Abstract

We study the dynamics of a gimbal-mounted gyroscope, which has a vertical external gimbal axis and is equipped with a synchronous electric drive that rotates a gyroscope (electric motor rotor). A mathematical model of the electric motor that includes differential equations for electric currents in the rotor windings is used. Both friction and control torques on the gimbal axes are assumed to be absent. Friction torque on the rotor axis is a nonlinear function of the angular velocity of its rotation relative to the stator.The differential equations of motion for such an electromechanical system have an infinite set of stationary (steady-state) solutions that describe its stationary motions, namely, regular precessions and uniform rotations of the rotor. Due to the verticality of the outer axis of the gimbal suspension, the considered system of differential equations has a first integral, which expresses the constancy of the projection of the angular momentum of the device onto the vertical axis.In the present article we have proved that the presence of an isolated minimum of the total reduced potential energy is the necessary and sufficient condition for the stability of stationary (steady-state) motions. Thus, the previously obtained result using a simplified non-current model of a synchronous electric motor is generalized to the case of the multi-current model.Content of the study is presented in two articles. In the first part, we consider the cases when the mechanical reduced potential energy is not identical constant with respect to the internal gimbal angle. In the second part, we analyze the case of constancy for the mechanical reduced potential energy.

中文翻译：

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同步云台固定陀螺仪多电流模型方程平稳解的稳定性判据。第1部分

摘要

我们研究了装有万向节的陀螺仪的动力学特性，该陀螺仪具有垂直的外万向节轴，并配备有同步电驱动器，该驱动器使陀螺仪（电动机转子）旋转。使用电动机的数学模型，该数学模型包括转子绕组中电流的微分方程。万向轴上的摩擦力矩和控制力矩均假定为不存在。转子轴上的摩擦转矩是其相对于定子的旋转角速度的非线性函数。此类机电系统的运动微分方程具有描述其静止运动的无穷组静止（稳态）解，也就是说，转子有规律地进动和均匀旋转。由于云台悬架的外轴是垂直的，所考虑的微分方程系统具有一个第一积分，该积分表示设备的角动量在垂直轴上的投影的恒定性。在本文中，我们证明了总的减小的势能存在一个孤立的最小值是稳定（稳态）运动稳定的必要和充分条件。因此，先前使用同步电动机的简化非电流模型获得的结果被推广到多电流模型的情况。研究内容分两篇。在第一部分中，我们考虑了相对于内部万向节角而言，机械减小的势能不是恒定不变的情况。在第二部分中，我们分析了机械减小的势能恒定的情况。