On the Forced Motion of a Gyrointegrator of Linear Accelerations
Abstract Generated abstract
This paper analyzes the forced motion of a gyrointegrator of linear accelerations when its base undergoes angular oscillations about an axis perpendicular to the instrument’s outer axis. Starting from the equations of motion, the authors transform the disturbance terms into components at combination frequencies involving the base oscillation frequency and the gyrointegrator rotation frequency, then derive expressions for the resulting forced oscillations. The analysis indicates that the instrument can oscillate at the difference and sum frequencies, as well as at their multiples, and that inadmissibly large amplitudes may occur when forcing frequencies coincide with the integrator rotation frequency or satisfy the derived resonance condition.
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PROCEEDINGS
OF THE TOMSK ORDER OF THE OCTOBER REVOLUTION AND ORDER
OF THE RED BANNER OF LABOR
POLYTECHNIC INSTITUTE named after S. M. KIROV
Volume 228 1974
ON THE FORCED MOTION OF A GYROINTEGRATOR OF LINEAR ACCELERATIONS
Yu. M. KAMASHEV, L. P. DEMIDKO, V. K. DEMIDKO
(Presented by the scientific-technical seminar of the Department of Gyroscopic Instruments and Devices)
The influence of angular oscillations of the base about an axis perpendicular to the outer axis of the gyrointegrator on its forced motion is considered.
The equations of motion [1] of the gyrointegrator (Fig. 1) under oscillations of the base about the transverse axis with velocity \(\omega_\eta = Aq \sin qt\) may be represented in the form:
Fig. 1
\[ I_{\mathrm{B}}\ddot{\vartheta}+H\dot{\psi}-M_g = -I_y^{\Gamma} A q^2 \cos qt \cos \nu t + \left(I_y^{\Gamma}-I_{\mathrm{r}}\right)Aq\nu \cdot \sin qt \sin \nu t; \]
\[ I_c\ddot{\psi}-H\dot{\vartheta}-m_0\vartheta = HAq\sin qt\cos \nu t+ \]
\[ +\left(I_x^{\Gamma}-I_y^{\Gamma}\right)\cdot A^2q^2\sin^2 qt\sin \nu t\cos \nu t, \]
where \(I_x^{\Gamma},\ I_y^{\Gamma},\ I_z^{\Gamma}\) are, respectively, the moments of inertia of the gyro assembly about the axes \(x, y, z\);
\(H\) is the angular momentum of the gyrointegrator;
\(I_{\mathrm{r}}\) is the moment of inertia of the gyro assembly;
\(M_g = mgL\) is the moment due to the static unbalance of the gyro assembly;
\(m_0\vartheta\) is the moment of the stabilizing motor.
\[ I_b = I_y + I_e + mL^2; \]
\[ I_c = I_n + I_z + I_e. \]
\(I_y, I_z\) are, respectively, the moments of inertia of the gyro chamber about the axes \(y, z\);
\(mL^2\) is the moment of inertia of the gyro weight about the suspension axis;
\(L\) is the displacement of the center of gravity;
\(I_n\) is the moment of inertia of the outer gimbal ring about the axis \(CC\);
\(I_e\) is the equatorial moment of inertia of the rotor;
\(\psi\) is the angle of rotation of the outer gimbal ring together with the gyro weight about the axis \(CC\);
\(\vartheta\) is the small angle of rotation of the gyro weight about the axis \(BB\);
\(A, q\) are the amplitude and frequency of oscillations of the bases;
\(\dot{\psi}=\nu\) is the angular frequency of rotation of the gyrointegrator about the axis \(CC\).
After trigonometric transformations and the introduction of new notation,
\[ A_1 = I_y^r A q^2 + (I_y^r - I_n)\cdot A q \nu, \]
\[ A_2 = H A q \]
\[ A_3 = (I_x^r - I_y^r)\cdot A^2 q^2 \]
we represent equations (1) in the form:
\[ I_b \ddot{\vartheta}+H\dot{\psi}-M_g = \frac{1}{2}A_1\cos(q-\nu)t + \frac{1}{2}A_1\cos(q+\nu)t; \]
\[ I_c\ddot{\psi}-H\dot{\vartheta}-m_0\vartheta = \frac{1}{2}A_2\sin(q-\nu)t + \frac{1}{2}A_2\sin(q+\nu)t - \]
\[ -\frac{1}{8}A_3\sin 2(q-\nu)t - \frac{1}{8}A_3\sin 2(q+\nu)t. \]
The forced oscillations of the gyrointegrator with the disturbance frequency \((q-\nu)\) have the form:
\[ \psi = \frac{1}{2}\frac{I_b m_0}{\Delta} \left[A_1 I_c(q-\nu)^2 + A_2H(q-\nu)\right]\cos(q-\nu)t + \]
\[ + \frac{1}{2\Delta(q-\nu)} \left[ -I_b^4 I_c A_2(q-\nu)^5 - I_b^2 I_c H A_1(q-\nu)^4 + \right. \]
\[ \left. + I_b^3H^2A_2(q-\nu)^3 - I_bH^3A_1(q-\nu)^2 + Hm_0^2A_1 \right]\sin(q-\nu)t, \]
where
\[ \Delta = I_b^3 I_c^2(q-\nu)^6 - 2I_b^2 I_cH^2(q-\nu)^4 + H^4I_b(q-\nu)^2 + H^2m_0^2. \]
An analysis of the amplitudes of the forced oscillations of the gyrointegrator makes it possible to draw the following conclusions:
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When the base oscillates about an axis perpendicular to the outer axis of the integrator, the instrument executes forced oscillations with frequencies \((q-\nu)\), \((q+\nu)\), and their multiples.
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When the forcing frequency is equal to the frequency \(\nu\) of rotation of the integrator about the outer axis, the amplitudes of the forced oscillations may reach inadmissibly large values.
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Solving the equation \(\Delta=0\) makes it possible to find the values of the frequencies \((q+\nu)\), \((q-\nu)\), \(2(q+\nu)\), \(2(q-\nu)\), at which the amplitudes of the forced oscillations also assume inadmissibly large values.
References
- S. S. Rivkin, Theory of Gyroscopic Devices. Parts 1–2, “Sudostroenie,” 1962–1964.