Test Bench for Testing Onboard Gyroscopic Devices
Abstract Generated abstract
The paper addresses laboratory testing of onboard gyroscopic devices under angular motions representative of aircraft and ship operation, where errors are strongly affected by random rocking rather than purely harmonic motion. It proposes a three-axis gimbal test bench whose platform and rings reproduce yaw, pitch, and roll as stationary Gaussian random processes with prescribed correlation functions matching those of the moving object. The command signals are generated by filtering white noise through linear filters derived from the desired spectral densities, with parameters adjustable to simulate different rocking regimes or, when needed, harmonic oscillations. The design combines these generators with tracking drives and angle feedback to approximate full-scale random motion conditions for gyroscopic-device evaluation.
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TEST BENCH FOR TESTING ONBOARD GYROSCOPIC DEVICES
A. M. MALYSHENKO
(Presented by the scientific seminar of the Department of Automation and Telemechanics)
The errors of gyroscopic devices (inertial navigation and control systems, gyrocompasses, gyrostabilizers, etc.) when operating on moving objects such as aircraft and ships are due primarily to the angular oscillations of the object [1].
At present, in laboratory conditions, the influence of an object’s rocking on the behavior of gyroscopic devices (GD) is investigated on rocking benches of the Scorsby type with a mechanical oscillation generator. Scorsby-type benches make it possible to investigate the operation of GD only when the rocking of the object is harmonic in character. It is known, however [2], that the rocking of moving objects of the indicated types constitutes random processes.
For this reason, in order to approximate laboratory test conditions to full-scale conditions, it is necessary to have a bench with a random character of rocking of its platform. At the same time, the statistical characteristics of the angular oscillations of the bench platform must correspond to the statistical characteristics of the angular oscillations of the object. In addition, the device must permit these characteristics to be varied within certain limits in order to simulate various rocking regimes. The bench described in the present work, proposed by the author, satisfies all these requirements.
To characterize the angular oscillations of an object, we shall attach to it rigidly (Fig. 1), originating at the object’s center of gravity, an orthogonal trihedron of axes $Oxyz$ and orient it as follows: the axis $Ox$ coincides with the longitudinal axis of symmetry of the object and is directed toward its nose; the axis $Oz$ is perpendicular to the deck (the plane of the wings) of the object and is directed upward; and the axis $Oy$, together with the axes $Ox$ and $Oz$, forms a right-handed trihedron of axes. We shall denote by $O\xi\eta\zeta$ the Darboux trihedron associated with the point of the object’s location, with the axis $O\zeta$ coinciding with the local vertical, while the axes $O\xi$ and $O\eta$ lie in the plane of the horizon and are oriented respectively toward North and West. We assume that the center of the Darboux trihedron coincides with the object’s center of gravity.
The position of the trihedron of axes $Oxyz$ relative to the Darboux trihedron can be determined by the angles $K + \gamma$, $\beta$, and $\alpha$, where $K$ is the course
motion of the object, $\gamma$ is the yaw angle, $\beta$ is the trim (pitch) angle, and $\alpha$ is the roll angle of the object. The characterization of the object’s angular oscillations by these angles is the most convenient [2] of all possible variants.
The oscillations of the object can be simulated by a platform mounted in a three-degree-of-freedom gimbal suspension according to the scheme of Fig. 2, with a vertical axis of the outer ring. In this case the angle $\gamma_{\text{п}}$ of rotation of the outer ring of the gimbal suspension relative to the base must be equal to the object’s course and yaw angle, the angle $\beta_{\text{п}}$ of rotation of the inner ring relative to the outer ring must be equal to the angle $\beta$ of trim (pitch), and the angle $\alpha_{\text{п}}$ of rotation of the platform relative to the inner ring must be equal to the angle $\alpha$ of the object’s onboard roll.
Fig. 1
On the basis of processing full-scale records [2, 3], it has been established that the rolling and yawing of moving objects, for example sea vessels and aircraft, are stationary random processes with a normal (Gaussian) distribution. The mathematical expectations of the object’s roll and yaw angles in the general case may differ from zero. The correlation functions of the random processes $\alpha(t)$, $\beta(t)$, $\gamma(t)$ are approximated by one-type formulas of the form
\[ R(\tau)=Ae^{-\mu|\tau|}\left(\cos \nu\tau+\frac{\mu}{\nu}\sin \nu\tau\right), \tag{1} \]
where \(\mu\) is the damping coefficient of the correlation function, characterizing the degree of irregularity of the rocking;
\(\nu\) is the frequency of variation of the correlation function, determining the predominant rocking frequency of the object;
\(A\) is the variance of the corresponding rocking angle of the object.
To obtain a law of motion of the stand platform analogous to the law of motion of the object, it is necessary that the rotation angles of the platform and of the gimbal rings also be stationary
Fig. 2
random processes with a normal distribution, correlation functions of the form (1), and mathematical expectations \(\bar{\alpha}_{\text{п}}, \bar{\beta}_{\text{п}}, \bar{\gamma}_{\text{п}}\), equal respectively to the mathematical expectations \(\bar{\alpha}, \bar{\beta}, \bar{\gamma}\) of the motion of the object.
When gyroscopic devices with carefully balanced masses are tested on the platform with respect to all axes of its suspension, deviations of the mathematical expectations \(\bar{\alpha}_{\text{п}}, \bar{\beta}_{\text{п}}\), and \(\bar{\gamma}_{\text{п}}\) from zero do not affect the dynamics of operation of the GD; therefore one can often assume
\[ \bar{\alpha}_{\text{п}} = \bar{\beta}_{\text{п}} = \bar{\gamma}_{\text{п}} = 0. \]
The correlation functions of the angles \(\alpha_{\text{п}}(t)\), \(\beta_{\text{п}}(t)\), \(\gamma_{\text{п}}(t)\) must correspond to the correlation functions of the rocking and yaw angles of the object, that is, the equalities must be ensured
\[ R_{\alpha_\mathrm{p}}(\tau)=R_\alpha(\tau); \]
\[ R_{\beta_\mathrm{p}}(\tau)=R_\beta(\tau); \]
\[ R_{\gamma_\mathrm{p}}(\tau)=R_\gamma(\tau). \]
Such similar motion of the test-bench platform can be provided by means of three tracking drive systems for the platform and gimbal rings, which reproduce the signals of random-process generators with probabilistic characteristics corresponding to the probabilistic characteristics of the object’s roll and yaw angles.
The random-process generators that set the oscillation regime of the platform may have different technical implementations. In particular, they may be devices reproducing actual records of the object’s oscillations. However, this variant cannot be considered optimal, since for testing GDs under various oscillation regimes it is necessary to have a large number of actual records of object oscillations.
The generators may also be constructed on the principle of linear filtering [4] of white noise, i.e., of a random process with a spectral density uniform over the entire frequency range. In this case the transfer function \(W(p)\) of the filter is determined by the spectral characteristic \(S(\omega)\) of the desired output process. The relationship between them is described in [4] by the equation
\[ W(p)W(-p)=S(\omega), \tag{2} \]
where
\[ p=j\omega \quad (j=\sqrt{-1}). \]
Consequently, the transfer function of the filter can be determined by factorizing the spectral characteristic of the desired output signal. The transfer function must not have zeros or poles in the right half-plane of the complex root plane. In this case we obtain a linear stable filter of minimum-phase type.
For stationary random processes with correlation function (1), the spectral density is
\[ S(\omega)=\frac{2A\mu}{\pi}\cdot \frac{\mu^2+\nu^2}{(\omega^2+\mu^2-\nu^2)^2+4\mu^2\nu^2}. \tag{3} \]
Factorization of \(S(\omega)\) according to formula (2) gives the following expression for the filter transfer function:
\[ W(p)=\frac{a_0}{p^2+b_1p+b_0}, \tag{4} \]
where
\[ a_0=\sqrt{\frac{2A\mu}{\pi}(\mu^2+\nu^2)}; \]
\[ b_1=2\mu; \]
\[ b_0=\mu^2+\nu^2. \]
Sometimes the object’s roll, pitch, and yaw are approximated by a stationary random process with correlation function
\[ R(\tau)=Ae^{-\mu|\tau|}\cos\nu\tau. \tag{5} \]
To obtain such a random process, the transfer function of the filter must be equal to
\[ W(p)=\frac{a_1p+a_0}{p^2+b_1p+b_0}, \tag{6} \]
where
\[ a_1=\sqrt{\frac{2A\mu}{\pi}}; \]
\[ a_0=\sqrt{\frac{2A\mu}{\pi}(\mu^2+\nu^2)}; \]
\[ b_1=2\mu; \]
\[ b_0=\mu^2+\nu^2. \]
Filters with the transfer functions (4) and (6) can be implemented in various ways, for example from passive \(R, L, C\) elements or on Miller integrators used in analog computers.
These filters are low-frequency filters, since the predominant pitch and yaw frequencies of the object usually lie in the range \(0.1\)—\(21/\mathrm{sec}\). Therefore, when these filters are implemented from passive \(R, L, C\) elements, large capacitances and inductances will be required.
A circuit for assembling a filter with transfer function (4) on integrators is shown in Fig. 3. The circuit for assembling filter (6) differs from the circuit for assembling filter (4) only by the additional chain, indicated in Fig. 3 by a dashed line.
It should be noted that, for prolonged simulation of the object’s pitching, the filter integrators must have a small zero drift. This condition is easily satisfied when using integrators with automatic zero stabilization.
When implementing the filter on integrators, changes in the modes of random pitching of the platform are easily carried out by changing the parameters of the model filter. In addition, the use of the model makes it possible, when necessary, to rock the bench platform according to a harmonic law. For this purpose, a conservative element with an oscillation frequency equal to the desired rocking frequency of the platform should be assembled on the model integrators. By changing the initial conditions in the integrators, various phase shifts can be set between the oscillations of the platform about the different axes of its suspension.
White-noise generators may be used to obtain white noise [5]. It is not possible to ensure constancy of the spectral density over an infinite frequency range; therefore, the random process generated by these devices, strictly speaking, is not white noise. However, the low natural frequency of the filters used in the device permits filtering only of the infralow-frequency components of the spectrum. For this reason, the white-noise generators used in the device under consideration are required to ensure constant spectral density only in the infralow-frequency region. A description of some types of such generators is given in [6, 7]. When such generators are used, the random process implemented at the filter output will be close to the desired one.
Figure 4 shows a diagram of the bench that makes it possible to test onboard gyroscopic devices under conditions of random pitching.
Fig. 3. Filter-selection circuit using operational amplifiers.
Fig. 4. Schematic of a bench for testing onboard gyroscopic devices in the random rocking mode.
Visible labels in the diagram: \(DU_r\), \(SD\), \(ChUD\), \(GU\), \(VK\), \(P\), \(NK\), \(ID_k\), \(ID_q\), \(ID_p\), \(DU_q\), \(DU_k\), \(U_k\), \(U_q\), \(U_p\), \(FB\), \(F_k\), \(F_q\), \(F_p\), \(PB\), \(O\).
The bench includes a platform \(P\) in a three-degree-of-freedom gimbal suspension, on which the tested gyroscopic device \(GU\) is installed. The drive of the platform and of the gimbal rings is carried out by means of three similar tracking systems \((SS)\), which consist of actuating motors \(ID\), installed on the axes of the gimbal suspension, amplifying devices \(U\), rotation-angle sensors \(DU\), and comparison elements \(OS\).
The tracking systems provide angular oscillations of the bench platform in accordance with the law of variation of the output signals of the filters \(F\), which transform the signals of white-noise generators \(GBSh\). The output signals of the filters must be statistically equivalent to the object’s pitch and yaw angles. Feedback in the \(SS\) is provided by feeding the signals from the angle sensors \(DU\) of the relative rotations of the platform and the gimbal rings to the comparison elements. There they are compared with the output signals of the shaping filters.
The tracking system of the drive of the outer gimbal ring \(NK\) must provide its motions relative to the base \(O\) according to the law of the object’s yaw. The tracking system of the drive of the inner ring \(VK\) must operate in the mode of reproducing the object’s trim (pitch), and the \(SS\) of the platform drive—in the mode of reproducing the object’s roll.
In this connection, the filter parameters of the command signals for the first \(SS\) are determined from the parameters of the correlation function of the object’s yaw angles. Accordingly, the parameters of the filters of the other two \(SS\) are determined from the parameters of the correlation functions of trim (pitch) and roll. In Fig. 4 all elements of the tracking systems for reproducing the object’s yaw, trim, and roll have, respectively, designations with the indices \(r, d, k\).
If necessary, the platform can be turned relative to the base through an angle corresponding to the selected course \(K\) of the object’s motion, by turning through the angle \(K\) the housing of the angle sensor \(DU_r\) of the outer ring rotation angles. Course setting can be performed through a reducer either manually or by means of a servodrive comprising a servomotor \(SD\) and a device for its control \(USD\). In an analogous manner and by analogous devices, the constant components of the platform rotation angles, corresponding to the mathematical expectations of the trim and roll angles, can be set if necessary.
It should be noted that the tracking systems of the gimbal-suspension drive are themselves filters, and therefore it is impossible to ensure exact reproduction of the roll-generator signals by the tracking systems. However, the reproduction error in this case will be small, since in the infralow-frequency region corresponding to the object’s rolling motion the tracking systems will have an almost constant transmission coefficient close to 1, i.e., they will reproduce the input signals with sufficient accuracy.
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