation of the gyrostabilizer drift rate, according to (5), is equal, in the first approximation, to
\[ \begin{aligned} m_\Omega(t)=&\ \frac{1}{H}m_{\Omega_\varepsilon}(t)+\frac{\mu}{H^2}m_{M_\theta}(t)+\frac{\mu}{H}m_{\Omega_y}(t)- \\ &-\frac{\mu k}{H^3}\int_0^t e^{\frac{t_1-t}{T}}m_{M_\theta}(t_1)\,dt_1 +\frac{1}{H}\int_0^t e^{\frac{t_1-t}{T}}\bigl[R_{\Omega_x M_\theta}(t,t_1)+ \\ &\quad +m_{\Omega_x}(t_1)\cdot m_{M_\theta}(t_1)\bigr]\,dt_1 -\frac{\mu k}{H^2}\int_0^t e^{\frac{t_1-t}{T}}\bigl[R_{\Omega_x\Omega_y}(t,t_1)+ \\ &\quad +m_{\Omega_x}(t)\cdot m_{\Omega_y}(t_1)\bigr]\,dt_1 . \end{aligned} \tag{28} \]
If it is assumed that the oscillations of the base and the moments of the disturbing forces are stationary random processes with zero mathematical expectations, then
\[ m_\Omega(t)=\int_0^t e^{\frac{t_1-t}{T}}\left[R_{\Omega_x\Omega_y}(t,t_1)+\frac{1}{H}R_{\Omega_x M_\theta}(t,t_1)\right]dt_1 . \tag{29} \]
In formulas (28) and (29), the symbol \(m_i(t)\) denotes the mathematical expectation of the \(i\)-th process, and \(R_{ij}(t,t_1)\) is the mutual correlation function of the \(i\)-th and \(j\)-th processes.
Refinement of the value of the mathematical expectation of the gyrostabilizer drift rate on the basis of the second approximate solution of the equations of motion for the above-indicated disturbance regimes leads to the following expression for \(m_\Omega(t)\):
\[ \begin{aligned} m_\Omega(t)=&\int_0^t e^{\frac{t_1-t}{T}} \left[R_{\Omega_x\Omega_y}(t,t_1)+\frac{1}{H}R_{\Omega_x M_\theta}(t,t_1)\right]dt_1- \\ &-\frac{\mu^2 k}{H^3}\int_0^t \left[ \frac{1}{H}R_{\Omega_x M_\varepsilon}(t,t_1) +\frac{\mu}{H^2}R_{\Omega_x M_\theta}(t,t_1) +\frac{\mu}{H}R_{\Omega_x\Omega_y}(t,t_1)- \right.\\ &\qquad\qquad\left. -R_{\Omega_x\Omega_z}(t,t_1) \right]dt_1 . \end{aligned} \tag{30} \]
From this it may be concluded that, under the adopted conditions, systematic drifts of the gyrostabilizer will occur in those cases when its base is subjected to angular oscillations about the \(x\) axis and, simultaneously, to correlated angular oscillations about the \(y\) or \(z\) axes, or when, simultaneously with \(\Omega_x\), disturbing-force moments \(M_\theta\) or \(M_\varepsilon\) correlated with \(\Omega_x\) act on the gyrostabilizer.